C ALGORITHM 811, COLLECTED ALGORITHMS FROM ACM. C THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE, C VOL. 27,NO. 2, June, 2001, P. 193--213. #! /bin/sh # This is a shell archive, meaning: # 1. Remove everything above the #! /bin/sh line. # 2. Save the resulting text in a file. # 3. Execute the file with /bin/sh (not csh) to create the files: # Doc/ # Doc/makefile # Doc/pbun.txt # Doc/pmin.txt # Doc/pnew.txt # Doc/pvar.txt # Doc/readme.txt # Doc/routinedoc.tex # Fortran/ # Fortran/Dp/ # Fortran/Dp/Drivers/ # Fortran/Dp/Drivers/data1 # Fortran/Dp/Drivers/driver1.f # Fortran/Dp/Drivers/driver2.f # Fortran/Dp/Drivers/driver3.f # Fortran/Dp/Drivers/driver4.f # Fortran/Dp/Drivers/driver5.f # Fortran/Dp/Drivers/driver6.f # Fortran/Dp/Drivers/driver7.f # Fortran/Dp/Drivers/driver8.f # Fortran/Dp/Drivers/res1 # Fortran/Dp/Drivers/res2 # Fortran/Dp/Drivers/res3 # Fortran/Dp/Drivers/res4 # Fortran/Dp/Drivers/res5 # Fortran/Dp/Drivers/res6 # Fortran/Dp/Drivers/res7 # Fortran/Dp/Drivers/res8 # Fortran/Dp/Drivers/subs.f # Fortran/Dp/Src/ # Fortran/Dp/Src/src.f # This archive created: Tue Nov 13 09:58:10 2001 export PATH; PATH=/bin:$PATH if test ! -d 'Doc' then mkdir 'Doc' fi cd 'Doc' if test -f 'makefile' then echo shar: will not over-write existing file "'makefile'" else cat << "SHAR_EOF" > 'makefile' OBJS = src.o subs.o all: res1 res2 res3 res4 res5 res6 res7 res8 res2: driver2.o ${OBJS} ${FC} ${FFLAGS} tbunu.o ${OBJS} a.out < data1 > $@ res1: driver1.o ${OBJS} ${FC} ${FFLAGS} tbunl.o ${OBJS} a.out > $@ res4: driver4.o ${OBJS} ${FC} ${FFLAGS} tminu.o ${OBJS} a.out < data1 > $@ res3: driver3.o ${OBJS} ${FC} ${FFLAGS} tminl.o ${OBJS} a.out > $@ res6: driver6.o ${OBJS} ${FC} ${FFLAGS} tnewu.o ${OBJS} a.out < data1 > $@ res5: driver5.o ${OBJS} ${FC} ${FFLAGS} tnewl.o ${OBJS} a.out > $@ res8: driver8.o ${OBJS} ${FC} ${FFLAGS} tnewu.o ${OBJS} a.out < data1 > $@ res7: driver7.o ${OBJS} ${FC} ${FFLAGS} tnewl.o ${OBJS} a.out > $@ SHAR_EOF fi # end of overwriting check if test -f 'pbun.txt' then echo shar: will not over-write existing file "'pbun.txt'" else cat << "SHAR_EOF" > 'pbun.txt' *********************************************************************** * * * PBUN - A PROXIMAL BUNDLE ALGORITHM FOR NONSMOOTH * * OPTIMIZATION. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PBUN is designed to find a close approximation to a local minimum of a nonlinear nonsmooth function F(X) with simple bounds on variables and general linear constraints. Here X is a vector of N variables and F(X), is assumed to be a locally Lipschitz continuous function. Simple bounds are assumed in the form X(I) unbounded if IX(I) = 0, XL(I) <= X(I) if IX(I) = 1, X(I) <= XU(I) if IX(I) = 2, XL(I) <= X(I) <= XU(I) if IX(I) = 3, XL(I) = X(I) = XU(I) if IX(I) = 5, where 1 <= I <= N. General linear constraints are assumed in the form C(I) unbounded if IC(I) = 0, CL(I) <= C(I) if IC(I) = 1, C(I) <= CU(I) if IC(I) = 2, CL(I) <= C(I) <= CU(I) if IC(I) = 3, CL(I) = C(I) = CU(I) if IC(I) = 5, where C(I) = A_I * X, 1 <= I <= NC, are linear functions. To simplify user's work, three additional easy to use subroutines are added. They call the basic general subroutine PBUN: PBUNU - unconstrained nonsmooth optimization, PBUNS - nonsmooth optimization with simple bounds, PBUNL - nonsmooth optimization with simple bounds and general linear constraints. All subroutines contain a description of formal parameters and extensive comments. Furthermore, two test programs TBUNU and TBUNL are included, which contain several test problems (see [4]). These test programs serve as examples for using the subroutines, verify their correctness and demonstrate their efficiency. In this short guide, we describe all subroutines which can be called from the user's program. A detailed description of methods is given in [2] and [3]. In the description of formal parameters, we introduce a type of the argument that specifies whether the argument must have a value defined on entry to the subroutine (I), whether it is a value which will be returned (O), or both (U), or whether it is an auxiliary value (A). Note that the arguments of the type I can be changed on output under some circumstances, especially if improper input values were given. Besides formal parameters, we can use a COMMON /STAT/ block containing statistical information. This block, used in each subroutine, has the following form: COMMON /STAT/ NDECF,NRES,NRED,NREM,NADD,NIT,NFV,NFG,NFH The arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- NDECF O Positive INTEGER variable that indicates the number of matrix decompositions. NRES O Positive INTEGER variable that indicates the number of restarts. NRED O Positive INTEGER variable that indicates the number of reductions. NREM O Positive INTEGER variable that indicates the number of constraint deletions during the QP solutions. NADD O Positive INTEGER variable that indicates the number of constraint additions during the QP solutions. NIT O Positive INTEGER variable that indicates the number of iterations. NFV O Positive INTEGER variable that indicates the number of function evaluations. NFG O Positive INTEGER variable that specifies the number of gradient evaluations. NFH O Positive INTEGER variable that specifies the number of Hessian evaluations. 2. Subroutines PBUNU, PBUNS, PBUNL: ----------------------------------- The calling sequences are CALL PBUNU(NF,NA,X,IA,RA,IPAR,RPAR,FP,GMAX,ITERM) CALL PBUNS(NF,NA,NB,X,IX,XL,XU,IA,RA,IPAR,RPAR,FP,GMAX,ITERM) CALL PBUNL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,IA,RA,IPAR, & RPAR,FP,GMAX,ITERM) The arguments have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. NA I Nonnegative INTEGER variable that specifies the maximum bundle dimension. The choice NA=0 causes that the default value NA=NF+3 will be taken. NB I Nonnegative INTEGER variable that specifies whether the simple bounds are suppressed (NB=0) or accepted (NB>0). NC I Nonnegative INTEGER variable that specifies the number of linear constraints; if NC=0 the linear constraints are suppressed. X(NF) U On input, DOUBLE PRECISION vector with the initial estimate to the solution. On output, the approximation to the minimum. IX(NF) I On input (significant only if NB>0) INTEGER vector containing the simple bounds types: IX(I)=0 - the variable X(I) is unbounded, IX(I)=1 - the lower bound X(I) >= XL(I), IX(I)=2 - the upper bound X(I) <= XU(I), IX(I)=3 - the two side bound XL(I) <= X(I) <= XU(I), IX(I)=5 - the variable X(I) is fixed (given by its initial estimate). XL(NF) I DOUBLE PRECISION vector with lower bounds for variables (significant only if NB>0). XU(NF) I DOUBLE PRECISION vector with upper bounds for variables (significant only if NB>0). CF(NC) A DOUBLE PRECISION vector which contains values of constraint functions (only if NC>0). IC(NC) I On input (significant only if NC>0) INTEGER vector which contains constraint types: IC(K)=0 - the constraint CF(K) is not used, IC(K)=1 - the lower constraint CF(K) >= CL(K), IC(K)=2 - the upper constraint CF(K) <= CU(K), IC(K)=3 - the two side constraint CL(K) <= CF(K) <= CU(K), IC(K)=5 - the equality constraint CF(K) = CL(K). CL(NC) I DOUBLE PRECISION vector with lower bounds for constraint functions (significant only if NC>0). CU(NC) I DOUBLE PRECISION vector with upper bounds for constraint functions (significant only if NC>0). CG(NF*NC) I DOUBLE PRECISION matrix whose columns are normals of the linear constraints (significant only if NC>0). IA(NIA) A INTEGER working array of the dimension of at least NIA=NF+NA+1. RA(NRA) A DOUBLE PRECISION working array of the dimension of at least NRA=NF*(NF+1)/2+NF*(NA+5)+5*NA+4. IPAR(7) A INTEGER parameters: IPAR(1)=MET, IPAR(2)=MES, IPAR(3)=MTESX, IPAR(4)=MTESF, IPAR(5)=MIT, IPAR(6)=MFV, IPAR(7)=IPRNT. Parameters MET, MES, MTESX, MTESF, MIT, MFV, IPRNT are described in Section 3 together with other parameters of the subroutine PBUN. RPAR(9) A DOUBLE PRECISION parameters: RPAR(1)=TOLX, RPAR(2)=TOLF, RPAR(3)=TOLB, RPAR(4)=TOLG, RPAR(5)=TOLD, RPAR(6)=TOLS, RPAR(7)=TOLP. RPAR(8)=ETA, RPAR(9)=XMAX. Parameters TOLX, TOLF, TOLB, TOLG, TOLD, TOLS, TOLP, ETA, XMAX are described in Section 3 together with other parameters of the subroutine PBUN. FP O DOUBLE PRECISION value of the objective function at the solution X. GMAX O DOUBLE PRECISION maximum absolute value of a partial derivative of the Lagrangian function. ITERM O INTEGER variable that indicates the cause of termination: ITERM= 1 - if |X - XO| was less than or equal to TOLX in MTESX subsequent iterations, ITERM= 2 - if |F - FO| was less than or equal to TOLF in MTESF subsequent iterations, ITERM= 3 - if F is less than or equal to TOLB, ITERM= 4 - if GMAX is less than or equal to TOLG, ITERM=11 - if NFV exceeded MFV, ITERM=12 - if NIT exceeded MIT, ITERM< 0 - if the method failed. The subroutines PBUNU, PBUNS, PBUNL require the user supplied subroutine FUNDER that defines the objective function and its subgradient and has the form SUBROUTINE FUNDER(NF,X,F,G) The arguments of the user supplied subroutine have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. X(NF) I DOUBLE PRECISION an estimate to the solution. F O DOUBLE PRECISION value of the objective function at the point X. G(NF) O DOUBLE PRECISION subgradient of the objective function at the point X. 3. Subroutine PBUN: ------------------- This general subroutine is called from all the subroutines described in Section 2. The calling sequence is CALL PBUN(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,AF,IA,AFD,AG, & IAA,AR,AZ,G,H,S,XO,GO,XS,GS,TOLX,TOLF,TOLB,TOLG,TOLD,TOLS,TOLP, & ETA,XMAX,GMAX,FP,MET,MES,MTESX,MTESF,MIT,MFV,IPRNT,ITERM). The arguments NF, NA, NB, NC, X, IX, XL, XU, CF, IC, CL, CU, CG, GMAX, FP, ITERM, have the same meaning as in Section 2. Other arguments have the following meaning: Argument Type Significance --------------------------------------------------------------------- AF(4*NA) A DOUBLE PRECISION vector of bundle function values. IA(NA) A INTEGER vector containing types of bundle functions. AFD(NA) A DOUBLE PRECISION vector of bundle function increments. AG(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle gradients. IAA(NA) A INTEGER vector containing indices of active functions. AR(NAR) A DOUBLE PRECISION matrix containing triangular decomposition of the orthogonal projection kernel (NAR is equal to NF*(NF+1)/2). AZ(NF+1) A DOUBLE PRECISION vector of Lagrange multipliers. G(NF) A DOUBLE PRECISION subgradient of the objective function. H(NF) A DOUBLE PRECISION diagonal matrix of weight parameters. S(NF+1) A DOUBLE PRECISION direction vector. XO(NF) A DOUBLE PRECISION vector which contains increments of variables. GO(NF+1) A DOUBLE PRECISION gradient of the Lagrangian function. XS(NF) A DOUBLE PRECISION auxiliary vector. GS(NF) A DOUBLE PRECISION auxiliary vector. TOLX I DOUBLE PRECISION tolerance for the change of the coordinate vector X; the choice TOLX=0 causes that the default value TOLX=1.0D-16 will be taken. TOLF I DOUBLE PRECISION tolerance for the change of function values; the choice TOLF=0 causes that the default value TOLF=1.0D-8 will be taken. TOLB I DOUBLE PRECISION minimum acceptable function value; the choice TOLB=0 causes that the default value TOLB=-1.0D60 will be taken. TOLG I DOUBLE PRECISION tolerance for the Lagrangian function gradient; the choice TOLG=0 causes that the default value TOLG=1.0D-6 will be taken. TOLD I DOUBLE PRECISION tolerance for a descent direction; the choice TOLD= 0 causes that the default value TOLD=1.0D-4 will be taken. TOLS I DOUBLE PRECISION tolerance parameter for a function decrease in a line search; the choice TOLS=0 causes that the default value TOLS=1.0D-2 will be taken. TOLP I DOUBLE PRECISION tolerance parameter for a significant modification of the next line search direction; the choice TOLP=0 causes that the default value TOLP=0.5D0 will be taken. ETA I DOUBLE PRECISION distance measure parameter. XMAX I DOUBLE PRECISION maximum stepsize; the choice XMAX=0 causes that the default value 1.0D3 will be taken. MET I INTEGER variable that specifies the weight updating method: MET=1 - quadratic interpolation, MET=2 - local minimization, MET=3 - quasi-Newton condition. The choice MET=0 causes that the default value MET=1 will be taken. MES I INTEGER variable that specifies the interpolation method selection in a line search: MES=1 - bisection, MES=2 - two point quadratic interpolation, MES=3 - three point quadratic interpolation, MES=4 - three point cubic interpolation. The choice MES=0 causes that the default value MES=1 will be taken. MTESX I INTEGER variable that specifies the maximum number of iterations with changes of the coordinate vector X smaller than TOLX; the choice MTESX=0 causes that the default value MTESX=20 will be taken. MTESF I INTEGER variable that specifies the maximum number of iterations with changes of function values smaller than TOLF; the choice MTESF=0 causes that the default value MTESF=2 will be taken. MIT I INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 200 will be taken. MFV I INTEGER variable that specifies the maximum number of function evaluations; the choice |MFV|=0 causes that the default value 500 will be taken. IPRNT I INTEGER variable that specifies PRINT: IPRNT= 0 - print is suppressed, IPRNT= 1 - basic print of final results, IPRNT=-1 - extended print of final results, IPRNT= 2 - basic print of intermediate and final results, IPRNT=-2 - extended print of intermediate and final results. The subroutine PBUN has a modular structure. The following list contains its most important subroutines: PDDBQ1 - Determination of the descent direction using quadratic programming subroutine and bundle updating. PLQDF1 - Dual range space method for solving a quadratic programming subproblem with linear constraints (see [1]). PS1L05 - Line search using function values and derivatives. The subroutine PBUN requires the user supplied subroutine FUNDER which is described in Section 2. 4. Subroutine PLQDF1: --------------------- Since the dual range space method for special quadratic programming subproblems arising in bundle type nonsmooth optimization can be used separately in many applications (e.g. in minimax optimization), we describe the subroutine PLQDF1 in more details. The calling sequence is CALL PLQDF1(NF,NA,NC,X,IX,XL,XU,AF,AFD,IA,IAA,AG,AR,AZ, & CF,IC,CL,CU,CG,G,H,S,MFP,KBF,KBC,IDECF,ETA0,ETA2,ETA9, & EPS7,EPS9,XNORM,UMAX,GMAX,N,ITERQ) The arguments NF, NA, NC, X, IX, XL, XU, AF, CF, IC, CL, CU, CG have the same meaning as in Section 2 (only with the difference that the arguments X and AF are of the type (I), i.e. they must have a value defined on entry to PLQDF1 and they are not changed). The arguments AFD, IA, IAA, AG, AR, AZ have the same meaning as in Section 3 (only with the difference that the arguments AFD, IAA, AR, AZ are of the type (O), i.e. their values can be used subsequently). Other arguments have the following meaning: Argument Type Significance --------------------------------------------------------------------- G(NF+1) O DOUBLE PRECISION gradient of the Lagrangian function. H(NH) U DOUBLE PRECISION Choleski decomposition of the approximate Hessian (NH is equal to NF*(NF+1)/2). S(NF+1) O DOUBLE PRECISION direction vector. MFP I INTEGER variable that specifies the type of the computed point. MFP=1 - computation is terminated whenever an arbitrary feasible point is found, MFP=2 - computation is terminated whenever an optimum feasible point is found, MFP=3 - computation starts from the previously reached point and is terminated whenever an optimum feasible point is found. KBF I INTEGER variable that specifies simple bounds on variables. KBF=0 - simple bounds are suppressed, KBF=1 - one sided simple bounds, KBF=2 - two sided simple bounds. KBC I INTEGER variable that specifies general linear constraints. KBC=0 - linear constraints are suppressed, KBC=1 - one sided linear constraints, KBC=2 - two sided linear constraints. IDECF U INTEGER variable that specifies the type of matrix decomposition. IDECF= 0 - no decomposition, IDECF= 1 - Choleski decomposition, IDECF= 9 - inversion, IDECF=10 - diagonal matrix. ETA0 I DOUBLE PRECISION machine precision (the recommended value is 1.0D-15. ETA2 I DOUBLE PRECISION tolerance for positive definiteness in the Choleski decomposition. ETA9 I DOUBLE PRECISION maximum floating point number. EPS7 I DOUBLE PRECISION tolerance for linear independence of constraints (the recommended value is 1.0D-10). EPS9 I DOUBLE PRECISION tolerance for the definition of active constraints (the recommended value is 1.0D-8). XNORM O DOUBLE PRECISION value of the linearized minimax function. UMAX O DOUBLE PRECISION maximum absolute value of the negative Lagrange multiplier. GMAX O DOUBLE PRECISION infinity norm of the gradient of the Lagrangian function. N O INTEGER dimension of a manifold defined by active constraints. ITERQ O INTEGER variable that indicates the type of the computed feasible point. ITERQ= 1 - an arbitrary feasible point was found, ITERQ= 2 - the optimum feasible point was found, ITERQ=-1 - an arbitrary feasible point does not exist, ITERQ=-2 - the optimum feasible point does not exist. 5. Form of printed results: --------------------------- The form of printed results is specified by the parameter IPRNT as is described in Section 3. Here we demonstrate individual forms of printed results by the simple use of the program TBUNU described in the next section (with NEXT=9). If we set IPRNT=1, then the printed results will have the form NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4 If we set IPRNT=-1, then the printed results will have the form EXIT FROM PBUN : NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4 X= .1000000D+01 .0000000D+00 If we set IPRNT=2, then the printed results will have the form ENTRY TO PBUN : NIT= 0 NFV= 1 NFG= 1 F= .00000000D+00 G= .1000D+61 NIT= 1 NFV= 3 NFG= 3 F= -.37888889D+00 G= .8500D+01 NIT= 2 NFV= 4 NFG= 4 F= -.60615144D+00 G= .9333D+00 NIT= 3 NFV= 5 NFG= 5 F= -.60615144D+00 G= .8024D+00 NIT= 4 NFV= 6 NFG= 6 F= -.72848266D+00 G= .8024D+00 NIT= 5 NFV= 7 NFG= 7 F= -.72848266D+00 G= .3478D+00 NIT= 6 NFV= 8 NFG= 8 F= -.82757096D+00 G= .7222D+00 NIT= 7 NFV= 9 NFG= 9 F= -.84360358D+00 G= .1618D+00 NIT= 8 NFV= 10 NFG= 10 F= -.99860813D+00 G= .9839D-01 NIT= 9 NFV= 11 NFG= 11 F= -.99860813D+00 G= .1141D+00 NIT= 10 NFV= 12 NFG= 12 F= -.99928519D+00 G= .5014D+00 NIT= 11 NFV= 13 NFG= 13 F= -.99999999D+00 G= .5301D-01 NIT= 12 NFV= 14 NFG= 14 F= -.99999999D+00 G= .2704D-06 NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 EXIT FROM PBUN : NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4 If we set IPRNT=-2, then the printed results will have the form ENTRY TO PBUN : NIT= 0 NFV= 1 NFG= 1 F= .00000000D+00 G= .1000D+61 NIT= 1 NFV= 3 NFG= 3 F= -.37888889D+00 G= .8500D+01 NIT= 2 NFV= 4 NFG= 4 F= -.60615144D+00 G= .9333D+00 NIT= 3 NFV= 5 NFG= 5 F= -.60615144D+00 G= .8024D+00 NIT= 4 NFV= 6 NFG= 6 F= -.72848266D+00 G= .8024D+00 NIT= 5 NFV= 7 NFG= 7 F= -.72848266D+00 G= .3478D+00 NIT= 6 NFV= 8 NFG= 8 F= -.82757096D+00 G= .7222D+00 NIT= 7 NFV= 9 NFG= 9 F= -.84360358D+00 G= .1618D+00 NIT= 8 NFV= 10 NFG= 10 F= -.99860813D+00 G= .9839D-01 NIT= 9 NFV= 11 NFG= 11 F= -.99860813D+00 G= .1141D+00 NIT= 10 NFV= 12 NFG= 12 F= -.99928519D+00 G= .5014D+00 NIT= 11 NFV= 13 NFG= 13 F= -.99999999D+00 G= .5301D-01 NIT= 12 NFV= 14 NFG= 14 F= -.99999999D+00 G= .2704D-06 NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 EXIT FROM PBUN : NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4 X= .1000000D+01 .0000000D+00 6. Verification of the subroutines: ----------------------------------- Subroutine PBUNU can be verified and tested using the program TBUNU. This program calls the subroutines TIUD19 (initiation), TFFU19 (function evaluation) and TFGU19 (subgradient evaluation) containing 20 academic test problems with at most 50 variables [4]. The results obtained by the program TBUNU on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 42 NFV= 45 NFG= 45 F= .38117065D-06 G= .1135D-02 ITERM= 2 NIT= 18 NFV= 20 NFG= 20 F= -.22203912D-16 G= .8975D-08 ITERM= 2 NIT= 31 NFV= 33 NFG= 33 F= .19522245D+01 G= .3085D-03 ITERM= 2 NIT= 14 NFV= 16 NFG= 16 F= .20000000D+01 G= .1921D-06 ITERM= 2 NIT= 17 NFV= 19 NFG= 19 F= -.30000000D+01 G= .5564D-08 ITERM= 4 NIT= 13 NFV= 15 NFG= 15 F= .72000015D+01 G= .2212D-02 ITERM= 4 NIT= 11 NFV= 12 NFG= 12 F= -.14142136D+01 G= .1437D-04 ITERM= 4 NIT= 66 NFV= 68 NFG= 68 F= -.99999941D+00 G= .1089D-02 ITERM= 4 NIT= 13 NFV= 15 NFG= 15 F= -.10000000D+01 G= .9859D-07 ITERM= 4 NIT= 43 NFV= 46 NFG= 46 F= -.80000000D+01 G= .1282D-02 ITERM= 4 NIT= 43 NFV= 45 NFG= 45 F= -.43999999D+02 G= .3734D-02 ITERM= 2 NIT= 27 NFV= 29 NFG= 29 F= .22600162D+02 G= .1451D-03 ITERM= 4 NIT= 60 NFV= 62 NFG= 62 F= -.32348679D+02 G= .2190D-02 ITERM= 2 NIT= 117 NFV= 118 NFG= 118 F= -.29196928D+01 G= .1683D-02 ITERM= 2 NIT= 92 NFV= 93 NFG= 93 F= .55981567D+00 G= .8266D-03 ITERM= 4 NIT= 74 NFV= 75 NFG= 75 F= -.84140829D+00 G= .7236D-03 ITERM= 2 NIT= 157 NFV= 159 NFG= 159 F= .97857727D+01 G= .6510D-03 ITERM= 2 NIT= 89 NFV= 94 NFG= 94 F= .16703858D+02 G= .3694D-02 ITERM= 2 NIT= 150 NFV= 151 NFG= 151 F= .16712381D-06 G= .7782D-04 ITERM= 2 NIT= 39 NFV= 40 NFG= 40 F= .12440973D-12 G= .2969D-01 ITERM= 2 The rows corresponding to individual test problems contain the number of iterations NIT, the number of function evaluations NFV, the number of gradient evaluations NFG, the final value of the objective function F, the value of the criterion for the termination G and the cause of termination ITERM. Subroutine PBUNL can be verified and tested using the program TBUNL. This program calls the subroutines TIUD22 (initiation), TAFU22 (function evaluation), TAGU22 (subgradient evaluation) containing 10 academic test problems with at most 20 variables [4]. The results obtained by the program TBUNL on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 10 NFV= 11 NFG= 11 F= -.38965952D+00 G= .4532D-04 ITERM= 4 NIT= 4 NFV= 5 NFG= 5 F= -.33035714D+00 G= .3220D-14 ITERM= 4 NIT= 8 NFV= 10 NFG= 10 F= -.44891079D+00 G= .6982D-03 ITERM= 4 NIT= 79 NFV= 80 NFG= 80 F= -.42928061D+00 G= .7703D-05 ITERM= 2 NIT= 16 NFV= 17 NFG= 17 F= -.18596138D+01 G= .2017D-09 ITERM= 2 NIT= 16 NFV= 17 NFG= 17 F= .10183089D+00 G= .1272D-06 ITERM= 2 NIT= 43 NFV= 44 NFG= 44 F= .28724436D-08 G= .1966D-07 ITERM= 2 NIT= 74 NFV= 76 NFG= 76 F= .24306219D+02 G= .5835D-02 ITERM= 4 NIT= 140 NFV= 143 NFG= 143 F= .13372840D+03 G= .2872D-01 ITERM= 2 NIT= 65 NFV= 68 NFG= 68 F= .50694798D+00 G= .3576D-04 ITERM= 2 References: ----------- [1] Luksan L.: Dual Method for Solving a Special Problem of Quadratic Programming as a Subproblem at Linearly Constrained Nonlinear Minimax Approximation. Kybernetika 20 (1984) 445-457. [2] Vlcek J.: Bundle Algorithms for Nonsmooth Unconstrained Minimization. Research Report V-608, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 1995. [3] Luksan L., Vlcek J.: NDA: Algorithms for Nondifferentiable Optimization. Research Report V-797, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000. [4] Luksan L., Vlcek J.: Subroutines for Testing Nonsmooth Unconstrained and Linearly Constrained Optimization Problems. Research Report V-798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000.  SHAR_EOF fi # end of overwriting check if test -f 'pmin.txt' then echo shar: will not over-write existing file "'pmin.txt'" else cat << "SHAR_EOF" > 'pmin.txt' *********************************************************************** * * * PMIN - A RECURSUVE QUADRATIC PROGRAMMING VARIABLE * * METRIC ALGORITHM FOR MINIMAX OPTIMIZATION. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PMIN is designed to find a close approximation to a local minimum of a special objective function F(X) = MAX ( F_I(X) ) , 1 <= i <= NA with simple bounds on variables and general linear constraints. Here X is a vector of N variables and F_I(X), 1 <= I <= NA, are twice continuously differentiable functions. Simple bounds are assumed in the form X(I) unbounded if IX(I) = 0, XL(I) <= X(I) if IX(I) = 1, X(I) <= XU(I) if IX(I) = 2, XL(I) <= X(I) <= XU(I) if IX(I) = 3, XL(I) = X(I) = XU(I) if IX(I) = 5, where 1 <= I <= N. General linear constraints are assumed in the form C(I) unbounded if IC(I) = 0, CL(I) <= C(I) if IC(I) = 1, C(I) <= CU(I) if IC(I) = 2, CL(I) <= C(I) <= CU(I) if IC(I) = 3, CL(I) = C(I) = CU(I) if IC(I) = 5, where C(I) = A_I*X, 1 <= I <= NC, are linear functions. To simplify user's work, three additional easy to use subroutines are added. They call the basic general subroutine PMIN: PMINU - unconstrained minimax optimization, PMINS - minimax optimization with simple bounds, PMINL - minimax optimization with simple bounds and general linear constraints. All subroutines contain a description of formal parameters and extensive comments. Furthermore, two test programs TMINU and TMINL are included, which contain several test problems (see e.g. [4]). These test programs serve as examples for using the subroutines, verify their correctness and demonstrate their efficiency. In this short guide, we describe all subroutines which can be called from the user's program. A detailed description of methods is given in [2] and [3]. In the description of formal parameters, we introduce a type of the argument that specifies whether the argument must have a value defined on entry to the subroutine (I), whether it is a value which will be returned (O), or both (U), or whether it is an auxiliary value (A). Note that the arguments of the type I can be changed on output under some circumstances, especially if improper input values were given. Besides formal parameters, we can use a COMMON /STAT/ block containing statistical information. This block, used in each subroutine has the following form: COMMON /STAT/ NDECF,NRES,NRED,NREM,NADD,NIT,NFV,NFG,NFH The arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- NDECF O Positive INTEGER variable that indicates the number of matrix decompositions. NRES O Positive INTEGER variable that indicates the number of restarts. NRED O Positive INTEGER variable that indicates the number of reductions. NREM O Positive INTEGER variable that indicates the number of constraint deletions during the QP solutions. NADD O Positive INTEGER variable that indicates the number of constraint additions during the QP solutions. NIT O Positive INTEGER variable that indicates the number of iterations. NFV O Positive INTEGER variable that indicates the number of function evaluations. NFG O Positive INTEGER variable that specifies the number of gradient evaluations. NFH O Positive INTEGER variable that specifies the number of Hessian evaluations. 2. Subroutines PMINU, PMINS, PMINL: ----------------------------------- The calling sequences are CALL PMINU(NF,NA,X,AF,IA,RA,IPAR,RPAR,F,GMAX,IEXT,ITERM) CALL PMINS(NF,NA,NB,X,IX,XL,XU,AF,IA,RA,IPAR,RPAR,F, & GMAX,IEXT,ITERM) CALL PMINL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,AF, & IA,RA,IPAR,RPAR,F,GMAX,IEXT,ITERM) The arguments have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. NA I Nonnegative INTEGER variable that specifies the number of functions in the minimax criterion. NB I Nonnegative INTEGER variable that specifies whether the simple bounds are suppressed (NB=0) or accepted (NB>0). NC I Nonnegative INTEGER variable that specifies the number of linear constraints; if NC=0 the linear constraints are suppressed. X(NF) U On input, DOUBLE PRECISION vector with the initial estimate to the solution. On output, the approximation to the minimum. IX(NF) I On input (significant only if NB>0) INTEGER vector containing the simple bounds types: IX(I)=0 - the variable X(I) is unbounded, IX(I)=1 - the lower bound X(I) >= XL(I), IX(I)=2 - the upper bound X(I) <= XU(I), IX(I)=3 - the two side bound XL(I) <= X(I) <= XU(I), IX(I)=5 - the variable X(I) is fixed (given by its initial estimate). XL(NF) I DOUBLE PRECISION vector with lower bounds for variables (significant only if NB>0). XU(NF) I DOUBLE PRECISION vector with upper bounds for variables (significant only if NB>0). CF(NC) A DOUBLE PRECISION vector which contains values of constraint functions (only if NC>0). IC(NC) I On input (significant only if NC>0) INTEGER vector which contains constraint types: IC(K)=0 - the constraint CF(K) is not used, IC(K)=1 - the lower constraint CF(K) >= CL(K), IC(K)=2 - the upper constraint CF(K) <= CU(K), IC(K)=3 - the two side constraint CL(K) <= CF(K) <= CU(K), IC(K)=5 - the equality constraint CF(K) = CL(K). CL(NC) I DOUBLE PRECISION vector with lower bounds for constraint functions (significant only if NC>0). CU(NC) I DOUBLE PRECISION vector with upper bounds for constraint functions (significant only if NC>0). CG(NF*NC) I DOUBLE PRECISION matrix whose columns are normals of the linear constraints (significant only if NC>0). AF(NA) O DOUBLE PRECISION vector which contains values of functions in the minimax criterion. IA(NIA) A INTEGER working array of the dimension of at least NIA=NF+NA+1. RA(NRA) A DOUBLE PRECISION working array of the dimension of at least NRA=(NF+NA+8)*NF+2*NA+4. IPAR(7) A INTEGER parameters: IPAR(1)=MET, IPAR(2)=MES, IPAR(3)=MEC, IPAR(4)=MER, IPAR(5)=MIT, IPAR(6)=MFV, IPAR(7)=IPRNT. Parameters MET, MEC, MER, MES, MIT, MFV, IPRNT are described in Section 3 together with other parameters of the subroutine PMIN. RPAR(7) A DOUBLE PRECISION parameters: RPAR(1)=TOLX, RPAR(2)=TOLF, RPAR(3)=TOLB, RPAR(4)=TOLG, RPAR(5)=TOLD, RPAR(6)=TOLS, RPAR(7)=XMAX. Parameters TOLX, TOLF, TOLB, TOLG, TOLD, TOLS, XMAX are described in Section 3 together with other parameters of the subroutine PMIN. F O DOUBLE PRECISION value of the objective function at the solution X. GMAX O DOUBLE PRECISION maximum absolute value of a partial derivative of the Lagrangian function. IEXT I INTEGER variable that specifies the minimax criterion: IEXT < 0 - maximum of positive values, IEXT = 0 - maximum of absolute values, IEXT > 0 - maximum of negative values. ITERM O INTEGER variable that indicates the cause of termination: ITERM= 1 - if |X - XO| was less than or equal to TOLX in MTESX subsequent iterations, ITERM= 2 - if |F - FO| was less than or equal to TOLF in MTESF subsequent iterations, ITERM= 3 - if F is less than or equal to TOLB, ITERM= 4 - if GMAX is less than or equal to TOLG, ITERM=11 - if NFV exceeded MFV, ITERM=12 - if NIT exceeded MIT, ITERM< 0 - if the method failed. The subroutines PMINU, PMINS, PMINL require the user supplied subroutines FUN and DER that define the values and the gradients of the functions in the minimax criterion and have the form SUBROUTINE FUN(NF,KA,X,FA) SUBROUTINE DER(NF,KA,X,GA) The arguments of user supplied subroutines have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. KA I Positive INTEGER variable that specifies the index of a function in the minimax criterion. X(NF) I DOUBLE PRECISION an estimate to the solution. FA O DOUBLE PRECISION value of a function with the index KA at the point X. GA(NF) O DOUBLE PRECISION gradient of a function with the index KA at the point X. 3. Subroutine PMIN: ------------------- This general subroutine is called from all the subroutines described in Section 2. The calling sequence is CALL PMIN(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,AF,IA,AFO,AFD, & GA,AG,IAA,AR,AZ,G,H,S,XO,GO,TOLX,TOLF,TOLB,TOLG,TOLD,TOLS, & XMAX,GMAX,F,IEXT,MET,MEC,MER,MES,MIT,MFV,IPRNT,ITERM). The arguments NF, NA, NB, NC, X, IX, XL, XU, CF, IC, CL, CU, CG, AF, GMAX, F, IEXT, ITERM, have the same meaning as in Section 2. Other arguments have the following meaning: Argument Type Significance --------------------------------------------------------------------- IA(NA) A INTEGER vector containing types of functions in the minimax criterion. AFO(NA) A DOUBLE PRECISION vector of saved values of functions in the minimax criterion. AFD(NA) A DOUBLE PRECISION vector of increments of functions in the minimax criterion. GA(NF) A DOUBLE PRECISION gradient of the selected function in the minimax criterion. AG(NF*NA) A DOUBLE PRECISION matrix whose columns are gradients of functions in the minimax criterion. IAA(NA) A INTEGER vector containing indices of active functions. AR(NAR) A DOUBLE PRECISION matrix containing triangular decomposition of the orthogonal projection kernel (NAR is equal to (NF+1)*(NF+2)/2). AZ(NF+1) A DOUBLE PRECISION vector of Lagrange multipliers. GF(NF) A DOUBLE PRECISION gradient of the Lagrangian function. H(NH) A DOUBLE PRECISION Hessian matrix of the Lagrangian function (NH is equal to NF*(NF+1)/2). S(NF+1) A DOUBLE PRECISION direction vector. XO(NF) A DOUBLE PRECISION vector which contains increments of variables. GO(NF+1) A DOUBLE PRECISION vector which contains increments of partial derivatives. TOLX I DOUBLE PRECISION tolerance for the change of the coordinate vector X; the choice TOLX=0 causes that the default value 1.0D-16 will be taken. TOLF I DOUBLE PRECISION tolerance for the change of function values; the choice TOLF=0 causes that the default value 1.0D-8 will be taken. TOLB I DOUBLE PRECISION minimum acceptable function value; the choice TOLB=0 causes that the default value -1.0D60 will be taken. TOLG I DOUBLE PRECISION tolerance for the Lagrangian function gradient; the choice TOLG=0 causes that the default value 1.0D-6 will be taken. TOLD I DOUBLE PRECISION tolerance for a descent direction; the choice TOLD= 0 causes that the default value 1.0D-4 will be taken. TOLS I DOUBLE PRECISION tolerance parameter for a function decrease in a line search; the choice TOLS=0 causes that the default value 1.0D-2 will be taken. XMAX I DOUBLE PRECISION maximum stepsize; the choice XMAX=0 causes that the default value 1.0D3 will be taken. MET I INTEGER variable that specifies self scaling for variable metric updates: MET=1 - self scaling is suppressed, MET=2 - self scaling is used only in the first iteration (initial self scaling), MET=3 - self scaling in controlled by a special procedure. The choice MET=0 causes that the default value MET=2 will be taken. MEC I INTEGER variable that specifies correction for variable metric updates if a negative curvature occurs: MEC=1 - correction is suppressed, MEC=2 - Powell's correction is used. The choice MEC=0 causes that the default value MEC=1 will be taken. MER I INTEGER variable that specifies restart after unsuccesfull variable metric updates: MER=0 - restart is suppressed, MER=1 - restart is performed. MES I INTEGER variable that specifies the interpolation method selection in a line search: MES=1 - bisection, MES=2 - two point quadratic interpolation, MES=3 - three point quadratic interpolation, MES=4 - three point cubic interpolation. The choice MES=0 causes that the default value MES=1 will be taken. MIT I INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 200 will be taken. MFV I INTEGER variable that specifies the maximum number of function evaluations; the choice |MFV|=0 causes that the default value 500 will be taken. IPRNT I INTEGER variable that specifies PRINT: IPRNT= 0 - print is suppressed, IPRNT= 1 - basic print of final results, IPRNT=-1 - extended print of final results, IPRNT= 2 - basic print of intermediate and final results, IPRNT=-2 - extended print of intermediate and final results. The subroutine PMIN has a modular structure. The following list contains its most important subroutines: PA1MX2 - Minimax criterion evaluation. PDDXQ1 - Determination of the descent direction using quadratic programming subroutine. PLQDF1 - Dual range space method for solving a quadratic programming subproblem with linear constraints (see [1]). PS0LA2 - Line search using only function values. PUDBG1 - The BFGS variable metric update applied to the Choleski decomposition of the approximate Hessian matrix. The subroutine PMIN requires the user supplied subroutines FUN and DER which are described in Section 2. 4. Subroutine PLQDF1: --------------------- Since the dual range space method for special quadratic programming subproblems arising in nonlinear minimax optimization can be used separately in many applications (e.g. in bundle-type methods for nonsmooth optimization), we describe the subroutine ULQDF1 in more details. The calling sequence is CALL PLQDF1(NF,NA,NC,X,IX,XL,XU,AF,AFD,IA,IAA,AG,AR,AZ, & CF,IC,CL,CU,CG,G,H,S,MFP,KBF,KBC,IDECF,ETA0,ETA2,ETA9, & EPS7,EPS9,XNORM,UMAX,GMAX,N,ITERQ) The arguments NF, NA, NC, X, IX, XL, XU, AF, CF, IC, CL, CU, CG have the same meaning as in Section 2 (only with the difference that the arguments X and AF are of the type (I), i.e. they must have a value defined on entry to ULQDF1 and they are not changed). The arguments AFD, IA, IAA, AG, AR, AZ have the same meaning as in Section 3 (only with the difference that the arguments AFD, IAA, AR, AZ are of the type (O), i.e. their values can be used subsequently). Other arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- G(NF+1) O DOUBLE PRECISION gradient of the Lagrangian function. H(NH) U DOUBLE PRECISION Choleski decomposition of the approximate Hessian (NH is equal to NF*(NF+1)/2). S(NF+1) O DOUBLE PRECISION direction vector. MFP I INTEGER variable that specifies the type of the computed point. MFP=1 - computation is terminated whenever an arbitrary feasible point is found, MFP=2 - computation is terminated whenever an optimum feasible point is found, MFP=3 - computation starts from the previously reached point and is terminated whenever an optimum feasible point is found. KBF I INTEGER variable that specifies simple bounds on variables. KBF=0 - simple bounds are suppressed, KBF=1 - one sided simple bounds, KBF=2 - two sided simple bounds. KBC I INTEGER variable that specifies general linear constraints. KBC=0 - linear constraints are suppressed, KBC=1 - one sided linear constraints, KBC=2 - two sided linear constraints. IDECF U INTEGER variable that specifies the type of matrix decomposition. IDECF= 0 - no decomposition, IDECF= 1 - Choleski decomposition, IDECF= 9 - inversion, IDECF=10 - diagonal matrix. ETA0 I DOUBLE PRECISION machine precision (the recommended value is 1.0D-15. ETA2 I DOUBLE PRECISION tolerance for positive definiteness in the Choleski decomposition. ETA9 I DOUBLE PRECISION maximum floating point number. EPS7 I DOUBLE PRECISION tolerance for linear independence of constraints (the recommended value is 1.0D-10). EPS9 I DOUBLE PRECISION tolerance for the definition of active constraints (the recommended value is 1.0D-8). XNORM O DOUBLE PRECISION value of the linearized minimax function. UMAX O DOUBLE PRECISION maximum absolute value of the negative Lagrange multiplier. GMAX O DOUBLE PRECISION infinity norm of the gradient of the Lagrangian function. N O INTEGER dimension of a manifold defined by active constraints. ITERQ O INTEGER variable that indicates the type of the computed feasible point. ITERQ= 1 - an arbitrary feasible point was found, ITERQ= 2 - the optimum feasible point was found, ITERQ=-1 - an arbitrary feasible point does not exist, ITERQ=-2 - the optimum feasible point does not exist. 5. Form of printed results: --------------------------- The form of printed results is specified by the parameter IPRNT as is described in Section 3. Here we demonstrate individual forms of printed results by the simple use of the program TMINL described in the next section (with NEXT=6). If we set IPRNT=1, then the printed results will have the form NIT= 15 NFV= 16 NFG= 15 F= .50694800D+00 G= .1488D-10 ITERM= 4 If we set IPRNT=-1, then the printed results will have the form EXIT FROM PMIN : NIT= 15 NFV= 16 NFG= 15 F= .50694800D+00 G= .1488D-10 ITERM= 4 X = .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.5069240D+00 If we set IPRNT=2, then the printed results will have the form ENTRY TO PMIN : NIT= 0 NFV= 1 NFG= 1 F= .21899000D+05 G= .1000D+61 NIT= 1 NFV= 2 NFG= 2 F= .13670000D+05 G= .2200D+02 NIT= 2 NFV= 3 NFG= 3 F= .35097538D+04 G= .1050D+02 NIT= 3 NFV= 4 NFG= 4 F= .90439182D+03 G= .2476D+01 NIT= 4 NFV= 5 NFG= 5 F= .21124136D+03 G= .4935D+00 NIT= 5 NFV= 6 NFG= 6 F= .36315848D+02 G= .1027D+00 NIT= 6 NFV= 7 NFG= 7 F= .33929080D+01 G= .3163D-01 NIT= 7 NFV= 8 NFG= 8 F= .82287170D+00 G= .1223D-01 NIT= 8 NFV= 9 NFG= 9 F= .73088967D+00 G= .4031D-01 NIT= 9 NFV= 10 NFG= 10 F= .69140770D+00 G= .6643D-01 NIT= 10 NFV= 11 NFG= 11 F= .56052270D+00 G= .1179D+00 NIT= 11 NFV= 13 NFG= 12 F= .53014436D+00 G= .1019D-01 NIT= 12 NFV= 14 NFG= 13 F= .52097640D+00 G= .2339D-01 NIT= 13 NFV= 15 NFG= 14 F= .50698339D+00 G= .8925D-03 NIT= 14 NFV= 16 NFG= 15 F= .50694800D+00 G= .2328D-05 EXIT FROM PMIN : NIT= 15 NFV= 16 NFG= 15 F= .50694800D+00 G= .1488D-10 ITERM= 4 If we set IPRNT=-2, then the printed results will have the form ENTRY TO PMIN : NIT= 0 NFV= 1 NFG= 1 F= .21899000D+05 G= .1000D+61 NIT= 1 NFV= 2 NFG= 2 F= .13670000D+05 G= .2200D+02 NIT= 2 NFV= 3 NFG= 3 F= .35097538D+04 G= .1050D+02 NIT= 3 NFV= 4 NFG= 4 F= .90439182D+03 G= .2476D+01 NIT= 4 NFV= 5 NFG= 5 F= .21124136D+03 G= .4935D+00 NIT= 5 NFV= 6 NFG= 6 F= .36315848D+02 G= .1027D+00 NIT= 6 NFV= 7 NFG= 7 F= .33929080D+01 G= .3163D-01 NIT= 7 NFV= 8 NFG= 8 F= .82287170D+00 G= .1223D-01 NIT= 8 NFV= 9 NFG= 9 F= .73088967D+00 G= .4031D-01 NIT= 9 NFV= 10 NFG= 10 F= .69140770D+00 G= .6643D-01 NIT= 10 NFV= 11 NFG= 11 F= .56052270D+00 G= .1179D+00 NIT= 11 NFV= 13 NFG= 12 F= .53014436D+00 G= .1019D-01 NIT= 12 NFV= 14 NFG= 13 F= .52097640D+00 G= .2339D-01 NIT= 13 NFV= 15 NFG= 14 F= .50698339D+00 G= .8925D-03 NIT= 14 NFV= 16 NFG= 15 F= .50694800D+00 G= .2328D-05 EXIT FROM PMIN: NIT= 15 NFV= 16 NFG= 15 F= .50694800D+00 G= .1488D-10 ITERM= 4 X = .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 .5000000D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.4166693D+00 -.5069240D+00 6. Verification of the subroutines: ----------------------------------- Subroutine PMINU can be verified and tested using the program TMINU. This program calls the subroutines TIUD06 (initiation), TAFU06 (function evaluation) and TAGU06 (subgradient evaluation) containing 25 academic test problems with at most 20 variables [4]. The results obtained by the program TMINU on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 7 NFV= 8 NFG= 8 F= .19522245D+01 G= .1041D-07 ITERM= 4 NIT= 7 NFV= 8 NFG= 8 F= .97131735D-09 G= .1066D-13 ITERM= 4 NIT= 93 NFV= 195 NFG= 94 F= .27312458D-10 G= .6955D-06 ITERM= 4 NIT= 13 NFV= 15 NFG= 14 F= .35997193D+01 G= .2498D-07 ITERM= 4 NIT= 11 NFV= 16 NFG= 12 F= -.44000000D+02 G= .2507D-06 ITERM= 4 NIT= 12 NFV= 21 NFG= 13 F= -.44000000D+02 G= .8543D-06 ITERM= 4 NIT= 8 NFV= 9 NFG= 9 F= .42021427D-02 G= .6568D-09 ITERM= 4 NIT= 5 NFV= 6 NFG= 6 F= .50816327D-01 G= .1502D-06 ITERM= 4 NIT= 10 NFV= 12 NFG= 11 F= .80843684D-02 G= .1874D-08 ITERM= 4 NIT= 11 NFV= 11 NFG= 11 F= .11570644D+03 G= .7077D-08 ITERM= 4 NIT= 35 NFV= 113 NFG= 36 F= .26359735D-02 G= .1488D-07 ITERM= 4 NIT= 34 NFV= 86 NFG= 35 F= .20160756D-02 G= .1581D-08 ITERM= 4 NIT= 7 NFV= 8 NFG= 8 F= .99665116D-05 G= .4525D-06 ITERM= 4 NIT= 6 NFV= 8 NFG= 7 F= .12237126D-03 G= .6839D-07 ITERM= 4 NIT= 16 NFV= 37 NFG= 16 F= .22340496D-01 G= .1960D-13 ITERM= 4 NIT= 21 NFV= 53 NFG= 22 F= .34904927D-01 G= .2522D-07 ITERM= 4 NIT= 11 NFV= 15 NFG= 12 F= .19729063D+00 G= .8187D-07 ITERM= 4 NIT= 17 NFV= 86 NFG= 18 F= .61852848D-02 G= .3254D-06 ITERM= 4 NIT= 19 NFV= 33 NFG= 20 F= .68063006D+03 G= .5743D-06 ITERM= 4 NIT= 13 NFV= 19 NFG= 14 F= .24306209D+02 G= .9366D-07 ITERM= 4 NIT= 19 NFV= 26 NFG= 20 F= .13372828D+03 G= .5547D-06 ITERM= 4 NIT= 37 NFV= 53 NFG= 38 F= .54598150D+02 G= .1320D-06 ITERM= 4 NIT= 21 NFV= 31 NFG= 22 F= .26108258D+03 G= .7934D-06 ITERM= 4 NIT= 18 NFV= 20 NFG= 19 F= .91150394D-07 G= .5473D-06 ITERM= 4 NIT= 80 NFV= 327 NFG= 81 F= .48028792D-01 G= .9071D-06 ITERM= 4 The rows corresponding to individual test problems contain the number of iterations NIT, the number of function evaluations NFV, the number of gradient evaluations NFG, the final value of the objective function F, the value of the criterion for the termination G and the cause of termination ITERM. Subroutines PMINL can be verified and tested using the program TMINL. This program calls the subroutines TIUD22 (initiation), TAFU22 (function evaluation), TAGU22 (subgradient evaluation) containing 15 academic test problems with at most 20 variables [4]. The results obtained by the program TMINL on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 6 NFV= 7 NFG= 7 F= -.38965952D+00 G= .6129D-08 ITERM= 4 NIT= 5 NFV= 5 NFG= 5 F= -.33035714D+00 G= .0000D+00 ITERM= 4 NIT= 8 NFV= 8 NFG= 8 F= -.44891079D+00 G= .2032D-10 ITERM= 4 NIT= 75 NFV= 75 NFG= 75 F= -.42928061D+00 G= .4449D-10 ITERM= 4 NIT= 9 NFV= 9 NFG= 9 F= -.18596187D+01 G= .8299D-12 ITERM= 4 NIT= 7 NFV= 9 NFG= 8 F= .10183089D+00 G= .8211D-06 ITERM= 4 NIT= 7 NFV= 10 NFG= 8 F= .10658141D-13 G= .6600D-06 ITERM= 4 NIT= 15 NFV= 23 NFG= 16 F= .24306209D+02 G= .3499D-06 ITERM= 4 NIT= 21 NFV= 29 NFG= 22 F= .13372828D+03 G= .1829D-06 ITERM= 4 NIT= 15 NFV= 16 NFG= 15 F= .50694800D+00 G= .1488D-10 ITERM= 4 NIT= 15 NFV= 20 NFG= 16 F= .27608127D-03 G= .2149D-09 ITERM= 4 NIT= 154 NFV= 850 NFG= 155 F= -.17688070D+04 G= .1304D-07 ITERM= 4 NIT= 15 NFV= 22 NFG= 16 F= .12272261D+04 G= .1706D-06 ITERM= 4 NIT= 148 NFV= 273 NFG= 149 F= .70492480D+04 G= .4700D-07 ITERM= 4 NIT= 62 NFV= 107 NFG= 63 F= .17478699D+03 G= .7868D-07 ITERM= 4 References: ----------- [1] Luksan L.: Dual Method for Solving a Special Problem of Quadratic Programming as a Subproblem at Linearly Constrained Nonlinear Minimax Approximation. Kybernetika 20 (1984) 445-457. [2] Luksan L.: An Implementation of Recursive Quadratic Programming Variable Metric Methods for Linearly Constrained Nonlinear Minimax Approximation. Kybernetika 21 (1985) 22-40. [3] Luksan L., Vlcek J.: NDA: Algorithms for Nondifferentiable Optimization. Research Report V-797, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000. [4] Luksan L., Vlcek J.: Subroutines for Testing Nonsmooth Unconstrained and Linearly Constrained Optimization Problems. Research Report V-798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000.  SHAR_EOF fi # end of overwriting check if test -f 'pnew.txt' then echo shar: will not over-write existing file "'pnew.txt'" else cat << "SHAR_EOF" > 'pnew.txt' *********************************************************************** * * * PNEW - A BUNDLE-NEWTON ALGORITHM FOR NONSMOOTH * * UNCONSTRAINED OPTIMIZATION. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PNEW is designed to find a close approximation to a local minimum of a nonlinear nonsmooth function F(X) with simple bounds on variables and general linear constraints. Here X is a vector of N variables and F(X), is assumed to be a locally Lipschitz continuous function. Simple bounds are assumed in the form X(I) unbounded if IX(I) = 0, XL(I) <= X(I) if IX(I) = 1, X(I) <= XU(I) if IX(I) = 2, XL(I) <= X(I) <= XU(I) if IX(I) = 3, XL(I) = X(I) = XU(I) if IX(I) = 5, where 1 <= I <= N. General linear constraints are assumed in the form C(I) unbounded if IC(I) = 0, CL(I) <= C(I) if IC(I) = 1, C(I) <= CU(I) if IC(I) = 2, CL(I) <= C(I) <= CU(I) if IC(I) = 3, CL(I) = C(I) = CU(I) if IC(I) = 5, where C(I) = A_I*X, 1 <= I <= NC, are linear functions. To simplify user's work, three additional easy to use subroutines are added. They call the basic general subroutine PNEW: PNEWU - unconstrained nonsmooth optimization, PNEWS - nonsmooth optimization with simple bounds, PNEWL - nonsmooth optimization with simple bounds and general linear constraints. All subroutines contain a description of formal parameters and extensive comments. Furthermore, two test programs TNEWU and TNEWL are included, which contain several test problems (see e.g. [4]). These test programs serve as examples for using the subroutines, verify their correctness and demonstrate their efficiency. In this short guide, we describe all subroutines which can be called from the user's program. A detailed description of methods is given in [2] and [3]. In the description of formal parameters, we introduce a type of the argument that specifies whether the argument must have a value defined on entry to the subroutine (I), whether it is a value which will be returned (O), or both (U), or whether it is an auxiliary value (A). Note that the arguments of the type I can be changed on output under some circumstances, especially if improper input values were given. Besides formal parameters, we can use a COMMON /STAT/ block containing statistical information. This block, used in each subroutine has the following form: COMMON /STAT/ NDECF,NRES,NRED,NREM,NADD,NIT,NFV,NFG,NFH The arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- NDECF O Positive INTEGER variable that indicates the number of matrix decompositions. NRES O Positive INTEGER variable that indicates the number of restarts. NRED O Positive INTEGER variable that indicates the number of reductions. NREM O Positive INTEGER variable that indicates the number of constraint deletions during the QP solutions. NADD O Positive INTEGER variable that indicates the number of constraint additions during the QP solutions. NIT O Positive INTEGER variable that indicates the number of iterations. NFV O Positive INTEGER variable that indicates the number of function evaluations. NFG O Positive INTEGER variable that specifies the number of gradient evaluations. NFH O Positive INTEGER variable that specifies the number of Hessian evaluations. 2. Subroutines PNEWU, PNEWS, PNEWL: ----------------------------------- The calling sequences are CALL PNEWU(NF,NA,X,IA,RA,IPAR,RPAR,FP,GMAX,IHES,ITERM) CALL PNEWS(NF,NA,NB,X,IX,XL,XU,IA,RA,IPAR,RPAR,FP,GMAX,IHES, & ITERM) CALL PNEWL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,IA,RA,IPAR, & RPAR,FP,GMAX,IHES,ITERM) The arguments have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. NA I Nonnegative INTEGER variable that specifies the maximum bundle dimension. The choice NA=0 causes that the default value NA=NF+3 will be taken. NB I Nonnegative INTEGER variable that specifies whether the simple bounds are suppressed (NB=0) or accepted (NB>0). NC I Nonnegative INTEGER variable that specifies the number of linear constraints; if NC=0 the linear constraints are suppressed. X(NF) U On input, DOUBLE PRECISION vector with the initial estimate to the solution. On output, the approximation to the minimum. IX(NF) I On input (significant only if NB>0) INTEGER vector containing the simple bounds types: IX(I)=0 - the variable X(I) is unbounded, IX(I)=1 - the lower bound X(I) >= XL(I), IX(I)=2 - the upper bound X(I) <= XU(I), IX(I)=3 - the two side bound XL(I) <= X(I) <= XU(I), IX(I)=5 - the variable X(I) is fixed (given by its initial estimate). XL(NF) I DOUBLE PRECISION vector with lower bounds for variables (significant only if NB>0). XU(NF) I DOUBLE PRECISION vector with upper bounds for variables (significant only if NB>0). CF(NC) A DOUBLE PRECISION vector which contains values of constraint functions (only if NC>0). IC(NC) I On input (significant only if NC>0) INTEGER vector which contains constraint types: IC(K)=0 - the constraint CF(K) is not used, IC(K)=1 - the lower constraint CF(K) >= CL(K), IC(K)=2 - the upper constraint CF(K) <= CU(K), IC(K)=3 - the two side constraint CL(K) <= CF(K) <= CU(K), IC(K)=5 - the equality constraint CF(K) = CL(K). CL(NC) I DOUBLE PRECISION vector with lower bounds for constraint functions (significant only if NC>0). CU(NC) I DOUBLE PRECISION vector with upper bounds for constraint functions (significant only if NC>0). CG(NF*NC) I DOUBLE PRECISION matrix whose columns are normals of the linear constraints (significant only if NC>0). IA(NIA) A INTEGER working array of the dimension of at least NIA=NF+NA+1. RA(NRA) A DOUBLE PRECISION working array of the dimension of at least NRA=NF*(NF+1)*(NA+3)/2+NF*(NA+6)+5*NA+4. IPAR(7) A INTEGER parameters: IPAR(1)=MOS, IPAR(2)=MES, IPAR(3)=MTESX, IPAR(4)=MTESF, IPAR(5)=MIT, IPAR(6)=MFV, IPAR(7)=IPRNT. Parameters MOS, MES, MTESX, MTESF, MIT, MFV, IPRNT are described in Section 3 together with other parameters of the subroutine PNEW. RPAR(9) A DOUBLE PRECISION parameters: RPAR(1)=TOLX, RPAR(2)=TOLF, RPAR(3)=TOLB, RPAR(4)=TOLG, RPAR(5)=TOLD, RPAR(6)=TOLS, RPAR(7)=TOLP. RPAR(8)=ETA, RPAR(9)=XMAX. Parameters TOLX, TOLF, TOLB, TOLG, TOLD, TOLS, TOLP, ETA, XMAX are described in Section 3 together with other parameters of the subroutine PNEW. FP O DOUBLE PRECISION value of the objective function at the solution X. GMAX O DOUBLE PRECISION maximum absolute value of a partial derivative of the Lagrangian function. IHES I INTEGER variable that specifies a way for computing second derivatives: IHES=0 - numerical computation, IHES=1 - analytical computation by the user supplied subroutine HES. ITERM O INTEGER variable that indicates the cause of termination: ITERM= 1 - if |X - XO| was less than or equal to TOLX in MTESX subsequent iterations, ITERM= 2 - if |F - FO| was less than or equal to TOLF in MTESF subsequent iterations, ITERM= 3 - if F is less than or equal to TOLB, ITERM= 4 - if GMAX is less than or equal to TOLG, ITERM=11 - if NFV exceeded MFV, ITERM=12 - if NIT exceeded MIT, ITERM< 0 - if the method failed. The subroutines PNEWU, PNEWS, PNEWL require user supplied subroutines FUNDER and HES that defines the objective function, its subgradient and has the form SUBROUTINE FUNDER(NF,X,F,G) SUBROUTINE HES(NF,X,H) The arguments of user supplied subroutines have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. X(NF) I DOUBLE PRECISION an estimate to the solution. F O DOUBLE PRECISION value of the objective function at the point X. G(NF) O DOUBLE PRECISION subgradient of the objective function at the point X. H(NH) O DOUBLE PRECISION matrix containing the second order information at the point X (NH is equal to NF*(NF+1)/2). 3. Subroutine PNEW: ------------------- This general subroutine is called from all the subroutines described in Section 2. The calling sequence is CALL PNEW(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,AF,IA,AFD,AG, & IAA,AR,AZ,G,H,HF,AH,S,SO,XO,GO,TOLX,TOLF,TOLB,TOLG,TOLD,TOLS, & TOLP,ETA,XMAX,GMAX,FP,MOS,MES,MTESX,MTESF,MIT,MFV,IPRNT,IHES, & ITERM). The arguments NF, NA, NB, NC, X, IX, XL, XU, CF, IC, CL, CU, CG, GMAX, FP, ITERM, have the same meaning as in Section 2. Other arguments have the following meaning M is equal to NF*(NF+1)/2): Argument Type Significance --------------------------------------------------------------------- AF(5*NA) A DOUBLE PRECISION vector of bundle function values. IA(NA) A INTEGER vector containing types of bundle functions. AFD(NA) A DOUBLE PRECISION vector of bundle function increments. AG(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle gradients. IAA(NA) A INTEGER vector containing indices of active functions. AR(NAR) A DOUBLE PRECISION matrix containing triangular decomposition of the orthogonal projection kernel (NAR is equal to (NF+1)*(NF+2)/2). AZ(NF+1) A DOUBLE PRECISION vector of Lagrange multipliers. G(NF) A DOUBLE PRECISION subgradient of the objective function. H(NH) A DOUBLE PRECISION aggregate Hessian matrix (NH is equal to NF*(NF+1)/2). HF(NH) A DOUBLE PRECISION Hessian matrix of the objective function. AH(NA*NH) A DOUBLE PRECISION Bundle of Hessian matrices. S(NF+1) A DOUBLE PRECISION direction vector. SO(NF) A DOUBLE PRECISION auxiliary vector. XO(NF) A DOUBLE PRECISION vector which contains increments of variables. GO(NF+1) A DOUBLE PRECISION gradient of the Lagrangian function. TOLX I DOUBLE PRECISION tolerance for the change of the coordinate vector X; the choice TOLX=0 causes that the default value TOLX=1.0D-16 will be taken. TOLF I DOUBLE PRECISION tolerance for the change of function values; the choice TOLF=0 causes that the default value TOLF=1.0D-8 will be taken. TOLB I DOUBLE PRECISION minimum acceptable function value; the choice TOLB=0 causes that the default value TOLB=-1.0D60 will be taken. TOLG I DOUBLE PRECISION tolerance for the Lagrangian function gradient; the choice TOLG=0 causes that the default value TOLG=1.0D-6 will be taken. TOLD I DOUBLE PRECISION tolerance for a descent direction; the choice TOLD= 0 causes that the default value TOLD=1.0D-4 will be taken. TOLS I DOUBLE PRECISION tolerance parameter for a function decrease in a line search; the choice TOLS=0 causes that the default value TOLS=1.0D-2 will be taken. TOLP I DOUBLE PRECISION tolerance parameter for a significant modification of the next line search direction; the choice TOLP=0 causes that the default value TOLP=0.5D0 will be taken. ETA I DOUBLE PRECISION distance measure parameter. XMAX I DOUBLE PRECISION maximum stepsize; the choice XMAX=0 causes that the default value 1.0D3 will be taken. MOS I INTEGER distance measure exponent (MOS=1 or MOS=2). The choice MOS=0 causes that the default value MOS=1 will be taken. MES I INTEGER variable that specifies the interpolation method selection in a line search: MES=1 - bisection, MES=2 - two point quadratic interpolation, MES=3 - three point quadratic interpolation, MES=4 - three point cubic interpolation. The choice MES=0 causes that the default value MES=1 will be taken. MTESX I INTEGER variable that specifies the maximum number of iterations with changes of the coordinate vector X smaller than TOLX; the choice MTESX=0 causes that the default value MTESX=20 will be taken. MTESF I INTEGER variable that specifies the maximum number of iterations with changes of function values smaller than TOLF; the choice MTESF=0 causes that the default value MTESF=2 will be taken. MIT I INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 200 will be taken. MFV I INTEGER variable that specifies the maximum number of function evaluations; the choice |MFV|=0 causes that the default value 500 will be taken. IPRNT I INTEGER variable that specifies PRINT: IPRNT= 0 - print is suppressed, IPRNT= 1 - basic print of final results, IPRNT=-1 - extended print of final results, IPRNT= 2 - basic print of intermediate and final results, IPRNT=-2 - extended print of intermediate and final results. The subroutine PNEW has a modular structure. The following list contains its most important subroutines: PF1HS1 - Numerical computation of the Hessian matrix. PDDBQ2 - Determination of the descent direction using quadratic programming subroutine and bundle updating. PLQDF1 - Dual range space method for solving a quadratic programming subproblem with linear constraints (see [1]). PS1L05 - Line search using function values and derivatives. The subroutine PNEW requires the user supplied subroutine FUNDER which is described in Section 2. 4. Subroutine PLQDF1: --------------------- Since the dual range space method for special quadratic programming subproblems arising in bundle type nonsmooth optimization can be used separately in many applications (e.g. in minimax optimization), we describe the subroutine PLQDF1 in more details. The calling sequence is CALL PLQDF1(NF,NA,NC,X,IX,XL,XU,AF,AFD,IA,IAA,AG,AR,AZ, & CF,IC,CL,CU,CG,G,H,S,MFP,KBF,KBC,IDECF,ETA0,ETA2,ETA9, & EPS7,EPS9,XNORM,UMAX,GMAX,N,ITERQ) The arguments NF, NA, NC, X, IX, XL, XU, AF, CF, IC, CL, CU, CG have the same meaning as in Section 2 (only with the difference that the arguments X and AF are of the type (I), i.e. they must have a value defined on entry to PLQDF1 and they are not changed). The arguments AFD, IA, IAA, AG, AR, AZ have the same meaning as in Section 3 (only with the difference that the arguments AFD, IAA, AR, AZ are of the type (O), i.e. their values can be used subsequently). Other arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- G(NF+1) O DOUBLE PRECISION gradient of the Lagrangian function. H(NH) U DOUBLE PRECISION Choleski decomposition of the approximate Hessian matrix (NH is equal to NF*(NF+1)/2). S(NF+1) O DOUBLE PRECISION direction vector. MFP I INTEGER variable that specifies the type of the computed point. MFP=1 - computation is terminated whenever an arbitrary feasible point is found, MFP=2 - computation is terminated whenever an optimum feasible point is found, MFP=3 - computation starts from the previously reached point and is terminated whenever an optimum feasible point is found. KBF I INTEGER variable that specifies simple bounds on variables. KBF=0 - simple bounds are suppressed, KBF=1 - one sided simple bounds, KBF=2 - two sided simple bounds. KBC I INTEGER variable that specifies general linear constraints. KBC=0 - linear constraints are suppressed, KBC=1 - one sided linear constraints, KBC=2 - two sided linear constraints. IDECF U INTEGER variable that specifies the type of matrix decomposition. IDECF= 0 - no decomposition, IDECF= 1 - Choleski decomposition, IDECF= 9 - inversion, IDECF=10 - diagonal matrix. ETA0 I DOUBLE PRECISION machine precision (the recommended value is 1.0D-15. ETA2 I DOUBLE PRECISION tolerance for positive definiteness in the Choleski decomposition. ETA9 I DOUBLE PRECISION maximum floating point number. EPS7 I DOUBLE PRECISION tolerance for linear independence of constraints (the recommended value is 1.0D-10). EPS9 I DOUBLE PRECISION tolerance for the definition of active constraints (the recommended value is 1.0D-8). XNORM O DOUBLE PRECISION value of the linearized minimax function. UMAX O DOUBLE PRECISION maximum absolute value of the negative Lagrange multiplier. GMAX O DOUBLE PRECISION infinity norm of the gradient of the Lagrangian function. N O INTEGER dimension of a manifold defined by active constraints. ITERQ O INTEGER variable that indicates the type of the computed feasible point. ITERQ= 1 - an arbitrary feasible point was found, ITERQ= 2 - the optimum feasible point was found, ITERQ=-1 - an arbitrary feasible point does not exist, ITERQ=-2 - the optimum feasible point does not exist. 5. Form of printed results: --------------------------- The form of printed results is specified by the parameter IPRNT as is described in Section 3. Here we demonstrate individual forms of printed results by the simple use of the program TNEWU described in the next section (with NEXT=16). If we set IPRNT=1, then the printed results will have the form NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 ITERM= 4 If we set IPRNT=-1, then the printed results will have the form EXIT FROM PNEW : NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 ITERM= 4 X= -.1262566D+00 -.3437830D-01 -.6857198D-02 .2636066D-01 .6729492D-01 -.2783995D+00 .7421866D-01 .1385240D+00 .8403122D-01 .3858031D-01 If we set IPRNT=2, then the printed results will have the form ENTRY TO PNEW : NIT= 0 NFV= 1 NFG= 1 F= .00000000D+00 G= .1000D+61 NIT= 1 NFV= 3 NFG= 3 F= .53370664D+04 G= .1200D+05 NIT= 2 NFV= 4 NFG= 4 F= .66499712D+02 G= .2610D+02 NIT= 3 NFV= 5 NFG= 5 F= .33934270D+02 G= .3771D+02 NIT= 4 NFV= 6 NFG= 6 F= .12040341D+01 G= .3214D+01 NIT= 5 NFV= 7 NFG= 7 F= .51324695D+00 G= .1459D+01 NIT= 6 NFV= 8 NFG= 8 F= -.76915236D+00 G= .2347D+01 NIT= 7 NFV= 9 NFG= 9 F= -.83859100D+00 G= .2683D+00 NIT= 8 NFV= 10 NFG= 10 F= -.84140726D+00 G= .2491D-02 NIT= 9 NFV= 11 NFG= 11 F= -.84140726D+00 G= .3213D-03 NIT= 10 NFV= 12 NFG= 12 F= -.84140726D+00 G= .4862D-03 NIT= 11 NFV= 13 NFG= 13 F= -.84140726D+00 G= .5265D-05 NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 EXIT FROM PNEW : NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 ITERM= 4 If we set IPRNT=-2, then the printed results will have the form ENTRY TO PNEW : NIT= 0 NFV= 1 NFG= 1 F= .00000000D+00 G= .1000D+61 NIT= 1 NFV= 3 NFG= 3 F= .53370664D+04 G= .1200D+05 NIT= 2 NFV= 4 NFG= 4 F= .66499712D+02 G= .2610D+02 NIT= 3 NFV= 5 NFG= 5 F= .33934270D+02 G= .3771D+02 NIT= 4 NFV= 6 NFG= 6 F= .12040341D+01 G= .3214D+01 NIT= 5 NFV= 7 NFG= 7 F= .51324695D+00 G= .1459D+01 NIT= 6 NFV= 8 NFG= 8 F= -.76915236D+00 G= .2347D+01 NIT= 7 NFV= 9 NFG= 9 F= -.83859100D+00 G= .2683D+00 NIT= 8 NFV= 10 NFG= 10 F= -.84140726D+00 G= .2491D-02 NIT= 9 NFV= 11 NFG= 11 F= -.84140726D+00 G= .3213D-03 NIT= 10 NFV= 12 NFG= 12 F= -.84140726D+00 G= .4862D-03 NIT= 11 NFV= 13 NFG= 13 F= -.84140726D+00 G= .5265D-05 NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 EXIT FROM PNEW : NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 ITERM= 4 X= -.1262566D+00 -.3437830D-01 -.6857198D-02 .2636066D-01 .6729492D-01 -.2783995D+00 .7421866D-01 .1385240D+00 .8403122D-01 .3858031D-01 6. Verification of the subroutines: ----------------------------------- Subroutine PNEWU can be verified and tested using the program TNEWU. This program calls the subroutines TIUD19 (initiation), TFFU19 (function evaluation) and TFGU19 (subgradient evaluation) containing 20 academic test problems with at most 50 variables [4]. The results obtained by the program TNEWU on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 58 NFV= 59 NFG= 59 F= .22533111D-15 G= .8624D-05 ITERM= 2 NIT= 7 NFV= 8 NFG= 8 F= .16765701D-10 G= .5792D-05 ITERM= 4 NIT= 9 NFV= 10 NFG= 10 F= .19522245D+01 G= .2172D-05 ITERM= 4 NIT= 10 NFV= 11 NFG= 11 F= .20000068D+01 G= .2161D-04 ITERM= 4 NIT= 14 NFV= 15 NFG= 15 F= -.30000000D+01 G= .5398D-08 ITERM= 2 NIT= 4 NFV= 6 NFG= 6 F= .72000000D+01 G= .1445D-08 ITERM= 4 NIT= 16 NFV= 17 NFG= 17 F= -.14142136D+01 G= .5653D-07 ITERM= 4 NIT= 11 NFV= 13 NFG= 13 F= -.10000000D+01 G= .4158D-07 ITERM= 4 NIT= 10 NFV= 11 NFG= 11 F= -.10000000D+01 G= .4562D-06 ITERM= 4 NIT= 25 NFV= 26 NFG= 26 F= -.79999999D+01 G= .3813D-02 ITERM= 4 NIT= 13 NFV= 15 NFG= 15 F= -.44000000D+02 G= .4215D-05 ITERM= 4 NIT= 7 NFV= 8 NFG= 8 F= .22600173D+02 G= .1263D-02 ITERM= 4 NIT= 22 NFV= 24 NFG= 24 F= -.32348679D+02 G= .3409D-02 ITERM= 4 NIT= 76 NFV= 77 NFG= 77 F= -.29197002D+01 G= .1061D-02 ITERM= 4 NIT= 89 NFV= 91 NFG= 91 F= .55981330D+00 G= .1528D-05 ITERM= 4 NIT= 12 NFV= 14 NFG= 14 F= -.84140833D+00 G= .6734D-06 ITERM= 4 NIT= 52 NFV= 53 NFG= 53 F= .97857721D+01 G= .2964D-03 ITERM= 4 NIT= 40 NFV= 42 NFG= 42 F= .16703855D+02 G= .1778D+00 ITERM= 4 NIT= 36 NFV= 37 NFG= 37 F= .38373702D-08 G= .5758D-08 ITERM= 2 NIT= 24 NFV= 25 NFG= 25 F= .45289427D-08 G= .1100D-09 ITERM= 2 The rows corresponding to individual test problems contain the number of iterations NIT, the number of function evaluations NFV, the number of gradient evaluations NFG, the final value of the objective function F, the value of the criterion for the termination G and the cause of termination ITERM. Subroutine PNEWL can be verified and tested using the program TNEWL. This program calls the subroutines TIUD22 (initiation), TAFU22 (function evaluation), TAGU22 (subgradient evaluation) containing 6 academic test problems with at most 20 variables [4]. The results obtained by the program TNEWL on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 6 NFV= 7 NFG= 7 F= -.38965952D+00 G= .1650D-07 ITERM= 4 NIT= 2 NFV= 13 NFG= 13 F= -.33035714D+00 G= .1110D-15 ITERM= 4 NIT= 49 NFV= 50 NFG= 50 F= -.44891079D+00 G= .1014D-06 ITERM= 4 NIT= 10 NFV= 11 NFG= 11 F= -.42928061D+00 G= .1269D-03 ITERM= 4 NIT= 5 NFV= 6 NFG= 6 F= -.18141200D+01 G= .1618D-03 ITERM= 4 NIT= 20 NFV= 21 NFG= 21 F= .10183089D+00 G= .2697D-06 ITERM= 4 NIT= 93 NFV= 98 NFG= 98 F= .53905532D-03 G= .3304D-06 ITERM= 2 NIT= 6 NFV= 7 NFG= 7 F= .24852881D+03 G= .1167D-06 ITERM= 4 NIT= 10 NFV= 12 NFG= 12 F= .35240929D+03 G= .6705D-04 ITERM= 4 NIT= 94 NFV= 98 NFG= 98 F= .50694800D+00 G= .4083D-06 ITERM= 4 References: ----------- [1] Luksan L.: Dual Method for Solving a Special Problem of Quadratic Programming as a Subproblem at Linearly Constrained Nonlinear Minimax Approximation. Kybernetika 20 (1984) 445-457. [2] Luksan L., Vlcek J.: A Bundle-Newton Method for Nonsmooth Unconstrained Minimization. Mathematical Programming. [3] Luksan L., Vlcek J.: NDA: Algorithms for Nondifferentiable Optimization. Research Report V-797, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000. [4] Luksan L., Vlcek J.: Subroutines for Testing Nonsmooth Unconstrained and Linearly Constrained Optimization Problems. Research Report V-798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000.  SHAR_EOF fi # end of overwriting check if test -f 'pvar.txt' then echo shar: will not over-write existing file "'pvar.txt'" else cat << "SHAR_EOF" > 'pvar.txt' *********************************************************************** * * * PVAR - A PROXIMAL BUNDLE ALGORITHM FOR NONSMOOTH * * OPTIMIZATION. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PVAR is designed to find a close approximation to a local minimum of a nonlinear nonsmooth function F(X) with simple bounds on variables and general linear constraints. Here X is a vector of N variables and F(X), is assumed to be a locally Lipschitz continuous function. Simple bounds are assumed in the form X(I) unbounded if IX(I) = 0, XL(I) <= X(I) if IX(I) = 1, X(I) <= XU(I) if IX(I) = 2, XL(I) <= X(I) <= XU(I) if IX(I) = 3, XL(I) = X(I) = XU(I) if IX(I) = 5, where 1 <= I <= N. General linear constraints are assumed in the form C(I) unbounded if IC(I) = 0, CL(I) <= C(I) if IC(I) = 1, C(I) <= CU(I) if IC(I) = 2, CL(I) <= C(I) <= CU(I) if IC(I) = 3, CL(I) = C(I) = CU(I) if IC(I) = 5, where C(I) = A_I*X, 1 <= I <= NC, are linear functions. To simplify user's work, three additional easy to use subroutines are added. They call the basic general subroutine PVAR: PVARU - unconstrained nonsmooth optimization, PVARS - nonsmooth optimization with simple bounds, PVARL - nonsmooth optimization with simple bounds and general linear constraints. All subroutines contain a description of formal parameters and extensive comments. Furthermore, two test programs TVARU and TVARL are included, which contain several test problems (see e.g. [4]). These test programs serve as examples for using the subroutines, verify their correctness and demonstrate their efficiency. In this short guide, we describe all subroutines which can be called from the user's program. A detailed description of methods is given in [1], [2], [3]. In the description of formal parameters, we introduce a type of the argument that specifies whether the argument must have a value defined on entry to the subroutine (I), whether it is a value which will be returned (O), or both (U), or whether it is an auxiliary value (A). Note that the arguments of the type I can be changed on output under some circumstances, especially if improper input values were given. Besides formal parameters, we can use a COMMON /STAT/ block containing statistical information. This block, used in each subroutine has the following form: COMMON /STAT/ NDECF,NRES,NRED,NREM,NADD,NIT,NFV,NFG,NFH The arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- NDECF O Positive INTEGER variable that indicates the number of matrix decompositions. NRES O Positive INTEGER variable that indicates the number of restarts. NRED O Positive INTEGER variable that indicates the number of reductions. NREM O Positive INTEGER variable that indicates the number of constraint deletions during the QP solutions. NADD O Positive INTEGER variable that indicates the number of constraint additions during the QP solutions. NIT O Positive INTEGER variable that indicates the number of iterations. NFV O Positive INTEGER variable that indicates the number of function evaluations. NFG O Positive INTEGER variable that specifies the number of gradient evaluations. NFH O Positive INTEGER variable that specifies the number of Hessian evaluations. 2. Subroutines PVARU, PVARS, PVARL: ----------------------------------- The calling sequences are CALL PVARU(NF,NA,X,RA,IPAR,RPAR,F,GMAX,ITERM) CALL PVARS(NF,NA,NB,X,IX,XL,XU,IA,RA,IPAR,RPAR,FP,GMAX,ITERM) CALL PVARL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,IA,RA,IPAR, & RPAR,FP,GMAX,ITERM) The arguments have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. NA I Nonnegative INTEGER variable that specifies the maximum bundle dimension. The choice NA=0 causes that the default value NA=NF+3 will be taken. NB I Nonnegative INTEGER variable that specifies whether the simple bounds are suppressed (NB=0) or accepted (NB>0). NC I Nonnegative INTEGER variable that specifies the number of linear constraints; if NC=0 the linear constraints are suppressed. X(NF) U On input, DOUBLE PRECISION vector with the initial estimate to the solution. On output, the approximation to the minimum. IX(NF) I On input (significant only if NB>0) INTEGER vector containing the simple bounds types: IX(I)=0 - the variable X(I) is unbounded, IX(I)=1 - the lower bound X(I) >= XL(I), IX(I)=2 - the upper bound X(I) <= XU(I), IX(I)=3 - the two side bound XL(I) <= X(I) <= XU(I), IX(I)=5 - the variable X(I) is fixed (given by its initial estimate). XL(NF) I DOUBLE PRECISION vector with lower bounds for variables (significant only if NB>0). XU(NF) I DOUBLE PRECISION vector with upper bounds for variables (significant only if NB>0). CF(NC) A DOUBLE PRECISION vector which contains values of constraint functions (only if NC>0). IC(NC) I On input (significant only if NC>0) INTEGER vector which contains constraint types: IC(K)=0 - the constraint CF(K) is not used, IC(K)=1 - the lower constraint CF(K) >= CL(K), IC(K)=2 - the upper constraint CF(K) <= CU(K), IC(K)=3 - the two side constraint CL(K) <= CF(K) <= CU(K), IC(K)=5 - the equality constraint CF(K) = CL(K). CL(NC) I DOUBLE PRECISION vector with lower bounds for constraint functions (significant only if NC>0). CU(NC) I DOUBLE PRECISION vector with upper bounds for constraint functions (significant only if NC>0). CG(NF*NC) I DOUBLE PRECISION matrix whose columns are normals of the linear constraints (significant only if NC>0). IA(NIA) A INTEGER working array of the dimension of at least NIA=NF+NA+1. RA(NRA) A DOUBLE PRECISION working array of the dimension of at least NRA=NF*(NF+11)/2+NA*(NF+2). IPAR(7) A INTEGER parameters: IPAR(1)=MEX, IPAR(2)=MOS, IPAR(3)=MTESX, IPAR(4)=MTESF, IPAR(5)=MIT, IPAR(6)=MFV, IPAR(7)=IPRNT. Parameters MEX, MOS, MTESX, MTESF, MIT, MFV, IPRNT are described in Section 3 together with other parameters of the subroutine PVAR. RPAR(7) A DOUBLE PRECISION parameters: RPAR(1)=TOLX, RPAR(2)=TOLF, RPAR(3)=TOLB, RPAR(4)=TOLG, RPAR(5)=ETA, RPAR(6)=EPS, RPAR(7)=XMAX. Parameters TOLX, TOLF, TOLB, TOLG, ETA, EPS, XMAX are described in Section 3 together with other parameters of the subroutine PVAR. F O DOUBLE PRECISION value of the objective function at the solution X. GMAX O DOUBLE PRECISION maximum absolute value of a partial derivative of the Lagrangian function. ITERM O INTEGER variable that indicates the cause of termination: ITERM= 1 - if |X - XO| was less than or equal to TOLX in MTESX subsequent iterations, ITERM= 2 - if |F - FO| was less than or equal to TOLF in MTESF subsequent iterations, ITERM= 3 - if F is less than or equal to TOLB, ITERM= 4 - if GMAX is less than or equal to TOLG, ITERM=11 - if NFV exceeded MFV, ITERM=12 - if NIT exceeded MIT, ITERM< 0 - if the method failed. The subroutines PVARU, PVARS, PVARL require the user supplied subroutine FUNDER that defines the objective function and its subgradient and has the form SUBROUTINE FUNDER(NF,X,F,G) The arguments of the user supplied subroutine have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the objective function. X(NF) I DOUBLE PRECISION an estimate to the solution. F O DOUBLE PRECISION value of the objective function at the point X. G(NF) O DOUBLE PRECISION subgradient of the objective function at the point X. 3. Subroutine PVAR: ------------------- This general subroutine is called from all the subroutines described in Section 2. The calling sequence is CALL PVAR(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,ICA,CFD,CR,CZ, & AF,AX,AG,G,GN,H,S,SN,XO,GO,GP,GS,TOLX,TOLF,TOLB,TOLG,ETA,EPS, & XMAX,GMAX,F,MEX,MOS,MTESX,MTESF,MIT,MFV,IPRNT,ITERM). The arguments NF, NA, NB, NC, X, IX, XL, XU, CF, IC, CL, CU, CG, GMAX, F, ITERM, have the same meaning as in Section 2. Other arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- ICA(NC) A INTEGER vector containing indices of active constraints. CFD(NC) A DOUBLE PRECISION vector of constraint function increments. CR(NCR) A DOUBLE PRECISION matrix containing triangular decomposition of the orthogonal projection kernel (NCR is equal to NF*(NF+1)/2). CZ(NF*NF) A DOUBLE PRECISION matrix containing orthogonal basis of linear manifold defined by active constraints. AF(4*NA) A DOUBLE PRECISION vector of bundle function values. AX(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle points. AG(NF*NA) A DOUBLE PRECISION matrix whose columns are bundle gradients. G(NF) A DOUBLE PRECISION subgradient of the objective function. GN(NF) A DOUBLE PRECISION Reduced aggregated subgradient of the objective function. H(NH) A DOUBLE PRECISION variable metric approximation of the Hessian matrix. S(NF) A DOUBLE PRECISION direction vector. SN(NF) A DOUBLE PRECISION reduced direction vector. XO(NF) A DOUBLE PRECISION vector which contains increments of variables. GO(NF) A DOUBLE PRECISION vector which contains increments of gradients. GP(NF) A DOUBLE PRECISION Aggregated subgradient of the objective function. GS(NF) A DOUBLE PRECISION Auxiliary vector. TOLX I DOUBLE PRECISION tolerance for the change of the coordinate vector X; the choice TOLX=0 causes that the default value TOLX=1.0D-16 will be taken. TOLF I DOUBLE PRECISION tolerance for the change of function values; the choice TOLF=0 causes that the default value TOLF=1.0D-8 will be taken. TOLB I DOUBLE PRECISION minimum acceptable function value; the choice TOLB=0 causes that the default value TOLB=-1.0D60 will be taken. TOLG I DOUBLE PRECISION tolerance for the Lagrangian function gradient; the choice TOLG=0 causes that the default value TOLG=1.0D-6 will be taken. ETA I DOUBLE PRECISION distance measure parameter. EPS I DOUBLE PRECISION parameter for constraint deletion; the choice EPS=0 causes that the default value EPS=5.0D-1 will be taken. XMAX I DOUBLE PRECISION maximum stepsize; the choice XMAX=0 causes that the default value 1.0D3 will be taken. MEX I INTEGER version of nonsmooth variable metric method: MEX=0 - convex version, MET=1 - nonconvex version. MOS I INTEGER distance measure exponent (MOS=1 or MOS=2). The choice MOS=0 causes that the default value MOS=1 will be taken. MTESX I INTEGER variable that specifies the maximum number of iterations with changes of the coordinate vector X smaller than TOLX; the choice MTESX=0 causes that the default value MTESX=20 will be taken. MTESF I INTEGER variable that specifies the maximum number of iterations with changes of function values smaller than TOLF; the choice MTESF=0 causes that the default value MTESF=2 will be taken. MIT I INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 200 will be taken. MFV I INTEGER variable that specifies the maximum number of function evaluations; the choice |MFV|=0 causes that the default value 500 will be taken. IPRNT I INTEGER variable that specifies PRINT: IPRNT= 0 - print is suppressed, IPRNT= 1 - basic print of final results, IPRNT=-1 - extended print of final results, IPRNT= 2 - basic print of intermediate and final results, IPRNT=-2 - extended print of intermediate and final results. The subroutine PVAR has a modular structure. The following list contains its most important subroutines: PLLPB1 - Primal null space method for solving a linear programming subproblem with linear constraints. PS1L07 - Simplified line search using function values and derivatives. PS1L08 - Nonsmooth line search using function values and derivatives. PUDVI2 - Special nonsmooth variable metric updates. The subroutine PVAR requires the user supplied subroutine FUNDER which is described in Section 2. 4. Subroutine PLLPB1: --------------------- Since the primal null space method for linear programming subproblems can be used separately in many applications, we describe the subroutine PLLPB1 in more details. The calling sequence is CALL PLLPB1(NF,NC,X,IX,XO,XL,XU,CF,CFD,IC,ICA,CL,CU,CG,CR, & CZ,G,GO,S,MFP,KBF,KBC,ETA9,EPS7,EPS9,UMAX,GMAX,N,ITERL) The arguments NF, NC, X, IX, XL, XU, CF, IC, CL, CU, CG have the same meaning as in Section 2 (only with the difference that the arguments X and CF are of the type (I), i.e. they must have a value defined on entry to PLLPB1 and they are not changed). The arguments CFD, ICA, CR, CZ have the same meaning as in Section 3 (only with the difference that the arguments CFD, ICA, CR, CZ are of the type (O), i.e. their values can be used subsequently). Other arguments have the following meaning: Argument Type Significance --------------------------------------------------------------------- G(NF) I DOUBLE PRECISION gradient of the objective function. XO(NF) A DOUBLE PRECISION auxiliary vector. approximate Hessian (NH is equal to NF*(NF+1)/2). S(NF) O DOUBLE PRECISION direction vector. MFP I INTEGER variable that specifies the type of the computed point. MFP=1 - computation is terminated whenever an arbitrary feasible point is found, MFP=2 - computation is terminated whenever an optimum feasible point is found, MFP=3 - computation starts from the previously reached point and is terminated whenever an optimum feasible point is found. KBF I INTEGER variable that specifies simple bounds on variables. KBF=0 - simple bounds are suppressed, KBF=1 - one sided simple bounds, KBF=2 - two sided simple bounds. KBC I INTEGER variable that specifies general linear constraints. KBC=0 - linear constraints are suppressed, KBC=1 - one sided linear constraints, KBC=2 - two sided linear constraints. ETA9 I DOUBLE PRECISION maximum floating point number. EPS7 I DOUBLE PRECISION tolerance for linear independence of constraints (the recommended value is 1.0D-10). EPS9 I DOUBLE PRECISION tolerance for the definition of active constraints (the recommended value is 1.0D-8). UMAX O DOUBLE PRECISION maximum absolute value of the negative Lagrange multiplier. GMAX O DOUBLE PRECISION infinity norm of the gradient of the Lagrangian function. N O INTEGER dimension of the manifold defined by active constraints. ITERL O INTEGER variable that indicates the type of the computed feasible point. ITERL= 1 - an arbitrary feasible point was found, ITERL= 2 - the optimum feasible point was found, ITERL=-1 - an arbitrary feasible point does not exist, ITERL=-2 - the optimum feasible point does not exist. 5. Form of printed results: --------------------------- The form of printed results is specified by the parameter IPRNT as is described in Section 3. Here we demonstrate individual forms of printed results by the simple use of the program TVARU described in the next section (with NEXT=11). If we set IPRNT=1, then the printed results will have the form NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4 If we set IPRNT=-1, then the printed results will have the form EXIT FROM PVAR : NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4 X= -.1000072D+01 -.1728811D-09 If we set IPRNT=2, then the printed results will have the form ENTRY TO PVAR : NIT= 0 NFV= 1 NFG= 1 F= .60207973D+02 G= .1329D+02 NIT= 1 NFV= 2 NFG= 2 F= .43113601D+02 G= .1229D+02 NIT= 2 NFV= 3 NFG= 3 F= .26633740D+02 G= .1346D+02 NIT= 3 NFV= 4 NFG= 4 F= .11345282D+02 G= .1495D+02 NIT= 4 NFV= 5 NFG= 5 F= .37878798D+01 G= .1600D+02 NIT= 5 NFV= 6 NFG= 6 F= -.52179782D+01 G= .1600D+02 NIT= 6 NFV= 7 NFG= 7 F= -.52179782D+01 G= .6225D+01 NIT= 7 NFV= 8 NFG= 8 F= -.65520912D+01 G= .1600D+02 NIT= 8 NFV= 9 NFG= 9 F= -.77403014D+01 G= .1600D+02 NIT= 9 NFV= 10 NFG= 10 F= -.79743187D+01 G= .1600D+02 NIT= 10 NFV= 11 NFG= 11 F= -.79987188D+01 G= .1600D+02 NIT= 11 NFV= 12 NFG= 12 F= -.79999678D+01 G= .1600D+02 NIT= 12 NFV= 13 NFG= 13 F= -.79999994D+01 G= .1600D+02 NIT= 13 NFV= 14 NFG= 14 F= -.79999998D+01 G= .1600D+02 EXIT FROM PVAR : NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4 If we set IPRNT=-2, then the printed results will have the form ENTRY TO PVAR : NIT= 0 NFV= 1 NFG= 1 F= .60207973D+02 G= .1329D+02 NIT= 1 NFV= 2 NFG= 2 F= .43113601D+02 G= .1229D+02 NIT= 2 NFV= 3 NFG= 3 F= .26633740D+02 G= .1346D+02 NIT= 3 NFV= 4 NFG= 4 F= .11345282D+02 G= .1495D+02 NIT= 4 NFV= 5 NFG= 5 F= .37878798D+01 G= .1600D+02 NIT= 5 NFV= 6 NFG= 6 F= -.52179782D+01 G= .1600D+02 NIT= 6 NFV= 7 NFG= 7 F= -.52179782D+01 G= .6225D+01 NIT= 7 NFV= 8 NFG= 8 F= -.65520912D+01 G= .1600D+02 NIT= 8 NFV= 9 NFG= 9 F= -.77403014D+01 G= .1600D+02 NIT= 9 NFV= 10 NFG= 10 F= -.79743187D+01 G= .1600D+02 NIT= 10 NFV= 11 NFG= 11 F= -.79987188D+01 G= .1600D+02 NIT= 11 NFV= 12 NFG= 12 F= -.79999678D+01 G= .1600D+02 NIT= 12 NFV= 13 NFG= 13 F= -.79999994D+01 G= .1600D+02 NIT= 13 NFV= 14 NFG= 14 F= -.79999998D+01 G= .1600D+02 EXIT FROM PVAR : NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4 X= -.1000072D+01 -.1728811D-09 6. Verification of the subroutines: ----------------------------------- Subroutine PVARU can be verified and tested using the program TVARU. This program calls the subroutines TIUD19 (initiation), TFFU19 (function evaluation) and TFGU19 (subgradient evaluation) containing 20 academic test problems with at most 50 variables [4]. The results obtained by the program TVARU (with MEX=0) on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 33 NFV= 33 NFG= 33 F= .31998143D-07 G= .6222D-07 ITERM= 4 NIT= 15 NFV= 16 NFG= 16 F= .94894120D-10 G= .4745D-10 ITERM= 4 NIT= 17 NFV= 17 NFG= 17 F= .19522247D+01 G= .9348D-06 ITERM= 4 NIT= 17 NFV= 17 NFG= 17 F= .20000000D+01 G= .2918D-07 ITERM= 4 NIT= 20 NFV= 20 NFG= 20 F= -.29999997D+01 G= .4258D-06 ITERM= 4 NIT= 19 NFV= 19 NFG= 19 F= .72000001D+01 G= .1901D-06 ITERM= 4 NIT= 10 NFV= 10 NFG= 10 F= -.14142133D+01 G= .1414D-06 ITERM= 4 NIT= 55 NFV= 59 NFG= 59 F= -.99999247D+00 G= .4052D-06 ITERM= 4 NIT= 35 NFV= 35 NFG= 35 F= -.99999978D+00 G= .9370D-06 ITERM= 4 NIT= 14 NFV= 14 NFG= 14 F= -.79999998D+01 G= .4535D-06 ITERM= 4 NIT= 35 NFV= 36 NFG= 36 F= -.43999995D+02 G= .5050D-05 ITERM= 2 NIT= 30 NFV= 31 NFG= 31 F= .22600186D+02 G= .5327D-05 ITERM= 2 NIT= 46 NFV= 47 NFG= 47 F= -.32348674D+02 G= .9524D-05 ITERM= 2 NIT= 32 NFV= 32 NFG= 32 F= -.29197004D+01 G= .3444D-06 ITERM= 4 NIT= 74 NFV= 76 NFG= 76 F= .55981840D+00 G= .7330D-06 ITERM= 4 NIT= 89 NFV= 89 NFG= 89 F= -.84140570D+00 G= .3911D-06 ITERM= 4 NIT= 176 NFV= 176 NFG= 176 F= .97859847D+01 G= .9253D-06 ITERM= 4 NIT= 77 NFV= 78 NFG= 78 F= .16703846D+02 G= .1430D-04 ITERM= 2 NIT= 123 NFV= 123 NFG= 123 F= .14683216D-05 G= .9099D-06 ITERM= 4 NIT= 23 NFV= 23 NFG= 23 F= .00000000D+00 G= .3200D-22 ITERM= 4 The rows corresponding to individual test problems contain the number of iterations NIT, the number of function evaluations NFV, the number of gradient evaluations NFG, the final value of the objective function F, the value of the criterion for the termination G and the cause of termination ITERM. Subroutine PVARL can be verified and tested using the program TVARL. This program calls the subroutines TIUD22 (initiation), TAFU22 (function evaluation), TAGU22 (subgradient evaluation) containing 6 academic test problems with at most 20 variables [4]. The results obtained by the program TVARL on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 10 NFV= 10 NFG= 10 F= -.38965950D+00 G= .4188D-06 ITERM= 4 NIT= 4 NFV= 4 NFG= 4 F= -.33035714D+00 G= .8804D-32 ITERM= 4 NIT= 14 NFV= 14 NFG= 14 F= -.44891078D+00 G= .3474D-07 ITERM= 4 NIT= 88 NFV= 89 NFG= 89 F= -.42928061D+00 G= .2323D+01 ITERM= 2 NIT= 24 NFV= 24 NFG= 24 F= -.18596186D+01 G= .2939D-06 ITERM= 4 NIT= 25 NFV= 25 NFG= 25 F= .10183094D+00 G= .9428D-07 ITERM= 4 NIT= 93 NFV= 93 NFG= 93 F= .37907365D-05 G= .7469D-06 ITERM= 4 NIT= 137 NFV= 138 NFG= 138 F= .24306604D+02 G= .2434D-05 ITERM= 2 NIT= 119 NFV= 120 NFG= 120 F= .13372950D+03 G= .1969D-04 ITERM= 2 NIT= 138 NFV= 139 NFG= 139 F= .50695309D+00 G= .5476D-01 ITERM= 2 References: ----------- [1] Luksan L., Vlcek J.: Globally convergent variable metric method for convex nonsmooth unconstrained minimization. Journal of Optimization Theory and Applications Vol.102, 1999, pp.593-613. [2] Vlcek J., Luksan L.: Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization. Technical Report B 8/1999. Department of Mathematical Information Technology. University of Jyvaskyla, 1999. [3] Luksan L., Vlcek J.: NDA: Algorithms for Nondifferentiable Optimization. Research Report V-797, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000. [4] Luksan L., Vlcek J.: Subroutines for Testing Nonsmooth Unconstrained and Linearly Constrained Optimization Problems. Research Report V-798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000.  SHAR_EOF fi # end of overwriting check if test -f 'readme.txt' then echo shar: will not over-write existing file "'readme.txt'" else cat << "SHAR_EOF" > 'readme.txt' The NDA collection contains the following files: 1) File psubs.f containing optimization subroutines. 2) File mbsubs.f containing matrix subroutines. 3) File tbsubs.f containing test subroutines. 4) Files tminu.f, tminl.f, tbunu.f, tbunl.f, tnewu.f, tnewl.f, tvaru.f, tvarl.f, containing sample routines for the demonstration of the NDA subroutines. 5) Files pmin.txt, pbun.txt, pnew.txt pvar.txt containing descriptions of the basic NDA subroutines. 6) File com which is a batch file for starting the sample routines. SHAR_EOF fi # end of overwriting check if test -f 'routinedoc.tex' then echo shar: will not over-write existing file "'routinedoc.tex'" else cat << "SHAR_EOF" > 'routinedoc.tex' \documentclass{article} \begin{document} \section*{DESCRIPTION OF SUBROUTINES} \noindent In this section we describe easy-to-use subroutines which can be called from the user's program. In the description of formal parameters we introduce a type of the argument denoted by two letters. The first letter is either \texttt{I} for integer arguments or \texttt{R} for double precision real arguments. The second letter specifies whether the argument must have a value defined on the entry to the subroutine (\texttt{I}), whether it is a value which will be returned (\texttt{O}), or both (\texttt{U}), or whether it is an auxiliary value (\texttt{A}). Notice that the input type arguments can be changed on the output under some circumstances, especially if improper input values were given. Besides the formal parameters, we use a \verb|COMMON /STAT/| block containing statistical information. This block, used in each subroutine, has the following form: {\small \begin{verbatim} COMMON /STAT/ NDECF,NRES,NRED,NREM,NADD,NIT,NFV,NFG,NFH \end{verbatim} } Its elements have the following meanings: \vspace{2mm} {\small \noindent\parbox{20mm}{Element}\parbox{10mm}{$\!$Type}\parbox[t]{91mm} {Significance}\par\noindent\rule[1mm]{121mm}{.4pt} \par \noindent\parbox{20mm}{\texttt{NDECF}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of matrix decompositions.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NRES}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of restarts.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NRED}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of reductions.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NREM}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of constraint deletions during the QP solutions.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NADD}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of constraint additions during the QP solutions.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NIT}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of iterations.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NFV}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of function evaluations.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NFG}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of gradient evaluations.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NFH}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Number of Hessian evaluations.} \vspace{2mm} } \noindent Easy-to-use subroutines are called by the following statements: {\small \begin{verbatim} CALL PMINU(NF,NA,X,AF,IA,RA,IPAR,RPAR,F,GMAX,IEXT,ITERM) CALL PMINS(NF,NA,NB,X,IX,XL,XU,AF,IA,RA,IPAR,RPAR,F,GMAX,IEXT,ITERM) CALL PMINL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,AF,IA,RA,IPAR,RPAR, & F,GMAX,IEXT,ITERM) CALL PBUNU(NF,NA,X,IA,RA,IPAR,RPAR,F,GMAX,ITERM) CALL PBUNS(NF,NA,NB,X,IX,XL,XU,IA,RA,IPAR,RPAR,F,GMAX,ITERM) CALL PBUNL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,IA,RA,IPAR,RPAR, & F,GMAX,ITERM) CALL PNEWU(NF,NA,X,IA,RA,IPAR,RPAR,F,GMAX,IHES,ITERM) CALL PNEWS(NF,NA,NB,X,IX,XL,XU,IA,RA,IPAR,RPAR,F,GMAX,IHES,ITERM) CALL PNEWL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,IA,RA,IPAR,RPAR, & F,GMAX,IHES,ITERM) CALL PVARU(NF,NA,X,RA,IPAR,RPAR,F,GMAX,ITERM) CALL PVARS(NF,NA,NB,X,IX,XL,XU,RA,IPAR,RPAR,F,GMAX,ITERM) CALL PVARL(NF,NA,NB,NC,X,IX,XL,XU,CF,IC,CL,CU,CG,IA,RA,IPAR,RPAR, & F,GMAX,ITERM) \end{verbatim} } Their arguments have the following meanings: %\newpage \vspace{2mm} {\small \noindent\parbox{20mm}{Argument}\parbox{10mm}{$\!$Type}\parbox[t]{91mm} {Significance}\par\noindent\rule[1mm]{121mm}{.4pt} \par \noindent\parbox{20mm}{\texttt{NF}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Number of variables of the objective function.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NA}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Number of functions in the minimax criterion for subroutines {\tt PMINU}, {\tt PMINS}, {\tt PMINL} or the maximum bundle dimension for the other subroutines (choice $\texttt{NA}=0$ causes that the default value $\texttt{NA}=$\texttt{NF}+3 will be taken in a later case)} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NB}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Specification whether the simple bounds are suppressed ($\texttt{NB}=0$) or accepted ($\texttt{NB}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{NC}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Number of linear constraints; if $\texttt{NC}=0$ the linear constraints are suppressed.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{X(NF)}}\parbox{10mm}{\texttt{RU}}\parbox[t]{91mm}{ On input, vector with the initial estimate to the solution. On output, the approximation to the minimum.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IX(NF)}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Vector containing the simple bound types (significant only if $\texttt{NB}>0$):} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IX(I)}=0$:}\parbox[t]{71mm}{ the variable X(I) is unbounded,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IX(I)}=1$:}\parbox[t]{71mm}{ the lower bound $\texttt{X(I)}\ge\texttt{XL(I)}$,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IX(I)}=2$:}\parbox[t]{71mm}{ the upper bound $\texttt{X(I)}\le\texttt{XU(I)}$,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IX(I)}=3$:}\parbox[t]{71mm}{ the two-side bound $\texttt{XL(I)}\le\texttt{X(I)}\le\texttt{XU(I)}$,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IX(I)}=5$:}\parbox[t]{71mm}{ the variable \texttt{X(I)} is fixed (given by its initial estimate).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{XL(NF)}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Vector with lower bounds for variables (significant only if $\texttt{NB}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{XU(NF)}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Vector with upper bounds for variables (significant only if $\texttt{NB}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{CF(NC)}}\parbox{10mm}{\texttt{RA}}\parbox[t]{91mm}{ Vector which contains values of constraint functions (significant only if $\texttt{NC}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IC(NC)}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ INTEGER vector which contains constraint types (significant only if $\texttt{NC}>0$):} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IC(K)}=0$:}\parbox[t]{71mm}{ the constraint \texttt{CF(K)} is not used,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IC(K)}=1$:}\parbox[t]{71mm}{ the lower constraint $\texttt{CF(K)}\ge\texttt{CL(K)}$,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IC(K)}=2$:}\parbox[t]{71mm}{ the upper constraint $\texttt{CF(K)}\le\texttt{CU(K)}$,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IC(K)}=3$:}\parbox[t]{71mm}{ the two-side constraint $\texttt{CL(K)}\le\texttt{CF(K)}\le\texttt{CU(K)}$,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IC(K)}=5$:}\parbox[t]{71mm}{ the equality constraint $\texttt{CF(K)}=\texttt{CL(K)}$.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{CL(NC)}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Vector with lower bounds for constraint functions (significant only if $\texttt{NC}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{CU(NC)}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Vector with upper bounds for constraint functions (significant only if $\texttt{NC}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{CG(NF*NC)}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Matrix whose columns are normals of the linear constraints (significant only if $\texttt{NC}>0$).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{AF(NA)}}\parbox{10mm}{\texttt{RO}}\parbox[t]{91mm}{ Vector which contains the values of functions in the minimax criterion.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IA(NIA)}}\parbox{10mm}{\texttt{IA}}\parbox[t]{91mm}{ Working array of the dimension \texttt{NIA}, where at least \texttt{NIA}=\texttt{NF}+\texttt{NA}+1.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{RA(NRA)}}\parbox{10mm}{\texttt{RA}}\parbox[t]{91mm}{ Working array of the dimension \texttt{NRA}. The minimum values of \texttt{NRA} required in individual subroutines can be found in text files {\tt PMIN.TXT}, {\tt PBUN.TXT}, {\tt PNEW.TXT}, {\tt PVAR.TXT}.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IPAR(7)}}\parbox{10mm}{\texttt{IA}}\parbox[t]{91mm}{ Integer parameters (see Table 5.1).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{RPAR(7)}}\parbox{10mm}{\texttt{RA}}\parbox[t]{91mm}{ Real parameters (see Table 5.1).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{F}}\parbox{10mm}{\texttt{RO}}\parbox[t]{91mm}{ Value of the objective function at the solution \texttt{X}.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{GMAX}}\parbox{10mm}{\texttt{RO}}\parbox[t]{91mm}{ value indicating the termination ($\|g^k\|_{\infty}$ in {\tt PMIN} or $w^k$ in {\tt PBUN}, {\tt PNEW} and {\tt PVAR}).} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IEXT}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies the minimax criterion:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IEXT}<0$:}\parbox[t]{71mm}{ maximum of positive values,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IEXT}=0$:}\parbox[t]{71mm}{ maximum of absolute values,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IEXT}>0$:}\parbox[t]{71mm}{ maximum of negative values,} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IHES}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies a way for computing second derivatives:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IHES}=0$:}\parbox[t]{71mm}{ numerical computation,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IHES}=1$:}\parbox[t]{71mm}{ analytical computation by the user supplied subroutine HES.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{ITERM}}\parbox{10mm}{\texttt{IO}}\parbox[t]{91mm}{ Variable that indicates the cause of termination:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}=1$:}\parbox[t]{71mm}{ if $|x - x_{old}|$ was less than or equal to \texttt{TOLX} in \texttt{MTESX} subsequent iterations,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}=2$:}\parbox[t]{71mm}{ if $|F - F_{old}|$ was less than or equal to \texttt{TOLF} in \texttt{MTESF} subsequent iterations,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}=3$:}\parbox[t]{71mm}{ if \texttt{F} is less than or equal to \texttt{TOLB},} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}=4$:}\parbox[t]{71mm}{ if \texttt{GMAX} is less than or equal to \texttt{TOLG},} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}=11$:}\parbox[t]{71mm}{ if \texttt{NFV} exceeded \texttt{MFV},} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}=12$:}\parbox[t]{71mm}{ if \texttt{NIT} exceeded \texttt{MIT},} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{ITERM}<0$:}\parbox[t]{71mm}{ if the method failed ({$\texttt{ITERM}=-6$} if the required precision was not achieved, {$\texttt{ITERM}=-10$} if two consecutive restarts were required, {$\texttt{ITERM}=-12$} if the quadratic programming subroutine failed).} } The integer and real parameters are listed in the following table: \vspace{2mm} \begin{center} \begin{tabular}{c|llll} \hline Parameter & {\tt PMIN} & {\tt PBUN} & {\tt PNEW} & {\tt PVAR} \\ \hline {\tt IPAR(1)} & {\tt MET} & {\tt MOT} & {\tt MOS} & {\tt MEX} \\ {\tt IPAR(2)} & {\tt MEC} & {\tt MES} & {\tt MES} & {\tt MOS} \\ {\tt IPAR(3)} & {\tt MER} & {\tt MTESX} & {\tt MTESX} & {\tt MTESX} \\ {\tt IPAR(4)} & {\tt MES} & {\tt MTESF} & {\tt MTESF} & {\tt MTESF} \\ {\tt IPAR(5)} & {\tt MIT} & {\tt MIT} & {\tt MIT} & {\tt MIT} \\ {\tt IPAR(6)} & {\tt MFV} & {\tt MFV} & {\tt MFV} & {\tt MFV} \\ {\tt IPAR(7)} & {\tt IPRNT} & {\tt IPRNT} & {\tt IPRNT} & {\tt IPRNT} \\ \hline {\tt RPAR(1)} & {\tt TOLX} & {\tt TOLX} & {\tt TOLX} & {\tt TOLX} \\ {\tt RPAR(2)} & {\tt TOLF} & {\tt TOLF} & {\tt TOLF} & {\tt TOLF} \\ {\tt RPAR(3)} & {\tt TOLB} & {\tt TOLB} & {\tt TOLB} & {\tt TOLB} \\ {\tt RPAR(4)} & {\tt TOLG} & {\tt TOLG} & {\tt TOLG} & {\tt TOLG} \\ {\tt RPAR(5)} & {\tt TOLD} & {\tt TOLD} & {\tt TOLD} & {\tt ETA} \\ {\tt RPAR(6)} & {\tt TOLS} & {\tt TOLS} & {\tt TOLS} & {\tt EPS} \\ {\tt RPAR(7)} & {\tt XMAX} & {\tt TOLP }& {\tt TOLP} & {\tt XMAX} \\ {\tt RPAR(8)} & - & {\tt ETA} & {\tt ETA} & - \\ {\tt RPAR(9)} & - & {\tt XMAX} & {\tt XMAX} & - \\ \hline \end{tabular} \vspace{4mm} Table 5.2 - Integer and real parameters \end{center} \vspace{2mm} Integer and real parameters have the following meanings: \vspace{2mm} {\small \noindent\parbox{20mm}{Argument}\parbox{10mm}{$\!$Type}\parbox[t]{91mm} {Significance}\par\noindent\rule[1mm]{121mm}{.4pt} \par \noindent\parbox{20mm}{\texttt{MET}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies self-scaling for variable metric updates:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MET}=1$:}\parbox[t]{71mm}{ self-scaling is suppressed,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MET}=2$:}\parbox[t]{71mm}{ self-scaling is used only in the first iteration (initial self-scaling),} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MET}=3$:}\parbox[t]{71mm}{ self-scaling is controlled by a special procedure.} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox[t]{91mm}{The choice $\texttt{MET}=0$ causes that the default value $\texttt{MET}=3$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MEC}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies correction of variable metric updates if negative curvature occurs:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MEC}=1$:}\parbox[t]{71mm}{ correction is suppressed,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MEC}=2$:}\parbox[t]{71mm}{ Powell's correction is used.} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox[t]{91mm}{The choice $\texttt{MEC}=0$ causes that the default value $\texttt{MEC}=1$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MER}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies restart after unsuccessgful variable metric updates:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MER}=0$:}\parbox[t]{71mm}{ restart is suppressed,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MER}=1$:}\parbox[t]{71mm}{ variable metric method is restarted by using the unit matrix.} \par\vspace{1mm} \noindent\parbox{20mm}{\texttt{MEX}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies version of nonsmooth variable metric method:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MEX}=0$:}\parbox[t]{71mm}{ convex version is used,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MEX}=1$:}\parbox[t]{71mm}{ nonconvex version is used.} \par\vspace{1mm} \noindent\parbox{20mm}{\texttt{MOT}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies the weight updating method:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MOT}=1$:}\parbox[t]{71mm}{ quadratic interpolation,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MOT}=2$:}\parbox[t]{71mm}{ local minimization,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MOT}=3$:}\parbox[t]{71mm}{ quasi-Newton condition.} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox[t]{91mm}{The choice $\texttt{MOT}=0$ causes that the default value $\texttt{MOT}=1$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MOS}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Distance measure exponent $\omega$ (either $1$ or $2$). The choice $\texttt{MOS}=0$ causes that the default value $\texttt{MOS}=1$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MES}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Variable that specifies the interpolation method selection in a line search (until a sufficient function decrease is reached; then only bisection will be used):} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MES}=1$:}\parbox[t]{71mm}{ bisection,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{MES}=2$:}\parbox[t]{71mm}{ two-point quadratic interpolation.} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox[t]{91mm}{The choice $\texttt{MES}=0$ causes that the default value $\texttt{MES}=2$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MTESX}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Maximum number of iterations with changes of the coordinate vector \texttt{X} smaller than \texttt{TOLX}; the choice $\texttt{MTESX}=0$ causes that the default value $20$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MTESF}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Maximum number of iterations with changes of function values smaller than TOLF; the choice $\texttt{MTESF}=0$ causes that the default value $2$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MIT}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Maximum number of iterations; the choice $\texttt{MIT}=0$ causes that the default value $200$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{MFV}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Maximum number of function evaluations; the choice $\texttt{MFV}=0$ causes that the default value $500$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{IPRNT}}\parbox{10mm}{\texttt{II}}\parbox[t]{91mm}{ Print specification:} \par\vspace{1mm} \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IPRNT}= 0$:}\parbox[t]{71mm}{ print is suppressed,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IPRNT}= 1$:}\parbox[t]{71mm}{ basic print of final results,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IPRNT}=-1$:}\parbox[t]{71mm}{ extended print of final results,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IPRNT}= 2$:}\parbox[t]{71mm}{ basic print of intermediate and final results,} \par \noindent\parbox{30mm}{$\;$}\parbox{20mm}{$\texttt{IPRNT}=-2$:}\parbox[t]{71mm}{ extended print of intermediate and final results,} \par \noindent\parbox{20mm}{\texttt{TOLX}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Tolerance for the change of the coordinate vector \texttt{X}; the choice $\texttt{TOLX}=0$ causes that the default value $10^{-16}$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{TOLF}}\parbox{10mm}{\texttt{RI}}\parbox[t]{91mm}{ Tolerance for the change of the function value; the choice $\texttt{TOLF}=0$ causes that the default value $10^{-8}$ will be taken.} \par\vspace{2mm} \noindent\parbox{20mm}{\texttt{TOLB}}\parbox{10mm}{\texttt{RI}}\parbox