*********************************************************************** * * * PMAX - A PRIMAL INTERIOR-POINT VARIABLE METRIC AND DISCRETE NEWTON * * METHODS WITH DIRECT DECOMPOSITION LINE-SEARCH SUBALGORITHM * * FOR LARGE-SCALE MINIMAX OPTIMIZATION. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PMAX is designed to find a close approximation to a local minimum of a special nonsmooth objectine function, which has the form F(X) = MAX(FA_1(X), FA_2(X), ..., FA_NA(X)), if IEXT<0, or F(X) = MAX(|FA_1(X)|, |FA_2(X)|, ..., |FA_NA(X)|), if IEXT=0, or F(X) = MAX(-FA_1(X), -FA_2(X), ..., -FA_NA(X)), if IEXT>0. Here X is a vector of NF variables and FA_I(X), 1<=I<=NA, are twice continuously differentiable functions. We assume that NF and NA are large, but partial functions FA_I(X), 1<=I<=NA, depend on a small number of variables. This implies that the mapping AF(X) = [FA_1(X), FA_2(X), ..., FA_NA(X)] has a sparse Jacobian matrix, which will be denoted by AG(X) (it has NA rows and NF columns). The sparsity pattern of the Jacobian matrix is stored in the coordinate form if ISPAS=1 or in the standard compressed row format if ISPAS=2 using arrays IAG and JAG. For example, if the Jacobian matrix has the following pattern AG = | * * 0 * | | * * * 0 | | * 0 0 * | | 0 * * 0 | | * 0 * 0 | (asterisks denote nonzero elements) then arrays IAG and JAG contain elements IAG(1)=1, IAG(2)=1, IAG(3)=1, IAG(4)=2, IAG(5)=2, IAG(6)=2, IAG(7)=3, IAG(8)=3, IAG(9)=4, IAG(10)=4, IAG(11)=5, IAG(12)=5, JAG(1)=1, JAG(2)=2, JAG(3)=4, JAG(4)=1, JAG(5)=2, JAG(6)=3, JAG(7)=1, JAG(8)=4, JAG(9)=2, JAG(10)=3, JAG(11)=1, JAG(12)=3 if ISPAS=1 or IAG(1)=1, IAG(2)=4, IAG(3)=7, IAG(4)=9, IAG(5)=11, IAG(6)=13, JAG(1)=1, JAG(2)=2, JAG(3)=4, JAG(4)=1, JAG(5)=2, JAG(6)=3, JAG(7)=1, JAG(8)=4, JAG(9)=2, JAG(10)=3, JAG(11)=1, JAG(12)=3 if ISPAS=2. In the first case, nonzero elements can be sorted in an arbitrary order (not only by rows as in the above example). Arrays IAG and JAG have to be declared with lengths NA+MA and MA at least, respectively, where MA is the number of nonzero elements. In the second case, nonzero elements can be sorted only by rows. Components of IAG contain total numbers of nonzero elements in all previous rows increased by 1 and elements of JAG contain corresponding column indices (note that IAG has NA+1 elements and the last element is equal to MA+1). Arrays IAG and JAG have to be declared with length NA+1 and MA at least, respectively. To simplify user's work, an additional easy to use subroutine PMAXU is added. It calls the basic general subroutine PMAX. All subroutines contain a description of formal parameters and extensive comments. Furthermore, test program TMAXU is included, which contains several test problems (see e.g. [2]). This test program serves as an example for using the subroutine PMAXU, verifies its correctness and demonstrates its efficiency. In this short guide, we describe all subroutines which can be called from the user's program. A detailed description of the method is given in [1]. 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). 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/ NRES,NDEC,NIN,NIT,NFV,NFG,NFH The arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- NRES O Positive INTEGER variable that indicates the number of restarts. NDEC O Positive INTEGER variable that indicates the number of matrix decompositions. NIN O Positive INTEGER variable that indicates the number of inner iterations (for solving linear systems). 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 indicates the number of gradient evaluations. NFH O Positive INTEGER variable that indicates the number of Hessian evaluations. 2. Subroutine PMAXU: -------------------- The calling sequence is CALL PMAXU(NF,NA,MA,X,AF,IAG,JAG,IPAR,RPAR,F,GMAX,IEXT,ISPAS,IPRNT, & ITERM) The arguments have the following meaning. Argument Type Significance ---------------------------------------------------------------------- NF I Positive INTEGER variable that specifies the number of variables of the minimax objective function. NA I Positive INTEGER variable that specifies the number of partial functions. MA I Number of nonzero elements in the Jacobian matrix. This parameter is used as input only if ISPAS=1 (it defines dimensions of arrays IAG and JAG in this case). X(NF) U On input, DOUBLE PRECISION vector with the initial estimate to the solution. On output, the approximation to the minimum. AF(NA) O DOUBLE PRECISION vector which contains values of partial functions. IAG(NA+1) I INTEGER array which contains pointers of the first elements in rows of the Jacobian matrix. JAG(MA) I INTEGER array which contains column indices of the nonzero elements. IPAR(7) U INTEGER parameters: IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)=MFG, IPAR(4)=IEST, IPAR(5)=MED, IPAR(6)-unused, IPAR(7)=IFIL. Parameters MIT, MFV, MFG, IEST, MED are described in Section 3 together with other parameters of the subroutine PMAX. Parameter IFIL specifies a relative size of the space reserved for fill-in. The choice IFIL=0 causes that the default value IFIL=1 will be taken. RPAR(9) U DOUBLE PRECISION parameters: RPAR(1)=XMAX, RPAR(2)=TOLX, RPAR(3)=TOLF, RPAR(4)=TOLB, RPAR(5)=TOLG, RPAR(6)=FMIN, RPAR(7)-unused, RPAR(6)=ETA4, RPAR(9)=ETA5. Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN, ETA4, ETA5 are described in Section 3 together with other parameters of the subroutine PMAX. 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 objective function. IEXT I INTEGER variable that specifies the type of minimax: IEXT< 0 - minimization of the maximum positive value. IEXT= 0 - minimization of the maximum absolute value. IEXT> 0 - minimization of the maximum negative value. ISPAS I INTEGER variable that specifies sparse structure of the Jacobian matrix: ISPAS= 1 - the coordinate form is used, ISPAS= 2 - the standard row compresed format is used. 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. ITERM O INTEGER variable that indicates the cause of termination: ITERM= 1 - if |X - XO| was less than or equal to TOLX in two subsequent iterations, ITERM= 2 - if |F - FO| was less than or equal to TOLF in two 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= 6 - if termination criterion was not satisfied, but the solution is probably acceptable, ITERM=11 - if NIT exceeded MIT, ITERM=12 - if NFV exceeded MFV, ITERM=13 - if NFG exceeded MFG, ITERM< 0 - if the method failed. Values ITERM<=-40 detect a lack of space. In this case, parameter IPAR(7)=IFIL has to be increased (IFIL=2, IFIL=3, etc.). The subroutine PMAXU requires the user supplied subroutines FUN and DFUN that define partial functions and their gradients and have the form SUBROUTINE FUN(NF,KA,X,FA) SUBROUTINE DFUN(NF,KA,X,GA) The arguments of the 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 INTEGER index of the partial function. X(NF) I DOUBLE PRECISION an estimate to the solution. FA O DOUBLE PRECISION value of the KA-th partial function at the point X. GA(NF) O DOUBLE PRECISION gradient of the KA-th partial function at the point X. Note that only nonzero elements of this gradient have to be assigned. 3. Subroutine PMAX: ------------------- This general subroutine is called from all subroutines described in Section 2. The calling sequence is CALL PMAX(NF,NA,MMAX,X,IX,AF,AFO,AG,AGO,GA,AH,AZL,AZU,G,H,IH,JH, & IA,IAG,JAG,S,XO,GO,GS,GP,COL,PSL,PERM,INVP,WN11,WN12,WN13,WN14, & XMAX,TOLX,TOLF,TOLB,TOLG,FMIN,ETA4,ETA5,GMAX,F,MIT,MFV,MFG,IEST, & MED,IEXT,IPRNT,ITERM) The arguments NF, NA, X, AF, IAG, JAG, GMAX, F, IEXT, IPRNT, ITERM have the same meaning as in Section 2. Other arguments have the following meaning: Argument Type Significance ---------------------------------------------------------------------- MMAX I INTEGER size of array H. IX(NF) A INTEGER auxiliary array. AFO(NA) A DOUBLE PRECISION auxiliary array. AG(MA) A DOUBLE PRECISION nonzero elements of the Jacobian matrix AGO(MA) A DOUBLE PRECISION difference between the current and the old Jacobian matrices. This array is used only if MED=1. GA(NF) A DOUBLE PRECISION gradient of the partial function. AH(MH) A DOUBLE PRECISION approximation of the partitioned Hessian matrix. This array is used only if MED=1. AZL(NA) A DOUBLE PRECISION lower Lagrange multipliers. AZU(NA) A DOUBLE PRECISION upper Lagrange multipliers. G(NF) A DOUBLE PRECISION gradient of the objective function. H(MMAX) A DOUBLE PRECISION nonzero elements of the approximation of the Hessian matrix and nonzero elements of the Choleski factor. IH(NF+1) I INTEGER array which contains pointers of the diagonal elements in the upper part of the Hessian matrix. JH(MMAX) I INTEGER array which contains column indices of the nonzero elements and additional working space for the Choleski factor. IA(NA) A INTEGER auxiliary array. S(NF) A DOUBLE PRECISION direction vector. XO(NF) A DOUBLE PRECISION array which contains increments of variables. GO(NF) A DOUBLE PRECISION array which contains increments of gradients. GS(NF) A DOUBLE PRECISION auxiliary array. GP(NF) A DOUBLE PRECISION auxiliary array. COL(NF) A INTEGER auxiliary array. PSL(NF+1) A INTEGER pointer vector in the compact form of the Choleski factor. PERM(NF) A INTEGER permutation vector. INVP(NF) A INTEGER inverse permutation vector. WN11(NF+1) A INTEGER auxiliary array. WN12(NF+1) A INTEGER auxiliary array. WN13(NF+1) A INTEGER auxiliary array. WN14(NF+1) A INTEGER auxiliary array. XMAX U DOUBLE PRECISION maximum stepsize; the choice XMAX=0 causes that the default value 1.0D+16 will be taken. TOLX U 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 U DOUBLE PRECISION tolerance for the change of function values; the choice TOLF=0 causes that the default value TOLF=1.0D-14 will be taken. TOLB U DOUBLE PRECISION minimum acceptable function value; the choice TOLB=0 causes that the default value TOLB=FMIN+1.0D-16 will be taken. TOLG U DOUBLE PRECISION tolerance for the Lagrangian function gradient; the choice TOLG=0 causes that the default value TOLG=1.0D-6 will be taken. FMIN U DOUBLE PRECISION lower bound for the minimum function value. It is significant only if IEST=1. If IEST=0, the default value FMIN=-1.0D+60 will be taken. ETA4 U DOUBLE PRECISION coefficient for the barrier parameter decrease; the choice ETA4=0 causes that the default value ETA5=8.5D-1 will be taken. ETA5 U DOUBLE PRECISION minimum permitted value of the barrier parameter; the choice ETA5=0 causes that the default value ETA5=1.0D-10 will be taken. MIT U INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 9000 will be taken. MFV U INTEGER variable that specifies the maximum number of function evaluations; the choice MFV=0 causes that the default value 9000 will be taken. MFG U INTEGER variable that specifies the maximum number of gradient evaluations; the choice MFG=0 causes that the default value 10000 will be taken. IEST I INTEGER estimation of the minimum functiom value for the line search: IEST=0 - estimation is not used, IEST=1 - lower bound FMIN is used as an estimation for the minimum function value. MED I INTEGER variable that specifies the method used. MED=1 - partitioned variable metric method, MED=2 - safeguarded discrete Newton method. The choice of parameter XMAX can be sensitive in many cases. First, the objective function can be evaluated only in a relatively small region (if it contains exponentials) so that the maximum stepsize is necessary. Secondly, the problem can be very ill-conditioned far from the solution point so that large steps can be unsuitable. Finally, if the problem has more local solutions, a suitably chosen maximum stepsize can lead to obtaining a better local solution. A suitable choice of parameter ETA5 (larger than 1.0d-8) can sometimes improve the efficiency of the method. The subroutine PMAX requires the user supplied subroutines FUN and DFUN which are described in Section 2. 4. Verification of the subroutines: ----------------------------------- Subroutine PMAXU can be verified and tested using the program TMAXU. This program calls the subroutines TIUB14 (initiation), TAFU14 (function evaluation) and TAGU14 (gradient evaluation) containing 22 unconstrained test problems with at most 200 variables [2]. The results obtained by the program TMAXU on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 53 NFV= 66 NFG= 54 F= 0.260681654E-14 G= 0.120E-07 ITERM= 4 NIT= 107 NFV= 160 NFG= 108 F= 0.502441303E-12 G= 0.262E-06 ITERM= 4 NIT= 33 NFV= 41 NFG= 34 F= 0.535439934E-08 G= 0.814E-06 ITERM= 4 NIT= 51 NFV= 93 NFG= 52 F= 0.540217976 G= 0.383E-06 ITERM= 4 NIT= 23 NFV= 24 NFG= 24 F= 0.132910215E-08 G= 0.482E-06 ITERM= 4 NIT= 46 NFV= 52 NFG= 47 F= 0.216701250E-08 G= 0.436E-06 ITERM= 4 NIT= 48 NFV= 113 NFG= 49 F= 0.260162540 G= 0.430E-06 ITERM= 4 NIT= 21 NFV= 58 NFG= 22 F= 282.380956 G= 0.806E-06 ITERM= 4 NIT= 59 NFV= 146 NFG= 60 F= 0.185849706 G= 0.287E-06 ITERM= 4 NIT= 159 NFV= 215 NFG= 160 F=-0.251638288 G= 0.250E-06 ITERM= 4 NIT= 70 NFV= 96 NFG= 71 F= 0.538829394E-01 G= 0.139E-07 ITERM= 4 NIT= 136 NFV= 245 NFG= 137 F= 0.941962422 G= 0.207E-06 ITERM= 4 NIT= 2 NFV= 4 NFG= 3 F= 0.456380111E-19 G= 0.304E-11 ITERM= 3 NIT= 5 NFV= 6 NFG= 6 F= 0.162409086E-08 G= 0.671E-06 ITERM= 4 NIT= 116 NFV= 120 NFG= 117 F= 0.199003538E-01 G= 0.436E-06 ITERM= 4 NIT= 110 NFV= 214 NFG= 111 F=-0.388943896E-02 G= 0.518E-05 ITERM= 2 NIT= 33 NFV= 49 NFG= 34 F=-0.110417781E-06 G= 0.764E-06 ITERM= 4 NIT= 62 NFV= 77 NFG= 63 F= 0.744234285E-09 G= 0.611E-07 ITERM= 4 NIT= 9 NFV= 23 NFG= 10 F= 42.6746811 G= 0.373E-09 ITERM= 4 NIT= 25 NFV= 32 NFG= 26 F=-0.497512435E-02 G= 0.422E-06 ITERM= 4 NIT= 16 NFV= 23 NFG= 17 F= 0.298487756E-01 G= 0.595E-06 ITERM= 4 NIT= 32 NFV= 82 NFG= 33 F= 0.577532726E-02 G= 0.972E-07 ITERM= 4 NITER = 1216 NFVAL = 1939 NSUCC = 22 TIME= 0:00:00.75 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 norm of gradient G and the cause of termination ITERM. References: ----------- [1] Luksan L., Matonoha C., Vlcek J.: LSA: Algorithms for large-scale unconstrained and box constrained optimization. Research Report V-896, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2004. [2] Luksan L., Vlcek J.: Sparse and partially separable test problems for unconstrained and equality constrained optimization. Research Report V-767, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 1998. 