*********************************************************************** * * * PNED - A DISCRETE NEWTON METHOD WITH DIRECT DECOMPOSITION * * TRUST-REGION SUBALGORITHMS FOR LARGE-SCALE OPTIMIZATION * * PROBLEMS WITH SPARSE HESSIAN MATRICES. * * * *********************************************************************** 1. Introduction: ---------------- The double-precision FORTRAN 77 basic subroutine PNED is designed to find a close approximation to a local minimum of a nonlinear function F(X) with simple bounds on variables. Here X is a vector of NF variables and F(X) is a smooth function. We suppose that NF is large and the sparsity pattern of the Hessian matrix is known. 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 <= NF. The sparsity pattern of the Hessian matrix (only the upper part) is stored in the coordinate form if ISPAS=1 or in the standard compressed row format if ISPAS=2 using arrays IH and JH. For example, if the Jacobian matrix has the following pattern H = | * * * 0 * | | * * 0 * 0 | | * 0 * 0 * | | 0 * 0 * 0 | | * 0 * 0 * | (asterisks denote nonzero elements) then arrays IH and JH contain elements IH(1)=1, IH(2)=1, IH(3)=1, IH(4)=1, IH(5)=2, IH(6)=2, IH(7)=3, IH(8)=3, IH(9)=4, IH(10)=5, JH(1)=1, JH(2)=2, JH(3)=3, JH(4)=5, JH(5)=2, JH(6)=4, JH(7)=3, JH(8)=5, JH(9)=4, JH(10)=5 if ISPAS=1 or IH(1)=1, IH(2)=5, IH(3)=7, IH(4)=9, IH(5)=10, IH(6)=11, JH(1)=1, JH(2)=2, JH(3)=3, JH(4)=5, JH(5)=2, JH(6)=4, JH(7)=3, JH(8)=5, JH(9)=4, JH(10)=5 if ISPAS=2. In the first case, nonzero elements in the upper part of the Hessian matrix can be sorted in an arbitrary order (not only by rows as in the above example) and arrays IH and JH have to be declared with lengths NF+MH at least, where MH is the number of nonzero elements. In the second case, nonzero elements can be sorted only by rows. Components of IH contain addresses of the diagonal elements in this sequence and components of JH contain corresponding column indices (note that IH has NF+1 elements and the last element is equal to MH+1). Arrays IH and JH have to be declared with lengths NF+1 and MH at least, respectively. To simplify user's work, two additional easy to use subroutines are added. They call the basic general subroutine PNED: PNEDU - unconstrained large-scale optimization, PNEDS - large-scale optimization with simple bounds. All subroutines contain a description of formal parameters and extensive comments. Furthermore, two test programs TNEDU and TNEDS are included, which contain several test problems (see e.g. [2]). 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 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. Subroutines PNEDU, PNEDS: ---------------------------- The calling sequences are CALL PNEDU(NF,MH,X,IH,JH,IPAR,RPAR,F,GMAX,ISPAS,IPRNT,ITERM) CALL PNEDS(NF,MH,X,IX,XL,XU,IH,JH,IPAR,RPAR,F,GMAX,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 objective function. MH I Number of nonzero elements in the upper part of the Hessian matrix. This parameter is used as input only if ISPAS=1 (it defines dimensions of arrays IH and JH 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. IX(NF) I On input (significant only for PNEDS) 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 for PNEDS). XU(NF) I DOUBLE PRECISION vector with upper bounds for variables (significant only for PNEDS). IH(NF+1) I INTEGER array which contains pointers of the diagonal elements in the upper part of the Hessian matrix. JH(MH) I INTEGER array which contains column indices of the nonzero elements and additional working space for the Choleski factor. IPAR(7) U INTEGER parameters: IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)=MFG, IPAR(4)=IEST, IPAR(5)=MOS, IPAR(6)-unused, IPAR(7)=IFIL. Parameters MIT, MFV, MFG, IEST, MOS are described in Section 3 together with other parameters of the subroutine PNED. 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)=XDEL, RPAR(8)-unused, RPAR(9)-unused. Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN, XDEL are described in Section 3 together with other parameters of the subroutine PNED. 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. ISPAS I INTEGER variable that specifies sparse structure of the Hessian 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 subroutines PNEDU, PNEDS require the user supplied subroutines OBJ and DOBJ that define the objective function and its gradient and have the form SUBROUTINE OBJ(NF,X,F) SUBROUTINE DOBJ(NF,X,G) 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. 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 gradient of the objective function at the point X. 3. Subroutine PNED: ------------------- This general subroutine is called from all subroutines described in Section 2. The calling sequence is CALL PNED(NF,NB,MMAX,X,IX,XL,XU,GF,HF,IH,JH,S,XO,GO,XS,COL,PSL, & PERM,INVP,WN11,WN12,WN13,WN14,XMAX,TOLX,TOLF,TOLB,TOLG,FMIN, & XDEL,GMAX,F,MIT,MFV,MFG,IEST,MOS,IPRNT,ITERM) The arguments NF, NB, X, IX, XL, XU, IH, JH, GMAX, F, 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. GF(NF) A DOUBLE PRECISION gradient of the objective function. HF(MMAX) A DOUBLE PRECISION nonzero elements of the original Hessian matrix and nonzero elements of the Choleski factor. 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. XS(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. XDEL U DOUBLE PRECISION trust region stepsize; the choice XDEL=0 causes that a suitable default value is computed. MIT U INTEGER variable that specifies the maximum number of iterations; the choice MIT=0 causes that the default value 5000 will be taken. MFV U INTEGER variable that specifies the maximum number of function evaluations; the choice MFV=0 causes that the default value 5000 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. MOS U INTEGER method for computing trust-region step: MOS=1 - double dog-leg method of Dennis and Mei, MOS=2 - method of More and Sorensen for obtaining optimum locally constrained step. The choice MOS=0 causes that the default value 2 will be taken. 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. The subroutine PNED requires the user supplied subroutines OBJ and DOBJ which are described in Section 2. 4. Verification of the subroutines: ----------------------------------- Subroutine PNEDU can be verified and tested using the program TNEDU. This program calls the subroutines TIUS14 (initiation), TFFU14 (function evaluation) and TFGU14 (gradient evaluation) containing 22 unconstrained test problems with at most 1000 variables [2]. The results obtained by the program TNEDU on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 1421 NFV= 1425 NFG= 5688 F= 0.465831486E-25 G= 0.418E-12 ITERM= 3 NIT= 39 NFV= 45 NFG= 200 F= 0.231406390E-14 G= 0.350E-06 ITERM= 4 NIT= 17 NFV= 18 NFG= 108 F= 0.839782900E-09 G= 0.933E-06 ITERM= 4 NIT= 24 NFV= 25 NFG= 100 F= 269.499543 G= 0.666E-10 ITERM= 4 NIT= 11 NFV= 12 NFG= 72 F= 0.795109456E-10 G= 0.473E-06 ITERM= 4 NIT= 13 NFV= 16 NFG= 196 F= 0.125944855E-10 G= 0.815E-06 ITERM= 4 NIT= 12 NFV= 13 NFG= 78 F= 336.937181 G= 0.300E-06 ITERM= 4 NIT= 4 NFV= 5 NFG= 90 F= 761774.954 G= 0.216E-06 ITERM= 4 NIT= 7 NFV= 9 NFG= 16 F= 316.436141 G= 0.146E-06 ITERM= 4 NIT= 69 NFV= 75 NFG= 630 F= -135.290000 G= 0.291E-11 ITERM= 4 NIT= 67 NFV= 68 NFG= 408 F= 10.7765879 G= 0.199E-06 ITERM= 4 NIT= 127 NFV= 128 NFG= 512 F= 982.273617 G= 0.495E-09 ITERM= 4 NIT= 6 NFV= 7 NFG= 28 F= 0.598998674E-10 G= 0.693E-06 ITERM= 4 NIT= 2 NFV= 3 NFG= 18 F= 0.129013604E-08 G= 0.792E-06 ITERM= 4 NIT= 9 NFV= 10 NFG= 40 F= 1.92401599 G= 0.414E-06 ITERM= 4 NIT= 7 NFV= 8 NFG= 48 F= -427.404476 G= 0.565E-07 ITERM= 4 NIT= 8 NFV= 9 NFG= 54 F=-0.379921091E-01 G= 0.314E-10 ITERM= 4 NIT= 7 NFV= 8 NFG= 48 F=-0.245741193E-01 G= 0.218E-09 ITERM= 4 NIT= 6 NFV= 7 NFG= 42 F= 59.5986241 G= 0.952E-08 ITERM= 4 NIT= 14 NFV= 15 NFG= 90 F= -1.00013520 G= 0.139E-08 ITERM= 4 NIT= 11 NFV= 12 NFG= 72 F= 2.13866377 G= 0.331E-08 ITERM= 4 NIT= 30 NFV= 34 NFG= 186 F= 1.00000000 G= 0.164E-08 ITERM= 4 NITER = 1911 NFVAL = 1952 NSUCC = 22 TIME= 0:00:03.00 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. Subroutine PNEDS can be verified and tested using the program TNEDS. This program calls the subroutines TIUS14 (initiation), TFFU14 (function evaluation), TFGU14 (gradient evaluation) containing 22 box constrained test problems with at most 1000 variables [2]. The results obtained by the program TNEDS on a PC computer with Microsoft Power Station Fortran compiler have the following form. NIT= 1420 NFV= 1424 NFG= 5680 F= 0.00000000 G= 0.000E+00 ITERM= 3 NIT= 128 NFV= 130 NFG= 640 F= 1980.05047 G= 0.911E-10 ITERM= 4 NIT= 17 NFV= 19 NFG= 108 F= 0.189355864E-09 G= 0.340E-06 ITERM= 4 NIT= 10 NFV= 12 NFG= 44 F= 269.522686 G= 0.328E-09 ITERM= 4 NIT= 13 NFV= 15 NFG= 84 F= 0.391905635E-12 G= 0.536E-06 ITERM= 4 NIT= 13 NFV= 14 NFG= 196 F= 0.136396633E-11 G= 0.901E-06 ITERM= 4 NIT= 30 NFV= 32 NFG= 186 F= 336.920046 G= 0.151E-05 ITERM= 2 NIT= 37 NFV= 38 NFG= 684 F= 761925.725 G= 0.119E-06 ITERM= 4 NIT= 507 NFV= 508 NFG= 1016 F= 428.056916 G= 0.347E-13 ITERM= 4 NIT= 109 NFV= 127 NFG= 990 F= -80.4518214 G= 0.639E-06 ITERM= 4 NIT= 6 NFV= 8 NFG= 42 F= 72291.4951 G= 0.178E-08 ITERM= 4 NIT= 519 NFV= 520 NFG= 2080 F= 4994.21410 G= 0.236E-06 ITERM= 4 NIT= 3 NFV= 4 NFG= 16 F= 0.660542076E-23 G= 0.363E-11 ITERM= 3 NIT= 2 NFV= 3 NFG= 18 F= 0.129013604E-08 G= 0.792E-06 ITERM= 4 NIT= 9 NFV= 10 NFG= 40 F= 1.92401599 G= 0.414E-06 ITERM= 4 NIT= 15 NFV= 18 NFG= 96 F= -427.391653 G= 0.342E-06 ITERM= 4 NIT= 8 NFV= 9 NFG= 54 F=-0.379921091E-01 G= 0.314E-10 ITERM= 4 NIT= 7 NFV= 8 NFG= 48 F=-0.245741193E-01 G= 0.218E-09 ITERM= 4 NIT= 13 NFV= 16 NFG= 84 F= 1654.94525 G= 0.174E-08 ITERM= 4 NIT= 14 NFV= 15 NFG= 90 F= -1.00013520 G= 0.139E-08 ITERM= 4 NIT= 9 NFV= 10 NFG= 60 F= 2.41354873 G= 0.388E-08 ITERM= 4 NIT= 30 NFV= 34 NFG= 186 F= 1.00000000 G= 0.164E-08 ITERM= 4 NITER = 2919 NFVAL = 2974 NSUCC = 22 TIME= 0:00:06.56 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. 