C ALGORITHM 829, COLLECTED ALGORITHMS FROM ACM. C THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE, C VOL. 29, NO. 4, December, 2003, P. 469--480. #! /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: # C/ # C/Dp/ # C/Dp/Drivers/ # C/Dp/Drivers/Makefile # C/Dp/Drivers/README # C/Dp/Drivers/deriv009.txt # C/Dp/Drivers/example.c # C/Dp/Drivers/lmin001 # C/Dp/Drivers/test001 # C/Dp/Drivers/test001.txt # C/Dp/Src/ # C/Dp/Src/gkls.c # C/Dp/Src/gkls.h # C/Dp/Src/rnd_gen.c # C/Dp/Src/rnd_gen.h # Doc/ # Doc/Readme.txt # Doc/UserManual.bib # Doc/UserManual.ps # Doc/UserManual.tex # This archive created: Fri Jan 23 09:55:41 2004 export PATH; PATH=/bin:$PATH if test ! -d 'C' then mkdir 'C' fi cd 'C' if test ! -d 'Dp' then mkdir 'Dp' fi cd 'Dp' if test ! -d 'Drivers' then mkdir 'Drivers' fi cd 'Drivers' if test -f 'Makefile' then echo shar: will not over-write existing file "'Makefile'" else cat << "SHAR_EOF" > 'Makefile' # Example of a UNIX makefile provided to # UNIX users for a simple compilation and linkage of separate files of # the application project. # # Authors: # # M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev # gkls : example.o gkls.o rnd_gen.o $(CC) $(CFLAGS) -lm example.o gkls.o rnd_gen.o -o gkls example.o : ../Src/gkls.h ../Src/rnd_gen.h example.c $(CC) -I. $(CFLAGS) -c example.c gkls.o : ../Src/gkls.h ../Src/rnd_gen.h ../Src/gkls.c $(CC) -I. $(CFLAGS) -c ../Src/gkls.c rnd_gen.o : ../Src/rnd_gen.h ../Src/rnd_gen.c $(CC) -I. $(CFLAGS) -c ../Src/rnd_gen.c clean: -rm *.o SHAR_EOF fi # end of overwriting check if test -f 'README' then echo shar: will not over-write existing file "'README'" else cat << "SHAR_EOF" > 'README' Line ** of example.c originally ran from 1 to 100. This generated 300+ output files so for the sake of brevity it has been set up to just generate 4 file for immediate checking. SHAR_EOF fi # end of overwriting check if test -f 'deriv009.txt' then echo shar: will not over-write existing file "'deriv009.txt'" else cat << "SHAR_EOF" > 'deriv009.txt' Gradients and Hessian matrix at the point with coordinates: x = ( 0.000000 0.000000 ) Vector of gradient of the D-type function number 9: grad D(x) = ( 1.400213 -0.616694 ) Vector of gradient of the D2-type function number 9: grad D2(x) = ( 1.422313 -0.703983 ) Hessian matrix of the D2-type function number 9: -10.261060 0.173888 0.173888 -10.902981 SHAR_EOF fi # end of overwriting check if test -f 'example.c' then echo shar: will not over-write existing file "'example.c'" else cat << "SHAR_EOF" > 'example.c' /*****************************************************************************/ /* An example of the use of the */ /* GKLS-Generator of Classes of ND (non-differentiable), */ /* D (continuously differentiable), and */ /* D2 (twice continuously differentiable) */ /* Test Functions for Global Optimization */ /* */ /* In this example, first, a class (with default parameters) of */ /* 100 D-type test functions is generated and the files with the function */ /* information and with the functions points (in the two-dimensional case) */ /* are created; */ /* second, a file with information about gradients and Hessian of specific */ /* D- and D2-type test functions at a given point is provided. */ /* */ /* Authors: */ /* */ /* M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev */ /* */ /* (C) 2002-2003 */ /* */ /*****************************************************************************/ #include #include #include #include "../Src/gkls.h" #include "../Src/rnd_gen.h" void print_error_msg (int); /* Print an error message */ void print_error_msg (int error_code) { switch( error_code ) { case GKLS_OK: printf("\nGKLS_OK: There is no error."); break; case GKLS_DIM_ERROR: printf("\nGKLS_DIM_ERROR: The problem dimension is out of the valid range [1,%u].", (unsigned int)NUM_RND); break; case GKLS_NUM_MINIMA_ERROR: printf("\nGKLS_NUM_MINIMA_ERROR: The number of local minima must be greater than 1."); break; case GKLS_FUNC_NUMBER_ERROR: printf("\nGKLS_FUNC_NUMBER_ERROR: The number of the test function to be generated is out of the range [1,100]."); break; case GKLS_BOUNDARY_ERROR: printf("\nGKLS_BOUNDARY_ERROR: The admissible region boundary vectors are not defined or ill-defined."); break; case GKLS_GLOBAL_MIN_VALUE_ERROR: printf("\nGKLS_GLOBAL_MIN_VALUE_ERROR: The global minimum value must be greater than %f.", (double)GKLS_PARABOLOID_MIN); break; case GKLS_GLOBAL_DIST_ERROR: printf("\nGKLS_GLOBAL_DIST_ERROR: The distance from the paraboloid vertex to the global minimizer is too great."); break; case GKLS_GLOBAL_RADIUS_ERROR: printf("\nGKLS_GLOBAL_RADIUS_ERROR: The radius of the attraction region of the global minimizer is too high."); break; case GKLS_MEMORY_ERROR: printf("\nGKLS_MEMORY_ERROR: There is not enough memory to allocate."); break; case GKLS_DERIV_EVAL_ERROR: printf("\nGKLS_DERIV_EVAL_ERROR: An error occurs during derivative evaluation."); break; case GKLS_FLOATING_POINT_ERROR: default : printf("\nUnknown error."); } printf("\n"); } /* print_error_msg() */ int main() { unsigned int i, j; /* cycle parameters */ int error_code; /* error codes variable */ int func_num; /* test function number within a class */ double *xx, dx1, dx2; /* for evaluation of the test function */ /* on a grid with steps dx1, dx2 */ double z; /* test function value */ double *g, **h; /* gradient and hessian matrix */ FILE *GKLS_info, *min_info, *fp, *fderiv; /* file pointers */ char filename[12]; /* name of files */ printf("\nGKLS-Generator of Classes of ND, D, and D2 Test Functions"); printf("\nfor Global Optimization,"); printf("\n(C) 2002-2003, M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev\n"); /*---------------------------------------------------------------------*/ /* First, generate a class (with default parameters) of the D-type */ /* test functions and create the files with the function information */ /* and with the functions points (in the two-dimensional case). */ /*---------------------------------------------------------------------*/ /* Set the input parameters */ if ((error_code=GKLS_set_default()) != GKLS_OK) { print_error_msg(error_code); return error_code; } /* Another way to set the input parameters is: */ /* GKLS_dim = 2; */ /* GKLS_num_minima = 10; */ /* if ((error_code = GKLS_domain_alloc()) != GKLS_OK) */ /* return error_code; */ /* GKLS_global_dist = 2.0/3.0; */ /* GKLS_global_radius = 0.5*GKLS_global_dist; */ /* GKLS_global_value = GKLS_GLOBAL_MIN_VALUE; */ /* if ((error_code = GKLS_parameters_check()) != GKLS_OK) */ /* return error_code; */ /* Allocate memory for the vector xx of feasible point */ if ( (xx=(double *)malloc((size_t)GKLS_dim*sizeof(double))) == NULL) return (-1); /* Generate the class of 100 D-type functions */ for (func_num=1; func_num <= 1; func_num++) { if((error_code=GKLS_arg_generate (func_num)) != GKLS_OK) { print_error_msg(error_code); return error_code; } /* Open files */ /* File of the test function information */ sprintf(filename,"test%03d.txt",func_num); if ((GKLS_info=fopen(filename,"wt")) == NULL) return (-1); if (GKLS_dim == 2) { /* File of points */ sprintf(filename,"test%03d",func_num); if ((fp=fopen(filename,"wt")) == NULL) return (-1); /* File of local minimizers */ sprintf(filename,"lmin%03d",func_num); if ((min_info=fopen(filename,"wt")) == NULL) return (-1); } printf("\nGenerating the function number %d\n", func_num); fprintf(GKLS_info,"D-type function number %d", func_num); fprintf(GKLS_info,"\nof the class with the following parameters:"); fprintf(GKLS_info,"\n problem dimension = %u;",GKLS_dim); fprintf(GKLS_info,"\n number of local minima = %u;",GKLS_num_minima); fprintf(GKLS_info,"\n global minimum value = %f;",GKLS_global_value); fprintf(GKLS_info,"\n radius of the g.m.attraction region = %f;", GKLS_global_radius); fprintf(GKLS_info,"\n distance from the paraboloid vertex to the global minimizer = %f.", GKLS_global_dist); /* Information about local minimizers */ fprintf(GKLS_info,"\n\nLocal minimizers:\n"); for (i=0; i=GKLS_MAX_VALUE-1000.0) /* An error: do something */; else fprintf(fp,"%f %f %f\n", dx1, dx2, z); } } /* Creating files for two-dimensional functions */ /* Close files */ if (GKLS_dim == 2) { fclose(fp); fclose(min_info); } fclose(GKLS_info); /* Deallocate memory */ GKLS_free(); } /* for func_num*/ /* Deallocate memory of the vector xx of feasible point */ free(xx); /*-----------------------------------------------------------------------*/ /* Second, create the file with information about gradients of specific */ /* D-type and D2-type test functions and about Hessian matrix of a */ /* specific D2-type test function at a given feasible point. */ /*-----------------------------------------------------------------------*/ /* Set the input parameters of the class */ if ((error_code=GKLS_set_default()) != GKLS_OK) { print_error_msg(error_code); return error_code; } /* Allocate memory for the vector xx of feasible point */ if ( (xx=(double *)malloc((size_t)GKLS_dim*sizeof(double))) == NULL) return (-1); /* Generate a specific function from the class with default parameters */ func_num = 9; /* test function number */ if((error_code=GKLS_arg_generate (func_num)) != GKLS_OK) { print_error_msg(error_code); return error_code; } /* Open the file of gradients and Hessian matrix at a given point */ sprintf(filename,"deriv%03d.txt",func_num); if ((fderiv=fopen(filename,"wt")) == NULL) return (-1); /* Allocate memory for the gradient vector and the Hessian matrix */ if ( (g=(double *)malloc((size_t)GKLS_dim*sizeof(double))) == NULL) return (-1); /* gradient vector */ if ( (h=(double **)malloc((size_t)GKLS_dim*sizeof(double *))) == NULL) return (-1); /* Hessian matrix */ for (i=0; i 'lmin001' -0.762614 0.597254 0.000000 -0.135523 0.823530 -1.000000 0.496543 -0.939405 0.655211 0.713418 0.627774 1.876545 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1.000000 -0.540000 4.375716 1.000000 -0.520000 4.345277 1.000000 -0.500000 4.309214 1.000000 -0.480000 4.267286 1.000000 -0.460000 4.224596 1.000000 -0.440000 4.182706 1.000000 -0.420000 4.141615 1.000000 -0.400000 4.101325 1.000000 -0.380000 4.061835 1.000000 -0.360000 4.020026 1.000000 -0.340000 3.948879 1.000000 -0.320000 3.845859 1.000000 -0.300000 3.715591 1.000000 -0.280000 3.562699 1.000000 -0.260000 3.391807 1.000000 -0.240000 3.207541 1.000000 -0.220000 3.014525 1.000000 -0.200000 2.817388 1.000000 -0.180000 2.620758 1.000000 -0.160000 2.429267 1.000000 -0.140000 2.247550 1.000000 -0.120000 2.080257 1.000000 -0.100000 1.932055 1.000000 -0.080000 1.807664 1.000000 -0.060000 1.711939 1.000000 -0.040000 1.650135 1.000000 -0.020000 1.627925 1.000000 0.000000 1.645593 1.000000 0.020000 1.694567 1.000000 0.040000 1.767962 1.000000 0.060000 1.861214 1.000000 0.080000 1.970393 1.000000 0.100000 2.091746 1.000000 0.120000 2.221579 1.000000 0.140000 2.356224 1.000000 0.160000 2.492024 1.000000 0.180000 2.625327 1.000000 0.200000 2.752486 1.000000 0.220000 2.869853 1.000000 0.240000 2.973784 1.000000 0.260000 3.060634 1.000000 0.280000 3.126761 1.000000 0.300000 3.168520 1.000000 0.320000 3.182269 1.000000 0.340000 3.172989 1.000000 0.360000 3.163099 1.000000 0.380000 3.154009 1.000000 0.400000 3.145719 1.000000 0.420000 3.138229 1.000000 0.440000 3.131538 1.000000 0.460000 3.125648 1.000000 0.480000 3.120558 1.000000 0.500000 3.116268 1.000000 0.520000 3.112778 1.000000 0.540000 3.110088 1.000000 0.560000 3.108197 1.000000 0.580000 3.107107 1.000000 0.600000 3.106817 1.000000 0.620000 3.107327 1.000000 0.640000 3.108637 1.000000 0.660000 3.110747 1.000000 0.680000 3.113656 1.000000 0.700000 3.117366 1.000000 0.720000 3.121876 1.000000 0.740000 3.127186 1.000000 0.760000 3.133296 1.000000 0.780000 3.140206 1.000000 0.800000 3.147916 1.000000 0.820000 3.156425 1.000000 0.840000 3.165735 1.000000 0.860000 3.175845 1.000000 0.880000 3.186755 1.000000 0.900000 3.198465 1.000000 0.920000 3.210975 1.000000 0.940000 3.224284 1.000000 0.960000 3.238394 1.000000 0.980000 3.253304 1.000000 1.000000 3.269014 SHAR_EOF fi # end of overwriting check if test -f 'test001.txt' then echo shar: will not over-write existing file "'test001.txt'" else cat << "SHAR_EOF" > 'test001.txt' D-type function number 1 of the class with the following parameters: problem dimension = 2; number of local minima = 10; global minimum value = -1.000000; radius of the g.m.attraction region = 0.333333; distance from the paraboloid vertex to the global minimizer = 0.666667. Local minimizers: f[1] = f( -0.763 0.597) = 0.000; rho[1] = 0.330. f[2] = f( -0.136 0.824) = -1.000; rho[2] = 0.333. f[3] = f( 0.497 -0.939) = 0.655; rho[3] = 0.677. f[4] = f( 0.713 0.628) = 1.877; rho[4] = 0.076. f[5] = f( -0.517 -0.605) = 0.933; rho[5] = 0.135. f[6] = f( -0.999 -0.460) = -0.044; rho[6] = 0.364. f[7] = f( 0.582 0.550) = 1.529; rho[7] = 0.076. f[8] = f( -0.474 -0.911) = 1.541; rho[8] = 0.171. f[9] = f( 0.974 -0.021) = 1.586; rho[9] = 0.348. f[10] = f( -0.244 -0.588) = 1.080; rho[10] = 0.135. There is one global minimizer. The number of the global minimizer is: 2 SHAR_EOF fi # end of overwriting check cd .. if test ! -d 'Src' then mkdir 'Src' fi cd 'Src' if test -f 'gkls.c' then echo shar: will not over-write existing file "'gkls.c'" else cat << "SHAR_EOF" > 'gkls.c' /*******************************************************************************/ /* GKLS-Generator of Classes of ND (non-differentiable), */ /* D (continuously differentiable), and */ /* D2 (twice continuously differentiable) */ /* Test Functions for Global Optimization */ /* */ /* Authors: */ /* */ /* M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev */ /* */ /* (C) 2002-2003 */ /* */ /* Each test class is defined by the following parameters: */ /* (1) problem dimension */ /* (2) number of local minima including paraboloid min and global min */ /* (3) global minimum value */ /* (3) distance from the paraboloid vertex to the global minimizer */ /* (4) radius of the attraction region of the global minimizer */ /*******************************************************************************/ /*******************************************************************************/ /* The following groups of subroutines are present in the software: */ /* -- parameters setting/checking subroutines: */ /* int GKLS_set_default (void); */ /* int GKLS_parameters_check (void); */ /* -- test classes generating subroutines: */ /* int GKLS_arg_generate (unsigned int); */ /* int GKLS_coincidence_check (void); */ /* int GKLS_set_basins (void); */ /* -- test functions calling subroutines: */ /* double GKLS_ND_func (double *); */ /* double GKLS_D_func (double *); */ /* double GKLS_D2_func (double *); */ /* -- subroutines of evaluation of the partial derivatives of test functions */ /* double GKLS_D_deriv (unsigned int, double *); */ /* double GKLS_D2_deriv1 (unsigned int, double *); */ /* double GKLS_D2_deriv2 (unsigned int, unsigned int, double *); */ /* -- subroutines of evaluation of the gradients and hessian matrix */ /* int GKLS_D_gradient (double *, double *); */ /* int GKLS_D2_gradient (double *, double *); */ /* int GKLS_D2_hessian (double *, double **); */ /* -- memory allocation/deallocation subroutines: */ /* int GKLS_domain_alloc (void); / void GKLS_domain_free (void); */ /* int GKLS_alloc (void); / void GKLS_free (void); */ /* -- auxiliary subroutines */ /* int GKLS_initialize_rnd (unsigned int, unsigned int, int); */ /* double GKLS_norm (double *, double *); */ /*******************************************************************************/ #include "gkls.h" #include "rnd_gen.h" #include #include #include /*---------------- Variables accessible by the user -------------------- */ double *GKLS_domain_left; /* left boundary vector of D */ /* D=[GKLS_domain_left; GKLS_domain_ight] */ double *GKLS_domain_right;/* right boundary vector of D */ unsigned int GKLS_dim; /* dimension of the problem, */ /* 2<=GKLS_dim=2 */ double GKLS_global_dist; /* distance from the paraboloid minimizer */ /* to the global minimizer */ double GKLS_global_radius;/* radius of the global minimizer */ /* attraction region */ double GKLS_global_value; /* global minimum value, */ /* GKLS_global_value < GKLS_PARABOLOID_MIN */ T_GKLS_Minima GKLS_minima; /* see the structures type description */ T_GKLS_GlobalMinima GKLS_glob; /*--------------------------- Global variables ----------------------*/ int isArgSet=0; /* isArgSet == 1 if all necessary parameters are set */ double delta; /* parameter using in D2-type function generation; */ /* it is chosen randomly from the */ /* open interval (0,GKLS_DELTA_MAX_VALUE) */ unsigned long rnd_counter; /* index of random array elements */ /*------------------ Auxiliary functions prototypes -----------------*/ double GKLS_norm(double *, double *); int GKLS_alloc(void); int GKLS_coincidence_check(void); int GKLS_set_basins(void); int GKLS_initialize_rnd(unsigned int, unsigned int, int); /*****************************************************************************/ /* Distance between two vectors in the Euclidean space R^(GKLS_dim) */ /* INPUT: */ /* x1, x2 -- arrays of the coordinates of the two vectors x1 and x2 */ /* of the dimension GKLS_dim */ /* RETURN VALUE: Euclidean norm ||x1-x2|| */ /*****************************************************************************/ double GKLS_norm(double *x1, double *x2) { unsigned int i; double norm = 0.0; for (i=0; i= NUM_RND) ) return GKLS_DIM_ERROR; /* problem dimension error */ if ((GKLS_domain_left = (double *)(malloc((size_t)GKLS_dim*sizeof(double)))) == NULL) return GKLS_MEMORY_ERROR; /* memory allocation error */ if ((GKLS_domain_right =(double *)(malloc((size_t)GKLS_dim*sizeof(double)))) == NULL) return GKLS_MEMORY_ERROR; /* memory allocation error */ /* Set the admissible region as [-1,1]^GKLS_dim */ for (i=0; i= NUM_RND) ) return GKLS_DIM_ERROR; /* problem dimension error */ if (GKLS_num_minima <= 1) return GKLS_NUM_MINIMA_ERROR; /* erroneous number of local minima */ if ((GKLS_minima.local_min =(double **)(malloc((size_t)GKLS_num_minima*sizeof(double *)))) == NULL) return GKLS_MEMORY_ERROR; /* memory allocation error */ for (i=0; i= NUM_RND)) return GKLS_DIM_ERROR; /* problem dimension errors */ if (GKLS_num_minima <= 1) /* number of local minima error */ return GKLS_NUM_MINIMA_ERROR; if ((GKLS_domain_left == NULL) || (GKLS_domain_right == NULL)) return GKLS_BOUNDARY_ERROR; /* the boundaries are not defined */ for (i=0; i= GKLS_domain_right[i] - GKLS_PRECISION) return GKLS_BOUNDARY_ERROR; /* the boundaries are erroneous */ if (GKLS_global_value >= GKLS_PARABOLOID_MIN - GKLS_PRECISION) return GKLS_GLOBAL_MIN_VALUE_ERROR; /* the global minimum value must */ /* be less than the paraboloid min */ /* Find min_side = min |b(i)-a(i)|, D=[a,b], and */ /* check the distance from paraboloid vertex to global minimizer */ min_side = GKLS_domain_right[0]-GKLS_domain_left[0]; for (i=1; i= 0.5*min_side - GKLS_PRECISION) || (GKLS_global_dist <= GKLS_PRECISION) ) return GKLS_GLOBAL_DIST_ERROR; /* global distance error */ if ( (GKLS_global_radius >= 0.5*GKLS_global_dist + GKLS_PRECISION) || (GKLS_global_radius <= GKLS_PRECISION) ) return GKLS_GLOBAL_RADIUS_ERROR; /* global minimizer attr. radius error */ return GKLS_OK; /* no errors */ } /* GKLS_parameters_check() */ /*****************************************************************************/ /* The subroutine checks possible coincidence of local minimizers */ /* The subroutine has no INPUT parameters */ /* RETURN VALUE: an error code (GKLS_OK if there are no errors): */ /* GKLS_PARABOLA_MIN_COINCIDENCE_ERROR - if some local minimizer coincides */ /* with the paraboloid minimizer */ /* GKLS_LOCAL_MIN_COINCIDENCE_ERRO - if there is a pair of identical */ /* local minimizers */ /*****************************************************************************/ int GKLS_coincidence_check() { unsigned int i, j; /* Check wether some local minimizer coincides with the paraboloid minimizer */ for (i=2; i GKLS_minima.rho[i] + GKLS_PRECISION) GKLS_minima.rho[i] = temp_min; } } /* Correct the radii by weight coefficients w(i) */ /* The weight coefficients can be chosen randomly; */ /* here they are defined by default as: */ /* w(i) = 0.99, i != 1 , and w(1) = 1.0 (global min index = 1) */ for (i=0; i= 2 */ /* Note that peak(i) is such that the function value f(i) is smaller*/ /* than min(GKLS_GLOBAL_MIN_VALUE, 2*rho(i)) */ temp_d1=GKLS_norm(GKLS_minima.local_min[0], GKLS_minima.local_min[i]); temp_min=(GKLS_minima.rho[i] - temp_d1)*(GKLS_minima.rho[i] - temp_d1)+ GKLS_minima.f[0]; /*the conditional minimum at the boundary*/ temp_d1 = (1.0 + rnd_num[rnd_counter])*GKLS_minima.rho[i]; temp_d2 = rnd_num[rnd_counter]*(temp_min - GKLS_global_value); /* temp_d1 := min(temp_d1, temp_d2) */ if (temp_d2 < temp_d1) temp_d1 = temp_d2; GKLS_minima.peak[i]= temp_d1; rnd_counter++; if (rnd_counter == NUM_RND) { ranf_array(rnd_num, NUM_RND); rnd_counter = 0L; } GKLS_minima.f[i] = temp_min - GKLS_minima.peak[i]; } /*********************************************************************/ /* Find all possible global minimizers and */ /* create a list of their indices among all the minimizers */ /* Note that the paraboloid minimum can not be the global one because*/ /* the global optimum value is set to be less than the paraboloid */ /* minimum value */ /*********************************************************************/ GKLS_glob.num_global_minima = 0; for (i=0; i= GKLS_global_value - GKLS_PRECISION) && (GKLS_minima.f[i] <= GKLS_global_value + GKLS_PRECISION)) { GKLS_glob.gm_index[GKLS_glob.num_global_minima] = i; GKLS_glob.num_global_minima ++; /* The first GKLS_glob.num_global_minima elements of the list */ /* contain the indices of the global minimizers */ } else GKLS_glob.gm_index[GKLS_num_minima-1-i+GKLS_glob.num_global_minima] = i; /* The remaining elements of the list */ /* contain the indices of local (non global) minimizers */ if (GKLS_glob.num_global_minima == 0) /* erroneous case: */ return GKLS_FLOATING_POINT_ERROR; /* some programmer's error */ return GKLS_OK; } /* GKLS_set_basins() */ /*****************************************************************************/ /* The subroutine initializes random sequence by generating a seed that */ /* depends on a specific test function number, on the number of local minima,*/ /* and on the problem dimension */ /* INPUT PARAMETERS: */ /* dim -- dimension of the problem (dim >= 2); */ /* nmin -- number of local minima (nmin >= 2); */ /* nf -- test function number (from 1 to 100) */ /* RETURN VALUE: normally GKLS_OK */ /*****************************************************************************/ int GKLS_initialize_rnd(unsigned int dim, unsigned int nmin, int nf) { long seed; /* seed number between 0 and 2^30-3 = 1,073,741,821*/ seed = (nf-1) + (nmin-1)*100 + dim*1000000L; /* If big values of nmin and dim are required, */ /* one must check wether seed <= 1073741821 */ ranf_start(seed); return GKLS_OK; } /* GKLS_initialize_rnd() */ /*****************************************************************************/ /* The main subroutine of the package that generates randomly the local and */ /* the global minimizers and function values at minimizers; */ /* it determines the radii of attraction regions of the minimizers */ /* INPUT PARAMETER: */ /* nf -- determines the number of test function, 1 <= nf <= 100 */ /* RETURN VALUE: */ /* an error code */ /* The boundaries vectors should be created and */ /* the parameters of the class should be defined first */ /*****************************************************************************/ int GKLS_arg_generate (unsigned int nf) { unsigned int i, j; int error; double sin_phi; /* for generating of the global minimizer coordinates */ /* by using the generalized spherical coordinates */ double gap = GKLS_global_radius; /* gap > 0 */ /* the minimal distance of any local minimizer to the attraction*/ /* region of the global minimizer M(1); the value */ /* GKLS_global_radius is given by default and can be changed, */ /* but it should not be too small. */ /* Check function number */ if ((nf < 1) || (nf > 100)) return GKLS_FUNC_NUMBER_ERROR; /* Check parameters */ if ( (error = GKLS_parameters_check()) != GKLS_OK) return error; /* Allocate memory */ if ( (error = GKLS_alloc()) != GKLS_OK) return error; /* Set random seed */ if ( (error = GKLS_initialize_rnd(GKLS_dim,GKLS_num_minima,nf)) != GKLS_OK) return error; ranf_array(rnd_num, NUM_RND); /* get random sequence */ rnd_counter = 0L; /* index of the random element from the sequence */ /* to be used as the next random number */ /* Set the paraboloid minimizer coordinates and */ /* the paraboloid minimum value */ for (i=0; i GKLS_domain_right[0] - GKLS_PRECISION) || (GKLS_minima.local_min[1][0] < GKLS_domain_left[0] + GKLS_PRECISION) ) GKLS_minima.local_min[1][0] = GKLS_minima.local_min[0][0] - GKLS_global_dist*cos(PI*rnd_num[rnd_counter]); sin_phi = sin(PI*rnd_num[rnd_counter]); rnd_counter++; /* Generate the remaining angles 0<=phi(i)<=2*PI, and */ /* the coordinates x(i), i=1,...,GKLS_dim-2 (not last!) */ for(j=1; j GKLS_domain_right[j] - GKLS_PRECISION) || (GKLS_minima.local_min[1][j] < GKLS_domain_left[j] + GKLS_PRECISION) ) GKLS_minima.local_min[1][j] = GKLS_minima.local_min[0][j] - GKLS_global_dist*cos(2.0*PI*rnd_num[rnd_counter])*sin_phi; sin_phi *= sin(2.0*PI*rnd_num[rnd_counter]); rnd_counter++; } /* Generate the last coordinate x(GKLS_dim-1) */ GKLS_minima.local_min[1][GKLS_dim-1] = GKLS_minima.local_min[0][GKLS_dim-1] + GKLS_global_dist*sin_phi; if ( (GKLS_minima.local_min[1][GKLS_dim-1] > GKLS_domain_right[GKLS_dim-1] - GKLS_PRECISION) || (GKLS_minima.local_min[1][GKLS_dim-1] < GKLS_domain_left[GKLS_dim-1] + GKLS_PRECISION) ) GKLS_minima.local_min[1][GKLS_dim-1] = GKLS_minima.local_min[0][GKLS_dim-1] - GKLS_global_dist*sin_phi; /* Set the global minimum value */ GKLS_minima.f[1] = GKLS_global_value; /* Set the weight coefficients w_rho(i) */ for (i=0; i GKLS_PRECISION ); i++; } error = GKLS_coincidence_check(); } while ( (error == GKLS_PARABOLA_MIN_COINCIDENCE_ERROR) || (error == GKLS_LOCAL_MIN_COINCIDENCE_ERROR) ); error = GKLS_set_basins(); if (error == GKLS_OK) isArgSet = 1; /* All the parameters are set */ /* and the user can evaluate a specific test function or */ /* its partial derivative by calling corresponding subroutines */ return error; } /* GKLS_arg_generate () */ /************************************************************************/ /* The subroutine evaluates the generated function */ /* of the ND-type (non-differentiable) */ /* */ /* INPUT PARAMETER: */ /* x -- a point of the (GKLS_dim)-dimensional euclidean space */ /* RETURN VALUE: */ /* a function value OR */ /* GKLS_MAX_VALUE if: (1) the vector x does not belong to D; */ /* (2) the user tries to call the function without */ /* parameter defining */ /************************************************************************/ double GKLS_ND_func(double *x) { unsigned int i, index; double norm, scal, a, rho; /* working variables */ if (!isArgSet) return GKLS_MAX_VALUE; for (i=0; i GKLS_domain_right[i]+GKLS_PRECISION) ) return GKLS_MAX_VALUE; /* Check wether x belongs to some basin of local minima, M(index) <> T */ /* Attention: number of local minima must be >= 2 */ index = 1; while ((index GKLS_minima.rho[index]) ) index++; if (index == GKLS_num_minima) { norm = GKLS_norm(GKLS_minima.local_min[0],x); /* Return the value of the paraboloid function */ return (norm * norm + GKLS_minima.f[0]); } /* Check wether x coincides with the local minimizer M(index) */ if ( GKLS_norm(x, GKLS_minima.local_min[index]) < GKLS_PRECISION ) return GKLS_minima.f[index]; norm = GKLS_norm(GKLS_minima.local_min[0],GKLS_minima.local_min[index]); a = norm * norm + GKLS_minima.f[0] - GKLS_minima.f[index]; rho = GKLS_minima.rho[index]; norm = GKLS_norm(GKLS_minima.local_min[index],x); scal = 0.0; for(i=0; i GKLS_domain_right[i]+GKLS_PRECISION) ) return GKLS_MAX_VALUE; /* Check wether x belongs to some basin of local minima, M(index) <> T */ /* Attention: number of local minima must be >= 2 */ index = 1; while ((index GKLS_minima.rho[index]) ) index++; if (index == GKLS_num_minima) { norm = GKLS_norm(GKLS_minima.local_min[0],x); /* Return the value of the paraboloid function */ return (norm * norm + GKLS_minima.f[0]); } /* Check wether x coincides with the local minimizer M(index) */ if ( GKLS_norm(x, GKLS_minima.local_min[index]) < GKLS_PRECISION ) return GKLS_minima.f[index]; norm = GKLS_norm(GKLS_minima.local_min[0],GKLS_minima.local_min[index]); a = norm * norm + GKLS_minima.f[0] - GKLS_minima.f[index]; rho = GKLS_minima.rho[index]; norm = GKLS_norm(GKLS_minima.local_min[index],x); scal = 0.0; for(i=0; i GKLS_domain_right[i] + GKLS_PRECISION) ) return GKLS_MAX_VALUE; /* Check wether x belongs to some basin of local minima */ index = 1; while ((index GKLS_minima.rho[index]) ) index++; if (index == GKLS_num_minima) { norm = GKLS_norm(GKLS_minima.local_min[0],x); /* Return the value of the paraboloid function */ return (norm * norm + GKLS_minima.f[0]); } /* Check wether x coincides with the local minimizer M(index) */ if ( GKLS_norm(x, GKLS_minima.local_min[index]) < GKLS_PRECISION ) return GKLS_minima.f[index]; norm = GKLS_norm(GKLS_minima.local_min[0],GKLS_minima.local_min[index]); a = norm * norm + GKLS_minima.f[0] - GKLS_minima.f[index]; rho = GKLS_minima.rho[index]; norm = GKLS_norm(GKLS_minima.local_min[index],x); scal = 0.0; for(i=0; i GKLS_dim) ) return GKLS_MAX_VALUE; else var_j = var_j - 1; /* to be used as an index of array */ if (!isArgSet) return GKLS_MAX_VALUE; for (i=0; i GKLS_domain_right[i]+GKLS_PRECISION) ) return GKLS_MAX_VALUE; /* Check wether x belongs to some basin of local minima, M(index) <> T */ /* Attention: number of local minima must be >= 2 */ index = 1; while ((index GKLS_minima.rho[index]) ) index++; if (index == GKLS_num_minima) { /* Return the value of the first order partial derivative of the paraboloid function */ return 2.0*(x[var_j]-GKLS_minima.local_min[0][var_j]); } /* Check wether x coincides with the local minimizer M(index) */ if ( GKLS_norm(x, GKLS_minima.local_min[index]) < GKLS_PRECISION ) return 0.0; norm = GKLS_norm(GKLS_minima.local_min[0],GKLS_minima.local_min[index]); a = norm * norm + GKLS_minima.f[0] - GKLS_minima.f[index]; rho = GKLS_minima.rho[index]; norm = GKLS_norm(GKLS_minima.local_min[index],x); scal = 0.0; for(i=0; i GKLS_dim) ) return GKLS_MAX_VALUE; else var_j = var_j - 1; /* to be used as an index of array */ if (!isArgSet) return GKLS_MAX_VALUE; for (i=0; i GKLS_domain_right[i]+GKLS_PRECISION) ) return GKLS_MAX_VALUE; /* Check wether x belongs to some basin of local minima, M(index) <> T */ /* Attention: number of local minima must be >= 2 */ index = 1; while ((index GKLS_minima.rho[index]) ) index++; if (index == GKLS_num_minima) { /* Return the value of the first order partial derivative of the paraboloid function */ return 2.0*(x[var_j]-GKLS_minima.local_min[0][var_j]); } /* Check wether x coincides with the local minimizer M(index) */ if ( GKLS_norm(x, GKLS_minima.local_min[index]) < GKLS_PRECISION ) return 0.0; norm = GKLS_norm(GKLS_minima.local_min[0],GKLS_minima.local_min[index]); a = norm * norm + GKLS_minima.f[0] - GKLS_minima.f[index]; rho = GKLS_minima.rho[index]; norm = GKLS_norm(GKLS_minima.local_min[index],x); scal = 0.0; for(i=0; i GKLS_dim) ) return GKLS_MAX_VALUE; if ( (var_k == 0) || (var_k > GKLS_dim) ) return GKLS_MAX_VALUE; the_same = (var_j == var_k); var_j = var_j - 1; var_k = var_k - 1; /* to be used as indexes of array */ if (!isArgSet) return GKLS_MAX_VALUE; for (i=0; i GKLS_domain_right[i]+GKLS_PRECISION) ) return GKLS_MAX_VALUE; /* Check wether x belongs to some basin of local minima, M(index) <> T */ /* Attention: number of local minima must be >= 2 */ index = 1; while ((index GKLS_minima.rho[index]) ) index++; if (index == GKLS_num_minima) { /* Return the value of the second order partial derivative of the paraboloid function */ if (the_same) return 2.0; else return 0.0; } /* Check wether x coincides with the local minimizer M(index) */ if ( GKLS_norm(x, GKLS_minima.local_min[index]) < GKLS_PRECISION ) { if (the_same) return delta; else return 0.0; } norm = GKLS_norm(GKLS_minima.local_min[0],GKLS_minima.local_min[index]); a = norm * norm + GKLS_minima.f[0] - GKLS_minima.f[index]; rho = GKLS_minima.rho[index]; norm = GKLS_norm(GKLS_minima.local_min[index],x); scal = 0.0; for(i=0; i= GKLS_MAX_VALUE-1000.0) error_code = GKLS_DERIV_EVAL_ERROR; } return error_code; } /* GKLS_D_gradient() */ /*******************************************************************************/ /* The subroutine evaluates the gradient of the D2-type test function */ /* INPUT PARAMETERS: */ /* x -- a point of the (GKLS_dim)-dimensional euclidean space */ /* g -- a pointer to the allocated array of the dimension GKLS_dim */ /* RETURN VALUES: */ /* an error code (that can be: GKLS_OK -- no error, or */ /* GKLS_DERIV_EVAL_ERROR -- otherwise) */ /* g -- a pointer to the array of gradient coordinates */ /*******************************************************************************/ int GKLS_D2_gradient (double *x, double *g) { unsigned int i; int error_code = GKLS_OK; if (!isArgSet) return GKLS_DERIV_EVAL_ERROR; if (g == NULL) return GKLS_DERIV_EVAL_ERROR; for (i=1; i<=GKLS_dim; i++) { g[i-1] = GKLS_D2_deriv1(i,x); if (g[i-1] >= GKLS_MAX_VALUE-1000.0) error_code = GKLS_DERIV_EVAL_ERROR; } return error_code; } /* GKLS_D2_gradient() */ /*******************************************************************************/ /* The subroutine evaluates the Hessian matrix of the D2-type test function */ /* INPUT PARAMETERS: */ /* x -- a point of the (GKLS_dim)-dimensional euclidean space */ /* h -- a pointer to the allocated matrix of dimension[GKLS_dim,GKLS_dim] */ /* RETURN VALUES: */ /* an error code (that can be: GKLS_OK -- no error, or */ /* GKLS_DERIV_EVAL_ERROR -- otherwise */ /* h -- a pointer to the Hessian matrix */ /*******************************************************************************/ int GKLS_D2_hessian (double *x, double **h) { unsigned int i, j; int error_code = GKLS_OK; if (!isArgSet) return GKLS_DERIV_EVAL_ERROR; if (h == NULL) return GKLS_DERIV_EVAL_ERROR; for (i=1; i<=GKLS_dim; i++) if (h[i-1] == NULL) return GKLS_DERIV_EVAL_ERROR; for (i=1; i<=GKLS_dim; i++) for (j=1; j<=GKLS_dim; j++) { h[i-1][j-1] = GKLS_D2_deriv2(i,j,x); if (h[i-1][j-1] >= GKLS_MAX_VALUE-1000.0) error_code = GKLS_DERIV_EVAL_ERROR; } return error_code; } /* GKLS_D2_hessian() */ /****************************** gkls.c ******************************/ SHAR_EOF fi # end of overwriting check if test -f 'gkls.h' then echo shar: will not over-write existing file "'gkls.h'" else cat << "SHAR_EOF" > 'gkls.h' /******************************************************************************/ /* GKLS-Generator of Classes of ND (non-differentiable), */ /* D (continuously differentiable), and */ /* D2 (twice continuously differentiable) */ /* Test Functions for Global Optimization */ /* */ /* Authors: */ /* */ /* M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev */ /* */ /* (C) 2002-2003 */ /* */ /* References: */ /* */ /* 1. M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev (2003), */ /* Software for Generation of Classes of Test Functions */ /* with Known Local and Global Minima for Global Optimization */ /* */ /* 2. D.Knuth (1997), The Art of Computer Programming, Vol. 2: */ /* Seminumerical Algorithms (Third Edition). Reading, Massachusetts: */ /* Addison-Wesley. */ /* */ /* The software constructs a convex quadratic function (paraboloid) and then*/ /* systematically distorts randomly selected parts of this function */ /* by polynomials in order to introduce local minima and to construct test */ /* functions which are non-differentiable (ND-type), continuously */ /* differentiable (D-type), and twice continuously differentiable (D2-type) */ /* at the feasible region. */ /* */ /* Each test class is defined by the following parameters: */ /* (1) problem dimension */ /* (2) number of local minima including the paraboloid min and the global min*/ /* (3) global minimum value */ /* (3) distance from the paraboloid vertex to the global minimizer */ /* (4) radius of the attraction region of the global minimizer */ /******************************************************************************/ #if !defined( __GKLS_H ) #define __GKLS_H /* Penalty value of the generated function if x is not in D */ #define GKLS_MAX_VALUE 1E+100 /* Value of the machine zero in the floating-point arithmetic */ #define GKLS_PRECISION 1.0E-10 /* Default value of the paraboloid minimum */ #define GKLS_PARABOLOID_MIN 0.0 /* Global minimum value: to be less than GKLS_PARABOLOID_MIN */ #define GKLS_GLOBAL_MIN_VALUE -1.0 /* Max value of the parameter delta for the D2-type class function */ /* The parameter delta is chosen randomly from (0, GKLS_DELTA_MAX_VALUE ) */ #define GKLS_DELTA_MAX_VALUE 10.0 /* Constant pi */ #ifndef PI #define PI 3.14159265 #endif /* Error codes */ #define GKLS_OK 0 #define GKLS_DIM_ERROR 1 #define GKLS_NUM_MINIMA_ERROR 2 #define GKLS_FUNC_NUMBER_ERROR 3 #define GKLS_BOUNDARY_ERROR 4 #define GKLS_GLOBAL_MIN_VALUE_ERROR 5 #define GKLS_GLOBAL_DIST_ERROR 6 #define GKLS_GLOBAL_RADIUS_ERROR 7 #define GKLS_MEMORY_ERROR 8 #define GKLS_DERIV_EVAL_ERROR 9 /* Reserved error codes */ #define GKLS_GREAT_DIM 10 #define GKLS_RHO_ERROR 11 #define GKLS_PEAK_ERROR 12 #define GKLS_GLOBAL_BASIN_INTERSECTION 13 /* Internal error codes */ #define GKLS_PARABOLA_MIN_COINCIDENCE_ERROR 14 #define GKLS_LOCAL_MIN_COINCIDENCE_ERROR 15 #define GKLS_FLOATING_POINT_ERROR 16 /* The next two structures define a list of all local minima and */ /* a list of the global minima. */ /* These lists are filled by the generator. */ /* The fields of the structures help to the user */ /* to study properties of a concrete generated test function */ /* The structure of type T_GKLS_Minima contains the following */ /* information about all local minima (including the paraboloid */ /* minimum and the global one): coordinates of local minimizers, */ /* local minima values, and attraction regions radii. */ typedef struct { double **local_min; /* list of local minimizers coordinates */ double *f; /* list of local minima values */ double *w_rho; /* list of radius weights */ double *peak; /* list of parameters gamma(i) = */ /* = local minimum value(i) - paraboloid */ /* minimum within attraction regions */ /* of local minimizer(i) */ double *rho; /* list of attraction regions radii */ } T_GKLS_Minima; /* The structure of type T_GKLS_GlobalMinima contains information */ /* about the number of global minimizers and their */ /* indices in the set of local minimizers */ typedef struct { unsigned int num_global_minima; /* number of global minima */ unsigned int *gm_index; /* list of indices of generated */ /* minimizers, which are the global ones (elements from 0 */ /* to (num_global_minima - 1) of the list) and the local ones */ /* (the resting elements of the list) */ } T_GKLS_GlobalMinima; /*-------------- Variables accessible by the user --------------------- */ extern double *GKLS_domain_left; /* left boundary vector of D */ /* D=[GKLS_domain_left; GKLS_domain_ight] */ extern double *GKLS_domain_right;/* right boundary vector of D */ extern unsigned int GKLS_dim; /* dimension of the problem, */ /* 2<=test_dim=2 */ extern double GKLS_global_dist; /* distance from the paraboloid minimizer */ /* to the global minimizer */ extern double GKLS_global_radius;/* radius of the global minimizer */ /* attraction region */ extern double GKLS_global_value; /* global minimum value, */ /* test_global_value < GKLS_PARABOLOID_MIN */ extern T_GKLS_Minima GKLS_minima; /* see the structures type description */ extern T_GKLS_GlobalMinima GKLS_glob; /*------------------------User function prototypes -------------------------*/ int GKLS_domain_alloc(void); /* allocate boundary vectors */ void GKLS_domain_free(void); /* deallocate boundary vectors */ int GKLS_set_default(void); /* set default values of the input parameters */ /* and allocate the boundary vectors if necessary */ void GKLS_free(void); /* deallocate memory needed for the generator */ int GKLS_parameters_check(void);/* test the validity of the input parameters*/ int GKLS_arg_generate (unsigned int); /* test function generator */ double GKLS_ND_func(double *); /* evaluation of an ND-typed test function */ double GKLS_D_func(double *); /* evaluation of a D-typed test function */ double GKLS_D2_func(double *); /* evaluation of a D2-type test function */ double GKLS_D_deriv(unsigned int, double *); /* first order partial derivative of the D-typed test function */ double GKLS_D2_deriv1(unsigned int, double *); /* first order partial derivative of the D2-typed test function */ double GKLS_D2_deriv2(unsigned int, unsigned int, double *); /* second order partial derivative of the D2-typed test function */ int GKLS_D_gradient (double *, double *); /* gradient of the D-type test function */ int GKLS_D2_gradient (double *, double *); /* gradient of the D2-type test function */ int GKLS_D2_hessian (double *, double **);/* Hessian of the D2-type test function */ #endif /* __GKLS_H */ SHAR_EOF fi # end of overwriting check if test -f 'rnd_gen.c' then echo shar: will not over-write existing file "'rnd_gen.c'" else cat << "SHAR_EOF" > 'rnd_gen.c' /****************************************************************************/ /* */ /* Copyright (c) 2000 by Donald E. Knuth */ /* It is explained in Seminumerical Algorithms, 3rd edition, Section 3.6 */ /* (or in the errata to the 2nd edition --- see */ /* http://www-cs-faculty.stanford.edu/~knuth/taocp.html */ /* in the changes to pages 171 and following of Volume 2). */ /* */ /* The functions of this file implement a random number generator. */ /* There are two functions: ranf_array() and ranf_start(). */ /* The logical-and operation '&' is used for efficiency, so this */ /* random generator is not strictly portable unless the computer uses */ /* two's complement representation for integer. It does not limit */ /* application of the package because almost all modern computers are */ /* based on two's complement arithmetic. */ /* */ /* The function ranf_start (long} seed) */ /* initializes the generator when given a seed number between */ /* 0 and 2^30 = 1,073,741,821. */ /* The function ranf_array (double aa[], int n) generates n new random */ /* numbers and places them into a given array aa, using the recurrence */ /* X(j) = [X(j-100) - X(j-37)] mod 2^30, */ /* that is particularly well studied to modern computers. The value n must */ /* be at least 100; larger values like 1000 are recommended, by default, */ /* n = NUM_RND = 1009. */ /* (see also the header file rnd_gen.h) */ /* */ /*********** See the book of D.Knuth for explanations and caveats! **********/ #include "rnd_gen.h" double rnd_num[NUM_RND]; /* array of random numbers */ double ran_u[KK]; /* the generator state */ void ranf_array(double aa[], int n) /* put n new random fractions in aa */ /* double *aa - destination */ /* int n - array length (must be at least KK) */ { int i, j; for (j=0;j=1.0) ss-=1.0-2*ulp; /* cyclic shift of 51 bits */ } for (;j0;j--) ul[j+j]=ul[j],u[j+j]=u[j]; /* "square" */ for (j=KK+KK-2;j>KK-LL;j-=2) ul[KK+KK-1-j]=0.0,u[KK+KK-1-j]=u[j]-ul[j]; for (j=KK+KK-2;j>=KK;j--) if(ul[j]) { ul[j-(KK-LL)]=ulp-ul[j-(KK-LL)], u[j-(KK-LL)]=mod_sum(u[j-(KK-LL)],u[j]); ul[j-KK]=ulp-ul[j-KK],u[j-KK]=mod_sum(u[j-KK],u[j]); } if (is_odd(s)) { /* "multiply by z" */ for (j=KK;j>0;j--) ul[j]=ul[j-1],u[j]=u[j-1]; ul[0]=ul[KK],u[0]=u[KK]; /* shift the buffer cyclically */ if (ul[KK]) ul[LL]=ulp-ul[LL],u[LL]=mod_sum(u[LL],u[KK]); } if (s) s>>=1; else t--; } for (j=0;j 'rnd_gen.h' /************************************************************************/ /* The header file for linkage to the file rnd_gen.c */ /************************************************************************/ #if !defined ( __RND_GEN_H ) #define __RND_GEN_H #define KK 100 /* the long lag */ #define LL 37 /* the short lag */ #define mod_sum(x,y) (((x)+(y))-(int)((x)+(y))) /* (x+y) mod 1.0 */ #define TT 70 /* guaranteed separation between streams */ #define is_odd(s) ((s)&1) #define QUALITY 1009 /* recommended quality level for high-res use */ #define NUM_RND 1009 /* size of the array of random numbers */ extern double rnd_num[NUM_RND]; /* array of random numbers */ /* For rnd_gen.c */ void ranf_array(double aa[], int n); /* put n new random fractions in aa */ /* double *aa - destination */ /* int n - array length (must be at least KK) */ void ranf_start(long seed); /* do this before using ranf_array */ /* long seed - selector for different streams */ #endif /* __RND_GEN_H */ SHAR_EOF fi # end of overwriting check cd .. cd .. cd .. if test ! -d 'Doc' then mkdir 'Doc' fi cd 'Doc' if test -f 'Readme.txt' then echo shar: will not over-write existing file "'Readme.txt'" else cat << "SHAR_EOF" > 'Readme.txt' /********************************************************************************/ /* GKLS-Generator of Classes of ND (non-differentiable), */ /* D (continuously differentiable), and */ /* D2 (twice continuously differentiable) */ /* Test Functions for Global Optimization */ /* */ /* Authors: */ /* */ /* M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev */ /* */ /* (C) 2002-2003 */ /* */ /* References: */ /* */ /* 1. M.Gaviano, D.E.Kvasov, D.Lera, and Ya.D.Sergeyev (2003), */ /* Software for Generation of Classes of Test Functions */ /* with Known Local and Global Minima for Global Optimization */ /* */ /* 2. D.Knuth (1997), The Art of Computer Programming, Vol. 2: */ /* Seminumerical Algorithms (Third Edition). Reading, Massachusetts: */ /* Addison-Wesley. */ /* */ /* The software constructs a convex quadratic function (paraboloid) and then */ /* systematically distorts randomly selected parts of this function */ /* by polynomials in order to introduce local minima and to construct test */ /* functions which are non-differentiable (ND-type), continuously */ /* differentiable (D-type), and twice continuously differentiable (D2-type) */ /* at the feasible region. */ /* */ /* Each test class is defined by the following parameters: */ /* (1) problem dimension */ /* (2) number of local minima including the paraboloid min and the global min */ /* (3) global minimum value */ /* (3) distance from the paraboloid vertex to the global minimizer */ /* (4) radius of the attraction region of the global minimizer */ /********************************************************************************/ This version is available also from the following WWW-address: http://wwwinfo.deis.unical.it/~yaro It includes: \Package -- Directory containing the C code of the generator. The package consists of the following files: gkls.c -- the main file; gkls.h -- the header file that users should include in their application projects in order to call subroutines from the file gkls.c; rnd_gen.c -- the file containing the uniform random number generator proposed by D.Knuth (1997); rnd_gen.h -- the header file for linkage to the file rnd_gen.c; example.c -- an example of the GKLS-generator usage; Makefile -- an example of a UNIX makefile provided to UNIX users for a simple compilation and linkage of separate files of the application project. UserManual.pdf -- A User Manual. \Tex -- LaTeX-files of the User Manual. SHAR_EOF fi # end of overwriting check if test -f 'UserManual.bib' then echo shar: will not over-write existing file "'UserManual.bib'" else cat << "SHAR_EOF" > 'UserManual.bib' % UserManual.bib %% bibtexfile{ %% author = "M. Gaviano and D. E. Kvasov and D. Lera and Ya. D. Sergeyev", %% date = "17 August 2003", %% filename = "UserManual.bib" } @article{Gaviano:et:al.(2003), author = "M. Gaviano and D. E. Kvasov and D. Lera and {Ya}. D. Sergeyev", title = "Generation of Classes of Test Functions with Known Local Minima", journal = "ACM Trans. Math. Soft.", year = 2003, volume = "V", number = "N", pages = "PP--PP"} @book{Knuth(1997), author = "D. Knuth", title = "The Art of Computer Programming, Vol. 2: Seminumerical Algorithms", edition = "third", year = 1997, publisher = "Addison-Wesley, Reading", address = "Massachusetts"} @misc{Knuth:HomePage, author = "D. Knuth", year = 2002, title = "Home page at: http://sunburn.stanford.edu/ $\tilde{ }$ knuth/"} SHAR_EOF fi # end of overwriting check if test -f 'UserManual.ps' then echo shar: will not over-write existing file "'UserManual.ps'" else cat << "SHAR_EOF" > 'UserManual.ps' %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: UserManual.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips UserManual %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2003.08.28:1743 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if 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y(can)g(\014nd)i(an)e(example)g(of)h(a)f(generated)g(test)g (function)g(also.)900 2534 y Fr(2.)166 b(STRUCTURE)57 b(OF)e(THE)g(P)-14 b(A)-5 b(CKA)g(GE)900 2799 y Fo(The)37 b(GKLS-generator)g(pac)-5 b(k)c(age)37 b(has)g(b)5 b(een)37 b(written)f(in)i(ANSI)e(Standard)h(C)g(and)h(successfully)900 2999 y(tested)g(on)h(Windo)-5 b(ws)39 b(and)g(UNIX)e(platforms.)69 b(Our)39 b(implemen)-5 b(tation)39 b(follo)-5 b(ws)40 b(the)e(pro)5 b(cedure)900 3198 y(describ)g(ed)56 b(in)f(Section)h(3)f (of)g(the)g Fi(p)-8 b(ap)g(er)p Fo(.)1066 3397 y(The)55 b(pac)-5 b(k)c(age)55 b(implemen)-5 b(ting)57 b(the)e(generator)g (includes)h(the)f(follo)-5 b(wing)57 b(\014les:)900 3671 y Fp(gkls.c)83 b Fo({)55 b(the)g(main)h(\014le;)900 3898 y Fp(gkls.h)83 b Fo({)62 b(the)h(header)f(\014le)h(that)f(users)i (should)g(include)f(in)g(their)f(application)i(pro)9 b(jects)62 b(in)1066 4097 y(order)56 b(to)e(call)i(subroutines)h(from)e (the)g(\014le)h Fp(gkls.c)p Fo(;)900 4323 y Fp(rnd)p 1202 4323 58 7 v 69 w(gen.c)83 b Fo({)66 b(the)f(\014le)h(con)-5 b(taining)67 b(the)e(uniform)i(random)g(n)-5 b(um)g(b)5 b(er)67 b(generator)f(prop)5 b(osed)1066 4522 y(in)56 b(Kn)-5 b(uth)56 b([1997;)g(2002];)900 4749 y Fp(rnd)p 1202 4749 V 69 w(gen.h)83 b Fo({)55 b(the)g(header)h(\014le)f(for)g (link)-9 b(age)55 b(to)g(the)g(\014le)h Fp(rnd)p 4889 4749 V 68 w(gen.c)p Fo(;)900 4975 y Fp(example.c)81 b Fo({)55 b(an)h(example)f(of)g(the)g(GKLS-generator)h(usage;)900 5201 y Fp(Mak)-5 b(e\014le)81 b Fo({)60 b(an)g(example)g(of)g(a)h(UNIX) d(mak)-5 b(e\014le)61 b(pro)-5 b(vided)61 b(to)e(UNIX)g(users)i(for)f (a)g(simple)1066 5401 y(compilation)c(and)g(link)-9 b(age)55 b(of)g(separate)g(\014les)h(of)g(the)f(application)g(pro)9 b(ject.)1066 5675 y(F)-14 b(or)79 b(implemen)-5 b(tation)79 b(details)g(the)f(user)g(can)h(consult)g(the)e(C)i(co)5 b(des.)142 b(Note)77 b(that)h(the)900 5874 y(random)62 b(n)-5 b(um)g(b)5 b(er)61 b(generator)g(in)f Fp(rnd)p 3369 5874 V 69 w(gen.c)h Fo(uses)g(the)f(logical-and)i(op)5 b(eration)60 b(`&')h(for)g(ef-)900 6073 y(\014ciency)-14 b(,)68 b(so)e(it)f(is)i(not)e(strictly)g(p)5 b(ortable)65 b(unless)i(the)e(computer)h(uses)g(t)-5 b(w)g(o's)67 b(complemen)-5 b(t)900 6273 y(represen)g(tation)79 b(for)f(in)-5 b(teger.)143 b(It)78 b(do)5 b(es)78 b(not)h(limit)f(p)5 b(ortabilit)-5 b(y)78 b(of)g(the)g(pac)-5 b(k)c(age)79 b(b)5 b(ecause)900 6472 y(almost)56 b(all)g(mo)5 b(dern)55 b(computers)h(are)f(based)i(on)e(t)-5 b(w)g(o's)56 b(complemen)-5 b(t)56 b(arithmetic.)900 6813 y Fr(3.)166 b(CALLING)36 b(SEQUENCE)i(F)-5 b(OR)37 b(GENERA)-14 b(TION)37 b(AND)f(USA)-5 b(GE)37 b(OF)g(THE)f(TESTS)i(CLASSES)900 7078 y Fo(Here)58 b(w)-5 b(e)59 b(describ)5 b(e)59 b(ho)-5 b(w)59 b(to)f(generate)g(and)i (use)f(classes)h(of)e(non-di\013eren)-5 b(tiable)60 b(\(ND-t)-5 b(yp)5 b(e\),)900 7277 y(di\013eren)-5 b(tiable)65 b(\(D-t)-5 b(yp)5 b(e\),)67 b(and)e(t)-5 b(wice)65 b(con)-5 b(tin)g(uously)67 b(di\013eren)-5 b(tiable)65 b(\(D2-t)-5 b(yp)5 b(e\))64 b(test)g(func-)900 7477 y(tions.)144 b(Again,)84 b(w)-5 b(e)79 b(concen)-5 b(trate)78 b(on)g(the)h(D-t)-5 b(yp)5 b(e)78 b(functions.)143 b(The)79 b(op)5 b(erations)79 b(for)f(the)900 7676 y(remaining)56 b(t)-5 b(w)g(o)56 b(t)-5 b(yp)5 b(es)55 b(are)g(analogous.)1066 7875 y(T)-14 b(o)55 b(utilize)h(the)f(GKLS-generator)g(the)g(user)h(m)-5 b(ust)57 b(p)5 b(erform)55 b(the)g(follo)-5 b(wing)56 b(steps:)900 8149 y Fp(Step)64 b(1.)83 b Fo(Input)56 b(of)f(the)g(parameters)h(de\014ning)g(a)f(sp)5 b(eci\014c)56 b(test)e(class.)900 8376 y Fp(Step)64 b(2.)83 b Fo(Generating)56 b(a)f(sp)5 b(eci\014c)55 b(test)g(function)h(of)f(the)g(de\014ned)h (test)e(class.)900 8602 y Fp(Step)64 b(3.)83 b Fo(Ev)-9 b(aluation)80 b(of)h(the)f(generated)g(test)g(function)g(and,)88 b(if)80 b(necessary)-14 b(,)88 b(its)80 b(partial)1066 8801 y(deriv)-9 b(ativ)k(es.)900 9028 y Fp(Step)64 b(4.)83 b Fo(Memory)56 b(deallo)5 b(cating.)900 9302 y(Let)54 b(us)j(consider)f(these)f(steps)h(in)g(turn.)900 9622 y Fr(3.1)166 b(Input)55 b(of)g(the)g(pa)-5 b(rameters)53 b(de\014ning)i(a)g(sp)5 b(eci\014c)55 b(test)f(class)900 9887 y Fo(This)48 b(step)f(is)h(sub)5 b(divided)48 b(in)-5 b(to:)70 b(\(1\))46 b(de\014ning)i(the)f(parameters)h(of)e(the)h(test)g (class,)i(\(2\))e(de\014n-)900 10087 y(ing)56 b(the)f(admissible)i (region)f(\012,)f(and)h(\(3\))e(c)-5 b(hec)g(king)56 b(\(if)f(necessary\).)p eop %%Page: 2 3 2 2 bop 900 499 a Fl(2)282 b Fj(\001)316 b Fl(M.)51 b(Gaviano,)g(D.)g (E.)g(Kvasov,)f(D.)h(Lera,)g(and)h(Y)-13 b(a.)50 b(D.)h(Sergey)l(ev) 1066 931 y Fo(3.1.1)170 b Fi(De\014ning)50 b(the)f(p)-8 b(ar)g(ameters)47 b(of)h(the)h(test)g(class.)83 b Fo(The)43 b(parameters)h(to)f(b)5 b(e)43 b(de\014ned)g(b)-5 b(y)900 1130 y(the)47 b(user)i(determine)f(a)g(sp)5 b(eci\014c)48 b(class)g(\(of)g(the)f(ND-,)i(D-)f(or)g(D2-t)-5 b(yp)5 b(e\))47 b(of)h(100)g(test)f(functions)900 1329 y(\(a)f(sp)5 b(eci\014c)46 b(function)g(is)h(retriev)-5 b(ed)46 b(b)-5 b(y)46 b(its)h(n)-5 b(um)g(b)5 b(er\).)71 b(There)46 b(are)g(the)g(follo)-5 b(wing)47 b(parameters:)900 1624 y Fi(GKLS)p 1363 1624 51 7 v 62 w(dim)82 b Fo({)73 b(\()p Fp(unsigned)83 b(in)-5 b(t)p Fo(\))72 b(dimension)j Fh(N)91 b Fo(\(from)73 b(\(1\)\))f(of)g(test)h(functions;)82 b Fh(N)94 b Fg(\025)76 b Fo(2)1066 1824 y(\(since)54 b(m)-5 b(ultidimensional)56 b(problems)f(are)e(considered)h(in)g (\(1\)\))f(and)h Fh(N)64 b(<)54 b Fo(NUM)p 6263 1824 50 7 v 59 w(RND)f(in)1066 2023 y Fp(rnd)p 1368 2023 58 7 v 69 w(gen.h)p Fo(;)66 b(this)e(v)-9 b(alue)62 b(is)h(limited)g(b)-5 b(y)63 b(the)g(p)5 b(o)-5 b(w)g(er)63 b(of)f Fp(unsigned)i(in)-5 b(t)o Fo(-represen)g(tation;)1066 2222 y(default)55 b Fh(N)65 b Fo(=)46 b(2;)900 2454 y Fi(GKLS)p 1363 2454 51 7 v 62 w(num)p 1743 2454 V 61 w(minima)83 b Fo({)68 b(\()p Fp(unsigned)77 b(in)-5 b(t)p Fo(\))67 b(n)-5 b(um)g(b)5 b(er)69 b Fh(m)f Fo(\(from)g(\(4\)\))f(of)h(lo)5 b(cal)68 b(minima)h(in-)1066 2653 y(cluding)58 b(the)e(parab)5 b(oloid)58 b Fh(Z)69 b Fo(minim)-5 b(um)59 b(\(from)e(\(3\)\))f(and)h (the)f(global)i(minim)-5 b(um;)60 b Fh(m)49 b Fg(\025)f Fo(2;)1066 2852 y(the)74 b(upp)5 b(er)75 b(b)5 b(ound)75 b(of)f(this)g(parameter)h(is)g(limited)f(b)-5 b(y)75 b(the)f(p)5 b(o)-5 b(w)g(er)75 b(of)f Fp(unsigned)63 b(in)-5 b(t)o Fo(-)1066 3051 y(represen)g(tation;)56 b(default)f Fh(m)46 b Fo(=)g(10;)900 3283 y Fi(GKLS)p 1363 3283 V 62 w(glob)-8 b(al)p 1823 3283 V 62 w(value)84 b Fo({)70 b(\()p Fp(double)p Fo(\))f(global)j(minim)-5 b(um)72 b(v)-9 b(alue)71 b Fh(f)5040 3223 y Fq(\003)5187 3283 y Fo(of)f Fh(f)18 b Fo(\()p Fh(x)p Fo(\);)77 b(condition)71 b(\(16\))1066 3482 y(m)-5 b(ust)62 b(b)5 b(e)60 b(satis\014ed;)65 b(the)60 b(default)h(v)-9 b(alue)60 b(is)h Fg(\000)p Fo(1)p Fh(:)p Fo(0)g(\(de\014ned)g(in)g(the)g(\014le)g Fp(gkls.h)f Fo(as)h(a)g(con-)1066 3681 y(stan)-5 b(t)56 b(GKLS)p 1952 3681 50 7 v 60 w(GLOBAL)p 2722 3681 V 59 w(MIN)p 3118 3681 V 59 w(V)-18 b(ALUE\);)900 3913 y Fi(GKLS)p 1363 3913 51 7 v 62 w(glob)-8 b(al)p 1823 3913 V 62 w(dist)83 b Fo({)67 b(\()p Fp(double)p Fo(\))f(distance)i Fh(r)3857 3853 y Fq(\003)4000 3913 y Fo(from)g(the)f(parab)5 b(oloid)68 b(v)-5 b(ertex)67 b Fh(T)90 b Fo(in)68 b(\(3\))f(to)1066 4112 y(the)47 b(global)h(minimizer)g Fh(x)2687 4052 y Fq(\003)2809 4112 y Fg(2)e Fo(\012)h(of)g Fh(f)18 b Fo(\()p Fh(x)p Fo(\);)50 b(condition)d(\(17\))g(m)-5 b(ust)48 b(b)5 b(e)47 b(satis\014ed;)k(the)c(default)1066 4311 y(v)-9 b(alue)55 b(is)2346 4604 y(GKLS)p 2811 4604 50 7 v 60 w(global)p 3304 4604 V 61 w(dist)3724 4510 y Ff(def)3751 4604 y Fo(=)217 b(min)4009 4712 y Ff(1)p Fq(\024)p Fe(j)7 b Fq(\024)p Fe(N)4490 4604 y Fg(j)p Fh(b)p Fo(\()p Fh(j)j Fo(\))36 b Fg(\000)h Fh(a)p Fo(\()p Fh(j)10 b Fo(\))p Fg(j)p Fh(=)28 b Fo(3)p Fh(;)1066 4986 y Fo(where)55 b(the)g(v)-5 b(ectors)55 b Fh(a)g Fo(and)h Fh(b)g Fo(determine)f(the)g (admissible)j(region)d(\012)g(in)h(\(2\);)900 5217 y Fi(GKLS)p 1363 5217 51 7 v 62 w(glob)-8 b(al)p 1823 5217 V 62 w(r)g(adius)82 b Fo({)c(\()p Fp(double)p Fo(\))f(radius)i Fh(\032)3928 5157 y Fq(\003)4083 5217 y Fo(of)f(the)f(attraction)h (region)g(of)g(the)g(global)1066 5416 y(minimizer)56 b Fh(x)1936 5356 y Fq(\003)2059 5416 y Fg(2)46 b Fo(\012)55 b(of)g Fh(f)18 b Fo(\()p Fh(x)p Fo(\);)55 b(condition)g(\(18\))g(m)-5 b(ust)56 b(b)5 b(e)55 b(satis\014ed;)i(the)e(default)g(v)-9 b(alue)55 b(is)2258 5763 y(GKLS)p 2723 5763 50 7 v 60 w(global)p 3216 5763 V 61 w(radius)3812 5669 y Ff(def)3839 5763 y Fo(=)217 b(min)4097 5871 y Ff(1)p Fq(\024)p Fe(j)7 b Fq(\024)p Fe(N)4577 5763 y Fg(j)p Fh(b)p Fo(\()p Fh(j)j Fo(\))37 b Fg(\000)g Fh(a)p Fo(\()p Fh(j)10 b Fo(\))p Fg(j)p Fh(=)28 b Fo(6)p Fh(:)900 6149 y Fo(The)55 b(user)g(ma)-5 b(y)56 b(call)f(subroutine)h Fi(GKLS)p 3541 6149 51 7 v 61 w(set)p 3801 6149 V 62 w(default)p Fo(\(\))f(to)f(set)h(the)f (default)h(v)-9 b(alues)55 b(of)g(these)900 6348 y(\014v)-5 b(e)55 b(v)-9 b(ariables.)1066 6647 y(3.1.2)170 b Fi(De\014ning)83 b(the)e(admissible)f(r)-8 b(e)g(gion.)83 b Fo(With)78 b Fh(N)97 b Fo(determined,)84 b(the)78 b(user)h(m)-5 b(ust)79 b(al-)900 6846 y(lo)5 b(cate)79 b(dynamic)h(arra)-5 b(ys)80 b Fi(GKLS)p 3089 6846 V 62 w(domain)p 3686 6846 V 60 w(left)h Fo(and)f Fi(GKLS)p 4851 6846 V 62 w(domain)p 5448 6846 V 61 w(right)g Fo(to)f(de\014ne)h(the)900 7045 y(b)5 b(oundary)56 b(of)f(the)g(h)-5 b(yp)5 b(errectangle)54 b(\012.)74 b(This)56 b(is)g(done)g(b)-5 b(y)56 b(calling)f(subroutine) 1066 7245 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 7245 V 62 w(domain)p 2409 7245 V 61 w(al)8 b(lo)-8 b(c)56 b Fo(\(\);)900 7444 y(whic)-5 b(h)56 b(has)g(no)g(parameters)g(and)f(returns)h(the)f (follo)-5 b(wing)56 b(error)f(co)5 b(des)56 b(de\014ned)g(in)f Fp(gkls.h)p Fo(:)900 7739 y Fp(GKLS)p 1432 7739 58 7 v 68 w(OK)84 b Fo({)55 b(no)h(errors;)900 7970 y Fp(GKLS)p 1432 7970 V 68 w(DIM)p 1899 7970 V 70 w(ERR)-5 b(OR)85 b Fo({)39 b(the)f(problem)i(dimension)h(is)e(out)g(of)g(range;)45 b(it)39 b(m)-5 b(ust)40 b(b)5 b(e)38 b(greater)1066 8169 y(than)55 b(or)h(equal)f(to)g(2)g(and)h(less)g(than)g(NUM)p 3865 8169 50 7 v 59 w(RND)f(de\014ned)h(in)g Fp(rnd)p 5411 8169 58 7 v 68 w(gen.h)p Fo(;)900 8401 y Fp(GKLS)p 1432 8401 V 68 w(MEMOR)-16 b(Y)p 2401 8401 V 71 w(ERR)-5 b(OR)85 b Fo({)55 b(there)g(is)h(not)f(enough)h(memory)f(to)g(allo)5 b(cate.)1066 8696 y(The)71 b(same)h(subroutine)h(de\014nes)f(the)f (admissible)i(region)f(\012.)121 b(The)72 b(default)f(v)-9 b(alue)71 b(\012)h(=)900 8895 y([)p Fg(\000)p Fo(1)p Fh(;)28 b Fo(1])1361 8835 y Fe(N)1543 8895 y Fo(is)55 b(set)h(b)-5 b(y)55 b Fi(GKLS)p 2662 8895 51 7 v 62 w(set)p 2923 8895 V 62 w(default)p Fo(\(\))o(.)1066 9194 y(3.1.3)170 b Fi(Che)-8 b(cking.)84 b Fo(The)71 b(follo)-5 b(wing)73 b(subroutine)f(allo)-5 b(ws)73 b(the)e(user)h(to)f(c)-5 b(hec)g(k)72 b(v)-9 b(alidit)k(y)71 b(of)900 9393 y(the)55 b(input)h(parameters:)1066 9592 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 9592 V 62 w(p)-8 b(ar)g(ameters)p 2664 9592 V 61 w(che)g(ck)56 b Fo(\(\).)900 9792 y(It)f(has)h(no)g(parameters)f(and)h (returns)g(the)f(follo)-5 b(wing)56 b(error)f(co)5 b(des)56 b(\(see)f Fp(gkls.h)p Fo(\):)900 10087 y Fp(GKLS)p 1432 10087 58 7 v 68 w(OK)84 b Fo({)55 b(no)h(errors;)p eop %%Page: 3 4 3 3 bop 2253 499 a Fl(GKLS-Generato)l(r)53 b(of)e(Classes)h(of)f(T)-13 b(est)52 b(F)l(unctions:)70 b Fk(User)53 b(Manual)317 b Fj(\001)283 b Fl(3)900 931 y Fp(GKLS)p 1432 931 58 7 v 68 w(DIM)p 1899 931 V 70 w(ERR)-5 b(OR)85 b Fo({)55 b(problem)h(dimension)h(error;)900 1156 y Fp(GKLS)p 1432 1156 V 68 w(NUM)p 1977 1156 V 70 w(MINIMA)p 2846 1156 V 71 w(ERR)-5 b(OR)84 b Fo({)55 b(n)-5 b(um)g(b)5 b(er)57 b(of)e(lo)5 b(cal)55 b(minima)i(error;)900 1380 y Fp(GKLS)p 1432 1380 V 68 w(BOUND)-5 b(AR)-16 b(Y)p 2631 1380 V 71 w(ERR)-5 b(OR)84 b Fo({)d(the)e(admissible)k(region)d(b)5 b(oundary)81 b(v)-5 b(ectors)80 b(are)1066 1580 y(ill-de\014ned;)900 1804 y Fp(GKLS)p 1432 1804 V 68 w(GLOBAL)p 2303 1804 V 70 w(MIN)p 2775 1804 V 70 w(V)-21 b(ALUE)p 3499 1804 V 69 w(ERR)-5 b(OR)84 b Fo({)64 b(the)g(global)h(minim)-5 b(um)66 b(v)-9 b(alue)64 b(is)h(not)1066 2004 y(less)71 b(than)f(the)g(parab)5 b(oloid)70 b(\(3\))f(minim)-5 b(um)73 b(v)-9 b(alue)69 b Fh(t)h Fo(de\014ned)h(in)f Fp(gkls.h)g Fo(as)g(a)g(constan)-5 b(t)1066 2203 y(GKLS)p 1531 2203 50 7 v 60 w(P)-14 b(ARABOLOID)p 2729 2203 V 59 w(MIN;)900 2427 y Fp(GKLS)p 1432 2427 58 7 v 68 w(GLOBAL)p 2303 2427 V 70 w(DIST)p 2830 2427 V 69 w(ERR)-5 b(OR)84 b Fo({)56 b(the)f(parameter)g Fh(r)4963 2367 y Fq(\003)5094 2427 y Fo(do)5 b(es)56 b(not)f(satisfy)g(\(17\);)900 2652 y Fp(GKLS)p 1432 2652 V 68 w(GLOBAL)p 2303 2652 V 70 w(RADIUS)p 3131 2652 V 70 w(ERR)-5 b(OR)84 b Fo({)48 b(the)f(parameter)g Fh(\032)5247 2592 y Fq(\003)5371 2652 y Fo(do)5 b(es)48 b(not)g(satisfy)f(\(18\).)900 2969 y Fr(3.2)166 b(Generating)55 b(a)g(sp)5 b(eci\014c)54 b(test)h(function)g(of)g(the)g(de\014ned)f(test)h(class)900 3234 y Fo(After)70 b(a)h(sp)5 b(eci\014c)71 b(test)g(class)h(has)g(b)5 b(een)70 b(c)-5 b(hosen)73 b(\(i.e.,)h(the)d(input)h(parameters)f(ha)-5 b(v)g(e)72 b(b)5 b(een)900 3434 y(determined\))68 b(the)f(user)i(can)f (generate)f(a)h(sp)5 b(eci\014c)68 b(function)g(that)g(b)5 b(elongs)68 b(to)g(the)g(c)-5 b(hosen)900 3633 y(class)56 b(of)f(100)h(test)f(functions.)74 b(This)56 b(is)g(done)g(b)-5 b(y)55 b(calling)h(subroutine)1066 3832 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 3832 51 7 v 62 w(ar)-8 b(g)p 2097 3832 V 61 w(gener)g(ate)57 b Fo(\()p Fp(unsigned)63 b(in)-5 b(t)54 b Fi(nf)p Fo(\);)900 4031 y(where)900 4299 y Fi(nf)83 b Fo({)56 b(the)f(n)-5 b(um)g(b)5 b(er)56 b(of)f(a)h(function)f(from)h (the)f(test)f(class)j(\(from)e(1)g(to)g(100\).)900 4567 y(This)48 b(subroutine)g(initializes)g(the)f(random)h(n)-5 b(um)g(b)5 b(er)48 b(generator,)h(c)-5 b(hec)g(ks)47 b(the)g(input)h(parame-)900 4766 y(ters,)c(allo)5 b(cates)40 b(dynamic)h(arra)-5 b(ys,)45 b(and)c(generates)f(a)h(test)f(function)h (follo)-5 b(wing)41 b(the)g(pro)5 b(cedure)900 4965 y(of)53 b(Section)g(3)g(of)g(the)f Fi(p)-8 b(ap)g(er)p Fo(.)72 b(It)53 b(returns)g(an)h(error)f(co)5 b(de)52 b(that)h(can)g(b)5 b(e)52 b(the)h(same)h(as)f(for)g(sub-)900 5165 y(routines)j Fi(GKLS)p 2001 5165 V 62 w(p)-8 b(ar)g(ameters)p 2853 5165 V 60 w(che)g(ck)p Fo(\(\))55 b(and)h Fi(GKLS)p 4255 5165 V 62 w(domain)p 4852 5165 V 61 w(al)8 b(lo)-8 b(c)p Fo(\(\),)55 b(or)h(additionally:)900 5432 y Fp(GKLS)p 1432 5432 58 7 v 68 w(FUNC)p 2054 5432 V 70 w(NUMBER)p 3005 5432 V 69 w(ERR)-5 b(OR)85 b Fo({)42 b(the)g(n)-5 b(um)g(b)5 b(er)43 b(of)f(a)h(test)e(function)h(to)g(generate)1066 5632 y(exceeds)55 b(100)g(or)h(it)f(is)h(less)g(than)f(1.)1066 5900 y Fi(GKLS)p 1529 5900 51 7 v 62 w(ar)-8 b(g)p 1814 5900 V 61 w(gener)g(ate)p Fo(\(\))74 b(generates)f(the)g(list)h(of)f (all)g(lo)5 b(cal)73 b(minima)i(and)f(the)f(list)h(of)f(the)900 6099 y(global)f(minima)g(as)g(parts)f(of)g(the)g(structures)g Fi(GKLS)p 4444 6099 V 62 w(minima)g Fo(and)g Fi(GKLS)p 5920 6099 V 62 w(glob)p Fo(,)76 b(resp)5 b(ec-)900 6298 y(tiv)-5 b(ely)-14 b(.)72 b(The)52 b(\014rst)g(structure)f(gathers)h (the)g(follo)-5 b(wing)52 b(information)g(ab)5 b(out)52 b(all)g(lo)5 b(cal)51 b(minima)900 6497 y(\(including)57 b(the)f(parab)5 b(oloid)57 b(minim)-5 b(um)59 b(and)e(the)f(global)g (one\):)76 b(co)5 b(ordinates)56 b(of)g(lo)5 b(cal)56 b(min-)900 6697 y(imizers,)75 b(lo)5 b(cal)70 b(minima)h(v)-9 b(alues,)75 b(and)70 b(attraction)g(regions)h(radii.)119 b(The)70 b(second)h(structure)900 6896 y(con)-5 b(tains)69 b(information)f(ab)5 b(out)68 b(the)f(n)-5 b(um)g(b)5 b(er)70 b(of)d(global)i(minimizers)g(and)g(their)e(indices)i(in)900 7095 y(the)55 b(set)g(of)g(lo)5 b(cal)55 b(minimizers.)76 b(It)55 b(has)h(the)f(follo)-5 b(wing)56 b(\014elds:)900 7363 y Fi(num)p 1228 7363 V 61 w(glob)-8 b(al)p 1687 7363 V 62 w(minima)83 b Fo({)55 b(\()p Fp(unsigned)63 b(in)-5 b(t)p Fo(\))54 b(total)h(n)-5 b(um)g(b)5 b(er)57 b(of)e(global)h(minima;)900 7588 y Fi(gm)p 1122 7588 V 61 w(index)84 b Fo({)47 b(\()p Fp(unsigned)54 b(in)-5 b(t)53 b(*)p Fo(\))47 b(list)h(of)f(indices)h(of)f(generated)g (minimizers,)j(whic)-5 b(h)48 b(are)f(the)1066 7787 y(global)h(ones)g (\(elemen)-5 b(ts)48 b(0)f(to)g(\()p Fi(num)p 3363 7787 V 61 w(glob)-8 b(al)p 3822 7787 V 62 w(minima)21 b Fg(\000)g Fo(1\))47 b(of)g(the)g(list\))h(and)g(the)f(lo)5 b(cal)47 b(ones)1066 7986 y(\(the)55 b(remaining)h(elemen)-5 b(ts)56 b(of)f(the)g(list\).)900 8254 y(The)g(elemen)-5 b(ts)56 b(of)f(the)g(list)g Fi(GKLS)p 3135 8254 V 62 w(glob)p Fo(.)p Fi(gm)p 3734 8254 V 62 w(index)h Fo(are)f(indices)h(to)f(a)g(sp) 5 b(eci\014c)55 b(minimizer)h(in)900 8453 y(the)f(\014rst)h(structure)f Fi(GKLS)p 2704 8453 V 62 w(minima)g Fo(c)-5 b(haracterized)55 b(b)-5 b(y)55 b(the)g(follo)-5 b(wing)56 b(\014elds:)900 8721 y Fi(lo)-8 b(c)g(al)p 1224 8721 V 61 w(min)83 b Fo({)56 b(\()p Fp(double)62 b(**)p Fo(\))56 b(list)g(of)f(lo)5 b(cal)55 b(minimizers)i(co)5 b(ordinates;)932 8946 y Fi(f)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(list)h(of)f(lo)5 b(cal)55 b(minima)i(v)-9 b(alues;)900 9170 y Fi(rho)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(list)h(of)f(attraction)f(regions)i(radii;)900 9395 y Fi(p)-8 b(e)g(ak)83 b Fo({)55 b(\()p Fp(double)62 b(*)p Fo(\))56 b(list)f(of)h(parameters)f Fh(\015)3737 9420 y Fe(i)3848 9395 y Fo(v)-9 b(alues)55 b(from)h(\(10\);)900 9620 y Fi(w)p 1020 9620 V 61 w(rho)83 b Fo({)56 b(\()p Fp(double)62 b(*)p Fo(\))55 b(list)h(of)f(parameters)h Fh(w)3876 9645 y Fe(i)3986 9620 y Fo(v)-9 b(alues)56 b(from)f(\(23\).)900 9887 y(The)68 b(\014elds)h(of)e(these)h (structures)h(can)f(b)5 b(e)67 b(useful)i(if)f(one)g(needs)g(to)g (study)g(prop)5 b(erties)68 b(of)g(a)900 10087 y(sp)5 b(eci\014c)55 b(generated)g(test)g(function)g(more)h(deeply)-14 b(.)p eop %%Page: 4 5 4 4 bop 900 499 a Fl(4)282 b Fj(\001)316 b Fl(M.)51 b(Gaviano,)g(D.)g (E.)g(Kvasov,)f(D.)h(Lera,)g(and)h(Y)-13 b(a.)50 b(D.)h(Sergey)l(ev)900 931 y Fr(3.3)166 b(Evaluation)56 b(of)f(a)g(generated)f(test)h (function)f(o)-5 b(r)56 b(its)g(pa)-5 b(rtial)56 b(derivatives)900 1197 y Fo(While)45 b(there)f(exists)g(a)h(structure)g Fi(GKLS)p 3528 1197 51 7 v 61 w(minima)g Fo(of)f(lo)5 b(cal)44 b(minima,)k(the)d(user)g(can)g(ev)-9 b(aluate)900 1397 y(a)64 b(test)g(function)g(\(or)g(partial)g(deriv)-9 b(ativ)k(es)64 b(of)g(D-)g(and)h(D2-t)-5 b(yp)5 b(e)63 b(functions\))h(that)g(is)h(deter-)900 1596 y(mined)56 b(b)-5 b(y)55 b(its)g(n)-5 b(um)g(b)5 b(er)56 b(\(a)e(parameter)h(to)f (the)h(subroutine)h Fi(GKLS)p 5232 1596 V 61 w(ar)-8 b(g)p 5516 1596 V 61 w(gener)g(ate)p Fo(\(\)\))55 b(within)900 1795 y(the)68 b(c)-5 b(hosen)70 b(test)e(class.)115 b(If)68 b(the)h(user)g(wishes)h(to)e(ev)-9 b(aluate)68 b(another)g(function)h (within)g(the)900 1994 y(same)48 b(class)g(he)g(should)g(deallo)5 b(cate)47 b(dynamic)h(arra)-5 b(ys)47 b(\(see)g(the)g(next)g (subsection\))h(and)g(recall)900 2194 y(the)f(generator)g Fi(GKLS)p 2372 2194 V 62 w(ar)-8 b(g)p 2657 2194 V 61 w(gener)g(ate)p Fo(\(\))48 b(\(passing)h(it)e(the)h(corresp)5 b(onding)48 b(function)g(n)-5 b(um)g(b)5 b(er\))900 2393 y(without)46 b(resetting)g(the)g(input)g(class)i(parameters)e(\(see)g (subsection)h(3.1\).)71 b(If)46 b(the)g(user)h(wishes)900 2592 y(to)55 b(c)-5 b(hange)55 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y Fe(N)3273 4309 y Fo(where)55 b(the)g(function)h(m)-5 b(ust)56 b(b)5 b(e)55 b(ev)-9 b(aluated.)900 4622 y(All)47 b(the)g(subroutines)j(return)d(a)h(test)f(function)g(v)-9 b(alue)47 b(corresp)5 b(onding)49 b(to)e(the)g(p)5 b(oin)-5 b(t)48 b Fh(x)p Fo(.)71 b(They)900 4821 y(return)f(the)f(v)-9 b(alue)70 b(GKLS)p 2632 4821 50 7 v 60 w(MAX)p 3094 4821 V 59 w(V)-18 b(ALUE)68 b(\(de\014ned)i(in)g Fp(gkls.h)p Fo(\))f(in)i(t)-5 b(w)g(o)70 b(cases:)104 b(\(a\))69 b(v)-5 b(ec-)900 5021 y(tor)58 b Fh(x)g Fo(do)5 b(es)58 b(not)g(b)5 b(elong)58 b(to)g(the)f(admissible)j(region)f(\012)f(and)g (\(b\))g(the)f(user)i(tries)f(to)g(call)g(the)900 5220 y(subroutines)f(without)e(generating)g(a)h(test)e(function.)1066 5420 y(The)60 b(follo)-5 b(wing)60 b(subroutines)i(are)e(pro)-5 b(vided)60 b(for)g(calculating)g(the)f(partial)h(deriv)-9 b(ativ)k(es)60 b(of)900 5619 y(the)55 b(test)g(functions)h(\(see)f(App) 5 b(endix)55 b(in)g(the)g Fi(p)-8 b(ap)g(er)p Fo(\).)1066 5819 y(Ev)f(aluation)58 b(of)g(the)f(\014rst)i(order)f(partial)g(deriv) -9 b(ativ)k(e)57 b(of)h(the)f(D-t)-5 b(yp)5 b(e)58 b(test)f(functions)i (with)900 6018 y(resp)5 b(ect)51 b(to)h(the)f(v)-9 b(ariable)52 b Fh(x)2662 6043 y Fe(j)2783 6018 y Fo(\(see)g(\(A.1\){\(A.2\))e(in)i (App)5 b(endix\))51 b(is)h(done)h(b)-5 b(y)52 b(calling)g(subrou-)900 6217 y(tine)1066 6417 y Fp(double)j Fi(GKLS)p 2139 6417 51 7 v 61 w(D)p 2325 6417 V 62 w(deriv)h Fo(\()p Fi(j)p Fo(,)f Fi(x)p Fo(\).)900 6617 y(Ev)-9 b(aluation)65 b(of)f(the)h (\014rst)g(order)g(partial)g(deriv)-9 b(ativ)k(e)64 b(of)h(the)g(D2-t) -5 b(yp)5 b(e)64 b(test)g(functions)i(with)900 6816 y(resp)5 b(ect)51 b(to)h(the)f(v)-9 b(ariable)52 b Fh(x)2662 6841 y Fe(j)2783 6816 y Fo(\(see)g(\(A.3\){\(A.4\))e(in)i(App)5 b(endix\))51 b(is)h(done)h(b)-5 b(y)52 b(calling)g(subrou-)900 7015 y(tine)1066 7215 y Fp(double)j Fi(GKLS)p 2139 7215 V 61 w(D2)p 2410 7215 V 62 w(deriv1)h Fo(\()p Fi(j)p Fo(,)e Fi(x)p Fo(\).)900 7414 y(Ev)-9 b(aluation)49 b(of)f(the)h 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b(of)h(the)g(class)g(with)g(the)g(same)g(parameters)900 7573 y(pro)5 b(duces)56 b(the)f(same)h(100)f(test)g(functions.)1066 7772 y(An)50 b(example)h(of)g(the)f(generation)g(and)h(use)h(of)e(some) h(of)g(the)f(test)g(classes)i(can)f(b)5 b(e)50 b(found)h(in)900 7971 y(the)k(\014le)g Fp(example.c)p Fo(.)900 8303 y Fc(REFERENCES)920 8536 y Fb(Ga)-12 b(viano,)56 b(M.)p Fa(,)e Fb(Kv)-12 b(aso)m(v,)55 b(D.)f(E.)p Fa(,)f Fb(Lera,)g(D.)p Fa(,)h Fb(and)g(Ser)m(geyev,)h(Y.)e(D.)h Fa(2003.)74 b(Gener)-7 b(ation)53 b(of)g(classes)h(of)1066 8702 y(test)e(functions) e(with)h(known)f(lo)-7 b(c)g(al)51 b(minima.)65 b Fm(A)l(CM)48 b(T)-12 b(rans.)47 b(Math.)g(Soft.)k(V,)g Fa(N,)g(PP{PP.)920 8884 y Fb(Knuth,)56 b(D.)f Fa(1997.)78 b Fm(The)51 b(Art)f(of)h (Computer)f(Programming,)e(V)-12 b(ol.)50 b(2:)h(Semin)l(umerical)d (Algorithms)p Fa(,)j(thir)-7 b(d)1066 9051 y(e)g(d.)66 b(A)-7 b(ddison-Wesley,)50 b(R)-7 b(e)g(ading,)49 b(Massachusetts.)920 9233 y Fb(Knuth,)j(D.)f Fa(2002.)67 b(Home)50 b(p)-7 b(age)49 b(at:)i(http://sunburn.stanfor)-7 b(d.e)g(du/)16 b Fm(~)d Fa(knuth/.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF SHAR_EOF fi # end of overwriting check if test -f 'UserManual.tex' then echo shar: will not over-write existing file "'UserManual.tex'" else cat << "SHAR_EOF" > 'UserManual.tex' \documentclass[acmtoms]{acmtrans2m} \usepackage{psfig} \usepackage{amssymb} % used for R in Real numbers % \acmVolume{} % \acmNumber{} % \acmYear{} % \acmMonth{} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \renewcommand{\footnoterule}{\hrule} \renewcommand{\firstfoot}{} \renewcommand{\runningfoot}{\em} \begin{document} \markboth{M.~Gaviano, D.~E.~Kvasov, D.~Lera, and Ya.~D.~Sergeyev}{GKLS-Generator of Classes of Test Functions: {\it User Manual}} \title{GKLS-Generator of Classes of Test Functions\\ with Known Local and Global Minima\\for Global Optimization:\\ USER MANUAL} \author{MARCO GAVIANO \\ Universit\`{a} di Cagliari, Italy \\ DMITRI E. KVASOV \\ Universit\`{a} di Roma ``La Sapienza'', Italy, and University of Nizhni Novgorod, Russia \\ DANIELA LERA \\ Universit\`{a} di Cagliari, Italy \and YAROSLAV D. SERGEYEV\footnote{Corresponding author, e-mail: yaro@si.deis.unical.it \newline Authors' addresses: M.~Gaviano and D.~Lera, Dipartimento di Matematica, Universit\`{a} di Cagliari, Via Ospedale, 72, 09100 Cagliari -- Italy; D.E.~Kvasov and Ya.D.~Sergeyev, DEIS, Universit\`a della Calabria, Via P.Bucci, Cubo 41C, 87036 Rende (CS) -- Italy. \newline This work has been partially supported by the following projects: FIRB RBAU01JYPN, FIRB RBNE01WBBB, and RFBR 01-01-00587.} \\ Universit\`{a} della Calabria, Italy, and University of Nizhni Novgorod, Russia} \maketitle \setcounter{page}{0} \vspace{4mm} \tableofcontents \newpage \section{Introduction} \label{sectionIntro} This {\it User Manual} describes an implementation of the GKLS-generator. First, the general structure of the package is presented, then instructions for using the test classes generator are described. The {\it User Manual} is strongly connected to the paper~\citeN{Gaviano:et:al.(2003)} (called hereafter {\it paper}). Particularly, any formulae referenced in the {\it User Manual} are to be found in the {\it paper}. In the {\it paper}, the user can find an example of a generated test function also. \section{Structure of the package} \label{sectionStructure} The GKLS-generator package has been written in ANSI Standard C and successfully tested on Windows and UNIX platforms. Our implementation follows the procedure described in Section~3 of the {\it paper}. The package implementing the generator includes the following files: \begin{itemize} \item[\bf gkls.c] -- the main file; \item[\bf gkls.h] -- the header file that users should include in their application projects in order to call subroutines from the file {\bf gkls.c}; \item[\bf rnd\_gen.c] -- the file containing the uniform random number generator proposed in Knuth~[\citeyearNP{Knuth(1997)}; \citeyearNP{Knuth:HomePage}]; \item[\bf rnd\_gen.h] -- the header file for linkage to the file {\bf rnd\_gen.c}; \item[\bf example.c] -- an example of the GKLS-generator usage; \item[\bf Makefile] -- an example of a UNIX makefile provided to UNIX users for a simple compilation and linkage of separate files of the application project. \end{itemize} For implementation details the user can consult the C codes. Note that the random number generator in {\bf rnd\_gen.c} uses the logical-and operation `\&' for efficiency, so it is not strictly portable unless the computer uses two's complement representation for integer. It does not limit portability of the package because almost all modern computers are based on two's complement arithmetic. \section{Calling sequence for generation and usage of the tests classes} \label{sectionInstructions} Here we describe how to generate and use classes of non-differentiable (ND-type), differentiable (D-type), and twice continuously differentiable (D2-type) test functions. Again, we concentrate on the D-type functions. The operations for the remaining two types are analogous. To utilize the GKLS-generator the user must perform the following steps: \begin{itemize} \item[\bf Step 1.] Input of the parameters defining a specific test class. \item[\bf Step 2.] Generating a specific test function of the defined test class. \item[\bf Step 3.] Evaluation of the generated test function and, if necessary, its partial derivatives. \item[\bf Step 4.] Memory deallocating. \end{itemize} Let us consider these steps in turn. \subsection{Input of the parameters defining a specific test class} \label{sectionInput} This step is subdivided into: (1)~defining the parameters of the test class, (2)~defining the admissible region $\Omega$, and (3)~checking (if necessary). \subsubsection{Defining the parameters of the test class} The parameters to be defined by the user determine a specific class (of the ND-, D- or D2-type) of 100 test functions (a specific function is retrieved by its number). There are the following parameters: \begin{itemize} \item[{\it GKLS\_dim}] -- ({\bf unsigned int}) dimension $N$ (from~(1)) of test functions; $N \geq 2$ (since multidimensional problems are considered in~(1)) and $N<$ NUM\_RND in {\bf rnd\_gen.h}; this value is limited by the power of \mbox{{\bf unsigned int}}-representation; default $N=2$; \item[{\it GKLS\_num\_minima}] -- ({\bf unsigned int}) number $m$ (from~(4)) of local minima including the paraboloid $Z$ minimum (from~(3)) and the global minimum; $m \geq 2$; the upper bound of this parameter is limited by the power of \mbox{{\bf unsigned int}}-representation; default $m=10$; \item[{\it GKLS\_global\_value}] -- ({\bf double}) global minimum value $f^*$ of $f(x)$; condition~(16) must be satisfied; the default value is $-1.0$ (defined in the file {\bf gkls.h} as a constant~GKLS\_GLOBAL\_MIN\_VALUE); \item[{\it GKLS\_global\_dist}] -- ({\bf double}) distance $r^*$ from the paraboloid vertex $T$ in~(3) to the global minimizer $x^* \in \Omega$ of $f(x)$; condition~(17) must be satisfied; the default value is $$ {\rm GKLS\_global\_dist}\ \stackrel{\rm def}{=} \ \min_{1 \leq j \leq N} | b(j) - a(j) | /\, 3, $$ where the vectors $a$ and $b$ determine the admissible region $\Omega$ in~(2); \item[{\it GKLS\_global\_radius}] -- ({\bf double}) radius $\rho ^*$ of the attraction region of the global minimizer $x^*\in \Omega$ of $f(x)$; condition~(18) must be satisfied; the default value is $$ {\rm GKLS\_global\_radius}\ \stackrel{\rm def}{=} \ \min_{1 \leq j \leq N} | b(j) - a(j) | /\, 6. $$ \end{itemize} The user may call subroutine \mbox{{\it GKLS\_set\_default}()} to set the default values of these five variables. \subsubsection{Defining the admissible region} \label{sectionOmega} With $N$ determined, the user must allocate dynamic arrays {\it GKLS\_domain\_left} and {\it GKLS\_domain\_right} to define the boundary of the hyperrectangle $\Omega$. This is done by calling subroutine {\bf int} {\it GKLS\_domain\_alloc} ();\\ which has no parameters and returns the following error codes defined in~{\bf gkls.h}: \begin{itemize} \item[\bf GKLS\_OK] -- no errors; \item[\bf GKLS\_DIM\_ERROR] -- the problem dimension is out of range; it must be greater than or equal to 2 and less than NUM\_RND defined in {\bf rnd\_gen.h}; \item[\bf GKLS\_MEMORY\_ERROR] -- there is not enough memory to allocate. \end{itemize} The same subroutine defines the admissible region $\Omega$. The default value $\Omega =[-1,1]^N$ is set by \mbox{{\it GKLS\_set\_default}()}. \subsubsection{Checking} \label{sectionChecking} The following subroutine allows the user to check validity of the input parameters: {\bf int} {\it GKLS\_parameters\_check} ().\\ It has no parameters and returns the following error codes (see {\bf gkls.h}): \begin{itemize} \item[\bf GKLS\_OK] -- no errors; \item[\bf GKLS\_DIM\_ERROR] -- problem dimension error; \item[\bf GKLS\_NUM\_MINIMA\_ERROR] -- number of local minima error; \item[\bf GKLS\_BOUNDARY\_ERROR] -- the admissible region boundary vectors are ill-defined; \item[\bf GKLS\_GLOBAL\_MIN\_VALUE\_ERROR] -- the global minimum value is not less than the paraboloid~(3) minimum value $t$ defined in {\bf gkls.h} as a constant GKLS\_PARABOLOID\_MIN; \item[\bf GKLS\_GLOBAL\_DIST\_ERROR] -- the parameter $r^*$ does not satisfy~(17); \item[\bf GKLS\_GLOBAL\_RADIUS\_ERROR] -- the parameter $\rho^*$ does not satisfy~(18). \end{itemize} \subsection{Generating a specific test function of the defined test class} \label{sectionGenerating} After a specific test class has been chosen (i.e., the input parameters have been determined) the user can generate a specific function that belongs to the chosen class of 100 test functions. This is done by calling subroutine {\bf int} {\it GKLS\_arg\_generate} ({\bf unsigned int} {\it nf}); \\ where \begin{itemize} \item[\it nf] -- the number of a function from the test class (from~1~to~100). \end{itemize} This subroutine initializes the random number generator, checks the input parameters, allocates dynamic arrays, and generates a test function following the procedure of Section~3 of the {\it paper}. It returns an error code that can be the same as for subroutines {\it GKLS\_parameters\_check}() and {\it GKLS\_domain\_alloc}(), or additionally: \begin{itemize} \item[\bf GKLS\_FUNC\_NUMBER\_ERROR] -- the number of a test function to generate exceeds 100 or it is less than 1. \end{itemize} {\it GKLS\_arg\_generate}() generates the list of all local minima and the list of the global minima as parts of the structures {\it GKLS\_minima} and {\it GKLS\_glob}, respectively. The first structure gathers the following information about all local minima (including the paraboloid minimum and the global one): coordinates of local minimizers, local minima values, and attraction regions radii. The second structure contains information about the number of global minimizers and their indices in the set of local minimizers. It has the following fields: \begin{itemize} \item[\it num\_global\_minima] -- ({\bf unsigned int}) total number of global minima; \item[\it gm\_index] -- ({\bf unsigned int *}) list of indices of generated minimizers, which are the global ones (elements 0 to ($\mbox{\it num\_global\_minima} - 1$) of the list) and the local ones (the remaining elements of the list). \end{itemize} The elements of the list {\it GKLS\_glob}.{\it gm\_index} are indices to a specific minimizer in the first structure {\it GKLS\_minima} characterized by the following fields: \begin{itemize} \item[\it local\_min] -- ({\bf double **}) list of local minimizers coordinates; \item[\it f] -- ({\bf double *}) list of local minima values; \item[\it rho] -- ({\bf double *}) list of attraction regions radii; \item[\it peak] -- ({\bf double *}) list of parameters $\gamma_i$ values from~(10); \item[\it w\_rho] -- ({\bf double *}) list of parameters $w_i$ values from~(23). \end{itemize} The fields of these structures can be useful if one needs to study properties of a specific generated test function more deeply. \subsection{Evaluation of a generated test function or its partial derivatives} \label{sectionEvaluation} While there exists a structure {\it GKLS\_minima} of local minima, the user can evaluate a test function (or partial derivatives of D- and D2-type functions) that is determined by its number (a parameter to the subroutine {\it GKLS\_arg\_generate}()) within the chosen test class. If the user wishes to evaluate another function within the same class he should deallocate dynamic arrays (see the next subsection) and recall the generator {\it GKLS\_arg\_generate}() (passing it the corresponding function number) without resetting the input class parameters (see subsection~\ref{sectionInput}). If the user wishes to change the test class properties he should reset also the input class parameters. Evaluation of an ND-type function is done by calling subroutine {\bf double} {\it GKLS\_ND\_func} ({\it x}).\\ Evaluation of a D-type function is done by calling subroutine {\bf double} {\it GKLS\_D\_func} ({\it x}).\\ Evaluation of a D2-type function is done by calling subroutine {\bf double} {\it GKLS\_D2\_func} ({\it x}).\\ All these subroutines have only one input parameter \begin{itemize} \item[\it x] -- ({\bf double *}) a point $x \in \mathbb{R} ^N$ where the function must be evaluated. \end{itemize} All the subroutines return a test function value corresponding to the point $x$. They return the value GKLS\_MAX\_VALUE (defined in {\bf gkls.h}) in two cases: (a) vector~$x$ does not belong to the admissible region $\Omega$ and (b) the user tries to call the subroutines without generating a test function. The following subroutines are provided for calculating the partial derivatives of the test functions (see Appendix in the {\it paper}). Evaluation of the first order partial derivative of the D-type test functions with respect to the variable $x_j$ (see~(A.1)--(A.2) in Appendix) is done by calling subroutine {\bf double} {\it GKLS\_D\_deriv} ({\it j}, {\it x}). \\ Evaluation of the first order partial derivative of the D2-type test functions with respect to the variable $x_j$ (see~(A.3)--(A.4) in Appendix) is done by calling subroutine {\bf double} {\it GKLS\_D2\_deriv1} ({\it j}, {\it x}). \\ Evaluation of the second order partial derivative of the D2-type test functions with respect to the variables $x_j$ and $x_k$ (see in Appendix the formulae~(A.5)--(A.6) for the case $j \neq k$ and~(A.7)--(A.8) for the case $j = k$) is done by calling subroutine {\bf double} {\it GKLS\_D2\_deriv2} ({\it j}, {\it k}, {\it x}). \\ Input parameters for these three subroutines are: \begin{itemize} \item[\it j, k] -- ({\bf unsigned int}) indices of the variables (that must be in the range from 1 to {\it GKLS\_dim}) with respect to which the partial derivative is evaluated; \item[\it x] -- ({\bf double *}) a point $x \in \mathbb{R} ^N$ where the derivative must be evaluated. \end{itemize} All subroutines return the value of a specific partial derivative corresponding to the point $x$ and to the given direction. They return the value GKLS\_MAX\_VALUE (defined in {\bf gkls.h}) in three cases: (a) index ($j$ or $k$) of a variable is out of the range [1,{\it GKLS\_dim}]; (b) vector $x$ does not belong to the admissible region~$\Omega$; (c) the user tries to call the subroutines without generating a test function. Subroutines for calculating the gradients of the D- and D2-type test functions and for calculating the Hessian matrix of the D2-type test functions at a given feasible point are also provided. These are {\bf int} {\it GKLS\_D\_gradient} ({\it x}, {\it g}), \\ {\bf int} {\it GKLS\_D2\_gradient} ({\it x}, {\it g}), \\ {\bf int} {\it GKLS\_D2\_hessian} ({\it x}, {\it h}). \\ Here \begin{itemize} \item[\it x] -- ({\bf double *}) a point $x \in \mathbb{R} ^N$ where the gradient or Hessian matrix must be evaluated; \item[\it g] -- ({\bf double *}) a pointer to the gradient vector calculated at {\it x}; \item[\it h] -- ({\bf double **}) a pointer to the Hessian matrix calculated at {\it x}. \end{itemize} Note that before calling these subroutines the user must allocate dynamic memory for the gradient vector {\it g} or the Hessian matrix {\it h} and pass the pointers {\it g} or {\it h} as parameters of the subroutines. These subroutines call the subroutines described above for calculating the partial derivatives and return an error code ({\bf GKLS\_DERIV\_EVAL\_ERROR} in the case of an error during evaluation of a particular component of the gradient or the Hessian matrix, or {\bf GKLS\_OK} if there are no errors). \subsection{Memory deallocating} \label{sectionMemory} When the user concludes his work with a test function he should deallocate dynamic arrays allocated by the generator. This is done by calling subroutine {\bf void} {\it GKLS\_free} ({\bf void});\\ with no parameters. When the user abandons the test class he should deallocate dynamic boundaries vectors {\it GKLS\_domain\_left} and {\it GKLS\_domain\_right} by calling subroutine {\bf void} {\it GKLS\_domain\_free} ({\bf void});\\ again with no parameters. It should be finally highlighted that if the user, after deallocating memory, wishes to return to the same class, generation of the class with the same parameters produces the same 100 test functions. An example of the generation and use of some of the test classes can be found in the file {\bf example.c}. \bibliographystyle{acmtrans} \bibliography{UserManual} \label{@lastpg} \end{document} SHAR_EOF fi # end of overwriting check cd .. # End of shell archive exit 0