/* spydual.c */ /*********************************************************************** * This code is part of GLPK (GNU Linear Programming Kit). * * Copyright (C) 2015 Andrew Makhorin, Department for Applied * Informatics, Moscow Aviation Institute, Moscow, Russia. All rights * reserved. E-mail: . * * GLPK is free software: you can redistribute it and/or modify it * under the terms of the GNU General Public License as published by * the Free Software Foundation, either version 3 of the License, or * (at your option) any later version. * * GLPK is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public * License for more details. * * You should have received a copy of the GNU General Public License * along with GLPK. If not, see . ***********************************************************************/ #include "glpenv.h" #include "simplex.h" #include "spxat.h" #include "spxnt.h" #include "spxprob.h" #include "spychuzc.h" #include "spychuzr.h" #define USE_AT 1 /* 1 - use A in row-wise format * 0 - use N in row-wise format */ /* (Looks like using A' provide more accuracy for dual simplex.) */ #define EXCL 1 /* 1 - exclude fixed non-basic variables * 0 - don't exclude variables */ #define SHIFT 1 /* 1 - shift bounds of variables toward zero * 0 - don't shift bounds of variables */ #define CHECK_ACCURACY 0 /* (for debugging) */ struct csa { /* common storage area */ SPXLP *lp; /* LP problem data and its (current) basis; this LP has m rows * and n columns */ int dir; /* original optimization direction: * +1 - minimization * -1 - maximization */ double *b; /* double b[1+m]; */ /* copy of original right-hand sides */ double *l; /* double l[1+n]; */ /* copy of original lower bounds */ double *u; /* double u[1+n]; */ /* copy of original upper bounds */ SPXAT *at; /* mxn-matrix A of constraint coefficients, in sparse row-wise * format (NULL if not used) */ SPXNT *nt; /* mx(n-m)-matrix N composed of non-basic columns of constraint * matrix A, in sparse row-wise format (NULL if not used) */ int phase; /* search phase: * 0 - not determined yet * 1 - searching for dual feasible solution * 2 - searching for optimal solution */ double *beta; /* double beta[1+m]; */ /* beta[i] is primal value of basic variable xB[i] */ int beta_st; /* status of the vector beta: * 0 - undefined * 1 - just computed * 2 - updated */ double *d; /* double d[1+n-m]; */ /* d[j] is reduced cost of non-basic variable xN[j] */ int d_st; /* status of the vector d: * 0 - undefined * 1 - just computed * 2 - updated */ SPYSE *se; /* dual projected steepest edge and Devex pricing data block * (NULL if not used) */ int num; /* number of eligible basic variables */ int *list; /* int list[1+m]; */ /* list[1], ..., list[num] are indices i of eligible basic * variables xB[i] */ int p; /* xB[p] is a basic variable chosen to leave the basis */ double *trow; /* double trow[1+n-m]; */ /* p-th (pivot) row of the simplex table */ int q; /* xN[q] is a non-basic variable chosen to enter the basis */ double *tcol; /* double tcol[1+m]; */ /* q-th (pivot) column of the simplex table */ double *work; /* double work[1+m]; */ /* working array */ double *work1; /* double work1[1+n-m]; */ /* another working array */ int p_stat, d_stat; /* primal and dual solution statuses */ /*--------------------------------------------------------------*/ /* control parameters (see struct glp_smcp) */ int msg_lev; /* message level */ int dualp; /* if this flag is set, report failure in case of instability */ int harris; /* dual ratio test technique: * 0 - textbook ratio test * 1 - Harris' two pass ratio test */ double tol_bnd, tol_bnd1; /* primal feasibility tolerances */ double tol_dj, tol_dj1; /* dual feasibility tolerances */ double tol_piv; /* pivot tolerance */ double obj_lim; /* objective limit */ int it_lim; /* iteration limit */ int tm_lim; /* time limit, milliseconds */ int out_frq; /* display output frequency, iterations */ int out_dly; /* display output delay, milliseconds */ /*--------------------------------------------------------------*/ /* working parameters */ double tm_beg; /* time value at the beginning of the search */ int it_beg; /* simplex iteration count at the beginning of the search */ int it_cnt; /* simplex iteration count; it increases by one every time the * basis changes */ int it_dpy; /* simplex iteration count at most recent display output */ int inv_cnt; /* basis factorization count since most recent display output */ }; /*********************************************************************** * check_flags - check correctness of active bound flags * * This routine checks that flags specifying active bounds of all * non-basic variables are correct. * * NOTE: It is important to note that if bounds of variables have been * changed, active bound flags should be corrected accordingly. */ static void check_flags(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; double *l = lp->l; double *u = lp->u; int *head = lp->head; char *flag = lp->flag; int j, k; for (j = 1; j <= n-m; j++) { k = head[m+j]; /* x[k] = xN[j] */ if (l[k] == -DBL_MAX && u[k] == +DBL_MAX) xassert(!flag[j]); else if (l[k] != -DBL_MAX && u[k] == +DBL_MAX) xassert(!flag[j]); else if (l[k] == -DBL_MAX && u[k] != +DBL_MAX) xassert(flag[j]); else if (l[k] == u[k]) xassert(!flag[j]); } return; } /*********************************************************************** * set_art_bounds - set artificial right-hand sides and bounds * * This routine sets artificial right-hand sides and artificial bounds * for all variables to minimize the sum of dual infeasibilities on * phase I. Given current reduced costs d = (d[j]) this routine also * sets active artificial bounds of non-basic variables to provide dual * feasibility (this is always possible because all variables have both * lower and upper artificial bounds). */ static void set_art_bounds(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; double *b = lp->b; double *l = lp->l; double *u = lp->u; int *head = lp->head; char *flag = lp->flag; double *d = csa->d; int i, j, k; /* set artificial right-hand sides */ for (i = 1; i <= m; i++) b[i] = 0.0; /* set artificial bounds depending on types of variables */ for (k = 1; k <= n; k++) { if (csa->l[k] == -DBL_MAX && csa->u[k] == +DBL_MAX) { /* force free variables to enter the basis */ l[k] = -1e3, u[k] = +1e3; } else if (csa->l[k] != -DBL_MAX && csa->u[k] == +DBL_MAX) l[k] = 0.0, u[k] = +1.0; else if (csa->l[k] == -DBL_MAX && csa->u[k] != +DBL_MAX) l[k] = -1.0, u[k] = 0.0; else l[k] = u[k] = 0.0; } /* set active artificial bounds for non-basic variables */ xassert(csa->d_st == 1); for (j = 1; j <= n-m; j++) { k = head[m+j]; /* x[k] = xN[j] */ flag[j] = (l[k] != u[k] && d[j] < 0.0); } /* invalidate values of basic variables, since active bounds of * non-basic variables have been changed */ csa->beta_st = 0; return; } /*********************************************************************** * set_orig_bounds - restore original right-hand sides and bounds * * This routine restores original right-hand sides and original bounds * for all variables. This routine also sets active original bounds for * non-basic variables; for double-bounded non-basic variables current * reduced costs d = (d[j]) are used to decide which bound (lower or * upper) should be made active. */ void set_orig_bounds(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; double *b = lp->b; double *l = lp->l; double *u = lp->u; int *head = lp->head; char *flag = lp->flag; double *d = csa->d; int j, k; /* restore original right-hand sides */ memcpy(b, csa->b, (1+m) * sizeof(double)); /* restore original bounds of all variables */ memcpy(l, csa->l, (1+n) * sizeof(double)); memcpy(u, csa->u, (1+n) * sizeof(double)); /* set active original bounds for non-basic variables */ xassert(csa->d_st == 1); for (j = 1; j <= n-m; j++) { k = head[m+j]; /* x[k] = xN[j] */ if (l[k] == -DBL_MAX && u[k] == +DBL_MAX) flag[j] = 0; else if (l[k] != -DBL_MAX && u[k] == +DBL_MAX) flag[j] = 0; else if (l[k] == -DBL_MAX && u[k] != +DBL_MAX) flag[j] = 1; else if (l[k] != u[k]) flag[j] = (d[j] < 0.0); else flag[j] = 0; } /* invalidate values of basic variables, since active bounds of * non-basic variables have been changed */ csa->beta_st = 0; return; } /*********************************************************************** * check_feas - check dual feasibility of basic solution * * This routine checks that reduced costs of all non-basic variables * d = (d[j]) have correct signs. * * Reduced cost d[j] is considered as having correct sign within the * specified tolerance depending on status of non-basic variable xN[j] * if one of the following conditions is met: * * xN[j] is free -eps <= d[j] <= +eps * * xN[j] has its lower bound active d[j] >= -eps * * xN[j] has its upper bound active d[j] <= +eps * * xN[j] is fixed d[j] has any value * * where eps = tol + tol1 * |cN[j]|, cN[j] is the objective coefficient * at xN[j]. (See also the routine spx_chuzc_sel.) * * The flag recov allows the routine to recover dual feasibility by * changing active bounds of non-basic variables. (For example, if * xN[j] has its lower bound active and d[j] < -eps, the feasibility * can be recovered by making xN[j] active on its upper bound.) * * If the basic solution is dual feasible, the routine returns zero. * If the basic solution is dual infeasible, but its dual feasibility * can be recovered (or has been recovered, if the flag recov is set), * the routine returns a negative value. Otherwise, the routine returns * the number j of some non-basic variable xN[j], whose reduced cost * d[j] is dual infeasible and cannot be recovered. */ static int check_feas(struct csa *csa, double tol, double tol1, int recov) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; double *c = lp->c; double *l = lp->l; double *u = lp->u; int *head = lp->head; char *flag = lp->flag; double *d = csa->d; int j, k, ret = 0; double eps; /* reduced costs should be just computed */ xassert(csa->d_st == 1); /* walk thru list of non-basic variables */ for (j = 1; j <= n-m; j++) { k = head[m+j]; /* x[k] = xN[j] */ if (l[k] == u[k]) { /* xN[j] is fixed variable; skip it */ continue; } /* determine absolute tolerance eps[j] */ eps = tol + tol1 * (c[k] >= 0.0 ? +c[k] : -c[k]); /* check dual feasibility of xN[j] */ if (d[j] > +eps) { /* xN[j] should have its lower bound active */ if (l[k] == -DBL_MAX || flag[j]) { /* but it either has no lower bound or its lower bound * is inactive */ if (l[k] == -DBL_MAX) { /* cannot recover, since xN[j] has no lower bound */ ret = j; break; } /* recovering is possible */ if (recov) flag[j] = 0; ret = -1; } } else if (d[j] < -eps) { /* xN[j] should have its upper bound active */ if (!flag[j]) { /* but it either has no upper bound or its upper bound * is inactive */ if (u[k] == +DBL_MAX) { /* cannot recover, since xN[j] has no upper bound */ ret = j; break; } /* recovering is possible */ if (recov) flag[j] = 1; ret = -1; } } } if (recov && ret) { /* invalidate values of basic variables, since active bounds * of non-basic variables have been changed */ csa->beta_st = 0; } return ret; } #if CHECK_ACCURACY /*********************************************************************** * err_in_vec - compute maximal relative error between two vectors * * This routine computes and returns maximal relative error between * n-vectors x and y: * * err_max = max |x[i] - y[i]| / (1 + |x[i]|). * * NOTE: This routine is intended only for debugginig purposes. */ static double err_in_vec(int n, const double x[], const double y[]) { int i; double err, err_max; err_max = 0.0; for (i = 1; i <= n; i++) { err = fabs(x[i] - y[i]) / (1.0 + fabs(x[i])); if (err_max < err) err_max = err; } return err_max; } #endif #if CHECK_ACCURACY /*********************************************************************** * err_in_beta - compute maximal relative error in vector beta * * This routine computes and returns maximal relative error in vector * of values of basic variables beta = (beta[i]). * * NOTE: This routine is intended only for debugginig purposes. */ static double err_in_beta(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; double err, *beta; beta = talloc(1+m, double); spx_eval_beta(lp, beta); err = err_in_vec(m, beta, csa->beta); tfree(beta); return err; } #endif #if CHECK_ACCURACY /*********************************************************************** * err_in_d - compute maximal relative error in vector d * * This routine computes and returns maximal relative error in vector * of reduced costs of non-basic variables d = (d[j]). * * NOTE: This routine is intended only for debugginig purposes. */ static double err_in_d(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; int j; double err, *pi, *d; pi = talloc(1+m, double); d = talloc(1+n-m, double); spx_eval_pi(lp, pi); for (j = 1; j <= n-m; j++) d[j] = spx_eval_dj(lp, pi, j); err = err_in_vec(n-m, d, csa->d); tfree(pi); tfree(d); return err; } #endif #if CHECK_ACCURACY /*********************************************************************** * err_in_gamma - compute maximal relative error in vector gamma * * This routine computes and returns maximal relative error in vector * of projected steepest edge weights gamma = (gamma[j]). * * NOTE: This routine is intended only for debugginig purposes. */ static double err_in_gamma(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; SPYSE *se = csa->se; int i; double err, *gamma; xassert(se != NULL); gamma = talloc(1+m, double); for (i = 1; i <= m; i++) gamma[i] = spy_eval_gamma_i(lp, se, i); err = err_in_vec(m, gamma, se->gamma); tfree(gamma); return err; } #endif #if CHECK_ACCURACY /*********************************************************************** * check_accuracy - check accuracy of basic solution components * * This routine checks accuracy of current basic solution components. * * NOTE: This routine is intended only for debugginig purposes. */ static void check_accuracy(struct csa *csa) { double e_beta, e_d, e_gamma; e_beta = err_in_beta(csa); e_d = err_in_d(csa); if (csa->se == NULL) e_gamma = 0.; else e_gamma = err_in_gamma(csa); xprintf("e_beta = %10.3e; e_d = %10.3e; e_gamma = %10.3e\n", e_beta, e_d, e_gamma); xassert(e_beta <= 1e-5 && e_d <= 1e-5 && e_gamma <= 1e-3); return; } #endif /*********************************************************************** * choose_pivot - choose xB[p] and xN[q] * * Given the list of eligible basic variables this routine first * chooses basic variable xB[p]. This choice is always possible, * because the list is assumed to be non-empty. Then the routine * computes p-th row T[p,*] of the simplex table T[i,j] and chooses * non-basic variable xN[q]. If the pivot T[p,q] is small in magnitude, * the routine attempts to choose another xB[p] and xN[q] in order to * avoid badly conditioned adjacent bases. */ static void choose_pivot(struct csa *csa) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; double *l = lp->l; int *head = lp->head; SPXAT *at = csa->at; SPXNT *nt = csa->nt; double *beta = csa->beta; double *d = csa->d; SPYSE *se = csa->se; int *list = csa->list; double *rho = csa->work; double *trow = csa->work1; int nnn, try, k, p, q, t; xassert(csa->beta_st); xassert(csa->d_st); /* initial number of eligible basic variables */ nnn = csa->num; /* nothing has been chosen so far */ csa->p = 0; try = 0; try: /* choose basic variable xB[p] */ xassert(nnn > 0); try++; if (se == NULL) { /* dual Dantzig's rule */ p = spy_chuzr_std(lp, beta, nnn, list); } else { /* dual projected steepest edge */ p = spy_chuzr_pse(lp, se, beta, nnn, list); } xassert(1 <= p && p <= m); /* compute p-th row of inv(B) */ spx_eval_rho(lp, p, rho); /* compute p-th row of the simplex table */ if (at != NULL) spx_eval_trow1(lp, at, rho, trow); else spx_nt_prod(lp, nt, trow, 1, -1.0, rho); /* choose non-basic variable xN[q] */ k = head[p]; /* x[k] = xB[p] */ if (!csa->harris) q = spy_chuzc_std(lp, d, beta[p] < l[k] ? +1. : -1., trow, csa->tol_piv, .30 * csa->tol_dj, .30 * csa->tol_dj1); else q = spy_chuzc_harris(lp, d, beta[p] < l[k] ? +1. : -1., trow, csa->tol_piv, .35 * csa->tol_dj, .35 * csa->tol_dj1); /* either keep previous choice or accept new choice depending on * which one is better */ if (csa->p == 0 || q == 0 || fabs(trow[q]) > fabs(csa->trow[csa->q])) { csa->p = p; memcpy(&csa->trow[1], &trow[1], (n-m) * sizeof(double)); csa->q = q; } /* check if current choice is acceptable */ if (csa->q == 0 || fabs(csa->trow[csa->q]) >= 0.001) goto done; if (nnn == 1) goto done; if (try == 5) goto done; /* try to choose other xB[p] and xN[q] */ /* find xB[p] in the list */ for (t = 1; t <= nnn; t++) if (list[t] == p) break; xassert(t <= nnn); /* move xB[p] to the end of the list */ list[t] = list[nnn], list[nnn] = p; /* and exclude it from consideration */ nnn--; /* repeat the choice */ goto try; done: /* the choice has been made */ return; } /*********************************************************************** * display - display search progress * * This routine displays some information about the search progress * that includes: * * search phase; * * number of simplex iterations performed by the solver; * * original objective value (only on phase II); * * sum of (scaled) dual infeasibilities for original bounds; * * number of dual infeasibilities (phase I) or primal infeasibilities * (phase II); * * number of basic factorizations since last display output. */ static void display(struct csa *csa, int spec) { SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; int *head = lp->head; char *flag = lp->flag; double *l = csa->l; /* original lower bounds */ double *u = csa->u; /* original upper bounds */ double *beta = csa->beta; double *d = csa->d; int j, k, nnn; double sum; /* check if the display output should be skipped */ if (csa->msg_lev < GLP_MSG_ON) goto skip; if (csa->out_dly > 0 && 1000.0 * xdifftime(xtime(), csa->tm_beg) < csa->out_dly) goto skip; if (csa->it_cnt == csa->it_dpy) goto skip; if (!spec && csa->it_cnt % csa->out_frq != 0) goto skip; /* display search progress depending on search phase */ switch (csa->phase) { case 1: /* compute sum and number of (scaled) dual infeasibilities * for original bounds */ sum = 0.0, nnn = 0; for (j = 1; j <= n-m; j++) { k = head[m+j]; /* x[k] = xN[j] */ if (d[j] > 0.0) { /* xN[j] should have lower bound */ if (l[k] == -DBL_MAX) { sum += d[j]; if (d[j] > +1e-7) nnn++; } } else if (d[j] < 0.0) { /* xN[j] should have upper bound */ if (u[k] == +DBL_MAX) { sum -= d[j]; if (d[j] < -1e-7) nnn++; } } } /* on phase I variables have artificial bounds which are * meaningless for original LP, so corresponding objective * function value is also meaningless */ xprintf(" %6d: %23s inf = %11.3e (%d)", csa->it_cnt, "", sum, nnn); break; case 2: /* compute sum of (scaled) dual infeasibilities */ sum = 0.0, nnn = 0; for (j = 1; j <= n-m; j++) { k = head[m+j]; /* x[k] = xN[j] */ if (d[j] > 0.0) { /* xN[j] should have its lower bound active */ if (l[k] == -DBL_MAX || flag[j]) sum += d[j]; } else if (d[j] < 0.0) { /* xN[j] should have its upper bound active */ if (l[k] != u[k] && !flag[j]) sum -= d[j]; } } /* compute number of primal infeasibilities */ nnn = spy_chuzr_sel(lp, beta, csa->tol_bnd, csa->tol_bnd1, NULL); xprintf("#%6d: obj = %17.9e inf = %11.3e (%d)", csa->it_cnt, (double)csa->dir * spx_eval_obj(lp, beta), sum, nnn); break; default: xassert(csa != csa); } if (csa->inv_cnt) { /* number of basis factorizations performed */ xprintf(" %d", csa->inv_cnt); csa->inv_cnt = 0; } xprintf("\n"); csa->it_dpy = csa->it_cnt; skip: return; } /*********************************************************************** * spy_dual - driver to dual simplex method * * This routine is a driver to the two-phase dual simplex method. * * On exit this routine returns one of the following codes: * * 0 LP instance has been successfully solved. * * GLP_EOBJLL * Objective lower limit has been reached (maximization). * * GLP_EOBJUL * Objective upper limit has been reached (minimization). * * GLP_EITLIM * Iteration limit has been exhausted. * * GLP_ETMLIM * Time limit has been exhausted. * * GLP_EFAIL * The solver failed to solve LP instance. */ static int dual_simplex(struct csa *csa) { /* dual simplex method main logic routine */ SPXLP *lp = csa->lp; int m = lp->m; int n = lp->n; double *l = lp->l; double *u = lp->u; int *head = lp->head; SPXNT *nt = csa->nt; double *beta = csa->beta; double *d = csa->d; SPYSE *se = csa->se; int *list = csa->list; double *trow = csa->trow; double *tcol = csa->tcol; double *pi = csa->work; int msg_lev = csa->msg_lev; double tol_bnd = csa->tol_bnd; double tol_bnd1 = csa->tol_bnd1; double tol_dj = csa->tol_dj; double tol_dj1 = csa->tol_dj1; int j, k, p_flag, refct, ret; check_flags(csa); loop: /* main loop starts here */ /* compute factorization of the basis matrix */ if (!lp->valid) { double cond; ret = spx_factorize(lp); csa->inv_cnt++; if (ret != 0) { if (msg_lev >= GLP_MSG_ERR) xprintf("Error: unable to factorize the basis matrix (%d" ")\n", ret); csa->p_stat = csa->d_stat = GLP_UNDEF; ret = GLP_EFAIL; goto fini; } /* check condition of the basis matrix */ cond = bfd_condest(lp->bfd); if (cond > 1.0 / DBL_EPSILON) { if (msg_lev >= GLP_MSG_ERR) xprintf("Error: basis matrix is singular to working prec" "ision (cond = %.3g)\n", cond); csa->p_stat = csa->d_stat = GLP_UNDEF; ret = GLP_EFAIL; goto fini; } if (cond > 0.001 / DBL_EPSILON) { if (msg_lev >= GLP_MSG_ERR) xprintf("Warning: basis matrix is ill-conditioned (cond " "= %.3g)\n", cond); } /* invalidate basic solution components */ csa->beta_st = csa->d_st = 0; } /* compute reduced costs of non-basic variables d = (d[j]) */ if (!csa->d_st) { spx_eval_pi(lp, pi); for (j = 1; j <= n-m; j++) d[j] = spx_eval_dj(lp, pi, j); csa->d_st = 1; /* just computed */ /* determine the search phase, if not determined yet (this is * performed only once at the beginning of the search for the * original bounds) */ if (!csa->phase) { j = check_feas(csa, 0.97 * tol_dj, 0.97 * tol_dj1, 1); if (j > 0) { /* initial basic solution is dual infeasible and cannot * be recovered */ /* start to search for dual feasible solution */ set_art_bounds(csa); csa->phase = 1; } else { /* initial basic solution is either dual feasible or its * dual feasibility has been recovered */ /* start to search for optimal solution */ csa->phase = 2; } } /* make sure that current basic solution is dual feasible */ j = check_feas(csa, tol_dj, tol_dj1, 0); if (j) { /* dual feasibility is broken due to excessive round-off * errors */ if (bfd_get_count(lp->bfd)) { /* try to provide more accuracy */ lp->valid = 0; goto loop; } if (msg_lev >= GLP_MSG_ERR) xprintf("Warning: numerical instability (dual simplex, p" "hase %s)\n", csa->phase == 1 ? "I" : "II"); if (csa->dualp) { /* do not continue the search; report failure */ csa->p_stat = csa->d_stat = GLP_UNDEF; ret = -1; /* special case of GLP_EFAIL */ goto fini; } /* try to recover dual feasibility */ j = check_feas(csa, 0.97 * tol_dj, 0.97 * tol_dj1, 1); if (j > 0) { /* dual feasibility cannot be recovered (this may happen * only on phase II) */ xassert(csa->phase == 2); /* restart to search for dual feasible solution */ set_art_bounds(csa); csa->phase = 1; } } } /* at this point the search phase is determined */ xassert(csa->phase == 1 || csa->phase == 2); /* compute values of basic variables beta = (beta[i]) */ if (!csa->beta_st) { spx_eval_beta(lp, beta); csa->beta_st = 1; /* just computed */ } /* reset the dual reference space, if necessary */ if (se != NULL && !se->valid) spy_reset_refsp(lp, se), refct = 1000; /* at this point the basis factorization and all basic solution * components are valid */ xassert(lp->valid && csa->beta_st && csa->d_st); check_flags(csa); #if CHECK_ACCURACY /* check accuracy of current basic solution components (only for * debugging) */ check_accuracy(csa); #endif /* check if the objective limit has been reached */ if (csa->phase == 2 && csa->obj_lim != DBL_MAX && spx_eval_obj(lp, beta) >= csa->obj_lim) { if (csa->beta_st != 1) csa->beta_st = 0; if (csa->d_st != 1) csa->d_st = 0; if (!(csa->beta_st && csa->d_st)) goto loop; display(csa, 1); if (msg_lev >= GLP_MSG_ALL) xprintf("OBJECTIVE %s LIMIT REACHED; SEARCH TERMINATED\n", csa->dir > 0 ? "UPPER" : "LOWER"); csa->num = spy_chuzr_sel(lp, beta, tol_bnd, tol_bnd1, list); csa->p_stat = (csa->num == 0 ? GLP_FEAS : GLP_INFEAS); csa->d_stat = GLP_FEAS; ret = (csa->dir > 0 ? GLP_EOBJUL : GLP_EOBJLL); goto fini; } /* check if the iteration limit has been exhausted */ if (csa->it_cnt - csa->it_beg >= csa->it_lim) { if (csa->beta_st != 1) csa->beta_st = 0; if (csa->d_st != 1) csa->d_st = 0; if (!(csa->beta_st && csa->d_st)) goto loop; display(csa, 1); if (msg_lev >= GLP_MSG_ALL) xprintf("ITERATION LIMIT EXCEEDED; SEARCH TERMINATED\n"); if (csa->phase == 1) { set_orig_bounds(csa); check_flags(csa); spx_eval_beta(lp, beta); } csa->num = spy_chuzr_sel(lp, beta, tol_bnd, tol_bnd1, list); csa->p_stat = (csa->num == 0 ? GLP_FEAS : GLP_INFEAS); csa->d_stat = (csa->phase == 1 ? GLP_INFEAS : GLP_FEAS); ret = GLP_EITLIM; goto fini; } /* check if the time limit has been exhausted */ if (1000.0 * xdifftime(xtime(), csa->tm_beg) >= csa->tm_lim) { if (csa->beta_st != 1) csa->beta_st = 0; if (csa->d_st != 1) csa->d_st = 0; if (!(csa->beta_st && csa->d_st)) goto loop; display(csa, 1); if (msg_lev >= GLP_MSG_ALL) xprintf("TIME LIMIT EXCEEDED; SEARCH TERMINATED\n"); if (csa->phase == 1) { set_orig_bounds(csa); check_flags(csa); spx_eval_beta(lp, beta); } csa->num = spy_chuzr_sel(lp, beta, tol_bnd, tol_bnd1, list); csa->p_stat = (csa->num == 0 ? GLP_FEAS : GLP_INFEAS); csa->d_stat = (csa->phase == 1 ? GLP_INFEAS : GLP_FEAS); ret = GLP_EITLIM; goto fini; } /* display the search progress */ display(csa, 0); /* select eligible basic variables */ switch (csa->phase) { case 1: csa->num = spy_chuzr_sel(lp, beta, 1e-8, 0.0, list); break; case 2: csa->num = spy_chuzr_sel(lp, beta, tol_bnd, tol_bnd1, list); break; default: xassert(csa != csa); } /* check for optimality */ if (csa->num == 0) { if (csa->beta_st != 1) csa->beta_st = 0; if (csa->d_st != 1) csa->d_st = 0; if (!(csa->beta_st && csa->d_st)) goto loop; /* current basis is optimal */ display(csa, 1); switch (csa->phase) { case 1: /* check for dual feasibility */ set_orig_bounds(csa); check_flags(csa); if (check_feas(csa, tol_dj, tol_dj1, 0) == 0) { /* dual feasible solution found; switch to phase II */ csa->phase = 2; xassert(!csa->beta_st); goto loop; } /* no dual feasible solution exists */ if (msg_lev >= GLP_MSG_ALL) xprintf("LP HAS NO DUAL FEASIBLE SOLUTION\n"); spx_eval_beta(lp, beta); csa->num = spy_chuzr_sel(lp, beta, tol_bnd, tol_bnd1, list); csa->p_stat = (csa->num == 0 ? GLP_FEAS : GLP_INFEAS); csa->d_stat = GLP_NOFEAS; ret = 0; goto fini; case 2: /* optimal solution found */ if (msg_lev >= GLP_MSG_ALL) xprintf("OPTIMAL LP SOLUTION FOUND\n"); csa->p_stat = csa->d_stat = GLP_FEAS; ret = 0; goto fini; default: xassert(csa != csa); } } /* choose xB[p] and xN[q] */ choose_pivot(csa); /* check for dual unboundedness */ if (csa->q == 0) { if (csa->beta_st != 1) csa->beta_st = 0; if (csa->d_st != 1) csa->d_st = 0; if (!(csa->beta_st && csa->d_st)) goto loop; display(csa, 1); switch (csa->phase) { case 1: /* this should never happen */ if (msg_lev >= GLP_MSG_ERR) xprintf("Error: dual simplex failed\n"); csa->p_stat = csa->d_stat = GLP_UNDEF; ret = GLP_EFAIL; goto fini; case 2: /* dual unboundedness detected */ if (msg_lev >= GLP_MSG_ALL) xprintf("LP HAS NO PRIMAL FEASIBLE SOLUTION\n"); csa->p_stat = GLP_NOFEAS; csa->d_stat = GLP_FEAS; ret = 0; goto fini; default: xassert(csa != csa); } } /* compute q-th column of the simplex table */ spx_eval_tcol(lp, csa->q, tcol); /* FIXME: tcol[p] and trow[q] should be close to each other */ xassert(tcol[csa->p] != 0.0); /* update values of basic variables for adjacent basis */ k = head[csa->p]; /* x[k] = xB[p] */ p_flag = (l[k] != u[k] && beta[csa->p] > u[k]); spx_update_beta(lp, beta, csa->p, p_flag, csa->q, tcol); csa->beta_st = 2; /* update reduced costs of non-basic variables for adjacent * basis */ if (spx_update_d(lp, d, csa->p, csa->q, trow, tcol) <= 1e-9) { /* successful updating */ csa->d_st = 2; } else { /* new reduced costs are inaccurate */ csa->d_st = 0; } /* update steepest edge weights for adjacent basis, if used */ if (se != NULL) { if (refct > 0) { if (spy_update_gamma(lp, se, csa->p, csa->q, trow, tcol) <= 1e-3) { /* successful updating */ refct--; } else { /* new weights are inaccurate; reset reference space */ se->valid = 0; } } else { /* too many updates; reset reference space */ se->valid = 0; } } /* update matrix N for adjacent basis, if used */ if (nt != NULL) spx_update_nt(lp, nt, csa->p, csa->q); /* change current basis header to adjacent one */ spx_change_basis(lp, csa->p, p_flag, csa->q); /* and update factorization of the basis matrix */ if (csa->p > 0) spx_update_invb(lp, csa->p, head[csa->p]); /* dual simplex iteration complete */ csa->it_cnt++; goto loop; fini: return ret; } int spy_dual(glp_prob *P, const glp_smcp *parm) { /* driver to dual simplex method */ struct csa csa_, *csa = &csa_; SPXLP lp; #if USE_AT SPXAT at; #else SPXNT nt; #endif SPYSE se; int ret, *map, *daeh; /* build working LP and its initial basis */ memset(csa, 0, sizeof(struct csa)); csa->lp = &lp; spx_init_lp(csa->lp, P, EXCL); spx_alloc_lp(csa->lp); map = talloc(1+P->m+P->n, int); spx_build_lp(csa->lp, P, EXCL, SHIFT, map); spx_build_basis(csa->lp, P, map); switch (P->dir) { case GLP_MIN: csa->dir = +1; break; case GLP_MAX: csa->dir = -1; break; default: xassert(P != P); } csa->b = talloc(1+csa->lp->m, double); memcpy(csa->b, csa->lp->b, (1+csa->lp->m) * sizeof(double)); csa->l = talloc(1+csa->lp->n, double); memcpy(csa->l, csa->lp->l, (1+csa->lp->n) * sizeof(double)); csa->u = talloc(1+csa->lp->n, double); memcpy(csa->u, csa->lp->u, (1+csa->lp->n) * sizeof(double)); #if USE_AT /* build matrix A in row-wise format */ csa->at = &at; csa->nt = NULL; spx_alloc_at(csa->lp, csa->at); spx_build_at(csa->lp, csa->at); #else /* build matrix N in row-wise format for initial basis */ csa->at = NULL; csa->nt = &nt; spx_alloc_nt(csa->lp, csa->nt); spx_init_nt(csa->lp, csa->nt); spx_build_nt(csa->lp, csa->nt); #endif /* allocate and initialize working components */ csa->phase = 0; csa->beta = talloc(1+csa->lp->m, double); csa->beta_st = 0; csa->d = talloc(1+csa->lp->n-csa->lp->m, double); csa->d_st = 0; switch (parm->pricing) { case GLP_PT_STD: csa->se = NULL; break; case GLP_PT_PSE: csa->se = &se; spy_alloc_se(csa->lp, csa->se); break; default: xassert(parm != parm); } csa->list = talloc(1+csa->lp->m, int); csa->trow = talloc(1+csa->lp->n-csa->lp->m, double); csa->tcol = talloc(1+csa->lp->m, double); csa->work = talloc(1+csa->lp->m, double); csa->work1 = talloc(1+csa->lp->n-csa->lp->m, double); /* initialize control parameters */ csa->msg_lev = parm->msg_lev; csa->dualp = (parm->meth == GLP_DUALP); switch (parm->r_test) { case GLP_RT_STD: csa->harris = 0; break; case GLP_RT_HAR: csa->harris = 1; break; default: xassert(parm != parm); } csa->tol_bnd = parm->tol_bnd; csa->tol_bnd1 = .001 * parm->tol_bnd; csa->tol_dj = parm->tol_dj; csa->tol_dj1 = .001 * parm->tol_dj; csa->tol_piv = parm->tol_piv; switch (P->dir) { case GLP_MIN: csa->obj_lim = + parm->obj_ul; break; case GLP_MAX: csa->obj_lim = - parm->obj_ll; break; default: xassert(parm != parm); } csa->it_lim = parm->it_lim; csa->tm_lim = parm->tm_lim; csa->out_frq = parm->out_frq; csa->out_dly = parm->out_dly; /* initialize working parameters */ csa->tm_beg = xtime(); csa->it_beg = csa->it_cnt = P->it_cnt; csa->it_dpy = -1; csa->inv_cnt = 0; /* try to solve working LP */ ret = dual_simplex(csa); /* return basis factorization back to problem object */ P->valid = csa->lp->valid; P->bfd = csa->lp->bfd; /* set solution status */ P->pbs_stat = csa->p_stat; P->dbs_stat = csa->d_stat; /* if the solver failed, do not store basis header and basic * solution components to problem object */ if (ret == GLP_EFAIL) goto skip; /* convert working LP basis to original LP basis and store it to * problem object */ daeh = talloc(1+csa->lp->n, int); spx_store_basis(csa->lp, P, map, daeh); /* compute simplex multipliers for final basic solution found by * the solver */ spx_eval_pi(csa->lp, csa->work); /* convert working LP solution to original LP solution and store * it to problem object */ spx_store_sol(csa->lp, P, SHIFT, map, daeh, csa->beta, csa->work, csa->d); tfree(daeh); /* save simplex iteration count */ P->it_cnt = csa->it_cnt; /* report auxiliary/structural variable causing unboundedness */ P->some = 0; if (csa->p_stat == GLP_NOFEAS && csa->d_stat == GLP_FEAS) { int k, kk; /* xB[p] = x[k] causes dual unboundedness */ xassert(1 <= csa->p && csa->p <= csa->lp->m); k = csa->lp->head[csa->p]; xassert(1 <= k && k <= csa->lp->n); /* convert to number of original variable */ for (kk = 1; kk <= P->m + P->n; kk++) { if (abs(map[kk]) == k) { P->some = kk; break; } } xassert(P->some != 0); } skip: /* deallocate working objects and arrays */ spx_free_lp(csa->lp); tfree(map); tfree(csa->b); tfree(csa->l); tfree(csa->u); if (csa->at != NULL) spx_free_at(csa->lp, csa->at); if (csa->nt != NULL) spx_free_nt(csa->lp, csa->nt); tfree(csa->beta); tfree(csa->d); if (csa->se != NULL) spy_free_se(csa->lp, csa->se); tfree(csa->list); tfree(csa->trow); tfree(csa->tcol); tfree(csa->work); tfree(csa->work1); /* return to calling program */ return ret >= 0 ? ret : GLP_EFAIL; } /* eof */