text: /* glpios05.c (Gomory's mixed integer cut generator) */
  
  /***********************************************************************
  *  This code is part of GLPK (GNU Linear Programming Kit).
  *
  *  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
  *  2009, 2010 Andrew Makhorin, Department for Applied Informatics,
  *  Moscow Aviation Institute, Moscow, Russia. All rights reserved.
  *  E-mail: <mao@gnu.org>.
  *
  *  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 <http://www.gnu.org/licenses/>.
  ***********************************************************************/
  
  #ifdef __clang__
  #pragma clang diagnostic ignored "-Wsign-conversion"
  #endif
  
  #include "glpios.h"
  
  /***********************************************************************
  *  NAME
  *
  *  ios_gmi_gen - generate Gomory's mixed integer cuts.
  *
  *  SYNOPSIS
  *
  *  #include "glpios.h"
  *  void ios_gmi_gen(glp_tree *tree, IOSPOOL *pool);
  *
  *  DESCRIPTION
  *
  *  The routine ios_gmi_gen generates Gomory's mixed integer cuts for
  *  the current point and adds them to the cut pool. */
  
  #define MAXCUTS 50
  /* maximal number of cuts to be generated for one round */
  
  struct worka
  {     /* Gomory's cut generator working area */
        int *ind; /* int ind[1+n]; */
        double *val; /* double val[1+n]; */
        double *phi; /* double phi[1+m+n]; */
  };
  
  #define f(x) ((x) - floor(x))
  /* compute fractional part of x */
  
  static void gen_cut(glp_tree *tree, struct worka *worka, int j)
  {     /* this routine tries to generate Gomory's mixed integer cut for
           specified structural variable x[m+j] of integer kind, which is
           basic and has fractional value in optimal solution to current
           LP relaxation */
        glp_prob *mip = tree->mip;
        int m = mip->m;
        int n = mip->n;
        int *ind = worka->ind;
        double *val = worka->val;
        double *phi = worka->phi;
        int i, k, len, kind, stat;
        double lb, ub, alfa, beta, ksi, phi1, rhs;
        /* compute row of the simplex tableau, which (row) corresponds
           to specified basic variable xB[i] = x[m+j]; see (23) */
        len = glp_eval_tab_row(mip, m+j, ind, val);
        /* determine beta[i], which a value of xB[i] in optimal solution
           to current LP relaxation; note that this value is the same as
           if it would be computed with formula (27); it is assumed that
           beta[i] is fractional enough */
        beta = mip->col[j]->prim;
        /* compute cut coefficients phi and right-hand side rho, which
           correspond to formula (30); dense format is used, because rows
           of the simplex tableau is usually dense */
        for (k = 1; k <= m+n; k++) phi[k] = 0.0;
        rhs = f(beta); /* initial value of rho; see (28), (32) */
        for (j = 1; j <= len; j++)
        {  /* determine original number of non-basic variable xN[j] */
           k = ind[j];
           xassert(1 <= k && k <= m+n);
           /* determine the kind, bounds and current status of xN[j] in
              optimal solution to LP relaxation */
           if (k <= m)
           {  /* auxiliary variable */
              GLPROW *row = mip->row[k];
              kind = GLP_CV;
              lb = row->lb;
              ub = row->ub;
              stat = row->stat;
           }
           else
           {  /* structural variable */
              GLPCOL *col = mip->col[k-m];
              kind = col->kind;
              lb = col->lb;
              ub = col->ub;
              stat = col->stat;
           }
           /* xN[j] cannot be basic */
           xassert(stat != GLP_BS);
           /* determine row coefficient ksi[i,j] at xN[j]; see (23) */
           ksi = val[j];
           /* if ksi[i,j] is too large in the magnitude, do not generate
              the cut */
           if (fabs(ksi) > 1e+05) goto fini;
           /* if ksi[i,j] is too small in the magnitude, skip it */
           if (fabs(ksi) < 1e-10) goto skip;
           /* compute row coefficient alfa[i,j] at y[j]; see (26) */
           switch (stat)
           {  case GLP_NF:
                 /* xN[j] is free (unbounded) having non-zero ksi[i,j];
                    do not generate the cut */
                 goto fini;
              case GLP_NL:
                 /* xN[j] has active lower bound */
                 alfa = - ksi;
                 break;
              case GLP_NU:
                 /* xN[j] has active upper bound */
                 alfa = + ksi;
                 break;
              case GLP_NS:
                 /* xN[j] is fixed; skip it */
                 goto skip;
              default:
                 xassert(stat != stat);
           }
           /* compute cut coefficient phi'[j] at y[j]; see (21), (28) */
           switch (kind)
           {  case GLP_IV:
                 /* y[j] is integer */
                 if (fabs(alfa - floor(alfa + 0.5)) < 1e-10)
                 {  /* alfa[i,j] is close to nearest integer; skip it */
                    goto skip;
                 }
                 else if (f(alfa) <= f(beta))
                    phi1 = f(alfa);
                 else
                    phi1 = (f(beta) / (1.0 - f(beta))) * (1.0 - f(alfa));
                 break;
              case GLP_CV:
                 /* y[j] is continuous */
                 if (alfa >= 0.0)
                    phi1 = + alfa;
                 else
                    phi1 = (f(beta) / (1.0 - f(beta))) * (- alfa);
                 break;
              default:
                 xassert(kind != kind);
           }
           /* compute cut coefficient phi[j] at xN[j] and update right-
              hand side rho; see (31), (32) */
           switch (stat)
           {  case GLP_NL:
                 /* xN[j] has active lower bound */
                 phi[k] = + phi1;
                 rhs += phi1 * lb;
                 break;
              case GLP_NU:
                 /* xN[j] has active upper bound */
                 phi[k] = - phi1;
                 rhs -= phi1 * ub;
                 break;
              default:
                 xassert(stat != stat);
           }
  skip:    ;
        }
        /* now the cut has the form sum_k phi[k] * x[k] >= rho, where cut
           coefficients are stored in the array phi in dense format;
           x[1,...,m] are auxiliary variables, x[m+1,...,m+n] are struc-
           tural variables; see (30) */
        /* eliminate auxiliary variables in order to express the cut only
           through structural variables; see (33) */
        for (i = 1; i <= m; i++)
        {  GLPROW *row;
           GLPAIJ *aij;
           if (fabs(phi[i]) < 1e-10) continue;
           /* auxiliary variable x[i] has non-zero cut coefficient */
           row = mip->row[i];
           /* x[i] cannot be fixed */
           xassert(row->type != GLP_FX);
           /* substitute x[i] = sum_j a[i,j] * x[m+j] */
           for (aij = row->ptr; aij != NULL; aij = aij->r_next)
              phi[m+aij->col->j] += phi[i] * aij->val;
        }
        /* convert the final cut to sparse format and substitute fixed
           (structural) variables */
        len = 0;
        for (j = 1; j <= n; j++)
        {  GLPCOL *col;
           if (fabs(phi[m+j]) < 1e-10) continue;
           /* structural variable x[m+j] has non-zero cut coefficient */
           col = mip->col[j];
           if (col->type == GLP_FX)
           {  /* eliminate x[m+j] */
              rhs -= phi[m+j] * col->lb;
           }
           else
           {  len++;
              ind[len] = j;
              val[len] = phi[m+j];
           }
        }
        if (fabs(rhs) < 1e-12) rhs = 0.0;
        /* if the cut inequality seems to be badly scaled, reject it to
           avoid numeric difficulties */
        for (k = 1; k <= len; k++)
        {  if (fabs(val[k]) < 1e-03) goto fini;
           if (fabs(val[k]) > 1e+03) goto fini;
        }
        /* add the cut to the cut pool for further consideration */
  #if 0
        ios_add_cut_row(tree, pool, GLP_RF_GMI, len, ind, val, GLP_LO,
           rhs);
  #else
        glp_ios_add_row(tree, NULL, GLP_RF_GMI, 0, len, ind, val, GLP_LO,
           rhs);
  #endif
  fini: return;
  }
  
  struct var { int j; double f; };
  
  static int fcmp(const void *p1, const void *p2)
  {     const struct var *v1 = p1, *v2 = p2;
        if (v1->f > v2->f) return -1;
        if (v1->f < v2->f) return +1;
        return 0;
  }
  
  void ios_gmi_gen(glp_tree *tree)
  {     /* main routine to generate Gomory's cuts */
        glp_prob *mip = tree->mip;
        int m = mip->m;
        int n = mip->n;
        struct var *var;
        int k, nv, j, size;
        struct worka _worka, *worka = &_worka;
        /* allocate working arrays */
        var = xcalloc(1+n, sizeof(struct var));
        worka->ind = xcalloc(1+n, sizeof(int));
        worka->val = xcalloc(1+n, sizeof(double));
        worka->phi = xcalloc(1+m+n, sizeof(double));
        /* build the list of integer structural variables, which are
           basic and have fractional value in optimal solution to current
           LP relaxation */
        nv = 0;
        for (j = 1; j <= n; j++)
        {  GLPCOL *col = mip->col[j];
           double frac;
           if (col->kind != GLP_IV) continue;
           if (col->type == GLP_FX) continue;
           if (col->stat != GLP_BS) continue;
           frac = f(col->prim);
           if (!(0.05 <= frac && frac <= 0.95)) continue;
           /* add variable to the list */
           nv++, var[nv].j = j, var[nv].f = frac;
        }
        /* order the list by descending fractionality */
        qsort(&var[1], nv, sizeof(struct var), fcmp);
        /* try to generate cuts by one for each variable in the list, but
           not more than MAXCUTS cuts */
        size = glp_ios_pool_size(tree);
        for (k = 1; k <= nv; k++)
        {  if (glp_ios_pool_size(tree) - size >= MAXCUTS) break;
           gen_cut(tree, worka, var[k].j);
        }
        /* free working arrays */
        xfree(var);
        xfree(worka->ind);
        xfree(worka->val);
        xfree(worka->phi);
        return;
  }
  
  /* eof */
  