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/* cpxbas.c (construct Bixby's initial LP basis) */
/***********************************************************************
* This code is part of GLPK (GNU Linear Programming Kit).
*
* Copyright (C) 2008-2018 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/>.
***********************************************************************/
#include "env.h"
#include "prob.h"
struct var
{ /* structural variable */
int j;
/* ordinal number */
double q;
/* penalty value */
};
static int CDECL fcmp(const void *ptr1, const void *ptr2)
{ /* this routine is passed to the qsort() function */
struct var *col1 = (void *)ptr1, *col2 = (void *)ptr2;
if (col1->q < col2->q) return -1;
if (col1->q > col2->q) return +1;
return 0;
}
static int get_column(glp_prob *lp, int j, int ind[], double val[])
{ /* Bixby's algorithm assumes that the constraint matrix is scaled
such that the maximum absolute value in every non-zero row and
column is 1 */
int k, len;
double big;
len = glp_get_mat_col(lp, j, ind, val);
big = 0.0;
for (k = 1; k <= len; k++)
if (big < fabs(val[k])) big = fabs(val[k]);
if (big == 0.0) big = 1.0;
for (k = 1; k <= len; k++) val[k] /= big;
return len;
}
static void cpx_basis(glp_prob *lp)
{ /* main routine */
struct var *C, *C2, *C3, *C4;
int m, n, i, j, jk, k, l, ll, t, n2, n3, n4, type, len, *I, *r,
*ind;
double alpha, gamma, cmax, temp, *v, *val;
xprintf("Constructing initial basis...\n");
/* determine the number of rows and columns */
m = glp_get_num_rows(lp);
n = glp_get_num_cols(lp);
/* allocate working arrays */
C = xcalloc(1+n, sizeof(struct var));
I = xcalloc(1+m, sizeof(int));
r = xcalloc(1+m, sizeof(int));
v = xcalloc(1+m, sizeof(double));
ind = xcalloc(1+m, sizeof(int));
val = xcalloc(1+m, sizeof(double));
/* make all auxiliary variables non-basic */
for (i = 1; i <= m; i++)
{ if (glp_get_row_type(lp, i) != GLP_DB)
glp_set_row_stat(lp, i, GLP_NS);
else if (fabs(glp_get_row_lb(lp, i)) <=
fabs(glp_get_row_ub(lp, i)))
glp_set_row_stat(lp, i, GLP_NL);
else
glp_set_row_stat(lp, i, GLP_NU);
}
/* make all structural variables non-basic */
for (j = 1; j <= n; j++)
{ if (glp_get_col_type(lp, j) != GLP_DB)
glp_set_col_stat(lp, j, GLP_NS);
else if (fabs(glp_get_col_lb(lp, j)) <=
fabs(glp_get_col_ub(lp, j)))
glp_set_col_stat(lp, j, GLP_NL);
else
glp_set_col_stat(lp, j, GLP_NU);
}
/* C2 is a set of free structural variables */
n2 = 0, C2 = C + 0;
for (j = 1; j <= n; j++)
{ type = glp_get_col_type(lp, j);
if (type == GLP_FR)
{ n2++;
C2[n2].j = j;
C2[n2].q = 0.0;
}
}
/* C3 is a set of structural variables having excatly one (lower
or upper) bound */
n3 = 0, C3 = C2 + n2;
for (j = 1; j <= n; j++)
{ type = glp_get_col_type(lp, j);
if (type == GLP_LO)
{ n3++;
C3[n3].j = j;
C3[n3].q = + glp_get_col_lb(lp, j);
}
else if (type == GLP_UP)
{ n3++;
C3[n3].j = j;
C3[n3].q = - glp_get_col_ub(lp, j);
}
}
/* C4 is a set of structural variables having both (lower and
upper) bounds */
n4 = 0, C4 = C3 + n3;
for (j = 1; j <= n; j++)
{ type = glp_get_col_type(lp, j);
if (type == GLP_DB)
{ n4++;
C4[n4].j = j;
C4[n4].q = glp_get_col_lb(lp, j) - glp_get_col_ub(lp, j);
}
}
/* compute gamma = max{|c[j]|: 1 <= j <= n} */
gamma = 0.0;
for (j = 1; j <= n; j++)
{ temp = fabs(glp_get_obj_coef(lp, j));
if (gamma < temp) gamma = temp;
}
/* compute cmax */
cmax = (gamma == 0.0 ? 1.0 : 1000.0 * gamma);
/* compute final penalty for all structural variables within sets
C2, C3, and C4 */
switch (glp_get_obj_dir(lp))
{ case GLP_MIN: temp = +1.0; break;
case GLP_MAX: temp = -1.0; break;
default: xassert(lp != lp);
}
for (k = 1; k <= n2+n3+n4; k++)
{ j = C[k].j;
C[k].q += (temp * glp_get_obj_coef(lp, j)) / cmax;
}
/* sort structural variables within C2, C3, and C4 in ascending
order of penalty value */
qsort(C2+1, n2, sizeof(struct var), fcmp);
for (k = 1; k < n2; k++) xassert(C2[k].q <= C2[k+1].q);
qsort(C3+1, n3, sizeof(struct var), fcmp);
for (k = 1; k < n3; k++) xassert(C3[k].q <= C3[k+1].q);
qsort(C4+1, n4, sizeof(struct var), fcmp);
for (k = 1; k < n4; k++) xassert(C4[k].q <= C4[k+1].q);
/*** STEP 1 ***/
for (i = 1; i <= m; i++)
{ type = glp_get_row_type(lp, i);
if (type != GLP_FX)
{ /* row i is either free or inequality constraint */
glp_set_row_stat(lp, i, GLP_BS);
I[i] = 1;
r[i] = 1;
}
else
{ /* row i is equality constraint */
I[i] = 0;
r[i] = 0;
}
v[i] = +DBL_MAX;
}
/*** STEP 2 ***/
for (k = 1; k <= n2+n3+n4; k++)
{ jk = C[k].j;
len = get_column(lp, jk, ind, val);
/* let alpha = max{|A[l,jk]|: r[l] = 0} and let l' be such
that alpha = |A[l',jk]| */
alpha = 0.0, ll = 0;
for (t = 1; t <= len; t++)
{ l = ind[t];
if (r[l] == 0 && alpha < fabs(val[t]))
alpha = fabs(val[t]), ll = l;
}
if (alpha >= 0.99)
{ /* B := B union {jk} */
glp_set_col_stat(lp, jk, GLP_BS);
I[ll] = 1;
v[ll] = alpha;
/* r[l] := r[l] + 1 for all l such that |A[l,jk]| != 0 */
for (t = 1; t <= len; t++)
{ l = ind[t];
if (val[t] != 0.0) r[l]++;
}
/* continue to the next k */
continue;
}
/* if |A[l,jk]| > 0.01 * v[l] for some l, continue to the
next k */
for (t = 1; t <= len; t++)
{ l = ind[t];
if (fabs(val[t]) > 0.01 * v[l]) break;
}
if (t <= len) continue;
/* otherwise, let alpha = max{|A[l,jk]|: I[l] = 0} and let l'
be such that alpha = |A[l',jk]| */
alpha = 0.0, ll = 0;
for (t = 1; t <= len; t++)
{ l = ind[t];
if (I[l] == 0 && alpha < fabs(val[t]))
alpha = fabs(val[t]), ll = l;
}
/* if alpha = 0, continue to the next k */
if (alpha == 0.0) continue;
/* B := B union {jk} */
glp_set_col_stat(lp, jk, GLP_BS);
I[ll] = 1;
v[ll] = alpha;
/* r[l] := r[l] + 1 for all l such that |A[l,jk]| != 0 */
for (t = 1; t <= len; t++)
{ l = ind[t];
if (val[t] != 0.0) r[l]++;
}
}
/*** STEP 3 ***/
/* add an artificial variable (auxiliary variable for equality
constraint) to cover each remaining uncovered row */
for (i = 1; i <= m; i++)
if (I[i] == 0) glp_set_row_stat(lp, i, GLP_BS);
/* free working arrays */
xfree(C);
xfree(I);
xfree(r);
xfree(v);
xfree(ind);
xfree(val);
return;
}
/***********************************************************************
* NAME
*
* glp_cpx_basis - construct Bixby's initial LP basis
*
* SYNOPSIS
*
* void glp_cpx_basis(glp_prob *lp);
*
* DESCRIPTION
*
* The routine glp_cpx_basis constructs an advanced initial basis for
* the specified problem object.
*
* The routine is based on Bixby's algorithm described in the paper:
*
* Robert E. Bixby. Implementing the Simplex Method: The Initial Basis.
* ORSA Journal on Computing, Vol. 4, No. 3, 1992, pp. 267-84. */
void glp_cpx_basis(glp_prob *lp)
{ if (lp->m == 0 || lp->n == 0)
glp_std_basis(lp);
else
cpx_basis(lp);
return;
}
/* eof */
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