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ipm_mc_pk.c
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ipm_mc_pk.c
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/*************************************************************************
* IPM_MC_PK (primal-dual predictor-corrector interior-point method *
* for basic SDP relaxation for Max-Cut) solves: *
* *
* max tr(LX), subject to diag(X) = e, X psd *
* min e'y, subject to Z = Diag(y) - L psd, y unconstrained *
* *
* input: L ... Laplacian matrix of the graph (1/4 of L for max-cut) *
* n ... size of the problem *
* print... print level *
* output: phi ... optimal value of SDP (value of the dual problem) *
* X ... optimal primal matrix *
* y ... optimal dual vector *
* Z ... optimal dual matrix *
*************************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include "admm.h"
/* NOTE: C uses row-major, but blas and lapack routines (written in Fortran)
* use column-major --> be careful when multiplying non-symmetric matrices) */
void ipm_mc_pk(double *L, int n, double *X, double *y, double *Z, double *phi, int print) {
/* variables for blas and lapack routines */
int inc = 1;
char up = 'U'; // for lapack take upper triangular part of matrix
char side = 'L'; // in matrix product AB, the left matrix is symmetric
int info; // test whether lapack function succeded
double alpha, beta; // scalars in linear combination (lapack)
/* other variables */
int i, j, k; // loop iter
double *p, *p2, *p3; // pointers in loops
double psi; // value of primal problem
double mu; // ZX = mu * I (parametrized optimality condition)
double alpha_p, alpha_d; // step lengths
double *b; // vector of ones
double *dX, *dy, *dZ;
double *Zi; // inv(Z)
double *M; // M * dy = rhs
double *dy1, *dX1;
double *tmp, *tmp2; // need non-symm matrices when computing for instance Zi*diag(dy)*X
/*************************************************
* initial positive definite matrices X, Z and y *
* primal,dual cost, gap *
*************************************************/
int nn = n*n;
/* set y to zero vector */
for (int i = 0; i < n; ++i)
y[i] = 0.0;
/* y = 1.1 + sum(abs(L))' */
for (i = 1, p = L, p2 = y; i <= nn; ++i, ++p) {
*p2 += fabs(*p);
if ((i % n) == 0) {
*p2 += 1.1;
++p2;
}
}
/* vector b of all ones */
alloc_vector(b, n, double);
for (i = 0; i < n; ++i)
b[i] = 1.0;
/* X = eye(n) = Diag(b) */
Diag(X, b, n);
/* Z = Diag(y) - L */
Diag(Z, y, n); /* Z = Diag(y) */
alpha = -1.0;
daxpy_(&nn,&alpha,L,&inc,Z,&inc); /* Z = Z - L */
/* phi = ones(n,1)'*y */ /* initial dual value */
*phi = ddot_(&n,b,&inc,y,&inc);
/* psi = L(:)'*X(:) */ /* initial primal value */
psi = ddot_(&nn,L,&inc,X,&inc);
/* mu = Z(:)'*X(:) / (2*n); */ /* initial complementarity */
mu = ddot_(&nn,Z,&inc,X,&inc) / (2.0 * n);
/* print output */
if (print) {
puts("iter log10(gap) primal dual");
puts("*******************************************");
}
/* allocate space */
alloc_matrix(dX, n, double);
alloc_vector(dy, n, double);
alloc_matrix(dZ, n, double);
alloc_vector(dy1, n, double);
alloc_matrix(dX1, n, double);
alloc_matrix(tmp, n, double);
alloc_matrix(tmp2, n, double);
alloc_matrix(Zi, n, double);
alloc_matrix(M, n, double);
/*************
* main loop *
*************/
for (i = 1; *phi - psi > 1e-4; ++i) {/* while duality gap too large */
/******** compute inverse of Z ********/
dcopy_(&nn,Z,&inc,Zi,&inc); /* copy Z to Zi */
dpotrf_(&up,&n,Zi,&n,&info); /* computes Cholesky factorization */
if (info != 0) {
fprintf(stderr, "%s: Problem with Cholesky factorization \
(line: %d).\n", __func__, __LINE__);
exit(EXIT_FAILURE);
}
dpotri_(&up,&n,Zi,&n,&info); /* computes Zi = inv(Z) */
if (info != 0) {
fprintf(stderr, "%s: Problem with computation of inverse matrix \
(line: %d).\n", __func__, __LINE__);
exit(EXIT_FAILURE);
}
/* NOTE: only upper triangular part of Zi is ok and can be used! */
/* copy strictly upper triangular part of Zi to strictly lower triangular part */
for (j = 0; j < n; ++j) {
for (k = 0; k < j; ++k)
Zi[j+n*k] = Zi[k+n*j];
}
/******** predictor step (mu = 0) solves: ********
******** Z * X + Diag(dy1) * X + Z * dX1 = 0 ********/
/* M = Zi .* X */
for (j = 0, p = M; j < nn; ++j, ++p)
*p = Zi[j] * X[j];
/* copy -b to dy1 */
alpha = -1.0;
dcopy_(&n,b,&inc,dy1,&inc); // copy
dscal_(&n,&alpha,dy1,&inc); // scale
/* solve the system: dy1 = (Zi .* X) \ (-e) */
dposv_(&up, &n, &inc, M, &n, dy1, &n, &info);
/* NOTE: M and dy1 are changed on exit! */
if (info != 0) {
fprintf(stderr, "%s: predictor step: problem in solving linear system \
(line: %d).\n", __func__, __LINE__);
exit(EXIT_FAILURE);
}
/* dX1 = -Zi*diag(dy1)*X - X */
// NOTE: be carefull to multiply in column-major ordering!
/* 1. step: tmp = -diag(dy1)*X */
/* = multiply j-th row of X by -dy1[j] (FORTRAN) */
/* = multiply j-th column of X by -dy[j] (C) */
p3 = tmp;
for (j = 0, p = X, p2 = dy1; j < nn; ++j, ++p, ++p3) {
*p3 = -(p2[j%n]) * (*p);
}
/* 2. step: Zi * tmp */
alpha = 1.0;
beta = 0.0;
dsymm_(&side,&up,&n,&n,&alpha,Zi,&n,tmp,&n,&beta,dX1,&n);
/* dX1 = dX1 - X */
alpha = -1.0;
daxpy_(&nn,&alpha,X,&inc,dX1,&inc);
/* symmetrize: dX1 = (dX1 + dX1')/2 */
for (j = 0; j < n; ++j) {
for (k = 0; k <= j; ++k)
dX1[k+n*j] = dX1[j+n*k] = 0.5 * (dX1[j+n*k] + dX1[k+n*j]);
}
/*************** corrector step solves: ******************/
/******** diag(dy2)*X + Z*dX2 - mu*I + diag(dy1)*dX1 = 0 **/
/* dy2 = M \ (mu*diag(Zi) - (Zi .* dX1)*dy1) */
/* dy = diag(Zi) */
diag(Zi,dy,n);
/* tmp = Zi .* dX1 */
for (j = 0, p = tmp; j < nn; ++j, ++p)
*p = Zi[j] * dX1[j];
/* dy = -tmp*dy1 + mu*dy */
alpha = -1.0;
dsymv_(&up,&n,&alpha,tmp,&n,dy1,&inc,&mu,dy,&inc);
// NOTE: M was changed during dposv
/* M = Zi .* X */
for (j = 0, p = M; j < nn; ++j, ++p)
*p = Zi[j] * X[j];
/* dy2 = M \ dy */
/* dy2 is saved in dy! */
dposv_(&up, &n, &inc, M, &n, dy, &n, &info);
if (info != 0) {
fprintf(stderr, "%s: corrector step: problem in solving linear system \
(line: %d).\n", __func__, __LINE__);
exit(EXIT_FAILURE);
}
/* dX2 = mu*Zi - Zi*( diag(dy2) * X + diag(dy1) * dX1) */
/* 1. step: tmp = -diag(dy2)*X */
/* = multiply j-th row of X by -dy2[j] (FORTRAN) */
/* = multiply j-th column of X by -dy2[j] (C) */
/* NOTE: dy2 = dy */
p3 = tmp;
for (j = 0, p = X, p2 = dy; j < nn; ++j, ++p, ++p3) {
*p3 = -(p2[j%n]) * (*p);
}
/* 2. step: tmp2 = -diag(dy1)*dX1 */
p3 = tmp2;
for (j = 0, p = dX1, p2 = dy1; j < nn; ++j, ++p, ++p3) {
*p3 = -(p2[j%n]) * (*p);
}
/* tmp = tmp + tmp2 */
alpha = 1.0;
daxpy_(&nn,&alpha,tmp2,&inc,tmp,&inc);
/* dX2 = Zi * tmp*/
/* NOTE: dX2 is stored in dX */
alpha = 1.0;
beta = 0.0;
dsymm_(&side,&up,&n,&n,&alpha,Zi,&n,tmp,&n,&beta,dX,&n);
/* dX = dX2 = mu*Zi + dX2 */
daxpy_(&nn,&mu,Zi,&inc,dX,&inc);
/**** final steps ****/
alpha = 1.0;
daxpy_(&n,&alpha,dy1,&inc,dy,&inc); /* dy = dy1 + dy */
daxpy_(&nn,&alpha,dX1,&inc,dX,&inc); /* dX = dX1 + dX */
/* symmetrize: dX = (dX + dX')/2 */
for (j = 0; j < n; ++j) {
for (k = 0; k <= j; ++k)
dX[k+n*j] = dX[j+n*k] = 0.5 * (dX[j+n*k] + dX[k+n*j]);
}
/* dZ = Diag(dy) */
Diag(dZ,dy,n);
/*********** find step lengths alpha_p and alpha_d ***********/
/* line search on primal: X = X + alpha_p * dX psd matrix */
alpha_p = 1.0;
info = 1;
while (info != 0) {
dcopy_(&nn,X,&inc,tmp,&inc); /* tmp = X */
daxpy_(&nn,&alpha_p,dX,&inc,tmp,&inc); /* tmp = alpha_p * dX + tmp */
dpotrf_(&up,&n,tmp,&n,&info);
if (info != 0)
alpha_p *= 0.8;
}
if (alpha_p < 1.0) /* stay away from boundary */
alpha_p *= 0.95;
/* line search on dual */
alpha_d = 1.0;
info = 1;
while (info != 0) {
dcopy_(&nn,Z,&inc,tmp,&inc); /* tmp = Z */
daxpy_(&nn,&alpha_d,dZ,&inc,tmp,&inc); /* tmp = alpha_d * dZ + tmp */
dpotrf_(&up,&n,tmp,&n,&info);
if (info != 0)
alpha_d *= 0.8;
}
if (alpha_d < 1.0) /* stay away from boundary */
alpha_d *= 0.95;
/******** update ********/
daxpy_(&nn,&alpha_p,dX,&inc,X,&inc); /* X = alpha_p * dX + X */
daxpy_(&n,&alpha_d,dy,&inc,y,&inc); /* y = alpha_d * dy + y */
daxpy_(&nn,&alpha_d,dZ,&inc,Z,&inc); /* Z = alpha_d * dZ + Z */
/* mu = Z(:)'*X(:) / (2*n); */
mu = ddot_(&nn,Z,&inc,X,&inc) / (2.0 * n);
/* speed up for long steps */
if (alpha_p + alpha_d > 1.6)
mu *= 0.5;
if (alpha_p + alpha_d > 1.9)
mu *= 0.2;
/***** objective values *****/
/* phi = ones(n,1)'*y */
*phi = ddot_(&n,b,&inc,y,&inc);
/* psi = L(:)'*X(:) */
psi = ddot_(&nn,L,&inc,X,&inc);
/* print output */
if (print)
printf("%3d %11.2f %14.5f %14.5f \n",i,log10(*phi-psi),psi,*phi);
} // end of main loop
if (print)
puts("*******************************************");
/************* free memory *************/
free(b);
free(dX);
free(dy);
free(dZ);
free(Zi);
free(M);
free(dy1);
free(dX1);
free(tmp);
free(tmp2);
}