UnconstrainedLocalSearch::ReturnCode UnconstrainedLocalSearch::minimize(const Vector& x0, Vector& xk, double eps, int max_iter) {
// parameter for the stopping criterion
this->eps = eps;
this->sigma=eps/::sqrt(n+1);
// parameter to update the trust region radius
double gamma00 = 0.05;
double gamma0 = 0.5;
double gamma2 = 2;
// parameters to measure the adequacy between
// the real function and its quadratic approximation.
// If >= mu, adequacy is good => the new iteration is kept
// If >= eta, adequacy is very good => the trust region is expanded.
// If <mu, the old iteration is kept but the approximation of the
// Hessian is updated (and more accurate so the next iteration is
// likely to succeed)
double mu = 0.25;
double eta = 0.75;
// could be initialized to something maybe more relevant
// for the case where x0 is invalid
xk = x0;
Vector xk1=x0; // next iterate
double fk1;
Vector gk1(n); // next gradient
niter = 0; //number of iteration
try {
// cout << " [minimize] xk= " << xk1 << endl;
// Initialize the quadratic approximation at the initial point x0
// like in the quasi-Newton algorithm
double fk=_mid(f.eval(xk1));
Vector gk=_mid(f.gradient(xk1));
Matrix Bk=Matrix::eye(n);
// cout << " [minimize] gk= " << gk << endl;
// initialize the current point
xk=xk1;
// initialization the trust region radius
double Delta = 0.1*gk.norm();
IntervalVector region(n);
BitSet I(BitSet::empty(n));
while ((niter<max_iter)&&(!stop(xk,gk))) {
niter++;
// cout << " [minimize] ITER= " << niter<<" fk= " << fk <<" ||gk||= " << gk.norm() << endl;
// cout << " [minimize] gk= " << gk << endl;
// creates the trust region (intersection with bounding box)
region=box & (IntervalVector(xk).inflate(Delta));
// Find the Generalized Cauchy Point
Vector x_gcp = find_gcp(gk, Bk, xk, region);
// Compute the active set I
I.clear();
// TODO: I should be retrieved from the last line search of find_gcp
for (int i =0; i<n;i++) {
if (fabs(x_gcp[i]-region[i].lb()) <sigma || fabs(x_gcp[i]-region[i].ub()) <sigma)
I.add(i);
}
// Compute the conjugate gradient
xk1 = conj_grad(gk,Bk,xk,x_gcp,region,I);
// Compute the ration of achieved to predicted reduction in the function
fk1 = _mid(f.eval(xk1));
// cout << " [minimize] xk1= " << xk1 <<" fk1 = "<<fk1<<" fk=" <<fk<< endl;
// computing m(xk1)-f(xk) = (xk1-xk)^T gk + 1/2 (xk1-xk)^T Bk (xk1-xzk)
Vector sk = xk1-xk;
double m =(sk*gk + 0.5* (sk*(Bk*sk)));
// cout << " [minimize] sk= " << sk <<" m= "<<m<< endl;
// warning if xk1=xk => sk =0 and m =0.
// In this case, xk = x_gcp = xk1. That means that we have converge
// but due to rounding error in the computation of the stopping criteria,
// we have not detected it. So, we stop the algorithm because
// it is impossible to reach the required precision eps.
if (m==0)
niter=max_iter;
else {
double rhok = (fk1-fk)/m;
// cout << " [minimize] rhok= " << rhok << endl;
// rhok can be <0 if we do not improve the criterion
// update x_k, f(x_k) and g(x_k)
if (rhok > mu) {
gk1 = _mid(f.gradient(xk1));
update_B_SR1(Bk,sk,gk,gk1);
fk = fk1;
xk = xk1;
gk = gk1;
}
// update Delta
if (rhok <= mu) {
if (Delta<sigma) niter = max_iter;
Delta = gamma0*Delta;
}
else if (rhok>=eta) Delta = gamma2*Delta;
// cout << " [minimize] Delta= " << Delta << endl;
}
}
} catch(InvalidPointException&) {
return INVALID_POINT;
}
return niter<max_iter ? SUCCESS : TOO_MANY_ITER;
}
int main(int argc, char *argv[])
{
int i, j, k, it;
double zeta;
double rnorm;
double norm_temp1, norm_temp2;
double t, mflops, tmax;
//char Class;
logical verified;
double zeta_verify_value, epsilon, err;
char *t_names[T_last];
//openmp environment setting
omp_set_dynamic(0);
omp_set_num_threads(8);
for (i = 0; i < T_last; i++) {
timer_clear(i);
}
timer_start(T_init);
firstrow = 0;
lastrow = NA-1;
firstcol = 0;
lastcol = NA-1;
zeta_verify_value = VALID_RESULT;
printf("\nCG start...\n\n");
printf(" Size: %11d\n", NA);
printf(" Iterations: %5d\n", NITER);
printf("\n");
naa = NA;
nzz = NZ;
//---------------------------------------------------------------------
// Inialize random number generator
//---------------------------------------------------------------------
tran = 314159265.0;
amult = 1220703125.0;
zeta = randlc(&tran, amult);
//---------------------------------------------------------------------
//
//---------------------------------------------------------------------
makea(naa, nzz, a, colidx, rowstr,
firstrow, lastrow, firstcol, lastcol,
arow,
(int (*)[NONZER+1])(void*)acol,
(double (*)[NONZER+1])(void*)aelt,
iv);
//---------------------------------------------------------------------
// Note: as a result of the above call to makea:
// values of j used in indexing rowstr go from 0 --> lastrow-firstrow
// values of colidx which are col indexes go from firstcol --> lastcol
// So:
// Shift the col index vals from actual (firstcol --> lastcol )
// to local, i.e., (0 --> lastcol-firstcol)
//---------------------------------------------------------------------
#pragma omp parallel for collapse(2)
for (j = 0; j < lastrow - firstrow + 1; j++) {
for (k = rowstr[j]; k < rowstr[j+1]; k++) {
colidx[k] = colidx[k] - firstcol;
}
}
//---------------------------------------------------------------------
// set starting vector to (1, 1, .... 1)
//---------------------------------------------------------------------
#pragma omp parallel for
for (i = 0; i < NA+1; i++) {
x[i] = 1.0;
}
#pragma omp parallel for
for (j = 0; j < lastcol - firstcol + 1; j++) {
q[j] = 0.0;
z[j] = 0.0;
r[j] = 0.0;
p[j] = 0.0;
}
zeta = 0.0;
//---------------------------------------------------------------------
//---->
// Do one iteration untimed to init all code and data page tables
//----> (then reinit, start timing, to niter its)
//---------------------------------------------------------------------
for (it = 1; it <= 1; it++) {
//---------------------------------------------------------------------
// The call to the conjugate gradient routine:
//---------------------------------------------------------------------
conj_grad(colidx, rowstr, x, z, a, p, q, r, &rnorm);
//---------------------------------------------------------------------
// zeta = shift + 1/(x.z)
// So, first: (x.z)
// Also, find norm of z
// So, first: (z.z)
//---------------------------------------------------------------------
norm_temp1 = 0.0;
norm_temp2 = 0.0;
#pragma omp parallel for reduction(+:norm_temp1, norm_temp2)
for (j = 0; j < lastcol - firstcol + 1; j++) {
norm_temp1 = norm_temp1 + x[j] * z[j];
norm_temp2 = norm_temp2 + z[j] * z[j];
}
norm_temp2 = 1.0 / sqrt(norm_temp2);
//---------------------------------------------------------------------
// Normalize z to obtain x
//---------------------------------------------------------------------
#pragma omp parallel for
for (j = 0; j < lastcol - firstcol + 1; j++) {
x[j] = norm_temp2 * z[j];
}
} // end of do one iteration untimed
//---------------------------------------------------------------------
// set starting vector to (1, 1, .... 1)
//---------------------------------------------------------------------
#pragma omp parallel for
for (i = 0; i < NA+1; i++) {
x[i] = 1.0;
}
zeta = 0.0;
timer_stop(T_init);
printf(" Initialization time = %15.3f seconds\n", timer_read(T_init));
timer_start(T_bench);
//---------------------------------------------------------------------
//---->
// Main Iteration for inverse power method
//---->
//---------------------------------------------------------------------
/* #pragma omp parallel for reduction(+:zeta) private(norm_temp1, norm_temp2) firstprivate(x, z, p, q) */
for (it = 1; it <= NITER; it++) {
//---------------------------------------------------------------------
// The call to the conjugate gradient routine:
//---------------------------------------------------------------------
if (timeron) timer_start(T_conj_grad);
conj_grad(colidx, rowstr, x, z, a, p, q, r, &rnorm);
if (timeron) timer_stop(T_conj_grad);
//---------------------------------------------------------------------
// zeta = shift + 1/(x.z)
// So, first: (x.z)
// Also, find norm of z
// So, first: (z.z)
//---------------------------------------------------------------------
norm_temp1 = 0.0;
norm_temp2 = 0.0;
#pragma omp parallel for reduction(+:norm_temp1, norm_temp2)
for (j = 0; j < lastcol - firstcol + 1; j++) {
norm_temp1 = norm_temp1 + x[j]*z[j];
norm_temp2 = norm_temp2 + z[j]*z[j];
}
norm_temp2 = 1.0 / sqrt(norm_temp2);
zeta = SHIFT + 1.0 / norm_temp1;
if (it == 1)
printf("\n iteration ||r|| zeta\n");
printf(" %5d %20.14E%20.13f\n", it, rnorm, zeta);
//---------------------------------------------------------------------
// Normalize z to obtain x
//---------------------------------------------------------------------
#pragma omp parallel for
for (j = 0; j < lastcol - firstcol + 1; j++) {
x[j] = norm_temp2 * z[j];
}
} // end of main iter inv pow meth
timer_stop(T_bench);
//---------------------------------------------------------------------
// End of timed section
//---------------------------------------------------------------------
t = timer_read(T_bench);
printf("\nComplete...\n");
epsilon = 1.0e-10;
err = fabs(zeta - zeta_verify_value) / zeta_verify_value;
if (err <= epsilon) {
verified = true;
printf(" VERIFICATION SUCCESSFUL\n");
printf(" Zeta is %20.13E\n", zeta);
printf(" Error is %20.13E\n", err);
} else {
verified = false;
printf(" VERIFICATION FAILED\n");
printf(" Zeta %20.13E\n", zeta);
printf(" The correct zeta is %20.13E\n", zeta_verify_value);
}
printf("\n\nExecution time : %lf seconds\n\n", t);
return 0;
}
int main(int argc, char **argv) {
int i, j, k, it;
int nthreads = 1;
double zeta;
double rnorm;
double norm_temp11;
double norm_temp12;
double t, mflops;
char cclass;
boolean verified;
double zeta_verify_value, epsilon;
firstrow = 1;
lastrow = NA;
firstcol = 1;
lastcol = NA;
if (NA == 1400 && NONZER == 7 && NITER == 15 && SHIFT == 10.0) {
cclass = 'S';
zeta_verify_value = 8.5971775078648;
} else if (NA == 7000 && NONZER == 8 && NITER == 15 && SHIFT == 12.0) {
cclass = 'W';
zeta_verify_value = 10.362595087124;
} else if (NA == 14000 && NONZER == 11 && NITER == 15 && SHIFT == 20.0) {
cclass = 'A';
zeta_verify_value = 17.130235054029;
} else if (NA == 75000 && NONZER == 13 && NITER == 75 && SHIFT == 60.0) {
cclass = 'B';
zeta_verify_value = 22.712745482631;
} else if (NA == 150000 && NONZER == 15 && NITER == 75 && SHIFT == 110.0) {
cclass = 'C';
zeta_verify_value = 28.973605592845;
} else {
cclass = 'U';
}
printf("\n\n NAS Parallel Benchmarks 2.3 OpenMP C version"
" - CG Benchmark\n");
printf(" Size: %10d\n", NA);
printf(" Iterations: %5d\n", NITER);
naa = NA;
nzz = NZ;
/*--------------------------------------------------------------------
c Initialize random number generator
c-------------------------------------------------------------------*/
tran = 314159265.0;
amult = 1220703125.0;
zeta = randlc( &tran, amult );
/*--------------------------------------------------------------------
c
c-------------------------------------------------------------------*/
makea(naa, nzz, a, colidx, rowstr, NONZER,
firstrow, lastrow, firstcol, lastcol,
RCOND, arow, acol, aelt, v, iv, SHIFT);
/*---------------------------------------------------------------------
c Note: as a result of the above call to makea:
c values of j used in indexing rowstr go from 1 --> lastrow-firstrow+1
c values of colidx which are col indexes go from firstcol --> lastcol
c So:
c Shift the col index vals from actual (firstcol --> lastcol )
c to local, i.e., (1 --> lastcol-firstcol+1)
c---------------------------------------------------------------------*/
#pragma omp parallel private(it,i,j,k)
{
#pragma omp for nowait
for (j = 1; j <= lastrow - firstrow + 1; j++) {
for (k = rowstr[j]; k < rowstr[j+1]; k++) {
colidx[k] = colidx[k] - firstcol + 1;
}
}
/*--------------------------------------------------------------------
c set starting vector to (1, 1, .... 1)
c-------------------------------------------------------------------*/
#pragma omp for nowait
for (i = 1; i <= NA+1; i++) {
x[i] = 1.0;
}
#pragma omp single
zeta = 0.0;
/*-------------------------------------------------------------------
c---->
c Do one iteration untimed to init all code and data page tables
c----> (then reinit, start timing, to niter its)
c-------------------------------------------------------------------*/
for (it = 1; it <= 1; it++) {
/*--------------------------------------------------------------------
c The call to the conjugate gradient routine:
c-------------------------------------------------------------------*/
conj_grad (colidx, rowstr, x, z, a, p, q, r, w, &rnorm);
/*--------------------------------------------------------------------
c zeta = shift + 1/(x.z)
c So, first: (x.z)
c Also, find norm of z
c So, first: (z.z)
c-------------------------------------------------------------------*/
#pragma omp single
{
norm_temp11 = 0.0;
norm_temp12 = 0.0;
} /* end single */
#pragma omp for reduction(+:norm_temp11,norm_temp12)
for (j = 1; j <= lastcol-firstcol+1; j++) {
norm_temp11 = norm_temp11 + x[j]*z[j];
norm_temp12 = norm_temp12 + z[j]*z[j];
}
#pragma omp single
norm_temp12 = 1.0 / sqrt( norm_temp12 );
/*--------------------------------------------------------------------
c Normalize z to obtain x
c-------------------------------------------------------------------*/
#pragma omp for
for (j = 1; j <= lastcol-firstcol+1; j++) {
x[j] = norm_temp12*z[j];
}
} /* end of do one iteration untimed */
/*--------------------------------------------------------------------
c set starting vector to (1, 1, .... 1)
c-------------------------------------------------------------------*/
#pragma omp for nowait
for (i = 1; i <= NA+1; i++) {
x[i] = 1.0;
}
#pragma omp single
zeta = 0.0;
} /* end parallel */
timer_clear( 1 );
timer_start( 1 );
/*--------------------------------------------------------------------
c---->
c Main Iteration for inverse power method
c---->
c-------------------------------------------------------------------*/
#pragma omp parallel private(it,i,j,k)
{
for (it = 1; it <= NITER; it++) {
/*--------------------------------------------------------------------
c The call to the conjugate gradient routine:
c-------------------------------------------------------------------*/
conj_grad(colidx, rowstr, x, z, a, p, q, r, w, &rnorm);
/*--------------------------------------------------------------------
c zeta = shift + 1/(x.z)
c So, first: (x.z)
c Also, find norm of z
c So, first: (z.z)
c-------------------------------------------------------------------*/
#pragma omp single
{
norm_temp11 = 0.0;
norm_temp12 = 0.0;
} /* end single */
#pragma omp for reduction(+:norm_temp11,norm_temp12)
for (j = 1; j <= lastcol-firstcol+1; j++) {
norm_temp11 = norm_temp11 + x[j]*z[j];
norm_temp12 = norm_temp12 + z[j]*z[j];
}
#pragma omp single
{
norm_temp12 = 1.0 / sqrt( norm_temp12 );
zeta = SHIFT + 1.0 / norm_temp11;
} /* end single */
#pragma omp master
{
if( it == 1 ) {
printf(" iteration ||r|| zeta\n");
}
printf(" %5d %20.14e%20.13e\n", it, rnorm, zeta);
} /* end master */
/*--------------------------------------------------------------------
c Normalize z to obtain x
c-------------------------------------------------------------------*/
#pragma omp for
for (j = 1; j <= lastcol-firstcol+1; j++) {
x[j] = norm_temp12*z[j];
}
} /* end of main iter inv pow meth */
#if defined(_OPENMP)
#pragma omp master
nthreads = omp_get_num_threads();
#endif /* _OPENMP */
} /* end parallel */
timer_stop( 1 );
/*--------------------------------------------------------------------
c End of timed section
c-------------------------------------------------------------------*/
t = timer_read( 1 );
printf(" Benchmark completed\n");
epsilon = 1.0e-10;
if (cclass != 'U') {
if (fabs(zeta - zeta_verify_value) <= epsilon) {
verified = TRUE;
printf(" VERIFICATION SUCCESSFUL\n");
printf(" Zeta is %20.12e\n", zeta);
printf(" Error is %20.12e\n", zeta - zeta_verify_value);
} else {
verified = FALSE;
printf(" VERIFICATION FAILED\n");
printf(" Zeta %20.12e\n", zeta);
printf(" The correct zeta is %20.12e\n", zeta_verify_value);
}
} else {
verified = FALSE;
printf(" Problem size unknown\n");
printf(" NO VERIFICATION PERFORMED\n");
}
if ( t != 0.0 ) {
mflops = (2.0*NITER*NA)
* (3.0+(NONZER*(NONZER+1)) + 25.0*(5.0+(NONZER*(NONZER+1))) + 3.0 )
/ t / 1000000.0;
} else {
mflops = 0.0;
}
c_print_results("CG", cclass, NA, 0, 0, NITER, nthreads, t,
mflops, " floating point",
verified, NPBVERSION, COMPILETIME,
CS1, CS2, CS3, CS4, CS5, CS6, CS7);
}
