int local(Trial &T, TBox &box, TBox &domain, double eps_cl, double *mgr,
Global &glob, int axis, RCRVector x_av
#ifdef NLOPT_UTIL_H
, nlopt_stopping *stop
#endif
) {
int n=box.GetDim();
RVector x(n);
double tmp, f;
x=T.xvals ;
#ifdef LS_DEBUG
cout << "Local Search, x=" << x << endl;
#endif
if (box.OutsideBox(x, domain) != 0) {
cout << "Starting point is not inside the boundary. Exiting...\n" ;
exit(1) ;
return LS_Out ;
}
// Check if we are close to a stationary point located previously
if (box.CloseToMin(x, &tmp, eps_cl)) {
#ifdef LS_DEBUG
cout << "Close to a previously located stationary point, exiting" << endl;
#endif
T.objval=tmp;
return LS_Old ;
}
#if 0
if (axis != -1) {
cout << "NLopt code only works with axis == -1, exiting...\n" ;
exit(EXIT_FAILURE);
}
f_local_data data;
data.glob = &glob;
data.maxgrad = *mgr;
data.stop = stop;
nlopt_result ret = nlopt_minimize(NLOPT_LOCAL_LBFGS, n, f_local, &data,
box.lb.raw_data(), box.ub.raw_data(),
x.raw_data(), &f,
stop->minf_max,
stop->ftol_rel, stop->ftol_abs,
stop->xtol_rel, stop->xtol_abs,
stop->maxeval - stop->nevals,
stop->maxtime - stop->start);
*mgr = data.maxgrad;
T.xvals=x ; T.objval=f ;
if (ret == NLOPT_MAXEVAL_REACHED || ret == NLOPT_MAXTIME_REACHED)
return LS_MaxEvalTime;
else if (ret > 0)
return LS_New;
else
return LS_Out; // failure
#else /* not using NLopt local optimizer ... use original STOgo BFGS code */
int k_max, info, outside = 0;
int k, i, good_enough, iTmp ;
double maxgrad, delta, f_new;
double alpha, gamma, beta, d2, s2, nom, den, ro ;
double nrm_sd, nrm_hn, snrm_hn, nrm_dl ;
RVector g(n), h_sd(n), h_dl(n), h_n(n), x_new(n), g_new(n) ;
RVector s(n),y(n),z(n),w(n) ; // Temporary vectors
RMatrix B(n), H(n) ; // Hessian and it's inverse
k_max = max_iter*n ;
// Initially B and H are equal to the identity matrix
B=0 ; H=0 ;
for (i=0 ; i<n ; i++) {
B(i,i)=1 ;
H(i,i)=1 ;
}
RVector g_av(x_av.GetLength());
if (axis==-1) {
f=glob.ObjectiveGradient(x,g,OBJECTIVE_AND_GRADIENT);
}
else {
x_av(axis)=x(0);
f=glob.ObjectiveGradient(x_av,g_av,OBJECTIVE_AND_GRADIENT);
g(0)=g_av(axis);
}
IF_NLOPT_CHECK_EVALS;
FC++;GC++;
if (axis == -1) {
// Skipping AV
#ifdef INI3
// Elaborate scheme to initalize delta
delta=delta_coef*norm2(g) ;
copy(g,z) ;
axpy(1.0,x,z) ;
if (!box.InsideBox(z)) {
if (box.Intersection(x,g,z)==TRUE) {
axpy(-1.0,x,z) ;
delta=min(delta,delta_coef*norm2(z)) ;
}
else {
// Algorithm broke down, use INI1
delta = (1.0/7)*box.ShortestSide(&iTmp) ;
}
}
#endif
#ifdef INI2
// Use INI2 scheme
delta = box.ClosestSide(x)*delta_coef ;
if (delta<MacEpsilon)
// Patch to avoid trust region with radius close to zero
delta = (1.0/7)*box.ShortestSide(&iTmp) ;
#endif
#ifdef INI1
delta = delta_coef*box.ShortestSide(&iTmp) ;
#endif
}
else {
// Use a simple scheme for the 1D minimization (INI1)
delta = (1.0/7.0)*box.ShortestSide(&iTmp) ;
}
k=0 ; good_enough = 0 ; info=LS_New ; outside=0 ;
maxgrad=*mgr ;
while (good_enough == 0) {
k++ ;
if (k>k_max) {
#ifdef LS_DEBUG
cout << "Maximum number of iterations reached\n" ;
#endif
info=LS_MaxIter ;
break ;
}
// Update maximal gradient value
maxgrad=max(maxgrad,normInf(g)) ;
// Steepest descent, h_sd = -g
copy(g,h_sd) ;
scal(-1.0,h_sd) ;
nrm_sd=norm2(h_sd) ;
if (nrm_sd < epsilon) {
// Stop criterion (gradient) fullfilled
#ifdef LS_DEBUG
cout << "Gradient small enough" << endl ;
#endif
good_enough = 1 ;
break ;
}
// Compute Newton step, h_n = -H*g
gemv('N',-1.0, H, g, 0.0, h_n) ;
nrm_hn = norm2(h_n) ;
if (nrm_hn < delta) {
// Pure Newton step
copy(h_n, h_dl) ;
#ifdef LS_DEBUG
cout << "[Newton step] " ;
#endif
}
else {
gemv('N',1.0,B,g,0.0,z) ;
tmp=dot(g,z) ;
if (tmp==0) {
info = LS_Unstable ;
break ;
}
alpha=(nrm_sd*nrm_sd)/tmp ; // Normalization (N38,eq. 3.30)
scal(alpha,h_sd) ;
nrm_sd=fabs(alpha)*nrm_sd ;
if (nrm_sd >= delta) {
gamma = delta/nrm_sd ; // Normalization (N38, eq. 3.33)
copy(h_sd,h_dl) ;
scal(gamma,h_dl) ;
#ifdef LS_DEBUG
cout << "[Steepest descent] " ;
#endif
}
else {
// Combination of Newton and SD steps
d2 = delta*delta ;
copy(h_sd,s) ;
s2=nrm_sd*nrm_sd ;
nom = d2 - s2 ;
snrm_hn=nrm_hn*nrm_hn ;
tmp = dot(h_n,s) ;
den = tmp-s2 + sqrt((tmp-d2)*(tmp-d2)+(snrm_hn-d2)*(d2-s2)) ;
if (den==0) {
info = LS_Unstable ;
break ;
}
// Normalization (N38, eq. 3.31)
beta = nom/den ;
copy(h_n,h_dl) ;
scal(beta,h_dl) ;
axpy((1-beta),h_sd,h_dl) ;
#ifdef LS_DEBUG
cout << "[Mixed step] " ;
#endif
}
}
nrm_dl=norm2(h_dl) ;
//x_new = x+h_dl ;
copy(x,x_new) ;
axpy(1.0,h_dl,x_new) ;
// Check if x_new is inside the box
iTmp=box.OutsideBox(x_new, domain) ;
if (iTmp == 1) {
#ifdef LS_DEBUG
cout << "x_new is outside the box " << endl ;
#endif
outside++ ;
if (outside>max_outside_steps) {
// Previous point was also outside, exit
break ;
}
}
else if (iTmp == 2) {
#ifdef LS_DEBUG
cout << " x_new is outside the domain" << endl ;
#endif
info=LS_Out ;
break ;
}
else {
outside=0 ;
}
// Compute the gain
if (axis==-1)
f_new=glob.ObjectiveGradient(x_new,g_new,OBJECTIVE_AND_GRADIENT);
else {
x_av(axis)=x_new(0);
f_new=glob.ObjectiveGradient(x_av,g_av,OBJECTIVE_AND_GRADIENT);
}
IF_NLOPT_CHECK_EVALS;
FC++; GC++;
gemv('N',0.5,B,h_dl,0.0,z);
ro = (f_new-f) / (dot(g,h_dl) + dot(h_dl,z)); // Quadratic model
if (ro > 0.75) {
delta = delta*2;
}
if (ro < 0.25) {
delta = delta/3;
}
if (ro > 0) {
// Update the Hessian and it's inverse using the BFGS formula
#if 0 // changed by SGJ to compute OBJECTIVE_AND_GRADIENT above
if (axis==-1)
glob.ObjectiveGradient(x_new,g_new,GRADIENT_ONLY);
else {
x_av(axis)=x_new(0);
glob.ObjectiveGradient(x_av,g_av,GRADIENT_ONLY);
g_new(0)=g_av(axis);
}
GC++;
IF_NLOPT_CHECK_EVALS;
#else
if (axis != -1)
g_new(0)=g_av(axis);
#endif
// y=g_new-g
copy(g_new,y);
axpy(-1.0,g,y);
// Check curvature condition
alpha=dot(y,h_dl);
if (alpha <= sqrt(MacEpsilon)*nrm_dl*norm2(y)) {
#ifdef LS_DEBUG
cout << "Curvature condition violated " ;
#endif
}
else {
// Update Hessian
gemv('N',1.0,B,h_dl,0.0,z) ; // z=Bh_dl
beta=-1/dot(h_dl,z) ;
ger(1/alpha,y,y,B) ;
ger(beta,z,z,B) ;
// Update Hessian inverse
gemv('N',1.0,H,y,0.0,z) ; // z=H*y
gemv('T',1.0,H,y,0.0,w) ; // w=y'*H
beta=dot(y,z) ;
beta=(1+beta/alpha)/alpha ;
// It should be possible to do this updating more efficiently, by
// exploiting the fact that (h_dl*y'*H) = transpose(H*y*h_dl')
ger(beta,h_dl,h_dl,H) ;
ger(-1/alpha,z,h_dl,H) ;
ger(-1/alpha,h_dl,w,H) ;
}
if (nrm_dl < norm2(x)*epsilon) {
// Stop criterion (iteration progress) fullfilled
#ifdef LS_DEBUG
cout << "Progress is marginal" ;
#endif
good_enough = 1 ;
}
// Check if we are close to a stationary point located previously
if (box.CloseToMin(x_new, &f_new, eps_cl)) {
// Note that x_new and f_new may be overwritten on exit from CloseToMin
#ifdef LS_DEBUG
cout << "Close to a previously located stationary point, exiting" << endl;
#endif
info = LS_Old ;
good_enough = 1 ;
}
// Update x, g and f
copy(x_new,x) ; copy(g_new,g) ; f=f_new ;
#ifdef LS_DEBUG
cout << " x=" << x << endl ;
#endif
}
else {
#ifdef LS_DEBUG
cout << "Step is no good, ro=" << ro << " delta=" << delta << endl ;
#endif
}
} // wend
// Make sure the routine returns correctly...
// Check if last iterate is outside the boundary
if (box.OutsideBox(x, domain) != 0) {
info=LS_Out; f=DBL_MAX;
}
if (info == LS_Unstable) {
cout << "Local search became unstable. No big deal but exiting anyway\n" ;
exit(1);
}
*mgr=maxgrad ;
T.xvals=x ; T.objval=f ;
if (outside>0)
return LS_Out ;
else
return info ;
#endif
}
