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#include "qlpsolve.hh"
#include "const.hh"
#include "debug.hh"
#include "choleski.hh"
const Real TOL=1e-2; // roughly 1/10 mm
String
Active_constraints::status() const
{
String s("Active|Inactive [");
for (int i=0; i< active.sz(); i++) {
s += String(active[i]) + " ";
}
s+="| ";
for (int i=0; i< inactive.sz(); i++) {
s += String(inactive[i]) + " ";
}
s+="]";
return s;
}
void
Active_constraints::OK() {
H.OK();
A.OK();
assert(active.sz() +inactive.sz() == opt->cons.sz());
assert(H.dim() == opt->dim());
assert(active.sz() == A.rows());
svec<int> allcons;
for (int i=0; i < opt->cons.sz(); i++)
allcons.add(0);
for (int i=0; i < active.sz(); i++) {
int j = active[i];
allcons[j]++;
}
for (int i=0; i < inactive.sz(); i++) {
int j = inactive[i];
allcons[j]++;
}
for (int i=0; i < allcons.sz(); i++)
assert(allcons[i] == 1);
}
Vector
Active_constraints::get_lagrange(Vector gradient)
{
Vector l(A*gradient);
return l;
}
void
Active_constraints::add(int k)
{
// add indices
int cidx=inactive[k];
active.add(cidx);
inactive.swap(k,inactive.sz()-1);
inactive.pop();
Vector a( opt->cons[cidx] );
// update of matrices
Vector Ha = H*a;
Real aHa = a*Ha;
if (ABS(aHa) > EPS) {
/*
a != 0, so if Ha = O(EPS), then
Ha * aH / aHa = O(EPS^2/EPS)
if H*a == 0, the constraints are dependent.
*/
H -= Matrix(Ha , Ha)/(aHa);
/*
sorry, don't know how to justify this. ..
*/
Vector addrow(Ha/(aHa));
A -= Matrix(A*a, addrow);
A.insert_row(addrow,A.rows());
}else
WARN << "degenerate constraints";
}
void
Active_constraints::drop(int k)
{
int q=active.sz()-1;
// drop indices
inactive.add(active[k]);
active.swap(k,q);
A.swap_rows(k,q);
active.pop();
Vector a(A.row(q));
if (a.norm() > EPS) {
/*
*/
H += Matrix(a,a)/(a*opt->quad*a);
A -= A*opt->quad*Matrix(a,a)/(a*opt->quad*a);
}else
WARN << "degenerate constraints";
Vector rem_row(A.row(q));
assert(rem_row.norm() < EPS);
A.delete_row(q);
}
Active_constraints::Active_constraints(Ineq_constrained_qp const *op)
: A(0,op->dim()),
H(op->dim()),
opt(op)
{
for (int i=0; i < op->cons.sz(); i++)
inactive.add(i);
Choleski_decomposition chol(op->quad);
H=chol.inverse();
}
/* Find the optimum which is in the planes generated by the active
constraints.
*/
Vector
Active_constraints::find_active_optimum(Vector g)
{
return H*g;
}
/****************************************************************/
int
min_elt_index(Vector v)
{
Real m=INFTY; int idx=-1;
for (int i = 0; i < v.dim(); i++){
if (v(i) < m) {
idx = i;
m = v(i);
}
assert(v(i) <= INFTY);
}
return idx;
}
///the numerical solving
Vector
Ineq_constrained_qp::solve(Vector start) const
{
Active_constraints act(this);
act.OK();
Vector x(start);
Vector gradient=quad*x+lin;
Vector last_gradient(gradient);
int iterations=0;
while (iterations++ < MAXITER) {
Vector direction= - act.find_active_optimum(gradient);
mtor << "gradient "<< gradient<< "\ndirection " << direction<<"\n";
if (direction.norm() > EPS) {
mtor << act.status() << '\n';
Real minalf = INFTY;
Inactive_iter minidx(act);
/*
we know the optimum on this "hyperplane". Check if we
bump into the edges of the simplex
*/
for (Inactive_iter ia(act); ia.ok(); ia++) {
if (ia.vec() * direction >= 0)
continue;
Real alfa= - (ia.vec()*x - ia.rhs())/
(ia.vec()*direction);
if (minalf > alfa) {
minidx = ia;
minalf = alfa;
}
}
Real unbounded_alfa = 1.0;
Real optimal_step = MIN(minalf, unbounded_alfa);
Vector deltax=direction * optimal_step;
x += deltax;
gradient += optimal_step * (quad * deltax);
mtor << "step = " << optimal_step<< " (|dx| = " <<
deltax.norm() << ")\n";
if (minalf < unbounded_alfa) {
/* bumped into an edge. try again, in smaller space. */
act.add(minidx.idx());
mtor << "adding cons "<< minidx.idx()<<'\n';
continue;
}
/*ASSERT: we are at optimal solution for this "plane"*/
}
Vector lagrange_mult=act.get_lagrange(gradient);
int m= min_elt_index(lagrange_mult);
if (m>=0 && lagrange_mult(m) > 0) {
break; // optimal sol.
} else if (m<0) {
assert(gradient.norm() < EPS) ;
break;
}
mtor << "dropping cons " << m<<'\n';
act.drop(m);
}
if (iterations >= MAXITER)
WARN<<"didn't converge!\n";
mtor << ": found " << x<<" in " << iterations <<" iterations\n";
assert_solution(x);
return x;
}
/** Mordecai Avriel, Nonlinear Programming: analysis and methods (1976)
Prentice Hall.
Section 13.3
This is a "projected gradient" algorithm. Starting from a point x
the next point is found in a direction determined by projecting
the gradient onto the active constraints. (well, not really the
gradient. The optimal solution obeying the active constraints is
tried. This is why H = Q^-1 in initialisation) )
*/
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