qlpsolve.cc

   1 #include "qlpsolve.hh"
   2 #include "const.hh"
   3 #include "debug.hh"
   4 #include "choleski.hh"
   5
   6 const Real TOL=1e-2;            // roughly 1/10 mm
   7
   8 String
   9 Active_constraints::status() const
  10 {
  11     String s("Active|Inactive [");
  12     for (int i=0; i< active.sz(); i++) {
  13         s += String(active[i]) + " ";
  14     }
  15
  16     s+="| ";
  17     for (int i=0; i< inactive.sz(); i++) {
  18         s += String(inactive[i]) + " ";
  19     }
  20     s+="]";
  21
  22     return s;
  23 }
  24
  25 void
  26 Active_constraints::OK() {
  27     H.OK();
  28     A.OK();
  29     assert(active.sz() +inactive.sz() == opt->cons.sz());
  30     assert(H.dim() == opt->dim());
  31     assert(active.sz() == A.rows());
  32     svec<int> allcons;
  33
  34     for (int i=0; i < opt->cons.sz(); i++)
  35         allcons.add(0);
  36     for (int i=0; i < active.sz(); i++) {
  37         int j = active[i];
  38         allcons[j]++;
  39     }
  40     for (int i=0; i < inactive.sz(); i++) {
  41         int j = inactive[i];
  42         allcons[j]++;
  43     }
  44     for (int i=0; i < allcons.sz(); i++)
  45         assert(allcons[i] == 1);
  46 }
  47
  48 Vector
  49 Active_constraints::get_lagrange(Vector gradient)
  50 {
  51     Vector l(A*gradient);
  52
  53     return l;
  54 }
  55
  56 void
  57 Active_constraints::add(int k)
  58 {
  59     // add indices
  60     int cidx=inactive[k];
  61     active.add(cidx);
  62
  63     inactive.swap(k,inactive.sz()-1);
  64     inactive.pop();
  65
  66     Vector a( opt->cons[cidx] );
  67     // update of matrices
  68     Vector Ha = H*a;
  69     Real aHa = a*Ha;
  70     if (ABS(aHa) > EPS) {
  71         /*
  72           a != 0, so if Ha = O(EPS), then
  73           Ha * aH / aHa = O(EPS^2/EPS)
  74
  75           if H*a == 0, the constraints are dependent.
  76           */
  77         H -= Matrix(Ha , Ha)/(aHa);
  78
  79
  80         /*
  81           sorry, don't know how to justify this. ..
  82           */
  83         Vector addrow(Ha/(aHa));
  84         A -= Matrix(A*a, addrow);
  85         A.insert_row(addrow,A.rows());
  86     }else
  87         WARN << "degenerate constraints";
  88 }
  89
  90 void
  91 Active_constraints::drop(int k)
  92 {
  93     int q=active.sz()-1;
  94
  95         // drop indices
  96     inactive.add(active[k]);
  97     active.swap(k,q);
  98     A.swap_rows(k,q);
  99     active.pop();
 100
 101     Vector a(A.row(q));
 102     if (a.norm() > EPS) {
 103         /*
 104
 105          */
 106         H += Matrix(a,a)/(a*opt->quad*a);
 107         A -= A*opt->quad*Matrix(a,a)/(a*opt->quad*a);
 108     }else
 109         WARN << "degenerate constraints";
 110     Vector rem_row(A.row(q));
 111     assert(rem_row.norm() < EPS);
 112     A.delete_row(q);
 113 }
 114
 115
 116 Active_constraints::Active_constraints(Ineq_constrained_qp const *op)
 117     :       A(0,op->dim()),
 118             H(op->dim()),
 119             opt(op)
 120 {
 121     for (int i=0; i < op->cons.sz(); i++)
 122         inactive.add(i);
 123     Choleski_decomposition chol(op->quad);
 124     H=chol.inverse();
 125 }
 126
 127 /* Find the optimum which is in the planes generated by the active
 128     constraints.
 129     */
 130 Vector
 131 Active_constraints::find_active_optimum(Vector g)
 132 {
 133     return H*g;
 134 }
 135
 136 /****************************************************************/
 137
 138 int
 139 min_elt_index(Vector v)
 140 {
 141     Real m=INFTY; int idx=-1;
 142     for (int i = 0; i < v.dim(); i++){
 143         if (v(i) < m) {
 144             idx = i;
 145             m = v(i);
 146         }
 147         assert(v(i) <= INFTY);
 148     }
 149     return idx;
 150 }
 151
 152 ///the numerical solving
 153 Vector
 154 Ineq_constrained_qp::solve(Vector start) const
 155 {
 156     Active_constraints act(this);
 157
 158
 159     act.OK();
 160
 161
 162     Vector x(start);
 163     Vector gradient=quad*x+lin;
 164
 165
 166     Vector last_gradient(gradient);
 167     int iterations=0;
 168
 169     while (iterations++ < MAXITER) {
 170         Vector direction= - act.find_active_optimum(gradient);
 171
 172         mtor << "gradient "<< gradient<< "\ndirection " << direction<<"\n";
 173
 174         if (direction.norm() > EPS) {
 175             mtor << act.status() << '\n';
 176
 177             Real minalf = INFTY;
 178
 179             Inactive_iter minidx(act);
 180
 181
 182             /*
 183     we know the optimum on this "hyperplane". Check if we
 184     bump into the edges of the simplex
 185     */
 186
 187             for (Inactive_iter ia(act); ia.ok(); ia++) {
 188
 189                 if (ia.vec() * direction >= 0)
 190                     continue;
 191                 Real alfa= - (ia.vec()*x - ia.rhs())/
 192                     (ia.vec()*direction);
 193
 194                 if (minalf > alfa) {
 195                     minidx = ia;
 196                     minalf = alfa;
 197                 }
 198             }
 199             Real unbounded_alfa = 1.0;
 200             Real optimal_step = MIN(minalf, unbounded_alfa);
 201
 202             Vector deltax=direction * optimal_step;
 203             x += deltax;
 204             gradient += optimal_step * (quad * deltax);
 205
 206             mtor << "step = " << optimal_step<< " (|dx| = " <<
 207                 deltax.norm() << ")\n";
 208
 209             if (minalf < unbounded_alfa) {
 210                 /* bumped into an edge. try again, in smaller space. */
 211                 act.add(minidx.idx());
 212                 mtor << "adding cons "<< minidx.idx()<<'\n';
 213                 continue;
 214             }
 215             /*ASSERT: we are at optimal solution for this "plane"*/
 216
 217
 218         }
 219
 220         Vector lagrange_mult=act.get_lagrange(gradient);
 221         int m= min_elt_index(lagrange_mult);
 222
 223         if (m>=0 && lagrange_mult(m) > 0) {
 224             break;              // optimal sol.
 225         } else if (m<0) {
 226             assert(gradient.norm() < EPS) ;
 227
 228             break;
 229         }
 230
 231         mtor << "dropping cons " << m<<'\n';
 232         act.drop(m);
 233     }
 234     if (iterations >= MAXITER)
 235         WARN<<"didn't converge!\n";
 236
 237     mtor <<  ": found " << x<<" in " << iterations <<" iterations\n";
 238     assert_solution(x);
 239     return x;
 240 }
 241
 242 /** Mordecai Avriel, Nonlinear Programming: analysis and methods (1976)
 243     Prentice Hall.
 244
 245     Section 13.3
 246
 247     This is a "projected gradient" algorithm. Starting from a point x
 248     the next point is found in a direction determined by projecting
 249     the gradient onto the active constraints.  (well, not really the
 250     gradient. The optimal solution obeying the active constraints is
 251     tried. This is why H = Q^-1 in initialisation) )
 252
 253
 254     */
 255