// Test KKT conditions on the solution.
// The function returns true if the tested KKT conditions are satisfied
// and false otherwise.
// The function assumes that the model currently extracted to CPLEX is fully
// described by obj, vars and rngs.
static bool
checkkkt (IloCplex& cplex, IloObjective const& obj, IloNumVarArray const& vars,
IloRangeArray const& rngs, IloIntArray const& cone, double tol)
{
IloEnv env = cplex.getEnv();
IloModel model = cplex.getModel();
IloNumArray x(env), dslack(env);
IloNumArray pi(env, rngs.getSize()), slack(env);
// Read primal and dual solution information.
cplex.getValues(x, vars);
cplex.getSlacks(slack, rngs);
// pi for second order cone constraints.
getsocpconstrmultipliers(cplex, vars, rngs, pi, dslack);
// pi for linear constraints.
for (IloInt i = 0; i < rngs.getSize(); ++i) {
IloRange r = rngs[i];
if ( !r.getQuadIterator().ok() )
pi[idx(r)] = cplex.getDual(r);
}
// Print out the data we just fetched.
streamsize oprec = env.out().precision(3);
ios_base::fmtflags oflags = env.out().setf(ios::fixed | ios::showpos);
env.out() << "x = [";
for (IloInt i = 0; i < x.getSize(); ++i)
env.out() << " " << x[i];
env.out() << " ]" << endl;
env.out() << "dslack = [";
for (IloInt i = 0; i < dslack.getSize(); ++i)
env.out() << " " << dslack[i];
env.out() << " ]" << endl;
env.out() << "pi = [";
for (IloInt i = 0; i < rngs.getSize(); ++i)
env.out() << " " << pi[i];
env.out() << " ]" << endl;
env.out() << "slack = [";
for (IloInt i = 0; i < rngs.getSize(); ++i)
env.out() << " " << slack[i];
env.out() << " ]" << endl;
env.out().precision(oprec);
env.out().flags(oflags);
// Test primal feasibility.
// This example illustrates the use of dual vectors returned by CPLEX
// to verify dual feasibility, so we do not test primal feasibility
// here.
// Test dual feasibility.
// We must have
// - for all <= constraints the respective pi value is non-negative,
// - for all >= constraints the respective pi value is non-positive,
// - the dslack value for all non-cone variables must be non-negative.
// Note that we do not support ranged constraints here.
for (IloInt i = 0; i < vars.getSize(); ++i) {
IloNumVar v = vars[i];
if ( cone[i] == NOT_IN_CONE && dslack[i] < -tol ) {
env.error() << "Dual multiplier for " << v << " is not feasible: "
<< dslack[i] << endl;
return false;
}
}
for (IloInt i = 0; i < rngs.getSize(); ++i) {
IloRange r = rngs[i];
if ( fabs (r.getLB() - r.getUB()) <= tol ) {
// Nothing to check for equality constraints.
}
else if ( r.getLB() > -IloInfinity && pi[i] > tol ) {
env.error() << "Dual multiplier " << pi[i] << " for >= constraint"
<< endl << r << endl
<< "not feasible"
<< endl;
return false;
}
else if ( r.getUB() < IloInfinity && pi[i] < -tol ) {
env.error() << "Dual multiplier " << pi[i] << " for <= constraint"
<< endl << r << endl
<< "not feasible"
<< endl;
return false;
}
}
// Test complementary slackness.
// For each constraint either the constraint must have zero slack or
// the dual multiplier for the constraint must be 0. We must also
// consider the special case in which a variable is not explicitly
// contained in a second order cone constraint.
for (IloInt i = 0; i < vars.getSize(); ++i) {
if ( cone[i] == NOT_IN_CONE ) {
if ( fabs(x[i]) > tol && dslack[i] > tol ) {
env.error() << "Invalid complementary slackness for " << vars[i]
<< ":" << endl
<< " " << x[i] << " and " << dslack[i]
<< endl;
return false;
}
}
}
for (IloInt i = 0; i < rngs.getSize(); ++i) {
if ( fabs(slack[i]) > tol && fabs(pi[i]) > tol ) {
env.error() << "Invalid complementary slackness for "
<< endl << rngs[i] << ":" << endl
<< " " << slack[i] << " and " << pi[i]
<< endl;
return false;
}
}
// Test stationarity.
// We must have
// c - g[i]'(X)*pi[i] = 0
// where c is the objective function, g[i] is the i-th constraint of the
// problem, g[i]'(x) is the derivate of g[i] with respect to x and X is the
// optimal solution.
// We need to distinguish the following cases:
// - linear constraints g(x) = ax - b. The derivative of such a
// constraint is g'(x) = a.
// - second order constraints g(x[1],...,x[n]) = -x[1] + |(x[2],...,x[n])|
// the derivative of such a constraint is
// g'(x) = (-1, x[2]/|(x[2],...,x[n])|, ..., x[n]/|(x[2],...,x[n])|
// (here |.| denotes the Euclidean norm).
// - bound constraints g(x) = -x for variables that are not explicitly
// contained in any second order cone constraint. The derivative for
// such a constraint is g'(x) = -1.
// Note that it may happen that the derivative of a second order cone
// constraint is not defined at the optimal solution X (this happens if
// X=0). In this case we just skip the stationarity test.
IloNumArray sum(env, vars.getSize());
for (IloExpr::LinearIterator it = obj.getLinearIterator(); it.ok(); ++it)
sum[idx(it.getVar())] = it.getCoef();
for (IloInt i = 0; i < vars.getSize(); ++i) {
IloNumVar v = vars[i];
if ( cone[i] == NOT_IN_CONE )
sum[i] -= dslack[i];
}
for (IloInt i = 0; i < rngs.getSize(); ++i) {
IloRange r = rngs[i];
if ( r.getQuadIterator().ok() ) {
// Quadratic (second order cone) constraint.
IloNum norm = 0.0;
for (IloExpr::QuadIterator q = r.getQuadIterator(); q.ok(); ++q) {
if ( q.getCoef() > 0 )
norm += x[idx(q.getVar1())] * x[idx(q.getVar1())];
}
norm = sqrt(norm);
if ( fabs(norm) <= tol ) {
// Derivative is not defined. Skip test.
env.warning() << "Cannot test stationarity at non-differentiable point."
<< endl;
return true;
}
else {
for (IloExpr::QuadIterator q = r.getQuadIterator(); q.ok(); ++q) {
if ( q.getCoef() < 0 )
sum[idx(q.getVar1())] -= pi[i];
else
sum[idx(q.getVar1())] += pi[i] * x[idx(q.getVar1())] / norm;
}
}
}
else {
// Linear constraint.
for (IloExpr::LinearIterator l = r.getLinearIterator(); l.ok(); ++l)
sum[idx(l.getVar())] -= pi[i] * l.getCoef();
}
}
// Now test that all elements in sum[] are 0.
for (IloInt i = 0; i < vars.getSize(); ++i) {
if ( fabs(sum[i]) > tol ) {
env.error() << "Invalid stationarity " << sum[i] << " for "
<< vars[i] << endl;
return false;
}
}
return true;
}
