// This function creates the following model:
// Minimize
// obj: x1 + x2 + x3 + x4 + x5 + x6
// Subject To
// c1: x1 + x2 + x5 = 8
// c2: x3 + x5 + x6 = 10
// q1: [ -x1^2 + x2^2 + x3^2 ] <= 0
// q2: [ -x4^2 + x5^2 ] <= 0
// Bounds
// x2 Free
// x3 Free
// x5 Free
// End
// which is a second order cone program in standard form.
// The function returns objective, variables and constraints in the
// values obj, vars and rngs.
// The function also sets up cone so that for a column j we have
// cone[j] >= 0 Column j is in a cone constraint and is the
// cone's head variable.
// cone[j] == NOT_CONE_HEAD Column j is in a cone constraint but is
// not the cone's head variable..
// cone[j] == NOT_IN_CONE Column j is not contained in any cone constraint.
static void
createmodel (IloModel& model, IloObjective &obj, IloNumVarArray &vars,
IloRangeArray &rngs, IloIntArray& cone)
{
// The indices we assign as user objects to the modeling objects.
// We define them as static data so that we don't have to worry about
// dynamic memory allocation/leakage.
static int indices[] = { 0, 1, 2, 3, 4, 5, 6 };
IloEnv env = model.getEnv();
// Create variables.
IloNumVar x1(env, 0, IloInfinity, "x1");
IloNumVar x2(env, -IloInfinity, IloInfinity, "x2");
IloNumVar x3(env, -IloInfinity, IloInfinity, "x3");
IloNumVar x4(env, 0, IloInfinity, "x4");
IloNumVar x5(env, -IloInfinity, IloInfinity, "x5");
IloNumVar x6(env, 0, IloInfinity, "x6");
// Create objective function and immediately store it in return value.
obj = IloMinimize(env, x1 + x2 + x3 + x4 + x5 + x6);
// Create constraints.
IloRange c1(env, 8, x1 + x2 + x5, 8, "c1");
IloRange c2(env, 10, x3 + x5 + x6, 10, "c2");
IloRange q1(env, -IloInfinity, -x1*x1 + x2*x2 + x3*x3, 0, "q1");
cone.add(2); // x1, cone head of constraint at index 2
cone.add(NOT_CONE_HEAD); // x2
cone.add(NOT_CONE_HEAD); // x3
IloRange q2(env, -IloInfinity, -x4*x4 + x5*x5, 0, "q2");
cone.add(3); // x4, cone head of constraint at index 3
cone.add(NOT_CONE_HEAD); // x5
cone.add(NOT_IN_CONE); // x6
// Setup model.
model.add(obj);
model.add(obj);
model.add(c1);
model.add(c2);
model.add(q1);
model.add(q2);
// Setup return values.
vars.add(x1);
vars.add(x2);
vars.add(x3);
vars.add(x4);
vars.add(x5);
vars.add(x6);
rngs.add(c1);
rngs.add(c2);
rngs.add(q1);
rngs.add(q2);
// We set the user object for each modeling object to its index in the
// respective array. This makes the code in checkkkt a little simpler.
for (IloInt i = 0; i < vars.getSize(); ++i)
vars[i].setObject(&indices[i]);
for (IloInt i = 0; i < rngs.getSize(); ++i)
rngs[i].setObject(&indices[i]);
}
// 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;
}
/** Extract the master block from <code>problem</code>.
* The constructor also sets up the solver for the newly created master
* block. The master block can only be extracted if all sub-blocks have
* already been extracted.
* @param problem The problem from which to extract the master.
* @param blocks The sub blocks that have already been extracted.
*/
BendersOpt::Block::Block(Problem const *problem, BlockVector const &blocks)
: env(), number(-1), vars(0), rows(0), cplex(0), cb(0)
{
IloNumVarArray problemVars = problem->getVariables();
IloRangeArray problemRanges = problem->getRows();
IloExpr masterObj(env);
IloNumVarArray masterVars(env);
IloRangeArray masterRows(env);
// Find columns that do not intersect block variables and
// copy them to the master block.
IdxMap idxMap;
RowSet rowSet;
for (IloInt j = 0; j < problemVars.getSize(); ++j) {
IloNumVar x = problemVars[j];
if ( problem->getBlock(x) < 0 ) {
// Column is not in a block. Copy it to the master.
IloNumVar v(env, x.getLB(), x.getUB(), x.getType(), x.getName());
varMap.insert(VarMap::value_type(v, x));
masterObj += problem->getObjCoef(x) * v;
idxMap[x] = masterVars.getSize();
masterVars.add(v);
}
else {
// Column is in a block. Collect all rows that intersect
// this column.
RowSet const &intersected = problem->getIntersectedRows(x);
for (RowSet::const_iterator it = intersected.begin();
it != intersected.end(); ++it)
rowSet.insert(*it);
idxMap[x] = -1;
}
}
// Pick up the rows that we need to copy.
// These are the rows that are only intersected by master variables,
// that is, the rows that are not in any block's rowset.
for (IloInt i = 0; i < problemRanges.getSize(); ++i) {
IloRange r = problemRanges[i];
if ( rowSet.find(r) == rowSet.end() ) {
IloRange masterRow(env, r.getLB(), r.getUB(), r.getName());
IloExpr lhs(env);
for (IloExpr::LinearIterator it = r.getLinearIterator(); it.ok(); ++it)
{
lhs += it.getCoef() * masterVars[idxMap[it.getVar()]];
}
masterRow.setExpr(lhs);
masterRows.add(masterRow);
}
}
// Adjust variable indices in blocks so that reference to variables
// in the original problem become references to variables in the master.
for (BlockVector::const_iterator b = blocks.begin(); b != blocks.end(); ++b) {
for (std::vector<FixData>::iterator it = (*b)->fixed.begin(); it != (*b)->fixed.end(); ++it)
it->col = idxMap[problemVars[it->col]];
}
// Create the eta variables, one for each block.
// See the comments at the top of this file for details about the
// eta variables.
IloInt const firsteta = masterVars.getSize();
for (BlockVector::size_type i = 0; i < blocks.size(); ++i) {
std::stringstream s;
s << "_eta" << i;
IloNumVar eta(env, 0.0, IloInfinity, s.str().c_str());
masterObj += eta;
masterVars.add(eta);
}
// Create model and solver instance
vars = masterVars;
rows = masterRows;
IloModel model(env);
model.add(obj = IloObjective(env, masterObj, problem->getObjSense()));
model.add(vars);
model.add(rows);
cplex = IloCplex(model);
cplex.use(cb = new (env) LazyConstraintCallback(env, this, blocks,
firsteta));
for (IloExpr::LinearIterator it = obj.getLinearIterator(); it.ok(); ++it)
objMap.insert(ObjMap::value_type(it.getVar(), it.getCoef()));
}
/** Create the dual of a linear program.
* The function can only dualize programs of the form
* <code>Ax <= b, x >= 0</code>. The data in <code>primalVars</code> and
* <code>dualRows</code> as well as in <code>primalRows</code> and
* <code>dualVars</code> is in 1-to-1-correspondence.
* @param primalObj Objective function of primal problem.
* @param primalVars Variables in primal problem.
* @param primalRows Rows in primal problem.
* @param dualObj Objective function of dual will be stored here.
* @param dualVars All dual variables will be stored here.
* @param dualRows All dual rows will be stored here.
*/
void BendersOpt::makeDual(IloObjective const &primalObj,
IloNumVarArray const &primalVars,
IloRangeArray const &primalRows,
IloObjective *dualObj,
IloNumVarArray *dualVars,
IloRangeArray *dualRows)
{
// To keep the code simple we only support problems
// of the form Ax <= b, b >= 0 here. We leave it as a reader's
// exercise to extend the function to something that can handle
// any kind of linear model.
for (IloInt j = 0; j < primalVars.getSize(); ++j)
if ( primalVars[j].getLB() != 0 ||
primalVars[j].getUB() < IloInfinity )
{
std::stringstream s;
s << "Cannot dualize variable " << primalVars[j];
throw s.str();
}
for (IloInt i = 0; i < primalRows.getSize(); ++i)
if ( primalRows[i].getLB() > -IloInfinity ||
primalRows[i].getUB() >= IloInfinity )
{
std::stringstream s;
s << "Cannot dualize constraint " << primalRows[i];
std::cerr << s.str() << std::endl;
throw s.str();
}
// The dual of
// min/max c^T x
// Ax <= b
// x >= 0
// is
// max/min y^T b
// y^T A <= c
// y <= 0
// We scale y by -1 to get >= 0
IloEnv env = primalVars.getEnv();
IloObjective obj(env, 0.0,
primalObj.getSense() == IloObjective::Minimize ?
IloObjective::Maximize : IloObjective::Minimize);
IloRangeArray rows(env);
IloNumVarArray y(env);
std::map<IloNumVar,IloInt,ExtractableLess<IloNumVar> > v2i;
for (IloInt j = 0; j < primalVars.getSize(); ++j) {
IloNumVar x = primalVars[j];
v2i.insert(std::map<IloNumVar,IloInt,ExtractableLess<IloNumVar> >::value_type(x, j));
rows.add(IloRange(env, -IloInfinity, 0, x.getName()));
}
for (IloExpr::LinearIterator it = primalObj.getLinearIterator(); it.ok(); ++it)
rows[v2i[it.getVar()]].setUB(it.getCoef());
for (IloInt i = 0; i < primalRows.getSize(); ++i) {
IloRange r = primalRows[i];
IloNumColumn col(env);
col += obj(-r.getUB());
for (IloExpr::LinearIterator it = r.getLinearIterator(); it.ok(); ++it)
col += rows[v2i[it.getVar()]](-it.getCoef());
y.add(IloNumVar(col, 0, IloInfinity, IloNumVar::Float, r.getName()));
}
*dualObj = obj;
*dualVars = y;
*dualRows = rows;
}
/** Extract sub block number <code>n</code> from <code>problem</code>.
* The constructor creates a representation of block number <code>n</code>
* as described in <code>problem</code>.
* The constructor will also connect the newly created block to a remote
* object solver instance.
* @param problem The problem from which the block is to be extracted.
* @param n Index of the block to be extracted.
* @param argc Argument for IloCplex constructor.
* @param argv Argument for IloCplex constructor.
* @param machines List of machines to which to connect. If the code is
* compiled for the TCP/IP transport then the block will
* be connected to <code>machines[n]</code>.
*/
BendersOpt::Block::Block(Problem const *problem, IloInt n, int argc, char const *const *argv,
std::vector<char const *> const &machines)
: env(), number(n), vars(0), rows(0), cplex(0), cb(0)
{
IloNumVarArray problemVars = problem->getVariables();
IloRangeArray problemRanges = problem->getRows();
// Create a map that maps variables in the original model to their
// respective index in problemVars.
std::map<IloNumVar,IloInt,ExtractableLess<IloNumVar> > origIdxMap;
for (IloInt j = 0; j < problemVars.getSize(); ++j)
origIdxMap.insert(std::map<IloNumVar,IloInt,ExtractableLess<IloNumVar> >::value_type(problemVars[j], j));
// Copy non-fixed variables from original problem into primal problem.
IloExpr primalObj(env);
IloNumVarArray primalVars(env);
IloRangeArray primalRows(env);
IdxMap idxMap; // Index of original variable in block's primal model
RowSet rowSet;
for (IloInt j = 0; j < problemVars.getSize(); ++j) {
IloNumVar x = problemVars[j];
if ( problem->getBlock(x) == number ) {
// Create column in block LP with exactly the same data.
if ( x.getType() != IloNumVar::Float ) {
std::stringstream s;
s << "Cannot create non-continuous block variable " << x;
std::cerr << s.str() << std::endl;
throw s.str();
}
IloNumVar v(env, x.getLB(), x.getUB(), x.getType(), x.getName());
// Normalize objective function to 'minimize'
double coef = problem->getObjCoef(x);
if ( problem->getObjSense() != IloObjective::Minimize )
coef *= -1.0;
primalObj += coef * v;
// Record the index that the copied variable has in the
// block model.
idxMap.insert(IdxMap::value_type(x, primalVars.getSize()));
primalVars.add(v);
// Mark the rows that are intersected by this column
// so that we can collect them later.
RowSet const &intersected = problem->getIntersectedRows(x);
for (RowSet::const_iterator it = intersected.begin();
it != intersected.end(); ++it)
rowSet.insert(*it);
}
else
idxMap.insert(IdxMap::value_type(x, -1));
}
// Now copy all rows that intersect block variables.
for (IloInt i = 0; i < problemRanges.getSize(); ++i) {
IloRange r = problemRanges[i];
if ( rowSet.find(r) == rowSet.end() )
continue;
// Create a copy of the row, normalizing it to '<='
double factor = 1.0;
if ( r.getLB() > -IloInfinity )
factor = -1.0;
IloRange primalR(env,
factor < 0 ? -r.getUB() : r.getLB(),
factor < 0 ? -r.getLB() : r.getUB(), r.getName());
IloExpr lhs(env);
for (IloExpr::LinearIterator it = r.getLinearIterator(); it.ok(); ++it)
{
IloNumVar v = it.getVar();
double const val = factor * it.getCoef();
if ( problem->getBlock(v) != number ) {
// This column is not explicitly in this block. This means
// that it is a column that will be fixed by the master.
// We collect all such columns so that we can adjust the
// dual objective function according to concrete fixings.
// Store information about variables in this block that
// will be fixed by master solves.
fixed.push_back(FixData(primalRows.getSize(), origIdxMap[v], -val));
}
else {
// The column is an ordinary in this block. Just copy it.
lhs += primalVars[idxMap[v]] * val;
}
}
primalR.setExpr(lhs);
primalRows.add(primalR);
lhs.end();
}
// Create the dual of the primal model we just created.
// Note that makeDual _always_ returns a 'maximize' objective.
IloObjective objective(env, primalObj, IloObjective::Minimize);
makeDual(objective, primalVars, primalRows,
&obj, &vars, &rows);
objective.end();
primalRows.endElements();
primalRows.end();
primalVars.endElements();
primalVars.end();
primalObj.end();
// Create a model.
IloModel model(env);
model.add(obj);
model.add(vars);
model.add(rows);
for (IloExpr::LinearIterator it = obj.getLinearIterator(); it.ok(); ++it)
objMap.insert(ObjMap::value_type(it.getVar(), it.getCoef()));
// Finally create the IloCplex instance that will solve
// the problems associated with this block.
char const **transargv = new char const *[argc + 3];
for (int i = 0; i < argc; ++i)
transargv[i] = argv[i];
#if defined(USE_MPI)
char extra[128];
sprintf (extra, "-remoterank=%d", static_cast<int>(number + 1));
transargv[argc++] = extra;
(void)machines;
#elif defined(USE_PROCESS)
char extra[128];
sprintf (extra, "-logfile=block%04d.log", static_cast<int>(number));
transargv[argc++] = extra;
(void)machines;
#elif defined(USE_TCPIP)
transargv[argc++] = machines[number];
#endif
cplex = IloCplex(model, TRANSPORT, argc, transargv);
delete[] transargv;
// Suppress output from this block's solver.
cplex.setOut(env.getNullStream());
cplex.setWarning(env.getNullStream());
}
