gtsam/gtsam_unstable/linear/LPSolver.h

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/**
* @file LPSolver.h
* @brief Class used to solve Linear Programming Problems as defined in LP.h
* @author Ivan Dario Jimenez
* @date 1/24/16
*/
#pragma once
namespace gtsam {
typedef std::map<Key, size_t> KeyDimMap;
typedef std::vector<std::pair<Key, Matrix> > TermsContainer;
class LPSolver {
const LP& lp_; //!< the linear programming problem
GaussianFactorGraph baseGraph_; //!< unchanged factors needed in every iteration
VariableIndex costVariableIndex_, equalityVariableIndex_,
inequalityVariableIndex_; //!< index to corresponding factors to build dual graphs
FastSet<Key> constrainedKeys_; //!< all constrained keys, will become factors in dual graphs
KeyDimMap keysDim_; //!< key-dim map of all variables in the constraints, used to create zero priors
public:
LPSolver(const LP& lp) :
lp_(lp) {
// Push back factors that are the same in every iteration to the base graph.
// Those include the equality constraints and zero priors for keys that are not
// in the cost
baseGraph_.push_back(lp_.equalities);
// Collect key-dim map of all variables in the constraints to create their zero priors later
keysDim_ = collectKeysDim(lp_.equalities);
KeyDimMap keysDim2 = collectKeysDim(lp_.inequalities);
keysDim_.insert(keysDim2.begin(), keysDim2.end());
// Create and push zero priors of constrained variables that do not exist in the cost function
baseGraph_.push_back(*createZeroPriors(lp_.cost.keys(), keysDim_));
// Variable index
equalityVariableIndex_ = VariableIndex(lp_.equalities);
inequalityVariableIndex_ = VariableIndex(lp_.inequalities);
constrainedKeys_ = lp_.equalities.keys();
constrainedKeys_.merge(lp_.inequalities.keys());
}
const LP& lp() const {
return lp_;
}
const KeyDimMap& keysDim() const {
return keysDim_;
}
//******************************************************************************
template<class LinearGraph>
KeyDimMap collectKeysDim(const LinearGraph& linearGraph) const {
KeyDimMap keysDim;
BOOST_FOREACH(const typename LinearGraph::sharedFactor& factor, linearGraph) {
if (!factor) continue;
BOOST_FOREACH(Key key, factor->keys())
keysDim[key] = factor->getDim(factor->find(key));
}
return keysDim;
}
//******************************************************************************
/**
* Create a zero prior for any keys in the graph that don't exist in the cost
*/
GaussianFactorGraph::shared_ptr createZeroPriors(const KeyVector& costKeys,
const KeyDimMap& keysDim) const {
GaussianFactorGraph::shared_ptr graph(new GaussianFactorGraph());
BOOST_FOREACH(Key key, keysDim | boost::adaptors::map_keys) {
if (find(costKeys.begin(), costKeys.end(), key) == costKeys.end()) {
size_t dim = keysDim.at(key);
graph->push_back(JacobianFactor(key, eye(dim), zero(dim)));
}
}
return graph;
}
//******************************************************************************
LPState iterate(const LPState& state) const {
static bool debug = false;
// Solve with the current working set
// LP: project the objective neggradient to the constraint's null space
// to find the direction to move
VectorValues newValues = solveWithCurrentWorkingSet(state.values,
state.workingSet);
// if (debug) state.workingSet.print("Working set:");
if (debug)
(newValues - state.values).print("New direction:");
// If we CAN'T move further
// LP: projection on the constraints' nullspace is zero: we are at a vertex
if (newValues.equals(state.values, 1e-7)) {
// Find and remove the bad ineq constraint by computing its lambda
// Compute lambda from the dual graph
// LP: project the objective's gradient onto each constraint gradient to obtain the dual scaling factors
// is it true??
if (debug)
cout << "Building dual graph..." << endl;
GaussianFactorGraph::shared_ptr dualGraph = buildDualGraph(
state.workingSet, newValues);
if (debug)
dualGraph->print("Dual graph: ");
VectorValues duals = dualGraph->optimize();
if (debug)
duals.print("Duals :");
// LP: see which ineq constraint has wrong pulling direction, i.e., dual < 0
int leavingFactor = identifyLeavingConstraint(state.workingSet, duals);
if (debug)
cout << "leavingFactor: " << leavingFactor << endl;
// If all inequality constraints are satisfied: We have the solution!!
if (leavingFactor < 0) {
// TODO If we still have infeasible equality constraints: the problem is over-constrained. No solution!
// ...
return LPState(newValues, duals, state.workingSet, true,
state.iterations + 1);
} else {
// Inactivate the leaving constraint
// LP: remove the bad ineq constraint out of the working set
InequalityFactorGraph newWorkingSet = state.workingSet;
newWorkingSet.at(leavingFactor)->inactivate();
return LPState(newValues, duals, newWorkingSet, false,
state.iterations + 1);
}
} else {
// If we CAN make some progress, i.e. p_k != 0
// Adapt stepsize if some inactive constraints complain about this move
// LP: projection on nullspace is NOT zero:
// find and put a blocking inactive constraint to the working set,
// otherwise the problem is unbounded!!!
double alpha;
int factorIx;
VectorValues p = newValues - state.values;
boost::tie(alpha, factorIx) = // using 16.41
computeStepSize(state.workingSet, state.values, p);
if (debug)
cout << "alpha, factorIx: " << alpha << " " << factorIx << " " << endl;
// also add to the working set the one that complains the most
InequalityFactorGraph newWorkingSet = state.workingSet;
if (factorIx >= 0)
newWorkingSet.at(factorIx)->activate();
// step!
newValues = state.values + alpha * p;
if (debug)
newValues.print("New solution:");
return LPState(newValues, state.duals, newWorkingSet, false,
state.iterations + 1);
}
}
//******************************************************************************
/**
* Create the factor ||x-xk - (-g)||^2 where xk is the current feasible solution
* on the constraint surface and g is the gradient of the linear cost,
* i.e. -g is the direction we wish to follow to decrease the cost.
*
* Essentially, we try to match the direction d = x-xk with -g as much as possible
* subject to the condition that x needs to be on the constraint surface, i.e., d is
* along the surface's subspace.
*
* The least-square solution of this quadratic subject to a set of linear constraints
* is the projection of the gradient onto the constraints' subspace
*/
GaussianFactorGraph::shared_ptr createLeastSquareFactors(
const LinearCost& cost, const VectorValues& xk) const {
GaussianFactorGraph::shared_ptr graph(new GaussianFactorGraph());
KeyVector keys = cost.keys();
for (LinearCost::const_iterator it = cost.begin(); it != cost.end(); ++it) {
size_t dim = cost.getDim(it);
Vector b = xk.at(*it) - cost.getA(it).transpose(); // b = xk-g
graph->push_back(JacobianFactor(*it, eye(dim), b));
}
return graph;
}
//******************************************************************************
VectorValues solveWithCurrentWorkingSet(const VectorValues& xk,
const InequalityFactorGraph& workingSet) const {
GaussianFactorGraph workingGraph = baseGraph_; // || X - Xk + g ||^2
workingGraph.push_back(*createLeastSquareFactors(lp_.cost, xk));
BOOST_FOREACH(const LinearInequality::shared_ptr& factor, workingSet) {
if (factor->active()) workingGraph.push_back(factor);
}
return workingGraph.optimize();
}
//******************************************************************************
/// Collect the Jacobian terms for a dual factor
template<typename FACTOR>
TermsContainer collectDualJacobians(Key key, const FactorGraph<FACTOR>& graph,
const VariableIndex& variableIndex) const {
TermsContainer Aterms;
if (variableIndex.find(key) != variableIndex.end()) {
BOOST_FOREACH(size_t factorIx, variableIndex[key]) {
typename FACTOR::shared_ptr factor = graph.at(factorIx);
if (!factor->active()) continue;
Matrix Ai = factor->getA(factor->find(key)).transpose();
Aterms.push_back(std::make_pair(factor->dualKey(), Ai));
}
}
return Aterms;
}
//******************************************************************************
JacobianFactor::shared_ptr createDualFactor(Key key,
const InequalityFactorGraph& workingSet, const VectorValues& delta) const {
// Transpose the A matrix of constrained factors to have the jacobian of the dual key
TermsContainer Aterms = collectDualJacobians<LinearEquality>(key,
lp_.equalities, equalityVariableIndex_);
TermsContainer AtermsInequalities = collectDualJacobians<LinearInequality>(
key, workingSet, inequalityVariableIndex_);
Aterms.insert(Aterms.end(), AtermsInequalities.begin(),
AtermsInequalities.end());
// Collect the gradients of unconstrained cost factors to the b vector
if (Aterms.size() > 0) {
Vector b = zero(delta.at(key).size());
Factor::const_iterator it = lp_.cost.find(key);
if (it != lp_.cost.end())
b = lp_.cost.getA(it).transpose();
return boost::make_shared < JacobianFactor > (Aterms, b); // compute the least-square approximation of dual variables
} else {
return boost::make_shared<JacobianFactor>();
}
}
//******************************************************************************
GaussianFactorGraph::shared_ptr buildDualGraph(
const InequalityFactorGraph& workingSet, const VectorValues& delta) const {
GaussianFactorGraph::shared_ptr dualGraph(new GaussianFactorGraph());
BOOST_FOREACH(Key key, constrainedKeys_) {
// Each constrained key becomes a factor in the dual graph
JacobianFactor::shared_ptr dualFactor = createDualFactor(key, workingSet,
delta);
if (!dualFactor->empty()) dualGraph->push_back(dualFactor);
}
return dualGraph;
}
//******************************************************************************
int identifyLeavingConstraint(const InequalityFactorGraph& workingSet,
const VectorValues& duals) const {
int worstFactorIx = -1;
// preset the maxLambda to 0.0: if lambda is <= 0.0, the constraint is either
// inactive or a good inequality constraint, so we don't care!
double max_s = 0.0;
for (size_t factorIx = 0; factorIx < workingSet.size(); ++factorIx) {
const LinearInequality::shared_ptr& factor = workingSet.at(factorIx);
if (factor->active()) {
double s = duals.at(factor->dualKey())[0];
if (s > max_s) {
worstFactorIx = factorIx;
max_s = s;
}
}
}
return worstFactorIx;
}
//******************************************************************************
std::pair<double, int> computeStepSize(const InequalityFactorGraph& workingSet,
const VectorValues& xk, const VectorValues& p) const {
static bool debug = false;
double minAlpha = std::numeric_limits<double>::infinity();
int closestFactorIx = -1;
for (size_t factorIx = 0; factorIx < workingSet.size(); ++factorIx) {
const LinearInequality::shared_ptr& factor = workingSet.at(factorIx);
double b = factor->getb()[0];
// only check inactive factors
if (!factor->active()) {
// Compute a'*p
double aTp = factor->dotProductRow(p);
// Check if a'*p >0. Don't care if it's not.
if (aTp <= 0)
continue;
// Compute a'*xk
double aTx = factor->dotProductRow(xk);
// alpha = (b - a'*xk) / (a'*p)
double alpha = (b - aTx) / aTp;
if (debug)
cout << "alpha: " << alpha << endl;
// We want the minimum of all those max alphas
if (alpha < minAlpha) {
closestFactorIx = factorIx;
minAlpha = alpha;
}
}
}
return std::make_pair(minAlpha, closestFactorIx);
}
//******************************************************************************
InequalityFactorGraph identifyActiveConstraints(
const InequalityFactorGraph& inequalities,
const VectorValues& initialValues, const VectorValues& duals) const {
InequalityFactorGraph workingSet;
BOOST_FOREACH(const LinearInequality::shared_ptr& factor, inequalities) {
LinearInequality::shared_ptr workingFactor(new LinearInequality(*factor));
double error = workingFactor->error(initialValues);
// TODO: find a feasible initial point for LPSolver.
// For now, we just throw an exception
if (error > 0) throw InfeasibleInitialValues();
if (fabs(error) < 1e-7) {
workingFactor->activate();
}
else {
workingFactor->inactivate();
}
workingSet.push_back(workingFactor);
}
return workingSet;
}
//******************************************************************************
/** Optimize with the provided feasible initial values
* TODO: throw exception if the initial values is not feasible wrt inequality constraints
*/
pair<VectorValues, VectorValues> optimize(const VectorValues& initialValues,
const VectorValues& duals = VectorValues()) const {
// Initialize workingSet from the feasible initialValues
InequalityFactorGraph workingSet = identifyActiveConstraints(lp_.inequalities,
initialValues, duals);
LPState state(initialValues, duals, workingSet, false, 0);
/// main loop of the solver
while (!state.converged) {
state = iterate(state);
}
return make_pair(state.values, state.duals);
}
//******************************************************************************
/**
* Optimize without initial values
* TODO: Find a feasible initial solution wrt inequality constraints
*/
// pair<VectorValues, VectorValues> optimize() const {
//
// // Initialize workingSet from the feasible initialValues
// InequalityFactorGraph workingSet = identifyActiveConstraints(
// lp_.inequalities, initialValues, duals);
// LPState state(initialValues, duals, workingSet, false, 0);
//
// /// main loop of the solver
// while (!state.converged) {
// state = iterate(state);
// }
//
// return make_pair(state.values, state.duals);
// }
};
}