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gtsam/gtsam_unstable/linear/QPSolver.cpp

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/*
* QPSolver.cpp
* @brief:
* @date: Apr 15, 2014
* @author: thduynguyen
*/
#include <gtsam/inference/Symbol.h>
#include <gtsam_unstable/linear/QPSolver.h>
#include <gtsam_unstable/linear/LPSolver.h>
#include <boost/foreach.hpp>
#include <boost/range/adaptor/map.hpp>
using namespace std;
namespace gtsam {
/* ************************************************************************* */
QPSolver::QPSolver(const GaussianFactorGraph& graph) :
graph_(graph), fullFactorIndices_(graph) {
// Split the original graph into unconstrained and constrained part
// and collect indices of constrained factors
for (size_t i = 0; i < graph.nrFactors(); ++i) {
// obtain the factor and its noise model
JacobianFactor::shared_ptr jacobian = toJacobian(graph.at(i));
if (jacobian && jacobian->get_model()
&& jacobian->get_model()->isConstrained()) {
constraintIndices_.push_back(i);
}
}
// Collect constrained variable keys
set<size_t> constrainedVars;
BOOST_FOREACH(size_t index, constraintIndices_) {
KeyVector keys = graph.at(index)->keys();
constrainedVars.insert(keys.begin(), keys.end());
}
// Collect unconstrained hessians of constrained vars to build dual graph
freeHessians_ = unconstrainedHessiansOfConstrainedVars(graph,
constrainedVars);
freeHessianFactorIndex_ = VariableIndex(*freeHessians_);
}
/* ************************************************************************* */
GaussianFactorGraph::shared_ptr QPSolver::unconstrainedHessiansOfConstrainedVars(
const GaussianFactorGraph& graph, const set<Key>& constrainedVars) const {
VariableIndex variableIndex(graph);
GaussianFactorGraph::shared_ptr hfg(new GaussianFactorGraph());
// Collect all factors involving constrained vars
FastSet<size_t> factors;
BOOST_FOREACH(Key key, constrainedVars) {
VariableIndex::Factors factorsOfThisVar = variableIndex[key];
BOOST_FOREACH(size_t factorIndex, factorsOfThisVar) {
factors.insert(factorIndex);
}
}
// Convert each factor into Hessian
BOOST_FOREACH(size_t factorIndex, factors) {
if (!graph[factorIndex])
continue;
// See if this is a Jacobian factor
JacobianFactor::shared_ptr jf = toJacobian(graph[factorIndex]);
if (jf) {
// Dealing with mixed constrained factor
if (jf->get_model() && jf->isConstrained()) {
// Turn a mixed-constrained factor into a factor with 0 information on the constrained part
Vector sigmas = jf->get_model()->sigmas();
Vector newPrecisions(sigmas.size());
bool mixed = false;
for (size_t s = 0; s < sigmas.size(); ++s) {
if (sigmas[s] <= 1e-9)
newPrecisions[s] = 0.0; // 0 info for constraints (both ineq and eq)
else {
newPrecisions[s] = 1.0 / sigmas[s];
mixed = true;
}
}
if (mixed) { // only add free hessians if it's mixed
JacobianFactor::shared_ptr newJacobian = toJacobian(jf->clone());
newJacobian->setModel(
noiseModel::Diagonal::Precisions(newPrecisions));
hfg->push_back(HessianFactor(*newJacobian));
}
} else { // unconstrained Jacobian
// Convert the original linear factor to Hessian factor
// TODO: This may fail and throw the following exception
// Assertion failed: (((!PanelMode) && stride==0 && offset==0) ||
// (PanelMode && stride>=depth && offset<=stride)), function operator(),
// file Eigen/Eigen/src/Core/products/GeneralBlockPanelKernel.h, line 1133.
// because of a weird error which might be related to clang
// See this: https://groups.google.com/forum/#!topic/ceres-solver/DYhqOLPquHU
// My current way to fix this is to compile both gtsam and my library in Release mode
hfg->add(HessianFactor(*jf));
}
} else { // If it's not a Jacobian, it should be a hessian factor. Just add!
hfg->push_back(graph[factorIndex]);
}
}
return hfg;
}
/* ************************************************************************* */
GaussianFactorGraph QPSolver::buildDualGraph(const GaussianFactorGraph& graph,
const VectorValues& x0, bool useLeastSquare) const {
static const bool debug = false;
// The dual graph to return
GaussianFactorGraph dualGraph;
// For each variable xi involving in some constraint, compute the unconstrained gradient
// wrt xi from the prebuilt freeHessian graph
// \grad f(xi) = \frac{\partial f}{\partial xi}' = \sum_j G_ij*xj - gi
if (debug)
freeHessianFactorIndex_.print("freeHessianFactorIndex_: ");
BOOST_FOREACH(const VariableIndex::value_type& xiKey_factors, freeHessianFactorIndex_) {
Key xiKey = xiKey_factors.first;
VariableIndex::Factors xiFactors = xiKey_factors.second;
// Find xi's dim from the first factor on xi
if (xiFactors.size() == 0)
continue;
GaussianFactor::shared_ptr xiFactor0 = freeHessians_->at(
*xiFactors.begin());
size_t xiDim = xiFactor0->getDim(xiFactor0->find(xiKey));
if (debug)
xiFactor0->print("xiFactor0: ");
if (debug)
cout << "xiKey: " << string(Symbol(xiKey)) << ", xiDim: " << xiDim
<< endl;
//++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++//
// Compute the b-vector for the dual factor Ax-b
// b = gradf(xi) = \frac{\partial f}{\partial xi}' = \sum_j G_ij*xj - gi
Vector gradf_xi = zero(xiDim);
BOOST_FOREACH(size_t factorIx, xiFactors) {
HessianFactor::shared_ptr factor = toHessian(freeHessians_->at(factorIx));
Factor::const_iterator xi = factor->find(xiKey);
// Sum over Gij*xj for all xj connecting to xi
for (Factor::const_iterator xj = factor->begin(); xj != factor->end();
++xj) {
// Obtain Gij from the Hessian factor
// Hessian factor only stores an upper triangular matrix, so be careful when i>j
Matrix Gij;
if (xi > xj) {
Matrix Gji = factor->info(xj, xi);
Gij = Gji.transpose();
} else {
Gij = factor->info(xi, xj);
}
// Accumulate Gij*xj to gradf
Vector x0_j = x0.at(*xj);
gradf_xi += Gij * x0_j;
}
// Subtract the linear term gi
gradf_xi += -factor->linearTerm(xi);
}
//++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++//
// Compute the Jacobian A for the dual factor Ax-b
// Obtain the jacobians for lambda variables from their corresponding constraints
// A = gradc_k(xi) = \frac{\partial c_k}{\partial xi}'
vector<pair<Key, Matrix> > lambdaTerms; // collection of lambda_k, and gradc_k
typedef pair<size_t, size_t> FactorIx_SigmaIx;
vector<FactorIx_SigmaIx> unconstrainedIndex; // pairs of factorIx,sigmaIx of unconstrained rows
BOOST_FOREACH(size_t factorIndex, fullFactorIndices_[xiKey]) {
JacobianFactor::shared_ptr factor = toJacobian(graph.at(factorIndex));
if (!factor || !factor->isConstrained())
continue;
// Gradient is the transpose of the Jacobian: A_k = gradc_k(xi) = \frac{\partial c_k}{\partial xi}'
// Each column for each lambda_k corresponds to [the transpose of] each constrained row factor
Matrix A_k = factor->getA(factor->find(xiKey)).transpose();
if (debug)
gtsam::print(A_k, "A_k = ");
// Deal with mixed sigmas: no information if sigma != 0
Vector sigmas = factor->get_model()->sigmas();
for (size_t sigmaIx = 0; sigmaIx < sigmas.size(); ++sigmaIx) {
// if it's either ineq (sigma<0) or unconstrained (sigma>0)
// we have no information about it
if (fabs(sigmas[sigmaIx]) > 1e-9) {
A_k.col(sigmaIx) = zero(A_k.rows());
// remember to add a zero prior on this lambda, otherwise the graph is under-determined
unconstrainedIndex.push_back(make_pair(factorIndex, sigmaIx));
}
}
// Use factorIndex as the lambda's key.
lambdaTerms.push_back(make_pair(factorIndex, A_k));
}
//++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++//
// Create and add factors to the dual graph
// If least square approximation is desired, use unit noise model.
if (debug)
cout << "Create dual factor" << endl;
if (useLeastSquare) {
if (debug)
cout << "use least square!" << endl;
dualGraph.push_back(
JacobianFactor(lambdaTerms, gradf_xi,
noiseModel::Unit::Create(gradf_xi.size())));
} else {
// Enforce constrained noise model so lambdas are solved with QR
// and should exactly satisfy all the equations
if (debug)
cout << gradf_xi << endl;
dualGraph.push_back(
JacobianFactor(lambdaTerms, gradf_xi,
noiseModel::Constrained::All(gradf_xi.size())));
}
// Add 0 priors on all lambdas of the unconstrained rows to make sure the graph is solvable
if (debug)
cout << "Create priors" << endl;
BOOST_FOREACH(FactorIx_SigmaIx factorIx_sigmaIx, unconstrainedIndex) {
size_t factorIx = factorIx_sigmaIx.first;
JacobianFactor::shared_ptr factor = toJacobian(graph.at(factorIx));
size_t dim = factor->get_model()->dim();
Matrix J = zeros(dim, dim);
size_t sigmaIx = factorIx_sigmaIx.second;
J(sigmaIx, sigmaIx) = 1.0;
// Use factorIndex as the lambda's key.
if (debug)
cout << "prior for factor " << factorIx << endl;
dualGraph.push_back(JacobianFactor(factorIx, J, zero(dim)));
}
}
return dualGraph;
}
/* ************************************************************************* */
pair<int, int> QPSolver::findWorstViolatedActiveIneq(
const VectorValues& lambdas) const {
int worstFactorIx = -1, worstSigmaIx = -1;
// preset the maxLambda to 0.0: if lambda is <= 0.0, the constraint is either
// inactive or a good ineq constraint, so we don't care!
double maxLambda = 0.0;
BOOST_FOREACH(size_t factorIx, constraintIndices_) {
Vector lambda = lambdas.at(factorIx);
Vector orgSigmas = toJacobian(graph_.at(factorIx))->get_model()->sigmas();
for (size_t j = 0; j < orgSigmas.size(); ++j)
// If it is a BAD active inequality, and lambda is larger than the current max
if (orgSigmas[j] < 0 && lambda[j] > maxLambda) {
worstFactorIx = factorIx;
worstSigmaIx = j;
maxLambda = lambda[j];
}
}
return make_pair(worstFactorIx, worstSigmaIx);
}
/* ************************************************************************* */bool QPSolver::updateWorkingSetInplace(
GaussianFactorGraph& workingGraph, int factorIx, int sigmaIx,
double newSigma) const {
if (factorIx < 0 || sigmaIx < 0)
return false;
Vector sigmas = toJacobian(workingGraph.at(factorIx))->get_model()->sigmas();
sigmas[sigmaIx] = newSigma; // removing it from the working set
toJacobian(workingGraph.at(factorIx))->setModel(true, sigmas);
return true;
}
/* ************************************************************************* */
/* We have to make sure the new solution with alpha satisfies all INACTIVE ineq constraints
* If some inactive ineq constraints complain about the full step (alpha = 1),
* we have to adjust alpha to stay within the ineq constraints' feasible regions.
*
* For each inactive ineq j:
* - We already have: aj'*xk - bj <= 0, since xk satisfies all ineq constraints
* - We want: aj'*(xk + alpha*p) - bj <= 0
* - If aj'*p <= 0, we have: aj'*(xk + alpha*p) <= aj'*xk <= bj, for all alpha>0
* it's good!
* - We only care when aj'*p > 0. In this case, we need to choose alpha so that
* aj'*xk + alpha*aj'*p - bj <= 0 --> alpha <= (bj - aj'*xk) / (aj'*p)
* We want to step as far as possible, so we should choose alpha = (bj - aj'*xk) / (aj'*p)
*
* We want the minimum of all those alphas among all inactive ineq.
*/
boost::tuple<double, int, int> QPSolver::computeStepSize(
const GaussianFactorGraph& workingGraph, const VectorValues& xk,
const VectorValues& p) const {
static bool debug = false;
double minAlpha = 1.0;
int closestFactorIx = -1, closestSigmaIx = -1;
BOOST_FOREACH(size_t factorIx, constraintIndices_) {
JacobianFactor::shared_ptr jacobian = toJacobian(workingGraph.at(factorIx));
Vector sigmas = jacobian->get_model()->sigmas();
Vector b = jacobian->getb();
for (size_t s = 0; s < sigmas.size(); ++s) {
// If it is an inactive inequality, compute alpha and update min
if (sigmas[s] < 0) {
// Compute aj'*p
double ajTp = 0.0;
for (Factor::const_iterator xj = jacobian->begin();
xj != jacobian->end(); ++xj) {
Vector pj = p.at(*xj);
Vector aj = jacobian->getA(xj).row(s);
ajTp += aj.dot(pj);
}
if (debug)
cout << "s, ajTp, b[s]: " << s << " " << ajTp << " " << b[s] << endl;
// Check if aj'*p >0. Don't care if it's not.
if (ajTp <= 0)
continue;
// Compute aj'*xk
double ajTx = 0.0;
for (Factor::const_iterator xj = jacobian->begin();
xj != jacobian->end(); ++xj) {
Vector xkj = xk.at(*xj);
Vector aj = jacobian->getA(xj).row(s);
ajTx += aj.dot(xkj);
}
if (debug)
cout << "b[s], ajTx: " << b[s] << " " << ajTx << " " << ajTp << endl;
// alpha = (bj - aj'*xk) / (aj'*p)
double alpha = (b[s] - ajTx) / ajTp;
if (debug)
cout << "alpha: " << alpha << endl;
// We want the minimum of all those max alphas
if (alpha < minAlpha) {
closestFactorIx = factorIx;
closestSigmaIx = s;
minAlpha = alpha;
}
}
}
}
return boost::make_tuple(minAlpha, closestFactorIx, closestSigmaIx);
}
/* ************************************************************************* */bool QPSolver::iterateInPlace(
GaussianFactorGraph& workingGraph, VectorValues& currentSolution,
VectorValues& lambdas) const {
static bool debug = false;
if (debug)
workingGraph.print("workingGraph: ");
// Obtain the solution from the current working graph
VectorValues newSolution = workingGraph.optimize();
if (debug)
newSolution.print("New solution:");
// If we CAN'T move further
if (newSolution.equals(currentSolution, 1e-5)) {
// Compute lambda from the dual graph
if (debug)
cout << "Building dual graph..." << endl;
GaussianFactorGraph dualGraph = buildDualGraph(workingGraph, newSolution);
if (debug)
dualGraph.print("Dual graph: ");
lambdas = dualGraph.optimize();
if (debug)
lambdas.print("lambdas :");
int factorIx, sigmaIx;
boost::tie(factorIx, sigmaIx) = findWorstViolatedActiveIneq(lambdas);
if (debug)
cout << "violated active ineq - factorIx, sigmaIx: " << factorIx << " "
<< sigmaIx << endl;
// Try to disactivate the weakest violated ineq constraints
// if not successful, i.e. all ineq constraints are satisfied: We have the solution!!
if (!updateWorkingSetInplace(workingGraph, factorIx, sigmaIx, -1.0))
return true;
} else {
// If we CAN make some progress
// Adapt stepsize if some inactive inequality constraints complain about this move
if (debug)
cout << "Computing stepsize..." << endl;
double alpha;
int factorIx, sigmaIx;
VectorValues p = newSolution - currentSolution;
boost::tie(alpha, factorIx, sigmaIx) = computeStepSize(workingGraph,
currentSolution, p);
if (debug)
cout << "alpha, factorIx, sigmaIx: " << alpha << " " << factorIx << " "
<< sigmaIx << endl;
// also add to the working set the one that complains the most
updateWorkingSetInplace(workingGraph, factorIx, sigmaIx, 0.0);
// step!
currentSolution = currentSolution + alpha * p;
// if (alpha <1e-5) {
// if (debug) cout << "Building dual graph..." << endl;
// GaussianFactorGraph dualGraph = buildDualGraph(workingGraph, newSolution);
// if (debug) dualGraph.print("Dual graph: ");
// lambdas = dualGraph.optimize();
// if (debug) lambdas.print("lambdas :");
// return true; // TODO: HACK HACK!!!
// }
}
return false;
}
/* ************************************************************************* */
pair<VectorValues, VectorValues> QPSolver::optimize(
const VectorValues& initials) const {
GaussianFactorGraph workingGraph = graph_.clone();
VectorValues currentSolution = initials;
VectorValues lambdas;
bool converged = false;
while (!converged) {
converged = iterateInPlace(workingGraph, currentSolution, lambdas);
}
return make_pair(currentSolution, lambdas);
}
/* ************************************************************************* */
pair<VectorValues, Key> QPSolver::initialValuesLP() const {
size_t firstSlackKey = 0;
BOOST_FOREACH(Key key, fullFactorIndices_ | boost::adaptors::map_keys) {
if (firstSlackKey < key)
firstSlackKey = key;
}
firstSlackKey += 1;
VectorValues initials;
// Create zero values for constrained vars
BOOST_FOREACH(size_t iFactor, constraintIndices_) {
JacobianFactor::shared_ptr jacobian = toJacobian(graph_.at(iFactor));
KeyVector keys = jacobian->keys();
BOOST_FOREACH(Key key, keys) {
if (!initials.exists(key)) {
size_t dim = jacobian->getDim(jacobian->find(key));
initials.insert(key, zero(dim));
}
}
}
// Insert initial values for slack variables
size_t slackKey = firstSlackKey;
BOOST_FOREACH(size_t iFactor, constraintIndices_) {
JacobianFactor::shared_ptr jacobian = toJacobian(graph_.at(iFactor));
Vector errorAtZero = jacobian->getb();
Vector slackInit = zero(errorAtZero.size());
Vector sigmas = jacobian->get_model()->sigmas();
for (size_t i = 0; i < sigmas.size(); ++i) {
if (sigmas[i] < 0) {
slackInit[i] = std::max(errorAtZero[i], 0.0);
} else if (sigmas[i] == 0.0) {
errorAtZero[i] = fabs(errorAtZero[i]);
} // if it has >0 sigma, i.e. normal Gaussian noise, initialize it at 0
}
initials.insert(slackKey, slackInit);
slackKey++;
}
return make_pair(initials, firstSlackKey);
}
/* ************************************************************************* */
VectorValues QPSolver::objectiveCoeffsLP(Key firstSlackKey) const {
VectorValues slackObjective;
for (size_t i = 0; i < constraintIndices_.size(); ++i) {
Key key = firstSlackKey + i;
size_t iFactor = constraintIndices_[i];
JacobianFactor::shared_ptr jacobian = toJacobian(graph_.at(iFactor));
size_t dim = jacobian->rows();
Vector objective = ones(dim);
/* We should not ignore unconstrained slack var dimensions (those rows with sigmas >0)
* because their values might be underdetermined in the LP. Since they will have only
* 1 constraint zi>=0, enforcing them in the min obj function won't harm the other constrained
* slack vars, but also makes them well defined: 0 at the minimum.
*/
slackObjective.insert(key, ones(dim));
}
return slackObjective;
}
/* ************************************************************************* */
pair<GaussianFactorGraph::shared_ptr, VectorValues> QPSolver::constraintsLP(
Key firstSlackKey) const {
// Create constraints and 0 lower bounds (zi>=0)
GaussianFactorGraph::shared_ptr constraints(new GaussianFactorGraph());
VectorValues slackLowerBounds;
for (size_t key = firstSlackKey;
key < firstSlackKey + constraintIndices_.size(); ++key) {
size_t iFactor = constraintIndices_[key - firstSlackKey];
JacobianFactor::shared_ptr jacobian = toJacobian(graph_.at(iFactor));
// Collect old terms to form a new factor
// TODO: it might be faster if we can get the whole block matrix at once
// but I don't know how to extend the current VerticalBlockMatrix
vector<pair<Key, Matrix> > terms;
for (Factor::iterator it = jacobian->begin(); it != jacobian->end(); ++it) {
terms.push_back(make_pair(*it, jacobian->getA(it)));
}
// Add the slack term to the constraint
// Unlike Nocedal06book, pg.473, we want ax-z <= b, since we always assume
// LE constraints ax <= b for sigma < 0.
size_t dim = jacobian->rows();
terms.push_back(make_pair(key, -eye(dim)));
constraints->push_back(
JacobianFactor(terms, jacobian->getb(), jacobian->get_model()));
// Add lower bound for this slack key
slackLowerBounds.insert(key, zero(dim));
}
return make_pair(constraints, slackLowerBounds);
}
/* ************************************************************************* */
pair<bool, VectorValues> QPSolver::findFeasibleInitialValues() const {
static const bool debug = false;
// Initial values with slack variables for the LP subproblem, Nocedal06book, pg.473
VectorValues initials;
size_t firstSlackKey;
boost::tie(initials, firstSlackKey) = initialValuesLP();
// Coefficients for the LP subproblem objective function, min \sum_i z_i
VectorValues objectiveLP = objectiveCoeffsLP(firstSlackKey);
// Create constraints and lower bounds of slack variables
GaussianFactorGraph::shared_ptr constraints;
VectorValues slackLowerBounds;
boost::tie(constraints, slackLowerBounds) = constraintsLP(firstSlackKey);
// Solve the LP subproblem
LPSolver lpSolver(objectiveLP, constraints, slackLowerBounds);
VectorValues solution = lpSolver.solve();
if (debug)
initials.print("Initials LP: ");
if (debug)
objectiveLP.print("Objective LP: ");
if (debug)
constraints->print("Constraints LP: ");
if (debug)
solution.print("LP solution: ");
// Remove slack variables from solution
double slackSum = 0.0;
for (Key key = firstSlackKey; key < firstSlackKey + constraintIndices_.size();
++key) {
slackSum += solution.at(key).cwiseAbs().sum();
solution.erase(key);
}
// Insert zero vectors for free variables that are not in the constraints
BOOST_FOREACH(Key key, fullFactorIndices_ | boost::adaptors::map_keys) {
if (!solution.exists(key)) {
GaussianFactor::shared_ptr factor = graph_.at(
*fullFactorIndices_[key].begin());
size_t dim = factor->getDim(factor->find(key));
solution.insert(key, zero(dim));
}
}
return make_pair(slackSum < 1e-5, solution);
}
/* ************************************************************************* */
pair<VectorValues, VectorValues> QPSolver::optimize() const {
bool isFeasible;
VectorValues initials;
boost::tie(isFeasible, initials) = findFeasibleInitialValues();
if (!isFeasible) {
throw runtime_error("LP subproblem is infeasible!");
}
return optimize(initials);
}
} /* namespace gtsam */
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