QPSolver in progress. Finish building dual graph, but not tested.

Use mixed constrained noise with sigma < 0 to denote inequalities.
Working set implements the active set method, turning inactive inequalities
to active one as equality constraints by setting their corresponding sigmas to 0
and vice versa. Dual graph now has to deal with mixed sigmas.

gtsam/base/Vector.cpp

@@ -327,12 +327,21 @@ double weightedPseudoinverse(const Vector& a, const Vector& weights,
   vector<bool> isZero;
   for (size_t i = 0; i < m; ++i) isZero.push_back(fabs(a[i]) < 1e-9);
 
-  // If there is a valid (a!=0) constraint (sigma==0) return the first one
-  for (size_t i = 0; i < m; ++i)
+  for (size_t i = 0; i < m; ++i) {
+    // If there is a valid (a!=0) constraint (sigma==0) return the first one
     if (weights[i] == inf && !isZero[i]) {
+      // Basically, instead of doing a normal QR step with the weighted
+      // pseudoinverse, we enforce the constraint by turning
+      // ax + AS = b into x + (A/a)S = b/a, for the first row where a!=0
       pseudo = delta(m, i, 1 / a[i]);
       return inf;
     }
+    // If there is a valid (a!=0) inequality constraint (sigma<0), ignore it by returning 0
+    else if (weights[i] < 0 && !isZero[i]) {
+      pseudo = zero(m);
+      return 0;
+    }
+  }
 
   // Form psuedo-inverse inv(a'inv(Sigma)a)a'inv(Sigma)
   // For diagonal Sigma, inv(Sigma) = diag(precisions)

gtsam/linear/JacobianFactor.cpp

@@ -642,6 +642,13 @@ void JacobianFactor::setModel(bool anyConstrained, const Vector& sigmas) {
     model_ = noiseModel::Diagonal::Sigmas(sigmas);
 }
 
+/* ************************************************************************* */
+void JacobianFactor::setModel(const noiseModel::Diagonal::shared_ptr& model) {
+  if ((size_t) model->dim() != this->rows())
+    throw InvalidNoiseModel(this->rows(), model->dim());
+  model_ = model;
+}
+
 /* ************************************************************************* */
 std::pair<boost::shared_ptr<GaussianConditional>,
     boost::shared_ptr<JacobianFactor> > EliminateQR(

gtsam/linear/JacobianFactor.h

@@ -298,6 +298,7 @@ namespace gtsam {
 
     /** set noiseModel correctly */
     void setModel(bool anyConstrained, const Vector& sigmas);
+    void setModel(const noiseModel::Diagonal::shared_ptr& model);
     
     /**
      * Densely partially eliminate with QR factorization, this is usually provided as an argument to

gtsam/linear/NoiseModel.cpp

@@ -428,20 +428,35 @@ SharedDiagonal Constrained::QR(Matrix& Ab) const {
 
   Vector pseudo(m); // allocate storage for pseudo-inverse
   Vector invsigmas = reciprocal(sigmas_);
+  // Obtain the signs of each elements.
+  // We use negative signs to denote inequality constraints
+  // TODO: might be slow!
+  Vector signs = ediv(sigmas_, sigmas_.cwiseAbs());
+  gtsam::print(invsigmas, "invsigmas: ");
   Vector weights = emul(invsigmas,invsigmas); // calculate weights once
+  // We use negative signs to denote inequality constraints
+  weights = emul(weights, signs);
+  gtsam::print(weights, "weights: ");
 
   // We loop over all columns, because the columns that can be eliminated
   // are not necessarily contiguous. For each one, estimate the corresponding
   // scalar variable x as d-rS, with S the separator (remaining columns).
   // Then update A and b by substituting x with d-rS, zero-ing out x's column.
+  gtsam::print(Ab, "Ab = ");
+  cout << " n = " << n << endl;
   for (size_t j=0; j<n; ++j) {
+    cout << "--------------------" << endl;
+    cout << "j: " << j << endl;
     // extract the first column of A
     Vector a = Ab.col(j);
+    gtsam::print(a, "a = ");
 
     // Calculate weighted pseudo-inverse and corresponding precision
     gttic(constrained_QR_weightedPseudoinverse);
     double precision = weightedPseudoinverse(a, weights, pseudo);
     gttoc(constrained_QR_weightedPseudoinverse);
+    cout << "precision: " << precision << endl;
+    gtsam::print(pseudo, "pseudo: ");
 
     // If precision is zero, no information on this column
     // This is actually not limited to constraints, could happen in Gaussian::QR
@@ -452,12 +467,16 @@ SharedDiagonal Constrained::QR(Matrix& Ab) const {
     // create solution [r d], rhs is automatically r(n)
     Vector rd(n+1); // uninitialized !
     rd(j)=1.0; // put 1 on diagonal
-    for (size_t j2=j+1; j2<n+1; ++j2) // and fill in remainder with dot-products
+    for (size_t j2=j+1; j2<n+1; ++j2) { // and fill in remainder with dot-products
+      Vector Abj2 = Ab.col(j2);
+      gtsam::print(Abj2, "Ab.col(j2): ");
       rd(j2) = pseudo.dot(Ab.col(j2));
+    }
     gttoc(constrained_QR_create_rd);
 
     // construct solution (r, d, sigma)
     Rd.push_back(boost::make_tuple(j, rd, precision));
+    gtsam::print(rd, "rd = ");
 
     // exit after rank exhausted
     if (Rd.size()>=maxRank) break;
@@ -466,6 +485,8 @@ SharedDiagonal Constrained::QR(Matrix& Ab) const {
     gttic(constrained_QR_update_Ab);
     Ab.middleCols(j+1,n-j) -= a * rd.segment(j+1, n-j).transpose();
     gttoc(constrained_QR_update_Ab);
+    gtsam::print(Ab, "Updated Ab = ");
+
   }
 
   // Create storage for precisions

gtsam/linear/tests/testQPSolver.cpp

@@ -0,0 +1,380 @@
+/* ----------------------------------------------------------------------------
+
+ * GTSAM Copyright 2010, Georgia Tech Research Corporation,
+ * Atlanta, Georgia 30332-0415
+ * All Rights Reserved
+ * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
+
+ * See LICENSE for the license information
+
+ * -------------------------------------------------------------------------- */
+
+/**
+ * @file testQPSolver.cpp
+ * @brief Test simple QP solver for a linear inequality constraint
+ * @date Apr 10, 2014
+ * @author Duy-Nguyen Ta
+ */
+
+#include <gtsam/inference/Symbol.h>
+#include <gtsam/linear/NoiseModel.h>
+#include <gtsam/linear/GaussianFactorGraph.h>
+#include <gtsam/linear/VectorValues.h>
+#include <gtsam/base/Testable.h>
+#include <CppUnitLite/TestHarness.h>
+
+using namespace std;
+using namespace gtsam;
+using namespace gtsam::symbol_shorthand;
+
+#ifdef __CDT_PARSER__
+#undef BOOST_FOREACH
+#define BOOST_FOREACH(a, b) for(a; ; )
+#endif
+
+class QPSolver {
+  const GaussianFactorGraph& graph_;   //!< the original graph, can't be modified!
+  GaussianFactorGraph workingGraph_;  //!< working set
+  VectorValues currentSolution_;
+  FastVector<size_t> constraintIndices_; //!< Indices of constrained factors in the original graph
+  GaussianFactorGraph::shared_ptr freeHessians_;
+  VariableIndex freeHessianVarIndex_;
+  VariableIndex fullFactorIndices_; //!< indices of factors involving each variable.
+                                    // gtsam calls it "VariableIndex", but I think FactorIndex
+                                    // makes more sense, because it really stores factor indices.
+
+public:
+  /// Constructor
+  QPSolver(const GaussianFactorGraph& graph, const VectorValues& initials) :
+      graph_(graph), workingGraph_(graph.clone()), currentSolution_(initials),
+      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
+    KeySet constrainedVars;
+    BOOST_FOREACH(size_t index, constraintIndices_) {
+      KeyVector keys = graph[index]->keys();
+      constrainedVars.insert(keys.begin(), keys.end());
+    }
+
+    // Collect unconstrained hessians of constrained vars to build dual graph
+    freeHessians_ = unconstrainedHessiansOfConstrainedVars(graph, constrainedVars);
+    freeHessianVarIndex_ = VariableIndex(*freeHessians_);
+  }
+
+  /// Return indices of all constrained factors
+  FastVector<size_t> constraintIndices() const { return constraintIndices_; }
+
+  /// Convert a Gaussian factor to a jacobian. return empty shared ptr if failed
+  JacobianFactor::shared_ptr toJacobian(const GaussianFactor::shared_ptr& factor) const {
+    JacobianFactor::shared_ptr jacobian(
+        boost::dynamic_pointer_cast<JacobianFactor>(factor));
+    return jacobian;
+  }
+
+  /// Convert a Gaussian factor to a Hessian. Return empty shared ptr if failed
+  HessianFactor::shared_ptr toHessian(const GaussianFactor::shared_ptr factor) const {
+    HessianFactor::shared_ptr hessian(boost::dynamic_pointer_cast<HessianFactor>(factor));
+    return hessian;
+  }
+
+  /// Return the Hessian factor graph of constrained variables
+  GaussianFactorGraph::shared_ptr freeHessiansOfConstrainedVars() const {
+    return freeHessians_;
+  }
+
+  /* ************************************************************************* */
+  GaussianFactorGraph::shared_ptr unconstrainedHessiansOfConstrainedVars(
+      const GaussianFactorGraph& graph, const KeySet& constrainedVars) const {
+    VariableIndex variableIndex(graph);
+    GaussianFactorGraph::shared_ptr hfg = boost::make_shared<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
+          hfg->push_back(HessianFactor(*graph[factorIndex]));
+        }
+      }
+      else { // If it's not a Jacobian, it should be a hessian factor. Just add!
+        hfg->push_back(graph[factorIndex]);
+      }
+
+    }
+    return hfg;
+  }
+
+  /* ************************************************************************* */
+  /**
+   * Build the dual graph to solve for the Lagrange multipliers.
+   *
+   * The Lagrangian function is:
+   *        L(X,lambdas) = f(X) - \sum_k lambda_k * c_k(X),
+   * where the unconstrained part is
+   *        f(X) = 0.5*X'*G*X - X'*g + 0.5*f0
+   * and the linear equality constraints are
+   *        c1(X), c2(X), ..., cm(X)
+   *
+   * Take the derivative of L wrt X at the solution and set it to 0, we have
+   *    \grad f(X) = \sum_k lambda_k * \grad c_k(X)   (*)
+   *
+   * For each set of rows of (*) corresponding to a variable xi involving in some constraints
+   * we have:
+   *    \grad f(xi) = \frac{\partial f}{\partial xi}' = \sum_j G_ij*xj - gi
+   *    \grad c_k(xi) = \frac{\partial c_k}{\partial xi}'
+   *
+   * Note: If xi does not involve in any constraint, we have the trivial condition
+   * \grad f(Xi) = 0, which should be satisfied as a usual condition for unconstrained variables.
+   *
+   * So each variable xi involving in some constraints becomes a linear factor A*lambdas - b = 0
+   * on the constraints' lambda multipliers, as follows:
+   *    - The jacobian term A_k for each lambda_k is \grad c_k(xi)
+   *    - The constant term b is \grad f(xi), which can be computed from all unconstrained
+   *    Hessian factors connecting to xi: \grad f(xi) = \sum_j G_ij*xj - gi
+   */
+  GaussianFactorGraph::shared_ptr buildDualGraph(const VectorValues& x0) const {
+    // The dual graph to return
+    GaussianFactorGraph::shared_ptr dualGraph = boost::make_shared<GaussianFactorGraph>();
+
+    // 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
+    BOOST_FOREACH(const VariableIndex::value_type& xiKey_factors, freeHessianVarIndex_) {
+      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(0);
+      size_t xiDim = xiFactor0->getDim(xiFactor0->find(xiKey));
+
+      // Compute 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);
+      }
+
+      // Obtain the jacobians for lambda variables from their corresponding constraints
+      // gradc_k(xi) = \frac{\partial c_k}{\partial xi}'
+      std::vector<std::pair<Key, Matrix> > lambdaTerms; // collection of lambda_k, and gradc_k
+      BOOST_FOREACH(size_t factorIndex, fullFactorIndices_[xiKey]) {
+        JacobianFactor::shared_ptr factor = toJacobian(workingGraph_.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();
+        // 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 (fabs(sigmas[sigmaIx]) > 1e-9) {    // if it's either ineq (sigma<0) or unconstrained (sigma>0)
+            A_k.col(sigmaIx) = zero(A_k.rows());
+          }
+        }
+        // Use factorIndex as the lambda's key.
+        lambdaTerms.push_back(make_pair(factorIndex, A_k));
+      }
+      // Enforce constrained noise model so lambda is solved with QR and exactly satisfies all the equation
+      dualGraph->push_back(JacobianFactor(lambdaTerms, gradf_xi, noiseModel::Constrained::All(gradf_xi.size())));
+    }
+    return dualGraph;
+  }
+
+  /// Find max lambda element
+  std::pair<int, int> findMostViolatedIneqConstraint(const VectorValues& lambdas) {
+    int worstFactorIx = -1, worstSigmaIx = -1;
+    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<lambda.size(); ++j)
+        // If it is an 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);
+  }
+
+  /** Iterate 1 step */
+//  bool iterate() {
+//    // Obtain the solution from the current working graph
+//    VectorValues newSolution = workingGraph_.optimize();
+//
+//    // If we CAN'T move further
+//    if (newSolution == currentSolution) {
+//      // Compute lambda from the dual graph
+//      GaussianFactorGraph dualGraph = buildDualGraph(graph, newSolution);
+//      VectorValues lambdas = dualGraph.optimize();
+//
+//      int factorIx, sigmaIx;
+//      boost::tie(factorIx, sigmaIx) = findMostViolatedIneqConstraint(lambdas);
+//      // If all constraints are satisfied: We have found the solution!!
+//      if (factorIx < 0) {
+//        return true;
+//      }
+//      else {
+//        // If some constraints are violated!
+//        Vector sigmas = toJacobian(workingGraph.at(factorIx))->get_model()->sigmas();
+//        sigmas[sigmaIx] = 0.0;
+//        toJacobian()->setModel(true, sigmas);
+//        // No need to update the currentSolution, since we haven't moved anywhere
+//      }
+//    }
+//    else {
+//      // If we CAN make some progress
+//      // Adapt stepsize if some inactive inequality constraints complain about this move
+//      // also add to the working set the one that complains the most
+//      VectorValues alpha = updateWorkingSetInplace();
+//      currentSolution_ = (1 - alpha) * currentSolution_ + alpha * newSolution;
+//    }
+//
+//    return false;
+//  }
+//
+//  VectorValues optimize() const {
+//    bool converged = false;
+//    while (!converged) {
+//      converged = iterate();
+//    }
+//  }
+
+};
+
+/* ************************************************************************* */
+// Create test graph according to Forst10book_pg171Ex5
+std::pair<GaussianFactorGraph, VectorValues> createTestCase() {
+  GaussianFactorGraph graph;
+
+  // Objective functions x1^2 - x1*x2 + x2^2 - 3*x1
+  // Note the Hessian encodes:
+  //        0.5*x1'*G11*x1 + x1'*G12*x2 + 0.5*x2'*G22*x2 - x1'*g1 - x2'*g2 + 0.5*f
+  // Hence, we have G11=2, G12 = -1, g1 = +3, G22 = 2, g2 = 0, f = 0
+  graph.push_back(
+      HessianFactor(X(1), X(2), 2.0*ones(1, 1), -ones(1, 1), 3.0*ones(1),
+          2.0*ones(1, 1), zero(1), 1.0));
+
+  // Inequality constraints
+  // x1 + x2 <= 2 --> x1 + x2 -2 <= 0, hence we negate the b vector
+  Matrix A1 = (Matrix(4, 1)<<1, -1, 0, 1);
+  Matrix A2 = (Matrix(4, 1)<<1, 0, -1, 0);
+  Vector b = -(Vector(4)<<2, 0, 0, 1.5);
+  // Special constrained noise model denoting <= inequalities with negative sigmas
+  noiseModel::Constrained::shared_ptr noise =
+      noiseModel::Constrained::MixedSigmas((Vector(4)<<-1, -1, -1, -1));
+  graph.push_back(JacobianFactor(X(1), A1, X(2), A2, b, noise));
+
+  // Initials values
+  VectorValues initials;
+  initials.insert(X(1), ones(1));
+  initials.insert(X(2), ones(1));
+
+  return make_pair(graph, initials);
+}
+
+TEST(QPSolver, constraintsAux) {
+  GaussianFactorGraph graph;
+  VectorValues initials;
+  boost::tie(graph, initials)= createTestCase();
+  QPSolver solver(graph, initials);
+  FastVector<size_t> constraintIx = solver.constraintIndices();
+  LONGS_EQUAL(1, constraintIx.size());
+  LONGS_EQUAL(1, constraintIx[0]);
+
+  VectorValues lambdas;
+  lambdas.insert(constraintIx[0], (Vector(4)<< -0.5, 0.0, 0.3, 0.1));
+  int factorIx, lambdaIx;
+  boost::tie(factorIx, lambdaIx) = solver.findMostViolatedIneqConstraint(lambdas);
+  LONGS_EQUAL(1, factorIx);
+  LONGS_EQUAL(2, lambdaIx);
+
+  VectorValues lambdas2;
+  lambdas2.insert(constraintIx[0], (Vector(4)<< -0.5, 0.0, -0.3, -0.1));
+  int factorIx2, lambdaIx2;
+  boost::tie(factorIx2, lambdaIx2) = solver.findMostViolatedIneqConstraint(lambdas2);
+  LONGS_EQUAL(-1, factorIx2);
+  LONGS_EQUAL(-1, lambdaIx2);
+
+  GaussianFactorGraph::shared_ptr freeHessian = solver.freeHessiansOfConstrainedVars();
+  GaussianFactorGraph expectedFreeHessian;
+  expectedFreeHessian.push_back(
+      HessianFactor(X(1), X(2), 2.0 * ones(1, 1), -ones(1, 1), 3.0 * ones(1),
+          2.0 * ones(1, 1), zero(1), 1.0));
+  EXPECT(expectedFreeHessian.equals(*freeHessian));
+
+  GaussianFactorGraph::shared_ptr dualGraph = solver.buildDualGraph(initials);
+  dualGraph->print("Dual graph: ");
+  VectorValues dual = dualGraph->optimize();
+  dual.print("Dual: ");
+}
+
+/* ************************************************************************* */
+int main() {
+  TestResult tr;
+  return TestRegistry::runAllTests(tr);
+}
+/* ************************************************************************* */
+