gtsam/gtsam_unstable/linear/LPSolver.h

/**
 * @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);
//  }
};
}