unrevised_lemke_solver.cc

#include "drake/solvers/unrevised_lemke_solver.h"

#include <algorithm>
#include <cmath>
#include <functional>
#include <iostream>
#include <limits>
#include <memory>
#include <sstream>
#include <stdexcept>
#include <utility>
#include <vector>

#include <Eigen/LU>

#include "drake/common/autodiff.h"
#include "drake/common/default_scalars.h"
#include "drake/common/drake_assert.h"
#include "drake/common/never_destroyed.h"
#include "drake/common/text_logging.h"
#include "drake/common/unused.h"

using drake::log;

namespace drake {
namespace solvers {

namespace {

// A linear system solver that accommodates the inability of the LU
// factorization to be AutoDiff'd (true in Eigen 3, at least). For double types,
// the faster LU factorization and solve is used. For other types, QR
// factorization and solve is used.
template <class T>
class LinearSolver {
 public:
  explicit LinearSolver(const MatrixX<T>& m);

  VectorX<T> Solve(const VectorX<T>& v) const;

 private:
  Eigen::ColPivHouseholderQR<MatrixX<T>> qr_;
  Eigen::PartialPivLU<MatrixX<double>> lu_;
};

template <class T>
LinearSolver<T>::LinearSolver(const MatrixX<T>& M) {
  if (M.rows() > 0)
    qr_ = Eigen::ColPivHouseholderQR<MatrixX<T>>(M);
}

template <>
LinearSolver<double>::LinearSolver(
    const MatrixX<double>& M) {
  if (M.rows() > 0)
    lu_ = Eigen::PartialPivLU<MatrixX<double>>(M);
}

template <class T>
VectorX<T> LinearSolver<T>::Solve(
    const VectorX<T>& v) const {
  if (v.rows() == 0) {
    DRAKE_DEMAND(qr_.rows() == 0);
    return VectorX<T>(0);
  }
  return qr_.solve(v);
}

template <>
VectorX<double> LinearSolver<double>::Solve(
    const VectorX<double>& v) const {
  if (v.rows() == 0) {
    DRAKE_DEMAND(lu_.rows() == 0);
    return VectorX<double>(0);
  }
  return lu_.solve(v);
}
}  // anonymous namespace

template <>
void UnrevisedLemkeSolver<AutoDiffXd>::DoSolve(
    const MathematicalProgram&, const Eigen::VectorXd&,
    const SolverOptions&, MathematicalProgramResult*) const {
  throw std::logic_error(
      "UnrevisedLemkeSolver cannot yet be used in a "
      "MathematicalProgram while templatized as an AutoDiff");
}

template <typename T>
void UnrevisedLemkeSolver<T>::DoSolve(
    const MathematicalProgram& prog,
    const Eigen::VectorXd& initial_guess,
    const SolverOptions& merged_options,
    MathematicalProgramResult* result) const {
  if (!prog.GetVariableScaling().empty()) {
    static const logging::Warn log_once(
      "UnrevisedLemkeSolver doesn't support the feature of variable scaling.");
  }

  unused(initial_guess);
  unused(merged_options);

  // Solve each individual LCP, writing the result back to the decision
  // variables through the binding and returning true iff all LCPs are
  // feasible.
  //
  // If any is infeasible, returns false and does not alter the decision
  // variables.

  // Create a dummy variable for the number of pivots used.
  int num_pivots = 0;
  const auto& bindings = prog.linear_complementarity_constraints();
  Eigen::VectorXd x_sol(prog.num_vars());
  for (const auto& binding : bindings) {
    Eigen::VectorXd constraint_solution(binding.GetNumElements());
    const std::shared_ptr<LinearComplementarityConstraint> constraint =
        binding.evaluator();
    bool solved = SolveLcpLemke(
        constraint->M(), constraint->q(), &constraint_solution, &num_pivots);
    if (!solved) {
      result->set_solution_result(SolutionResult::kUnknownError);
      return;
    }
    for (int i = 0; i < binding.evaluator()->num_vars(); ++i) {
      const int variable_index =
          prog.FindDecisionVariableIndex(binding.variables()(i));
      x_sol(variable_index) = constraint_solution(i);
    }
  }
  result->set_optimal_cost(0.0);
  result->set_x_val(x_sol);
  result->set_solution_result(SolutionResult::kSolutionFound);
}

// Utility function for copying an r-dimensional column vector v (designated by
// the indices in row_indices and col_index) from a matrix M to a target
// vector, `out`. Let M be the matrix `in` augmented with a single column of
// ones (i.e., the "covering vector"); put another way, M = | in 1 |, where "in"
// refers the n × m-dimensional matrix `in` and 1 is a column of ones (so M is
// n × (m+1)-dimensional). Let R be a r × n "row selection" matrix, constructed
// using `row_indices`. R is constructed using r = `row_indices.size()` as
// follows:
//
// Rᵢⱼ = { 1  if j = row_indices[i], ∀i ∈ 0..r-1, ∀j ∈ 0..n-1
//       { 0  otherwise
//
// Consider the following example with `in` set to the 3 × 3 identity matrix,
// `row_indices = { 2, 0}` and `col_index = 1`. This would make:
// R =  | 0 0 1 |      and R⋅M = | 0 0 1 1 |
//      | 1 0 0 |                | 1 0 0 1 |
//
// The second column (`col_index = 1`) of R⋅M is v = | 0 |
//                                                   | 0 |
//
// `row_indices` need not be in sorted order, and `out` need not be properly
// sized on entry (it will be resized as necessary). However, this method aborts
// if any element of `row_indices` is out of range, `col_index` is out of range,
// *or* if there are any duplicated elements in `row_indices`.
template <class T>
void UnrevisedLemkeSolver<T>::SelectSubColumnWithCovering(
    const MatrixX<T>& in,
    const std::vector<int>& row_indices,
    int col_index, VectorX<T>* out) {
  DRAKE_ASSERT(ValidateIndices(row_indices, in.rows()));

  const int num_rows = row_indices.size();
  out->resize(num_rows);

  // Look for the covering vector first.
  if (col_index == in.cols()) {
    out->setOnes();
    return;
  }

  DRAKE_DEMAND(0 <= col_index && col_index < in.cols());
  const auto in_column = in.col(col_index);
  for (int i = 0; i < num_rows; i++) {
    DRAKE_ASSERT(row_indices[i] < in_column.size());
    (*out)[i] = in_column[row_indices[i]];
  }
}

// Utility function for copying an r-dimensional column vector v (designated by
// the indices in `row_indices`) from n-dimensional vector `in` to a target
// vector, `out`. Let R be a r × n "row selection" matrix, constructed using
// r = `row_indices.size()` as follows:
//
// Rᵢⱼ = { 1  if j = row_indices[i], ∀i ∈ 0..r-1, ∀j ∈ 0..n-1
//       { 0  otherwise
//
// Consider the following example with `in` set to the vector [ 1 2 3 ]ᵀ,
// and `row_indices = { 2, 0}`. This would make:
// R =  | 0 0 1 |      and v = R⋅`in` = | 3 |
//      | 1 0 0 |                       | 1 |
//
// `row_indices` need not be in sorted order, and `out` need not be properly
// sized on entry (it will be resized as necessary). However, this method aborts
// if any element of `row_indices` is out of range  *or* if there are any
// duplicated elements in `row_indices`.
template <class T>
void UnrevisedLemkeSolver<T>::SelectSubVector(const VectorX<T>& in,
    const std::vector<int>& row_indices, VectorX<T>* out) {
  DRAKE_ASSERT(ValidateIndices(row_indices, in.size()));

  const int num_rows = row_indices.size();
  out->resize(num_rows);
  for (int i = 0; i < num_rows; i++) {
    DRAKE_ASSERT(row_indices[i] < in.rows());
    (*out)(i) = in(row_indices[i]);
  }
}

// Utility function for copying an r-dimensional column vector v (designated by
// the indices in `row_indices`) into n-dimensional vector `out`. Let R be the
// r × n "row selection" matrix defined in the documentation of
// SelectSubVector(). Then, the result of the operation is:
// `out` = `out` + Rᵀ⋅(`v` - R⋅`out`)
//
// Consider the following example with `v` set to the vector [ 3 1 ]ᵀ,
// `row_indices = { 2, 0}`, and `out` set to the vector [ 4 5 6 ]. This would
// make:
// R =  | 0 0 1 |      and `out` + Rᵀ⋅(`v` - R⋅`out`) = | 1 |
//      | 1 0 0 |                                       | 5 |
//                                                      | 3 |
//
// `row_indices` need not be in sorted order. This method aborts if any element
// of `row_indices` is out of the range of `out`, if `row_indices` is not the
// same size as `v`, or if there are any duplicated elements in `row_indices`.
template <class T>
void UnrevisedLemkeSolver<T>::SetSubVector(const VectorX<T>& v,
    const std::vector<int>& row_indices, VectorX<T>* out) {
  DRAKE_DEMAND(row_indices.size() == static_cast<size_t>(v.size()));
  DRAKE_ASSERT(ValidateIndices(row_indices, out->size()));
  for (size_t i = 0; i < row_indices.size(); ++i)
    (*out)[row_indices[i]] = v[i];
}

// Function for checking whether a set of indices that specify a view into
// a vector is valid. Returns `true` if row_indices are unique and each element
// lies in [0, vector_size-1] and `false` otherwise.
template <class T>
bool UnrevisedLemkeSolver<T>::ValidateIndices(
    const std::vector<int>& row_indices, int vector_size) {
  // Don't check anything for empty vectors.
  if (row_indices.empty())
    return true;

  // Sort the vector first.
  std::vector<int> sorted_row_indices = row_indices;
  std::sort(sorted_row_indices.begin(), sorted_row_indices.end());

  // Validate the maximum and minimum elements.
  if (sorted_row_indices.back() >= vector_size)
    return false;
  if (sorted_row_indices.front() < 0)
    return false;

  // Make sure that the vector is unique.
  return std::unique(sorted_row_indices.begin(), sorted_row_indices.end()) ==
         sorted_row_indices.end();
}

// Function for checking whether a set of indices that specify a view into
// a matrix is valid. Returns `true` if each element of row_indices is unique
// lies in [0, num_rows-1] and if each element of col_indices is unique and
// lies in [0, num_cols-1]. Returns `false` otherwise.
template <class T>
bool UnrevisedLemkeSolver<T>::ValidateIndices(
    const std::vector<int>& row_indices,
    const std::vector<int>& col_indices, int num_rows, int num_cols) {
  return ValidateIndices(row_indices, num_rows) &&
         ValidateIndices(col_indices, num_cols);
}

// Utility function for copying an r × c dimensional submatrix S (designated by
// the indices in row_indices and col_indices) from a matrix M to a target
// matrix, `out`. Let M be the matrix `in` augmented with a single column of
// ones (i.e., the "covering vector"); put another way, M = | in 1 |, where "in"
// refers the n × m-dimensional matrix `in` and 1 is a column of ones (so M is
// n × (m+1)-dimensional). Let R be a r × n "row selection" matrix, constructed
// using `row_indices` and C be a (m+1) × c-dimensional "column selection"
// matrix constructed using `col_indices`. R and C are constructed using
// r = `row_indices.size()` and c = `col_indices.size()` as follows:
//
// Rᵢⱼ = { 1  if j = row_indices[i], ∀i ∈ 0..r-1, ∀j ∈ 0..n-1
//       { 0  otherwise
// Cᵢⱼ = { 1  if col_indices[j] = i, ∀i ∈ 0..m, ∀j ∈ 0..c-1
//       { 0  otherwise
//
// Consider the following example with `in` set to the 3 × 3 identity matrix,
// `row_indices = { 2, 1, 0}` and `col_indices = { 1, 2, 3 }`. This would make:
// R =  | 0 0 1 |      C = | 0 0 0 |   and  R⋅M⋅C = | 0 1 1 |
//      | 0 1 0 |          | 1 0 0 |                | 1 0 1 |
//      | 1 0 0 |          | 0 1 0 |                | 0 0 1 |
//                         | 0 0 1 |
// `row_indices` and `col_indices` need not be in sorted order, and `out` need
// not be properly sized on entry (it will be resized as necessary). However,
// this method aborts if any element of `row_indices` or `col_indices` is out
// of range *or* if there are any duplicated elements.
template <class T>
void UnrevisedLemkeSolver<T>::SelectSubMatrixWithCovering(
    const MatrixX<T>& in,
    const std::vector<int>& row_indices,
    const std::vector<int>& col_indices, MatrixX<T>* out) {
  const int num_rows = row_indices.size();
  const int num_cols = col_indices.size();
  DRAKE_ASSERT(
      ValidateIndices(row_indices, col_indices, in.rows(), in.cols() + 1));
  out->resize(num_rows, num_cols);

  for (int i = 0; i < num_rows; i++) {
    const auto row_in = in.row(row_indices[i]);

    // `row_out` is a "view" into `out`: any modifications to row_out are
    // reflected in `out`.
    auto row_out = out->row(i);
    for (int j = 0; j < num_cols; j++) {
      if (col_indices[j] < in.cols()) {
        DRAKE_ASSERT(col_indices[j] >= 0);
        row_out(j) = row_in(col_indices[j]);
      } else {
        DRAKE_ASSERT(col_indices[j] == in.cols());
        row_out(j) = 1.0;
      }
    }
  }
}

// Determines the various index sets defined in Section 1.1 of [Dai 2018].
template <class T>
void UnrevisedLemkeSolver<T>::DetermineIndexSets() const {
  // Helper for determining index sets.
  auto DetermineIndexSetsHelper = [this](
      const std::vector<LCPVariable>& variables, bool is_z,
      std::vector<int>* variable_set,
      std::vector<int>* variable_set_prime) {
    variable_and_array_indices_.clear();
    for (int i = 0; i < static_cast<int>(variables.size()); ++i) {
      if (variables[i].is_z() == is_z)
        variable_and_array_indices_.emplace_back(variables[i].index(), i);
    }
    std::sort(variable_and_array_indices_.begin(),
              variable_and_array_indices_.end());

    // Construct the set and the primed set.
    for (const auto& variable_and_array_index_pair :
        variable_and_array_indices_) {
      variable_set->push_back(variable_and_array_index_pair.first);
      variable_set_prime->push_back(variable_and_array_index_pair.second);
    }
  };

  // Clear all sets.
  index_sets_.alpha.clear();
  index_sets_.alpha_bar.clear();
  index_sets_.alpha_prime.clear();
  index_sets_.alpha_bar_prime.clear();
  index_sets_.beta.clear();
  index_sets_.beta_bar.clear();
  index_sets_.beta_prime.clear();
  index_sets_.beta_bar_prime.clear();

  DetermineIndexSetsHelper(indep_variables_, false,
      &index_sets_.alpha, &index_sets_.alpha_prime);
  DetermineIndexSetsHelper(dep_variables_, false,
      &index_sets_.alpha_bar, &index_sets_.alpha_bar_prime);
  DetermineIndexSetsHelper(dep_variables_, true,
                           &index_sets_.beta, &index_sets_.beta_prime);
  DetermineIndexSetsHelper(indep_variables_, true,
                           &index_sets_.beta_bar, &index_sets_.beta_bar_prime);
}

// Verifies that each element of the pivoting set is unique. This is an
// expensive operation and should only be executed in Debug mode.
template <class T>
bool UnrevisedLemkeSolver<T>::IsEachUnique(
    const std::vector<LCPVariable>& vars) {
  // Copy the set.
  std::vector<LCPVariable> vars_copy = vars;
  std::sort(vars_copy.begin(), vars_copy.end());
  return (std::unique(vars_copy.begin(), vars_copy.end()) == vars_copy.end());
}

// Performs the pivoting operation, which is described in [Dai 2018].
// `M_prime_col` can be null, if the updated column of M' (pivoted version of M)
// is not needed and the driving_index corresponds to the artificial variable.
template <typename T>
bool UnrevisedLemkeSolver<T>::LemkePivot(
    const MatrixX<T>& M,
    const VectorX<T>& q,
    int driving_index,
    T zero_tol,
    VectorX<T>* M_prime_col,
    VectorX<T>* q_prime) const {
  DRAKE_DEMAND(q_prime);

  const int kArtificial = M.rows();
  DRAKE_DEMAND(driving_index >= 0 && driving_index <= kArtificial);

  // Verify that each member in the independent and dependent sets is unique.
  DRAKE_ASSERT(IsEachUnique(indep_variables_));
  DRAKE_ASSERT(IsEachUnique(dep_variables_));

  // If the driving index does not correspond to the artificial variable,
  // M_prime_col must be non-null.
  if (!IsArtificial(indep_variables_[driving_index]))
    DRAKE_DEMAND(M_prime_col);

  // Determine the sets.
  DetermineIndexSets();

  // Note: It is feasible to do a low-rank update to the factorization below,
  // since alpha and beta should change by no more than a single index between
  // consecutive pivots. Eigen only supports low-rank updates to Cholesky
  // factorizations at the moment, however.

  // Compute matrix and vector views.
  SelectSubMatrixWithCovering(M, index_sets_.alpha, index_sets_.beta,
                              &M_alpha_beta_);
  SelectSubMatrixWithCovering(M, index_sets_.alpha_bar, index_sets_.beta,
                  &M_alpha_bar_beta_);
  SelectSubVector(q, index_sets_.alpha, &q_alpha_);
  SelectSubVector(q, index_sets_.alpha_bar, &q_alpha_bar_);

  // Equation (2) from [Dai 2018].
  // Note: this equation, and those below, must be kept up-to-date with
  // [Dai 2018].
  LinearSolver<T> fMab(M_alpha_beta_);  // Factorized M_alpha_beta_.
  q_prime_beta_prime_ = -fMab.Solve(q_alpha_);

  // Check whether the solution is sufficiently close. We need to do this
  // because partial pivoting LU does not estimate rank (and, from prior
  // experience in solving LCPs), loss of rank need not lead to errors in
  // solving the LCP. We assume that if the factorization is good enough to
  // solve this linear system, it's good enough to solve the subsequent
  // linear system (below). NOTE: current unit tests do not exercise the
  // affirmative evaluation of the conditional (meaning that the "return false"
  // never gets called).
  // @TODO(edrumwri) Institute a unit test that exercises the affirmative
  //   evaluation branch of the conditional when such a LCP has been
  //   identified.
  if ((M_alpha_beta_ * q_prime_beta_prime_ + q_alpha_).norm() > zero_tol)
    return false;

  // Equation (3) from [Dai 2018].
  q_prime_alpha_bar_prime_ = M_alpha_bar_beta_ * q_prime_beta_prime_ +
      q_alpha_bar_;

  // Set the components of q'.
  SetSubVector(q_prime_beta_prime_, index_sets_.beta_prime, q_prime);
  SetSubVector(q_prime_alpha_bar_prime_, index_sets_.alpha_bar_prime, q_prime);

  DRAKE_LOGGER_DEBUG("q': {}", q_prime->transpose());

  // If it is not necessary to compute the column of M, quit now.
  if (!M_prime_col)
    return true;

  // Examine the driving variable.
  if (!indep_variables_[driving_index].is_z()) {
    DRAKE_LOGGER_DEBUG("Driving case #1: driving variable from w");
    // Case from Section 2.2.1.
    // Determine gamma by determining the position of the driving variable
    // in INDEPENDENT W (as defined in [Dai 2018]).
    const int n = static_cast<int>(indep_variables_.size());
    int gamma = 0;
    for (int i = 0; i < n; ++i) {
      if (!indep_variables_[i].is_z()) {
        if (indep_variables_[i].index() <
            indep_variables_[driving_index].index()) {
          ++gamma;
        }
      }
    }


    // From Equation (4) in [Dai 2018].
    DRAKE_ASSERT(index_sets_.alpha[gamma] ==
        indep_variables_[driving_index].index());

    // Set the unit vector.
    e_.setZero(index_sets_.beta.size());
    e_[gamma] = 1.0;

    // Equation (5).
    M_prime_driving_beta_prime_ = fMab.Solve(e_);

    // Equation (6).
    M_prime_driving_alpha_bar_prime_ = M_alpha_bar_beta_ *
        M_prime_driving_beta_prime_;
  } else {
    DRAKE_LOGGER_DEBUG("Driving case #2: driving variable from z");

    // Case from Section 2.2.2 of [Dai 2018].
    // Determine zeta.
    const int zeta = indep_variables_[driving_index].index();

    // Compute g_alpha and g_alpha_bar.
    SelectSubColumnWithCovering(M, index_sets_.alpha, zeta, &g_alpha_);
    SelectSubColumnWithCovering(M, index_sets_.alpha_bar, zeta, &g_alpha_bar_);

    // Equation (7).
    M_prime_driving_beta_prime_ = -fMab.Solve(g_alpha_);

    // Equation (8).
    M_prime_driving_alpha_bar_prime_ = g_alpha_bar_ +
        M_alpha_bar_beta_ * M_prime_driving_beta_prime_;
  }

  SetSubVector(M_prime_driving_beta_prime_, index_sets_.beta_prime,
               M_prime_col);
  SetSubVector(M_prime_driving_alpha_bar_prime_, index_sets_.alpha_bar_prime,
               M_prime_col);

  DRAKE_LOGGER_DEBUG("M' (driving): {}", M_prime_col->transpose());
  return true;
}

// Checks to see whether a given variable is the artificial variable.
template <class T>
bool UnrevisedLemkeSolver<T>::IsArtificial(const LCPVariable& v) const {
  const int n = static_cast<int>(dep_variables_.size());
  return v.is_z() && v.index() == n;
}

// Method for finding the index of the complement of an LCP variable in
// a tuple (strictly speaking, an unsorted vector) of independent variables.
// Aborts if the index is not found in the set or the variable is the artificial
// variable (it never should be).
template <class T>
int UnrevisedLemkeSolver<T>::FindComplementIndex(
    const LCPVariable& query) const {
  // Verify that the query is not the artificial variable.
  DRAKE_DEMAND(!IsArtificial(query));

  const auto iter = indep_variables_indices_.find(query.Complement());
  DRAKE_DEMAND(iter != indep_variables_indices_.end());
  return iter->second;
}

// Computes the solution using the current index sets. `z` must simply be
// non-null; it will be resized as necessary. Returns `true` if able to
// construct the solution and `false` if unable to find the solution to a
// necessary system of linear equations. Aborts (in LemkePivot()) if
// `artificial_index` does not correspond to the index of the artificial
// variable in the vector of independent_variables.
// @pre The artificial variable was the blocking variable, indicating that the
//      solution to the LCP can be obtained after a final pivoting operation.
// @pre `artificial_index` corresponds to the index of the artificial variable
//      in the vector of independent variables.
template <class T>
bool UnrevisedLemkeSolver<T>::ConstructLemkeSolution(
    const MatrixX<T>& M,
    const VectorX<T>& q,
    int artificial_index,
    T zero_tol,
    VectorX<T>* z) const {
  DRAKE_DEMAND(z);
  const int n = q.rows();

  // Compute the solution by pivoting the artificial variable, which was just
  // identified as the blocking variable, from the set of dependent variables
  // to the set of independent variables.
  VectorX<T> q_prime(n);
  if (!LemkePivot(M, q, artificial_index, zero_tol, nullptr, &q_prime))
    return false;

  z->setZero(n);
  for (int i = 0; i < static_cast<int>(dep_variables_.size()); ++i) {
    if (dep_variables_[i].is_z())
      (*z)[dep_variables_[i].index()] = q_prime[i];
  }
  return true;
}

// Computes the blocking index using the minimum ratio test. Returns `true`
// if successful, `false` if not (due to, e.g., the driving variable being
// "unblocked" or a cycle being detected). If `false`, `blocking_index` will
// set to -1 on return.
template <typename T>
bool UnrevisedLemkeSolver<T>::FindBlockingIndex(
    const T& zero_tol, const VectorX<T>& matrix_col, const VectorX<T>& ratios,
    int* blocking_index) const {
  DRAKE_DEMAND(blocking_index);
  DRAKE_DEMAND(ratios.size() == matrix_col.size());
  DRAKE_DEMAND(zero_tol > 0);

  const int n = matrix_col.size();
  T min_ratio = std::numeric_limits<double>::infinity();
  *blocking_index = -1;
  for (int i = 0; i < n; ++i) {
    if (matrix_col[i] < -zero_tol) {
      DRAKE_LOGGER_DEBUG("Ratio for index {}: {}", i, ratios[i]);
      if (ratios[i] < min_ratio) {
        min_ratio = ratios[i];
        *blocking_index = i;
      }
    }
  }

  if (*blocking_index < 0) {
    DRAKE_LOGGER_DEBUG("driving variable is unblocked- algorithm failed");
    return false;
  }

  // Determine all variables within the zero tolerance of the minimum ratio,
  // while simultaneously looking for the presence of the artificial variable
  // among the (possible multiple) minima.
  std::vector<int> blocking_indices;
  for (int i = 0; i < n; ++i) {
    if (matrix_col[i] < -zero_tol) {
      DRAKE_LOGGER_DEBUG("Ratio for index {}: {}", i, ratios[i]);
      if (ratios[i] < min_ratio + zero_tol) {
        if (IsArtificial(dep_variables_[i])) {
          // *Always* select the artificial variable, if multiple choices are
          // possible ([Cottle 1992] p. 280).
          *blocking_index = i;
          return true;
        }
        blocking_indices.push_back(i);
      }
    }
  }

  // If there are multiple blocking variables, replace the blocking index with
  // the cycling selection.
  if (blocking_indices.size() > 1) {
    auto& index = selections_[indep_variables_];

    // Verify that we have not run out of indices to select, which means that
    // cycling would be occurring, in spite of cycling prevention.
    if (index >= static_cast<int>(blocking_indices.size())) {
      DRAKE_LOGGER_DEBUG("Cycling detected- indicating failure.");
      *blocking_index = -1;
      return false;
    }
    *blocking_index = blocking_indices[index];
    ++index;
  }

  return true;
}

template <typename T>
bool UnrevisedLemkeSolver<T>::IsSolution(
    const MatrixX<T>& M, const VectorX<T>& q, const VectorX<T>& z,
    T zero_tol) {
  using std::abs;

  const T mod_zero_tol = (zero_tol > 0) ? zero_tol : ComputeZeroTolerance(M);

  // Find the minima of z and w.
  const T min_z = z.minCoeff();
  const auto w = M * z + q;
  const T min_w = w.minCoeff();

  // Compute the dot product of z and w.
  const T dot = w.dot(z);
  const int n = q.size();
  return (min_z > -mod_zero_tol && min_w > -mod_zero_tol &&
          abs(dot) < 10 * n * mod_zero_tol);
}

// Note: maintainers should read Section 4.4 - 4.4.5 of [Cottle 1992] to
// understand Lemke's Algorithm (and this function).
template <typename T>
bool UnrevisedLemkeSolver<T>::SolveLcpLemke(const MatrixX<T>& M,
                                     const VectorX<T>& q, VectorX<T>* z,
                                     int* num_pivots,
                                     const T& zero_tol) const {
  using std::max;
  using std::abs;
  DRAKE_DEMAND(num_pivots);

  DRAKE_LOGGER_DEBUG(
      "UnrevisedLemkeSolver::SolveLcpLemke() entered, M: {}, "
      "q: {}, ", M, q.transpose());

  const int n = q.size();
  const int max_pivots = 50 * n;  // O(n) pivots expected for solvable problems.

  if (M.rows() != n || M.cols() != n)
    throw std::logic_error("M's dimensions do not match that of q.");

  // Update the pivots.
  *num_pivots = 0;

  // Look for immediate exit.
  if (n == 0) {
    DRAKE_LOGGER_DEBUG("-- LCP is zero dimensional");
    z->resize(0);
    return true;
  }

  // Denote the index of the artificial variable (i.e., the variable denoted
  // z₀ in [Cottle 1992], p. 266). Because z₀ is prone to being confused with
  // the first dimension of z in 0-indexed languages like C++, we refer to the
  // artificial variable as zₙ in this implementation (it is denoted zₙ₊₁ in
  // [Dai 2018], as that document uses the 1-indexing prevalent in algorithmic
  // descriptions).
  const int kArtificial = n;

  // Compute a sensible value for zero tolerance if none is given.
  T mod_zero_tol = zero_tol;
  if (mod_zero_tol <= 0)
    mod_zero_tol = ComputeZeroTolerance(M);

  // Checks to see whether the trivial solution z = 0 to the LCP w = Mz + q
  // solves the LCP. This must be the case if q is non-negative, as w would then
  // be non-negative, z would be non-negative (zero), and w'z = 0.
  if (q.minCoeff() > -mod_zero_tol) {
    z->setZero(q.size());
    DRAKE_LOGGER_DEBUG(" -- trivial solution found");
    DRAKE_LOGGER_DEBUG("UnrevisedLemkeSolver::SolveLcpLemke() exited");
    return true;
  }

  // Clear the cycling selections.
  selections_.clear();

  // If 'n' is identical to the size of the last problem solved, try using the
  // indices from the last problem solved.
  if (static_cast<size_t>(n) == dep_variables_.size()) {
    // Verify that the last call found a solution (indicated by the presence
    // of the artificial variable (zn) in the independent set).
    int zn_index = -1;
    for (int i = 0;
         i < static_cast<int>(indep_variables_.size()) && zn_index < 0; ++i) {
      if (IsArtificial(indep_variables_[i]))
        zn_index = i;
    }

    if (zn_index >= 0) {
      // Compute the candidate solution.
      if (ConstructLemkeSolution(M, q, zn_index, mod_zero_tol, z)) {
        if (IsSolution(M, q, *z, mod_zero_tol)) {
          // If z truly is the solution, return now, indicating only one pivot
          // (in the solution construction) was performed.
          ++(*num_pivots);
          return true;
        }
      } else {
        DRAKE_LOGGER_DEBUG(
            "Failed to solve linear system implied by last solution");
      }
    }
  }

  // Set the LCP variables. Start with all z variables independent and all w
  // variables dependent.
  indep_variables_.resize(n+1);
  dep_variables_.resize(n);
  for (int i = 0; i < n; ++i) {
    dep_variables_[i] = LCPVariable(false, i);
    indep_variables_[i] = LCPVariable(true, i);
  }
  // z needs one more variable (the artificial variable), whose index we
  // denote as n to keep it from corresponding to any actual vector index.
  indep_variables_[n] = LCPVariable(true, n);

  // Compute zn*, the smallest value of the artificial variable zn for which
  // w = q + zn >= 0. Let blocking denote a component of w that equals
  // zero when zn = zn*.
  int blocking_index = -1;
  bool blocking_index_found = FindBlockingIndex(
      mod_zero_tol, q, q, &blocking_index);
  DRAKE_DEMAND(blocking_index_found);

  // Pivot blocking, artificial. Note that we rely upon the dependent variables
  // being ordered sequentially in both arrays.
  LCPVariable blocking = dep_variables_[blocking_index];
  int driving_index = blocking.index();
  std::swap(dep_variables_[blocking_index], indep_variables_[kArtificial]);
  DRAKE_LOGGER_DEBUG("First blocking variable {}{}",
                     ((blocking.is_z()) ? "z" : "w"), blocking.index());
  DRAKE_LOGGER_DEBUG("First driving variable (artificial)");

  // Initialize the independent variable indices. We do this after the initial
  // variable swap for simplicity.
  for (int i = 0; i < static_cast<int>(indep_variables_.size()); ++i)
    indep_variables_indices_[indep_variables_[i]] = i;

  // Output the independent and dependent variable tuples.
  auto to_string = [](const std::vector<LCPVariable>& vars) -> std::string {
    std::ostringstream oss;
    for (int i = 0; i < static_cast<int>(vars.size()); ++i)
      oss << ((vars[i].is_z()) ? "z" : "w") << vars[i].index() << " ";
    return oss.str();
  };
  unused(to_string);  // ... when in release mode.
  DRAKE_LOGGER_DEBUG("Independent set variables: {}",
      to_string(indep_variables_));
  DRAKE_LOGGER_DEBUG("Dependent set variables: {}",
      to_string(dep_variables_));

  // Pivot up to the maximum number of times.
  VectorX<T> q_prime(n), M_prime_col(n);
  while (++(*num_pivots) < max_pivots) {
    DRAKE_LOGGER_DEBUG("New driving variable {}{}",
                       ((indep_variables_[driving_index].is_z()) ? "z" : "w"),
                       indep_variables_[driving_index].index());

    // Compute the permuted q and driving column of the permuted M matrix.
    if (!LemkePivot(
        M, q, driving_index, mod_zero_tol, &M_prime_col, &q_prime)) {
      DRAKE_LOGGER_DEBUG("Linear system solve failed.");
      z->setZero(n);
      return false;
    }

    // Find the blocking variable.
    if (!FindBlockingIndex(
        mod_zero_tol, M_prime_col,
        -(q_prime.array() / M_prime_col.array()).matrix(), &blocking_index)) {
      z->setZero(n);
      return false;
    }
    blocking = dep_variables_[blocking_index];
    DRAKE_LOGGER_DEBUG("Blocking variable {}{}",
                       ((blocking.is_z()) ? "z" : "w"), blocking.index());

    // See whether the artificial variable blocks the driving variable.
    if (blocking.index() == kArtificial) {
      DRAKE_DEMAND(blocking.is_z());

      // Pivot zn with the driving variable.
      std::swap(dep_variables_[blocking_index],
                indep_variables_[driving_index]);

      // Compute the permuted q, and convert it into a solution.
      if (ConstructLemkeSolution(M, q, driving_index, mod_zero_tol, z)) {
        if (IsSolution(M, q, *z))
          return true;

        DRAKE_LOGGER_DEBUG("Solution not computed to requested tolerance");
        z->setZero(n);
        return false;
      }

      // Otherwise, indicate failure.
      DRAKE_LOGGER_DEBUG(
          "Linear system solver failed to construct Lemke solution");
      z->setZero(n);
      return false;
    }

    // Pivot the blocking variable and the driving variable.
    std::swap(dep_variables_[blocking_index],
              indep_variables_[driving_index]);

    // Update the index map.
    auto indep_variables_indices_iter = indep_variables_indices_.find(
        dep_variables_[blocking_index]);
    indep_variables_indices_.erase(indep_variables_indices_iter);
    indep_variables_indices_[indep_variables_[driving_index]] =
        driving_index;

    // Make the driving variable the complement of the blocking variable.
    driving_index = FindComplementIndex(blocking);

    DRAKE_LOGGER_DEBUG("Independent set variables: {}",
        to_string(indep_variables_));
    DRAKE_LOGGER_DEBUG("Dependent set variables: {}",
        to_string(dep_variables_));
  }

  // If here, the maximum number of pivots has been exceeded.
  z->setZero(n);
  DRAKE_LOGGER_DEBUG("Maximum number of pivots exceeded");
  return false;
}

template <typename T>
UnrevisedLemkeSolver<T>::UnrevisedLemkeSolver()
    : SolverBase(&id, &is_available, &is_enabled,
                 &ProgramAttributesSatisfied) {}

template <typename T>
UnrevisedLemkeSolver<T>::~UnrevisedLemkeSolver() = default;

SolverId UnrevisedLemkeSolverId::id() {
  static const never_destroyed<SolverId> singleton{"Unrevised Lemke"};
  return singleton.access();
}

template <typename T>
SolverId UnrevisedLemkeSolver<T>::id() {
  return UnrevisedLemkeSolverId::id();
}

template <typename T>
bool UnrevisedLemkeSolver<T>::is_available() {
  return true;
}

template <typename T>
bool UnrevisedLemkeSolver<T>::is_enabled() {
  return true;
}

template <typename T>
bool UnrevisedLemkeSolver<T>::ProgramAttributesSatisfied(
    const MathematicalProgram& prog) {
  // This solver imposes restrictions that its problem:
  //
  // (1) Contains only linear complementarity constraints,
  // (2) Has no element of any decision variable appear in more than one
  //     constraint, and
  // (3) Has every element of every decision variable in a constraint.
  //
  // Restriction 1 could reasonably be relaxed by reformulating other
  // constraint types that can be expressed as LCPs (eg, convex QLPs),
  // although this would also entail adding an output stage to convert
  // the LCP results back to the desired form.  See eg. @RussTedrake on
  // how to convert a linear equality constraint of n elements to an
  // LCP of 2n elements.
  //
  // There is no obvious way to relax restriction 2.
  //
  // Restriction 3 could reasonably be relaxed to simply let unbound
  // variables sit at 0.
  if (prog.required_capabilities() != ProgramAttributes({
        ProgramAttribute::kLinearComplementarityConstraint})) {
    return false;
  }

  // Check that the available LCPs cover the program and no two LCPs cover the
  // same variable.
  const auto& bindings = prog.linear_complementarity_constraints();
  for (int i = 0; i < static_cast<int>(prog.num_vars()); ++i) {
    int coverings = 0;
    for (const auto& binding : bindings) {
      if (binding.ContainsVariable(prog.decision_variable(i))) {
        coverings++;
      }
    }
    if (coverings != 1) {
      return false;
    }
  }

  return true;
}

}  // namespace solvers
}  // namespace drake

DRAKE_DEFINE_CLASS_TEMPLATE_INSTANTIATIONS_ON_DEFAULT_NONSYMBOLIC_SCALARS(
    class ::drake::solvers::UnrevisedLemkeSolver)