indexed_model.h

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// Copyright 2010-2021 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

// An index based C++ API for building optimization problems.
//
// TODO(b/188550843): move/rename to core/model_storage.h
//
// Models problems of the form:
//   min sum_{j in J} c_j * x_j + d
//   s.t. lb^c_i <= sum_{j in J} A_ij * x_j <= ub^c_i        for all i in I,
//        lb^v_j <= x_j <= ub^v_j                            for all j in J,
//        x_j integer                                        for all j in Z,
// where above:
//  * I: the set of linear constraints,
//  * J: the set of variables,
//  * Z: a subset of J, the integer variables,
//  * x: the decision variables (indexed by J),
//  * c: the linear objective, one double per variable,
//  * d: the objective offset, a double scalar,
//  * lb^c: the constraint lower bounds, one double per linear constraint,
//  * ub^c: the constraint upper bounds, one double per linear constraint,
//  * lb^v: the variable lower bounds, one double per variable,
//  * ub^v: the variable upper bounds, one double per variable,
//  * A: the linear constraint matrix, a double per variable/constraint pair.
//
// The min in the objective can also be changed to a max.
//
// A simple example:
//
// Model the problem:
//   max 2.0 * x + y
//   s.t. x + y <= 1.5
//            x in {0.0, 1.0}
//       0 <= y <= 2.5
//
//   using ::operations_research::math_opt::IndexedModel;
//   using ::operations_research::math_opt::VariableId;
//   using ::operations_research::math_opt::LinearConstraintId;
//   using ::operations_research::math_opt::ModelProto;
//   using ::operations_research::math_opt::ModelProtoUpdate;
//
//   IndexedModel model("my_model");
//   const VariableId x = model.AddVariable(0.0, 1.0, true, "x");
//   const VariableId y = model.AddVariable(0.0, 2.5, false, "y");
//   const LinearConstraintId c = model.AddLinearConstraint(
//       -std::numeric_limits<double>::infinity, 1.5, "c");
//   model.set_linear_constraint_coefficient(x, c, 1.0);
//   model.set_linear_constraint_coefficient(y, c, 1.0);
//   model.set_linear_objective_coefficient(x, 2.0);
//   model.set_linear_objective_coefficient(y, 1.0);
//   model.set_maximize();
//
// Now, export to a proto describing the model:
//
//   const ModelProto model_proto = model.ExportModel();
//
// Modify the problem and get a model update proto:
//
//   const std::unique_ptr<IndexedModel::UpdateTracker> update_tracker =
//     model.NewUpdateTracker();
//   c.set_upper_bound(2.0);
//   const absl::optional<ModelUpdateProto> update_proto =
//     update_tracker->ExportModelUpdate();
//   update_tracker->Checkpoint();
//
// Reading and writing model properties:
//
// Properties of the model (e.g. variable/constraint bounds) can be written
// and read in amortized O(1) time. Deleting a variable will take time
// O(#constraints containing the variable), and likewise deleting a constraint
// will take time O(#variables in the constraint). The constraint matrix is
// stored as hash map where the key is a {LinearConstraintId, VariableId}
// pair and the value is the coefficient. The nonzeros of the matrix are
// additionally stored by row and by column, but these indices generated lazily
// upon first use. Asking for the set of variables in a constraint, the
// constraints in a variable, deleting a variable or constraint, or requesting a
// ModelUpdate proto will all trigger these additional indices to be generated.
//
// Exporting the Model proto:
//
// The Model proto is an equivalent representation to IndexedModel. It has a
// smaller memory footprint and optimized for storage/transport, rather than
// efficient modification. It is also the format consumed by solvers in this
// library. The Model proto can be generated by calling
// IndexedModel::ExportModel().
//
// Incrementalism, the ModelUpdate proto, and Checkpoints:
//
// To update an existing model as specified by a Model proto, solvers consume a
// ModelUpdate proto, which describes the changes to a model (e.g. new variables
// or a change in a variable bound). IndexedModel::UpdateTracker tracks the
// changes made and produces a ModelUpdate proto describing these changes with
// the method UpdateTracker::ExportModelUpdate(). The changes returned will be
// the modifications since the previous call to
// UpdateTracker::Checkpoint(). Note that, for newly initialized models, before
// the first checkpoint, there is no additional memory overhead from tracking
// changes. See
// g3doc/ortools/math_opt/g3doc/model_building_complexity.md
// for details.
//
// On bad input:
//
// Using a bad variable id or constraint id (an id not in the current model,
// which includes ids that have been deleted) on any method will result in an
// immediate failure (either a CHECK failure or an exception, which is an
// implementation detail you should not rely on). We make no attempt to say if a
// model is invalid (e.g. a variable lower bound is infinite, exceeds an upper
// bound, or is NaN). The exported models are validated instead, see
// model_validator.h.

#ifndef OR_TOOLS_MATH_OPT_CORE_INDEXED_MODEL_H_
#define OR_TOOLS_MATH_OPT_CORE_INDEXED_MODEL_H_

#include <cstdint>
#include <limits>
#include <memory>
#include <optional>
#include <string>
#include <utility>
#include <vector>

#include "ortools/base/integral_types.h"
#include "absl/base/thread_annotations.h"
#include "absl/container/flat_hash_map.h"
#include "absl/container/flat_hash_set.h"
#include "absl/meta/type_traits.h"
#include "absl/strings/string_view.h"
#include "absl/synchronization/mutex.h"
#include "absl/types/span.h"
#include "ortools/base/map_util.h"
#include "ortools/base/int_type.h"
#include "ortools/math_opt/model.pb.h"
#include "ortools/math_opt/model_update.pb.h"
#include "ortools/math_opt/result.pb.h"
#include "ortools/math_opt/solution.pb.h"
#include "ortools/math_opt/sparse_containers.pb.h"

namespace operations_research {
namespace math_opt {

DEFINE_INT_TYPE(VariableId, int64_t);
DEFINE_INT_TYPE(LinearConstraintId, int64_t);

// A mathematical optimization model.
//
// Supports the efficient creation and modification of an optimization model,
// and the export of Model and ModelUpdate protos.
//
// All methods run in amortized O(1) (as amortized over calls to that exact
// function) unless otherwise specified.
class IndexedModel {
 public:
  // Tracks the changes of the model.
  //
  // For each update tracker we define a checkpoint that is the starting point
  // used to compute the ModelUpdateProto.
  //
  // Lifecycle: the IndexedModel must outlive all the UpdateTracker instances
  // since they keep a reference to it. They use this reference in their
  // destructor so it is not safe to destroy an UpdateTracker after the
  // destruction of the IndexedModel.
  //
  // Thread-safety: UpdateTracker methods must not be used while modifying the
  // IndexedModel. The user is expected to use proper synchronization primitives
  // to serialize changes to the model and the use of the update trackers. The
  // methods of different instances of UpdateTracker are safe to be called
  // concurrently (i.e. multiple trackers can be called concurrently on
  // ExportModelUpdate() or Checkpoint()).
  //
  // Example:
  //   IndexedModel model;
  //   ...
  //   ASSIGN_OR_RETURN(const auto solver,
  //                    Solver::New(solver_type, model.ExportModel(),
  //                                /*initializer=*/{}));
  //   const std::unique_ptr<IndexedModel::UpdateTracker> update_tracker =
  //     model.NewUpdatesTracker();
  //
  //   ASSIGN_OR_RETURN(const auto result_1,
  //                    solver->Solve(/*parameters=*/{});
  //
  //   model.AddVariable(0.0, 1.0, true, "y");
  //   model.set_maximize(true);
  //
  //   const absl::optional<ModelUpdateProto> update_proto =
  //     update_tracker.ExportModelUpdate();
  //   update_tracker.Checkpoint();
  //
  //   if (update_proto) {
  //     ASSIGN_OR_RETURN(const bool updated, solver->Update(*update_proto));
  //     if (!updated) {
  //       // The update is not supported by the solver, we create a new one.
  //       ASSIGN_OR_RETURN(const auto new_model_proto, model.ExportModel());
  //       ASSIGN_OR_RETURN(solver,
  //                        Solver::New(solver_type, new_model_proto,
  //                                    /*initializer=*/{}));
  //     }
  //   }
  //   ASSIGN_OR_RETURN(const auto result_2,
  //                    solver->Solve(/*parameters=*/{});
  //
  class UpdateTracker {
   public:
    ~UpdateTracker() ABSL_LOCKS_EXCLUDED(indexed_model_.update_trackers_lock_);

    // Returns a proto representation of the changes to the model since the most
    // recent checkpoint (i.e. last time Checkpoint() was called); nullopt if
    // the update would have been empty.
    absl::optional<ModelUpdateProto> ExportModelUpdate()
        ABSL_LOCKS_EXCLUDED(indexed_model_.update_trackers_lock_);

    // Uses the current model state as the starting point to calculate the
    // ModelUpdateProto next time ExportModelUpdate() is called.
    void Checkpoint() ABSL_LOCKS_EXCLUDED(indexed_model_.update_trackers_lock_);

   private:
    friend class IndexedModel;

    // Registers in update_trackers_ and calls Checkpoint() to synchronize the
    // checkpoint to the current state of the model.
    explicit UpdateTracker(IndexedModel& indexed_model)
        ABSL_LOCKS_EXCLUDED(indexed_model.update_trackers_lock_);

    // Same as Checkpoint() but the caller must have acquired the
    // update_trackers_lock_ mutex.
    void CheckpointLocked()
        ABSL_EXCLUSIVE_LOCKS_REQUIRED(indexed_model_.update_trackers_lock_);

    IndexedModel& indexed_model_;

    // All incremental updates that occurred since last checkpoint. It is
    // filled-in each time Checkpoint() is called on any update tracker. When an
    // ExportModelUpdate() is requested on a tracker, all these are merged along
    // with the remaining updates.
    std::vector<std::shared_ptr<const ModelUpdateProto>> updates_
        ABSL_GUARDED_BY(indexed_model_.update_trackers_lock_);
  };

  // Creates an empty minimization problem.
  explicit IndexedModel(absl::string_view name = "") : name_(name) {}

  // TODO(b/185892243): add `std::unique_ptr<IndexedModel> Clone()` method.
  IndexedModel(const IndexedModel&) = delete;
  IndexedModel& operator=(const IndexedModel&) = delete;

  inline const std::string& name() const { return name_; }

  // Variables

  // Adds a continuous unbounded variable to the model and returns its id.
  //
  // See AddVariable(double, double, bool, absl::string_view) for details.
  inline VariableId AddVariable(absl::string_view name = "");

  // Adds a variable to the model and returns its id.
  //
  // The returned ids begin at zero and increase by one with each call to
  // AddVariable. Deleted ids are NOT reused. If no variables are deleted,
  // the ids in the model will be consecutive.
  VariableId AddVariable(double lower_bound, double upper_bound,
                         bool is_integer, absl::string_view name = "");

  inline double variable_lower_bound(VariableId id) const;
  inline double variable_upper_bound(VariableId id) const;
  inline bool is_variable_integer(VariableId id) const;
  inline const std::string& variable_name(VariableId id) const;

  inline void set_variable_lower_bound(VariableId id, double lower_bound);
  inline void set_variable_upper_bound(VariableId id, double upper_bound);
  inline void set_variable_is_integer(VariableId id, bool is_integer);
  inline void set_variable_as_integer(VariableId id);
  inline void set_variable_as_continuous(VariableId id);

  // Removes a variable from the model.
  //
  // It is an error to use a deleted variable id as input to any subsequent
  // function calls on the model. Runs in O(#constraints containing the
  // variable).
  void DeleteVariable(VariableId id);

  // The number of variables in the model.
  //
  // Equal to the number of variables created minus the number of variables
  // deleted.
  inline int num_variables() const;

  // The returned id of the next call to AddVariable.
  //
  // Equal to the number of variables created.
  inline VariableId next_variable_id() const;

  // Returns true if this id has been created and not yet deleted.
  inline bool has_variable(VariableId id) const;

  // The VariableIds in use (not deleted), order not defined.
  std::vector<VariableId> variables() const;

  // Returns a sorted vector of all existing (not deleted) variables in the
  // model.
  //
  // Runs in O(n log(n)), where n is the number of variables returned.
  std::vector<VariableId> SortedVariables() const;

  // Linear Constraints

  // Adds a linear constraint to the model with a lower bound of -inf and an
  // upper bound of +inf and returns its id.
  //
  // See AddLinearConstraint(double, double, absl::string_view) for details.
  inline LinearConstraintId AddLinearConstraint(absl::string_view name = "");

  // Adds a linear constraint to the model returns its id.
  //
  // The returned ids begin at zero and increase by one with each call to
  // AddLinearConstraint. Deleted ids are NOT reused. If no linear
  // constraints are deleted, the ids in the model will be consecutive.
  LinearConstraintId AddLinearConstraint(double lower_bound, double upper_bound,
                                         absl::string_view name = "");

  inline double linear_constraint_lower_bound(LinearConstraintId id) const;
  inline double linear_constraint_upper_bound(LinearConstraintId id) const;
  inline const std::string& linear_constraint_name(LinearConstraintId id) const;

  inline void set_linear_constraint_lower_bound(LinearConstraintId id,
                                                double lower_bound);
  inline void set_linear_constraint_upper_bound(LinearConstraintId id,
                                                double upper_bound);

  // Removes a linear constraint from the model.
  //
  // It is an error to use a deleted linear constraint id as input to any
  // subsequent function calls on the model. Runs in O(#variables in the linear
  // constraint).
  void DeleteLinearConstraint(LinearConstraintId id);

  // The number of linear constraints in the model.
  //
  // Equal to the number of linear constraints created minus the number of
  // linear constraints deleted.
  inline int num_linear_constraints() const;

  // The returned id of the next call to AddLinearConstraint.
  //
  // Equal to the number of linear constraints created.
  inline LinearConstraintId next_linear_constraint_id() const;

  // Returns true if this id has been created and not yet deleted.
  inline bool has_linear_constraint(LinearConstraintId id) const;

  // The LinearConstraintsIds in use (not deleted), order not defined.
  std::vector<LinearConstraintId> linear_constraints() const;

  // Returns a sorted vector of all existing (not deleted) linear constraints in
  // the model.
  //
  // Runs in O(n log(n)), where n is the number of linear constraints returned.
  std::vector<LinearConstraintId> SortedLinearConstraints() const;

  // Linear constraint matrix

  // Returns 0.0 if the entry is not in matrix.
  inline double linear_constraint_coefficient(LinearConstraintId constraint,
                                              VariableId variable) const;
  inline bool is_linear_constraint_coefficient_nonzero(
      LinearConstraintId constraint, VariableId variable) const;

  // Setting a value to 0.0 will delete the {constraint, variable} pair from the
  // underlying sparse matrix representation (and has no effect if the pair is
  // not present).
  inline void set_linear_constraint_coefficient(LinearConstraintId constraint,
                                                VariableId variable,
                                                double value);

  // The {linear constraint, variable} pairs with nonzero linear constraint
  // matrix coefficients.
  inline const absl::flat_hash_map<std::pair<LinearConstraintId, VariableId>,
                                   double>&
  linear_constraint_matrix() const;

  // Returns the variables with nonzero coefficients in a linear constraint.
  //
  // Runs in O(1), but triggers allocations that are O(nnz) on first use through
  // a lazy initialization.
  inline const absl::flat_hash_set<VariableId>& variables_in_linear_constraint(
      LinearConstraintId constraint);

  // Returns the linear constraints with nonzero coefficients on a variable.
  //
  // Runs in O(1), but triggers allocations that are O(nnz) on first use through
  // a lazy initialization.
  inline const absl::flat_hash_set<LinearConstraintId>&
  linear_constraints_with_variable(VariableId variable);

  // Objective

  inline bool is_maximize() const;
  inline double objective_offset() const;
  // Returns 0.0 if this variable has no linear objective coefficient.
  inline double linear_objective_coefficient(VariableId variable) const;
  inline bool is_linear_objective_coefficient_nonzero(
      VariableId variable) const;

  inline void set_is_maximize(bool is_maximize);
  inline void set_maximize();
  inline void set_minimize();
  inline void set_objective_offset(double value);

  // Setting a value to 0.0 will delete the variable from the underlying sparse
  // representation (and has no effect if the variable is not present).
  inline void set_linear_objective_coefficient(VariableId variable,
                                               double value);

  // Equivalent to calling set_linear_objective_coefficient(v, 0.0) for every
  // variable with nonzero objective coefficient.
  //
  // Runs in O(#variables with nonzero objective coefficient).
  inline void clear_objective();

  // The variables with nonzero linear objective coefficients.
  inline const absl::flat_hash_map<VariableId, double>& linear_objective()
      const;

  // Returns a sorted vector of all variables in the model with nonzero linear
  // objective coefficients.
  //
  // Runs in O(n log(n)), where n is the number of variables returned.
  std::vector<VariableId> SortedLinearObjectiveNonzeroVariables() const;

  // Export

  // Returns a proto representation of the optimization model.
  ModelProto ExportModel() const;

  // Returns a tracker that can be used to generate a ModelUpdateProto with the
  // updates that happened since the last checkpoint. The tracker initial
  // checkpoint corresponds to the current state of the model.
  //
  // The returned UpdateTracker keeps a reference to this IndexedModel. Thus the
  // IndexedModel must not be destroyed before the destruction of all its
  // UpdateTracker instances.
  //
  // Thread-safety: this method must not be used while modifying the
  // IndexedModel. The user is expected to use proper synchronization primitive
  // to serialize changes to the model and the use of this method.
  std::unique_ptr<UpdateTracker> NewUpdateTracker();

 private:
  struct VariableData {
    double lower_bound = -std::numeric_limits<double>::infinity();
    double upper_bound = std::numeric_limits<double>::infinity();
    bool is_integer = false;
    std::string name = "";
  };
  struct LinearConstraintData {
    double lower_bound = -std::numeric_limits<double>::infinity();
    double upper_bound = std::numeric_limits<double>::infinity();
    std::string name = "";
  };

  template <typename T>
  void set_variable_property(VariableId id, T value, T VariableData::*field,
                             absl::flat_hash_set<VariableId>& dirty_set);

  inline void set_linear_constraint_property(
      const LinearConstraintId id, double value,
      double LinearConstraintData::*field,
      absl::flat_hash_set<LinearConstraintId>& dirty_set);

  // Initializes lazy_matrix_columns_ (column major storage of the linear
  // constraint matrix) if it is still empty and there is at least one variable
  // in the model.
  void EnsureLazyMatrixColumns();

  // Initializes lazy_matrix_rows_ (row major storage of the linear constraint
  // matrix) if it is still empty and there is at least one linear constraint in
  // the model.
  void EnsureLazyMatrixRows();

  // Export a single variable to proto.
  void AppendVariable(VariableId id, VariablesProto& variables_proto) const;

  // Export a single linear constraint to proto.
  void AppendLinearConstraint(
      LinearConstraintId id,
      LinearConstraintsProto& linear_constraints_proto) const;

  // If an element in entries is not found in linear_constraint_matrix_, it is
  // set to 0.0 in matrix. Entries should be sorted by constraint then by
  // variable.
  void ExportLinearConstraintMatrix(
      absl::Span<const std::pair<LinearConstraintId, VariableId>> entries,
      SparseDoubleMatrixProto& matrix) const;

  // Returns a proto representation of the changes to the model since the most
  // recent call to SharedCheckpoint() or nullopt if no changes happened.
  //
  // Thread-safety: this method must not be called concurrently (due to
  // EnsureLazyMatrixXxx() functions).
  absl::optional<ModelUpdateProto> ExportSharedModelUpdate()
      ABSL_EXCLUSIVE_LOCKS_REQUIRED(update_trackers_lock_);

  // Use the current model state as the starting point to calculate the
  // ModelUpdateProto next time ExportSharedModelUpdate() is called.
  void SharedCheckpoint() ABSL_EXCLUSIVE_LOCKS_REQUIRED(update_trackers_lock_);

  std::string name_;
  VariableId next_variable_id_ = VariableId(0);
  LinearConstraintId next_linear_constraint_id_ = LinearConstraintId(0);

  bool is_maximize_ = false;
  double objective_offset_ = 0.0;

  absl::flat_hash_map<VariableId, VariableData> variables_;
  absl::flat_hash_map<LinearConstraintId, LinearConstraintData>
      linear_constraints_;
  // The values of the map must never include zero.
  absl::flat_hash_map<VariableId, double> linear_objective_;
  // The values of the map must never include zero.
  absl::flat_hash_map<std::pair<LinearConstraintId, VariableId>, double>
      linear_constraint_matrix_;
  absl::flat_hash_map<VariableId, absl::flat_hash_set<LinearConstraintId>>
      lazy_matrix_columns_;
  absl::flat_hash_map<LinearConstraintId, absl::flat_hash_set<VariableId>>
      lazy_matrix_rows_;

  // Update information
  //
  // Implicitly, all data for variables and constraints added after the last
  // checkpoint are considered "new" and will NOT be stored in the "dirty" data
  // structures below.
  VariableId variables_checkpoint_ = VariableId(0);
  LinearConstraintId linear_constraints_checkpoint_ = LinearConstraintId(0);
  bool dirty_objective_direction_ = false;
  bool dirty_objective_offset_ = false;

  absl::flat_hash_set<VariableId> dirty_variable_deletes_;
  absl::flat_hash_set<VariableId> dirty_variable_lower_bounds_;
  absl::flat_hash_set<VariableId> dirty_variable_upper_bounds_;
  absl::flat_hash_set<VariableId> dirty_variable_is_integer_;

  absl::flat_hash_set<VariableId> dirty_linear_objective_coefficients_;

  absl::flat_hash_set<LinearConstraintId> dirty_linear_constraint_deletes_;
  absl::flat_hash_set<LinearConstraintId> dirty_linear_constraint_lower_bounds_;
  absl::flat_hash_set<LinearConstraintId> dirty_linear_constraint_upper_bounds_;

  // Only for pairs where both the variable and constraint are before the
  // checkpoint, i.e.
  //   var_id < variables_checkpoint_  &&
  //   lin_con_id < linear_constraints_checkpoint_
  absl::flat_hash_set<std::pair<LinearConstraintId, VariableId>>
      dirty_linear_constraint_matrix_keys_;

  // Lock used to serialize access to update_trackers_ and to the all fields of
  // UpdateTracker. We use only one lock since trackers are modified in group
  // (they share a chain of ModelUpdateProto and the update of one tracker
  // usually requires the update of some of the others).
  absl::Mutex update_trackers_lock_;

  // The UpdateTracker instances tracking the changes of to this model. This
  // collection is updated by UpdateTracker constructor and destructor. It is
  // used internally by UpdateTracker instances to know about other instances.
  absl::flat_hash_set<UpdateTracker*> update_trackers_
      ABSL_GUARDED_BY(update_trackers_lock_);
};

// A solution to the problem modeled in IndexedModel.
//
// IndexedPrimalSolution will satisfy (see top of file notation):
//   objective_value = c * variable_values + d
// In addition, if feasible (as is typical), it should additionally satisfy:
//   vlb <= variable_values <= vub
//   clb <= A * variable_values <= cub
//   variable_values_i integer for i in I.
//
// For details, see go/mathopt-solutions#primal-solution.
struct IndexedPrimalSolution {
  absl::flat_hash_map<VariableId, double> variable_values;
  double objective_value;
};

// A direction of improving objective value that maintains feasibility.
//
// Indexed primal ray will satisfy (see top of file notation):
//   * c * x < 0 for minimization problems, c * x > 0 for maximization problems.
//   * A_j * x <= 0 for finite cub_j
//   * A_j * x >= 0 for finite clb_j
//   * x_i <= 0 for finite vub_i
//   * x_i >= 0 for finite vlb_i
// where A_j are the row of matrix coefficients for the jth linear constraint.
//
//
// Note that this alone does not prove unboundedness, you also need a feasible
// point. This is a certificate of dual infeasibility for LPs, see
// go/mathopt-dual#dual-inf-cert.
//
// For details, see go/mathopt-solutions#primal-ray.
struct IndexedPrimalRay {
  absl::flat_hash_map<VariableId, double> variable_values;
};

// A solution to the dual of the problem modeled in IndexedModel. Applies only
// when the problem has a dual (e.g. LP).
//
// For details, see go/mathopt-dual#dual-solution.
// For an primal interpretation as objective-value/optimality certificates see:
// go/mathopt-solutions#opt-certificate
struct IndexedDualSolution {
  absl::flat_hash_map<LinearConstraintId, double> dual_values;
  absl::flat_hash_map<VariableId, double> reduced_costs;
  double objective_value;
};

// A direction of improving objective value that maintains feasibility for the
// dual problem. Applies only when the problem has a dual (e.g. LP).
//
// Note that this alone does not prove dual unboundedness, you also need a dual
// feasible point.
//
// Also, a certificate of primal infeasibility.
//
// For details, see
// go/mathopt-dual#dual-ray
// go/mathopt-solutions#primal-inf-cert
struct IndexedDualRay {
  absl::flat_hash_map<LinearConstraintId, double> dual_values;
  absl::flat_hash_map<VariableId, double> reduced_costs;
};

// A basis for a solution of the problem modeled in IndexedModel. Applies only
// when the problem is an LP.
//
// For details, see go/mathopt-basis#basis.
// TODO(b/171883688): consider improving consistency for matchers in
// solver_matchers.h/cc (e.g. removing PrimalDualPair from that file)
struct IndexedBasis {
  absl::flat_hash_map<LinearConstraintId, BasisStatus> constraint_status;
  absl::flat_hash_map<VariableId, BasisStatus> variable_status;
};

// The solution (or potentially multiple solutions) to an LP or MIP. For LP,
// if the problem is solved to completion (i.e. we don't have an error or hit a
// limit), then at least one of these should be populated.
struct IndexedSolutions {
  std::vector<IndexedPrimalSolution> primal_solutions;
  std::vector<IndexedPrimalRay> primal_rays;
  std::vector<IndexedDualSolution> dual_solutions;
  std::vector<IndexedDualRay> dual_rays;
  std::vector<IndexedBasis> basis;
};

IndexedSolutions IndexedSolutionsFromProto(
    const SolveResultProto& solve_result);

// Inlined function implementations

// Variables

VariableId IndexedModel::AddVariable(absl::string_view name) {
  return AddVariable(-std::numeric_limits<double>::infinity(),
                     std::numeric_limits<double>::infinity(), false, name);
}

double IndexedModel::variable_lower_bound(const VariableId id) const {
  return variables_.at(id).lower_bound;
}

double IndexedModel::variable_upper_bound(const VariableId id) const {
  return variables_.at(id).upper_bound;
}

bool IndexedModel::is_variable_integer(VariableId id) const {
  return variables_.at(id).is_integer;
}

const std::string& IndexedModel::variable_name(const VariableId id) const {
  return variables_.at(id).name;
}

template <typename T>
void IndexedModel::set_variable_property(
    const VariableId id, const T value, T VariableData::*const field,
    absl::flat_hash_set<VariableId>& dirty_set) {
  VariableData& var_data = variables_.at(id);
  if (var_data.*field != value) {
    var_data.*field = value;
    if (id < variables_checkpoint_) {
      dirty_set.insert(id);
    }
  }
}

void IndexedModel::set_variable_lower_bound(const VariableId id,
                                            const double lower_bound) {
  set_variable_property(id, lower_bound, &VariableData::lower_bound,
                        dirty_variable_lower_bounds_);
}

void IndexedModel::set_variable_upper_bound(const VariableId id,
                                            const double upper_bound) {
  set_variable_property(id, upper_bound, &VariableData::upper_bound,
                        dirty_variable_upper_bounds_);
}

void IndexedModel::set_variable_is_integer(const VariableId id,
                                           const bool is_integer) {
  set_variable_property(id, is_integer, &VariableData::is_integer,
                        dirty_variable_is_integer_);
}

void IndexedModel::set_variable_as_integer(VariableId id) {
  set_variable_is_integer(id, true);
}

void IndexedModel::set_variable_as_continuous(VariableId id) {
  set_variable_is_integer(id, false);
}

int IndexedModel::num_variables() const { return variables_.size(); }

VariableId IndexedModel::next_variable_id() const { return next_variable_id_; }

bool IndexedModel::has_variable(const VariableId id) const {
  return variables_.contains(id);
}

// Linear Constraints

LinearConstraintId IndexedModel::AddLinearConstraint(absl::string_view name) {
  return AddLinearConstraint(-std::numeric_limits<double>::infinity(),
                             std::numeric_limits<double>::infinity(), name);
}

double IndexedModel::linear_constraint_lower_bound(
    const LinearConstraintId id) const {
  return linear_constraints_.at(id).lower_bound;
}

double IndexedModel::linear_constraint_upper_bound(
    const LinearConstraintId id) const {
  return linear_constraints_.at(id).upper_bound;
}

const std::string& IndexedModel::linear_constraint_name(
    const LinearConstraintId id) const {
  return linear_constraints_.at(id).name;
}

void IndexedModel::set_linear_constraint_property(
    const LinearConstraintId id, const double value,
    double LinearConstraintData::*const field,
    absl::flat_hash_set<LinearConstraintId>& dirty_set) {
  LinearConstraintData& lin_con_data = linear_constraints_.at(id);
  if (lin_con_data.*field != value) {
    lin_con_data.*field = value;
    if (id < linear_constraints_checkpoint_) {
      dirty_set.insert(id);
    }
  }
}

void IndexedModel::set_linear_constraint_lower_bound(
    const LinearConstraintId id, const double lower_bound) {
  set_linear_constraint_property(id, lower_bound,
                                 &LinearConstraintData::lower_bound,
                                 dirty_linear_constraint_lower_bounds_);
}

void IndexedModel::set_linear_constraint_upper_bound(
    const LinearConstraintId id, const double upper_bound) {
  set_linear_constraint_property(id, upper_bound,
                                 &LinearConstraintData::upper_bound,
                                 dirty_linear_constraint_upper_bounds_);
}

int IndexedModel::num_linear_constraints() const {
  return linear_constraints_.size();
}

LinearConstraintId IndexedModel::next_linear_constraint_id() const {
  return next_linear_constraint_id_;
}

bool IndexedModel::has_linear_constraint(const LinearConstraintId id) const {
  return linear_constraints_.contains(id);
}

// Linear Constraint Matrix

double IndexedModel::linear_constraint_coefficient(
    LinearConstraintId constraint, VariableId variable) const {
  return gtl::FindWithDefault(linear_constraint_matrix_,
                              {constraint, variable});
}

bool IndexedModel::is_linear_constraint_coefficient_nonzero(
    LinearConstraintId constraint, VariableId variable) const {
  return linear_constraint_matrix_.contains({constraint, variable});
}

void IndexedModel::set_linear_constraint_coefficient(
    LinearConstraintId constraint, VariableId variable, double value) {
  bool was_updated = false;
  if (value == 0.0) {
    if (linear_constraint_matrix_.erase({constraint, variable}) > 0) {
      was_updated = true;
      if (!lazy_matrix_columns_.empty()) {
        lazy_matrix_columns_.at(variable).erase(constraint);
      }
      if (!lazy_matrix_rows_.empty()) {
        lazy_matrix_rows_.at(constraint).erase(variable);
      }
    }
  } else {
    const auto [iterator, inserted] =
        linear_constraint_matrix_.try_emplace({constraint, variable}, value);
    if (inserted) {
      was_updated = true;
    } else if (iterator->second != value) {
      iterator->second = value;
      was_updated = true;
    }
    if (!lazy_matrix_columns_.empty()) {
      lazy_matrix_columns_.at(variable).insert(constraint);
    }
    if (!lazy_matrix_rows_.empty()) {
      lazy_matrix_rows_.at(constraint).insert(variable);
    }
  }
  if (was_updated && constraint < linear_constraints_checkpoint_ &&
      variable < variables_checkpoint_) {
    dirty_linear_constraint_matrix_keys_.emplace(constraint, variable);
  }
}

const absl::flat_hash_map<std::pair<LinearConstraintId, VariableId>, double>&
IndexedModel::linear_constraint_matrix() const {
  return linear_constraint_matrix_;
}

const absl::flat_hash_set<VariableId>&
IndexedModel::variables_in_linear_constraint(LinearConstraintId constraint) {
  EnsureLazyMatrixRows();
  return lazy_matrix_rows_.at(constraint);
}

const absl::flat_hash_set<LinearConstraintId>&
IndexedModel::linear_constraints_with_variable(VariableId variable) {
  EnsureLazyMatrixColumns();
  return lazy_matrix_columns_.at(variable);
}

// Objective

bool IndexedModel::is_maximize() const { return is_maximize_; }

double IndexedModel::objective_offset() const { return objective_offset_; }

double IndexedModel::linear_objective_coefficient(VariableId variable) const {
  return gtl::FindWithDefault(linear_objective_, variable);
}

bool IndexedModel::is_linear_objective_coefficient_nonzero(
    VariableId variable) const {
  return linear_objective_.contains(variable);
}

void IndexedModel::set_is_maximize(bool is_maximize) {
  if (is_maximize_ != is_maximize) {
    dirty_objective_direction_ = true;
    is_maximize_ = is_maximize;
  }
}

void IndexedModel::set_maximize() { set_is_maximize(true); }

void IndexedModel::set_minimize() { set_is_maximize(false); }

void IndexedModel::set_objective_offset(double value) {
  if (value != objective_offset_) {
    dirty_objective_offset_ = true;
    objective_offset_ = value;
  }
}

void IndexedModel::set_linear_objective_coefficient(VariableId variable,
                                                    double value) {
  bool was_updated = false;
  if (value == 0.0) {
    if (linear_objective_.erase(variable) > 0) {
      was_updated = true;
    }
  } else {
    const auto [iterator, inserted] =
        linear_objective_.try_emplace(variable, value);
    if (inserted) {
      was_updated = true;
    } else if (iterator->second != value) {
      iterator->second = value;
      was_updated = true;
    }
  }
  if (was_updated && variable < variables_checkpoint_) {
    dirty_linear_objective_coefficients_.insert(variable);
  }
}

void IndexedModel::clear_objective() {
  set_objective_offset(0.0);
  while (!linear_objective_.empty()) {
    set_linear_objective_coefficient(linear_objective_.begin()->first, 0.0);
  }
}

const absl::flat_hash_map<VariableId, double>& IndexedModel::linear_objective()
    const {
  return linear_objective_;
}

}  // namespace math_opt
}  // namespace operations_research

#endif  // OR_TOOLS_MATH_OPT_CORE_INDEXED_MODEL_H_