interior_point.hpp

// Copyright (c) Sleipnir contributors
 
#pragma once
 
#include <algorithm>
#include <chrono>
#include <cmath>
#include <functional>
#include <span>
 
#include <Eigen/Core>
#include <Eigen/SparseCore>
#include <gch/small_vector.hpp>
 
#include "sleipnir/optimization/solver/exit_status.hpp"
#include "sleipnir/optimization/solver/interior_point_matrix_callbacks.hpp"
#include "sleipnir/optimization/solver/iteration_info.hpp"
#include "sleipnir/optimization/solver/options.hpp"
#include "sleipnir/optimization/solver/util/all_finite.hpp"
#include "sleipnir/optimization/solver/util/append_as_triplets.hpp"
#include "sleipnir/optimization/solver/util/feasibility_restoration.hpp"
#include "sleipnir/optimization/solver/util/filter.hpp"
#include "sleipnir/optimization/solver/util/fraction_to_the_boundary_rule.hpp"
#include "sleipnir/optimization/solver/util/is_locally_infeasible.hpp"
#include "sleipnir/optimization/solver/util/kkt_error.hpp"
#include "sleipnir/optimization/solver/util/regularized_ldlt.hpp"
#include "sleipnir/util/assert.hpp"
#include "sleipnir/util/print_diagnostics.hpp"
#include "sleipnir/util/profiler.hpp"
#include "sleipnir/util/scope_exit.hpp"
#include "sleipnir/util/symbol_exports.hpp"
 
// See docs/algorithms.md#Works_cited for citation definitions.
//
// See docs/algorithms.md#Interior-point_method for a derivation of the
// interior-point method formulation being used.
 
namespace slp {
 
template <typename Scalar>
ExitStatus interior_point(
    const InteriorPointMatrixCallbacks<Scalar>& matrix_callbacks,
    std::span<std::function<bool(const IterationInfo<Scalar>& info)>>
        iteration_callbacks,
    const Options& options,
#ifdef SLEIPNIR_ENABLE_BOUND_PROJECTION
    const Eigen::ArrayX<bool>& bound_constraint_mask,
#endif
    Eigen::Vector<Scalar, Eigen::Dynamic>& x) {
  using DenseVector = Eigen::Vector<Scalar, Eigen::Dynamic>;
 
  DenseVector s =
      DenseVector::Ones(matrix_callbacks.num_inequality_constraints);
  DenseVector y = DenseVector::Zero(matrix_callbacks.num_equality_constraints);
  DenseVector z =
      DenseVector::Ones(matrix_callbacks.num_inequality_constraints);
  Scalar μ = Scalar(0.1) * matrix_callbacks.scaling.f;
  int iterations = 0;
 
  return interior_point(matrix_callbacks, iteration_callbacks, options, false,
#ifdef SLEIPNIR_ENABLE_BOUND_PROJECTION
                        bound_constraint_mask,
#endif
                        x, s, y, z, μ, iterations);
}
 
template <typename Scalar>
ExitStatus interior_point(
    const InteriorPointMatrixCallbacks<Scalar>& matrix_callbacks,
    std::span<std::function<bool(const IterationInfo<Scalar>& info)>>
        iteration_callbacks,
    const Options& options, bool in_feasibility_restoration,
#ifdef SLEIPNIR_ENABLE_BOUND_PROJECTION
    const Eigen::ArrayX<bool>& bound_constraint_mask,
#endif
    Eigen::Vector<Scalar, Eigen::Dynamic>& x,
    Eigen::Vector<Scalar, Eigen::Dynamic>& s,
    Eigen::Vector<Scalar, Eigen::Dynamic>& y,
    Eigen::Vector<Scalar, Eigen::Dynamic>& z, Scalar& μ, int& iterations) {
  using DenseVector = Eigen::Vector<Scalar, Eigen::Dynamic>;
  using SparseMatrix = Eigen::SparseMatrix<Scalar>;
  using SparseVector = Eigen::SparseVector<Scalar>;
 
  struct Step {
    DenseVector p_x;
    DenseVector p_s;
    DenseVector p_y;
    DenseVector p_z;
  };
 
  using std::isfinite;
 
  const auto solve_start_time = std::chrono::steady_clock::now();
 
  gch::small_vector<SolveProfiler> solve_profilers;
  solve_profilers.emplace_back("solver");
  solve_profilers.emplace_back("↳ setup");
  solve_profilers.emplace_back("↳ iteration");
  solve_profilers.emplace_back("  ↳ feasibility check");
  solve_profilers.emplace_back("  ↳ callbacks");
  solve_profilers.emplace_back("  ↳ KKT matrix build");
  solve_profilers.emplace_back("  ↳ KKT matrix decomp");
  solve_profilers.emplace_back("  ↳ KKT system solve");
  solve_profilers.emplace_back("  ↳ line search");
  solve_profilers.emplace_back("    ↳ SOC");
  solve_profilers.emplace_back("  ↳ feas. restoration");
  solve_profilers.emplace_back("  ↳ f(x)");
  solve_profilers.emplace_back("  ↳ ∇f(x)");
  solve_profilers.emplace_back("  ↳ ∇²ₓₓL");
  solve_profilers.emplace_back("  ↳ ∇²ₓₓL_c");
  solve_profilers.emplace_back("  ↳ cₑ(x)");
  solve_profilers.emplace_back("  ↳ ∂cₑ/∂x");
  solve_profilers.emplace_back("  ↳ cᵢ(x)");
  solve_profilers.emplace_back("  ↳ ∂cᵢ/∂x");
 
  auto& solver_prof = solve_profilers[0];
  auto& setup_prof = solve_profilers[1];
  auto& inner_iter_prof = solve_profilers[2];
  auto& feasibility_check_prof = solve_profilers[3];
  auto& iter_callbacks_prof = solve_profilers[4];
  auto& kkt_matrix_build_prof = solve_profilers[5];
  auto& kkt_matrix_decomp_prof = solve_profilers[6];
  auto& kkt_system_solve_prof = solve_profilers[7];
  auto& line_search_prof = solve_profilers[8];
  auto& soc_prof = solve_profilers[9];
  auto& feasibility_restoration_prof = solve_profilers[10];
 
  // Set up profiled matrix callbacks
#ifndef SLEIPNIR_DISABLE_DIAGNOSTICS
  auto& f_prof = solve_profilers[11];
  auto& g_prof = solve_profilers[12];
  auto& H_prof = solve_profilers[13];
  auto& H_c_prof = solve_profilers[14];
  auto& c_e_prof = solve_profilers[15];
  auto& A_e_prof = solve_profilers[16];
  auto& c_i_prof = solve_profilers[17];
  auto& A_i_prof = solve_profilers[18];
 
  InteriorPointMatrixCallbacks<Scalar> matrices{
      matrix_callbacks.num_decision_variables,
      matrix_callbacks.num_equality_constraints,
      matrix_callbacks.num_inequality_constraints,
      [&](const DenseVector& x) -> Scalar {
        ScopedProfiler prof{f_prof};
        return matrix_callbacks.f(x);
      },
      [&](const DenseVector& x) -> SparseVector {
        ScopedProfiler prof{g_prof};
        return matrix_callbacks.g(x);
      },
      [&](const DenseVector& x, const DenseVector& y,
          const DenseVector& z) -> SparseMatrix {
        ScopedProfiler prof{H_prof};
        return matrix_callbacks.H(x, y, z);
      },
      [&](const DenseVector& x, const DenseVector& y,
          const DenseVector& z) -> SparseMatrix {
        ScopedProfiler prof{H_c_prof};
        return matrix_callbacks.H_c(x, y, z);
      },
      [&](const DenseVector& x) -> DenseVector {
        ScopedProfiler prof{c_e_prof};
        return matrix_callbacks.c_e(x);
      },
      [&](const DenseVector& x) -> SparseMatrix {
        ScopedProfiler prof{A_e_prof};
        return matrix_callbacks.A_e(x);
      },
      [&](const DenseVector& x) -> DenseVector {
        ScopedProfiler prof{c_i_prof};
        return matrix_callbacks.c_i(x);
      },
      [&](const DenseVector& x) -> SparseMatrix {
        ScopedProfiler prof{A_i_prof};
        return matrix_callbacks.A_i(x);
      },
      matrix_callbacks.scaling};
#else
  const auto& matrices = matrix_callbacks;
#endif
 
  solver_prof.start();
  setup_prof.start();
 
  Scalar f = matrices.f(x);
  SparseVector g = matrices.g(x);
  SparseMatrix H = matrices.H(x, y, z);
  DenseVector c_e = matrices.c_e(x);
  SparseMatrix A_e = matrices.A_e(x);
  DenseVector c_i = matrices.c_i(x);
  SparseMatrix A_i = matrices.A_i(x);
 
  // Ensure matrix callback dimensions are consistent
  slp_assert(g.rows() == matrices.num_decision_variables);
  slp_assert(H.rows() == matrices.num_decision_variables);
  slp_assert(H.cols() == matrices.num_decision_variables);
  slp_assert(c_e.rows() == matrices.num_equality_constraints);
  slp_assert(A_e.rows() == matrices.num_equality_constraints);
  slp_assert(A_e.cols() == matrices.num_decision_variables);
  slp_assert(c_i.rows() == matrices.num_inequality_constraints);
  slp_assert(A_i.rows() == matrices.num_inequality_constraints);
  slp_assert(A_i.cols() == matrices.num_decision_variables);
 
  DenseVector trial_x;
  DenseVector trial_s;
  DenseVector trial_y;
  DenseVector trial_z;
 
  Scalar trial_f;
  DenseVector trial_c_e;
  DenseVector trial_c_i;
 
  // Check for overconstrained problem
  if (matrices.num_equality_constraints > matrices.num_decision_variables) {
    if (options.diagnostics) {
      print_too_few_dofs_error(c_e);
    }
 
    return ExitStatus::TOO_FEW_DOFS;
  }
 
  // Check whether initial guess has finite cost, constraints, and derivatives
  if (!isfinite(f) || !all_finite(g) || !all_finite(H) || !c_e.allFinite() ||
      !all_finite(A_e) || !c_i.allFinite() || !all_finite(A_i)) {
    return ExitStatus::NONFINITE_INITIAL_GUESS;
  }
 
#ifdef SLEIPNIR_ENABLE_BOUND_PROJECTION
  // We set sʲ = cᵢʲ(x) for each bound inequality constraint index j
  s = bound_constraint_mask.select(c_i, s);
#endif
 
  // Barrier parameter minimum
  const Scalar μ_min =
      matrices.scaling.f * Scalar(options.tolerance) / Scalar(10);
 
  // Fraction-to-the-boundary rule scale factor minimum
  constexpr Scalar τ_min(0.99);
 
  // Fraction-to-the-boundary rule scale factor τ
  Scalar τ = τ_min;
 
  Filter<Scalar> filter{c_e.template lpNorm<1>() +
                        (c_i - s).template lpNorm<1>()};
 
  // This should be run when the error is below a desired threshold for the
  // current barrier parameter
  auto update_barrier_parameter_and_reset_filter = [&] {
    // Barrier parameter linear decrease power in "κ_μ μ". Range of (0, 1).
    constexpr Scalar κ_μ(0.2);
 
    // Barrier parameter superlinear decrease power in "μ^(θ_μ)". Range of (1,
    // 2).
    constexpr Scalar θ_μ(1.5);
 
    // Update the barrier parameter.
    //
    //   μⱼ₊₁ = max(εₜₒₗ/10, min(κ_μ μⱼ, μⱼ^θ_μ))
    //
    // See equation (7) of [2].
    using std::pow;
    μ = std::max(μ_min, std::min(κ_μ * μ, pow(μ, θ_μ)));
 
    // Update the fraction-to-the-boundary rule scaling factor.
    //
    //   τⱼ = max(τₘᵢₙ, 1 − μⱼ)
    //
    // See equation (8) of [2].
    τ = std::max(τ_min, Scalar(1) - μ);
 
    // Reset the filter when the barrier parameter is updated
    filter.reset();
  };
 
  // Kept outside the loop so its storage can be reused
  gch::small_vector<Eigen::Triplet<Scalar>> triplets;
 
  const int lhs_rows =
      matrices.num_decision_variables + matrices.num_equality_constraints;
  RegularizedLDLT<Scalar> solver{
      // Use sparse solver if lower triangle fills < 25% of system
      H.nonZeros() +
              (A_i.transpose() * A_i)
                  .template triangularView<Eigen::Lower>()
                  .eval()
                  .nonZeros() +
              A_e.nonZeros() <
          0.25 * lhs_rows * lhs_rows,
      matrices.num_decision_variables, matrices.num_equality_constraints,
      // Constraint regularization is forced to zero in feasibility restoration
      // because the equality constraint Jacobian cannot be rank-deficient
      in_feasibility_restoration ? Scalar(0) : Scalar(1e-10)};
 
  // Variables for determining when a step is acceptable
  constexpr Scalar α_reduction_factor(0.5);
  constexpr Scalar α_min(1e-7);
 
  int full_step_rejected_counter = 0;
 
  // Error
  Scalar E_0 = unscaled_kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
      matrices.scaling, g, A_e, c_e, A_i, c_i, s, y, z, Scalar(0));
 
  setup_prof.stop();
 
  // Prints final solver diagnostics when the solver exits
  scope_exit exit{[&] {
    if (options.diagnostics) {
      solver_prof.stop();
 
      if (in_feasibility_restoration) {
        return;
      }
 
      if (iterations > 0) {
        print_bottom_iteration_diagnostics();
      }
      print_solver_diagnostics(solve_profilers);
    }
  }};
 
  while (E_0 > Scalar(options.tolerance)) {
    ScopedProfiler inner_iter_profiler{inner_iter_prof};
    ScopedProfiler feasibility_check_profiler{feasibility_check_prof};
 
    // Check for local equality constraint infeasibility
    if (is_equality_locally_infeasible(A_e, c_e)) {
      if (options.diagnostics) {
        print_c_e_local_infeasibility_error(c_e);
      }
 
      return ExitStatus::LOCALLY_INFEASIBLE;
    }
 
    // Check for local inequality constraint infeasibility
    if (is_inequality_locally_infeasible(A_i, c_i)) {
      if (options.diagnostics) {
        print_c_i_local_infeasibility_error(c_i);
      }
 
      return ExitStatus::LOCALLY_INFEASIBLE;
    }
 
    // Check for diverging iterates
    if (x.template lpNorm<Eigen::Infinity>() > Scalar(1e10) || !x.allFinite() ||
        s.template lpNorm<Eigen::Infinity>() > Scalar(1e10) || !s.allFinite()) {
      return ExitStatus::DIVERGING_ITERATES;
    }
 
    feasibility_check_profiler.stop();
    ScopedProfiler iter_callbacks_profiler{iter_callbacks_prof};
 
    // Call iteration callbacks
    for (const auto& callback : iteration_callbacks) {
      if (callback({iterations, x, s, y, z, g, H, A_e, A_i})) {
        return ExitStatus::CALLBACK_REQUESTED_STOP;
      }
    }
 
    iter_callbacks_profiler.stop();
    ScopedProfiler kkt_matrix_build_profiler{kkt_matrix_build_prof};
 
    // S = diag(s)
    // Z = diag(z)
    // Σ = S⁻¹Z
    const SparseMatrix Σ{s.cwiseInverse().asDiagonal() * z.asDiagonal()};
 
    // lhs = [H + AᵢᵀΣAᵢ  Aₑᵀ]
    //       [    Aₑ       0 ]
    //
    // Don't assign upper triangle because solver only uses lower triangle.
    const SparseMatrix top_left =
        H + (A_i.transpose() * Σ * A_i).template triangularView<Eigen::Lower>();
    triplets.clear();
    triplets.reserve(top_left.nonZeros() + A_e.nonZeros());
    append_as_triplets(triplets, 0, 0, {top_left, A_e});
    SparseMatrix lhs(
        matrices.num_decision_variables + matrices.num_equality_constraints,
        matrices.num_decision_variables + matrices.num_equality_constraints);
    lhs.setFromSortedTriplets(triplets.begin(), triplets.end());
 
    // rhs = −[∇f − Aₑᵀy − Aᵢᵀ(−Σcᵢ + μS⁻¹e + z)]
    //        [               cₑ                ]
    DenseVector rhs{x.rows() + y.rows()};
    rhs.segment(0, x.rows()) =
        -g + A_e.transpose() * y +
        A_i.transpose() * (-Σ * c_i + μ * s.cwiseInverse() + z);
    rhs.segment(x.rows(), y.rows()) = -c_e;
 
    kkt_matrix_build_profiler.stop();
    ScopedProfiler kkt_matrix_decomp_profiler{kkt_matrix_decomp_prof};
 
    Step step;
    Scalar α_max(1);
    Scalar α(1);
    Scalar α_z(1);
    bool call_feasibility_restoration = false;
 
    // Solve the Newton-KKT system
    //
    // [H + AᵢᵀΣAᵢ  Aₑᵀ][ pˣ] = −[∇f − Aₑᵀy − Aᵢᵀ(−Σcᵢ + μS⁻¹e + z)]
    // [    Aₑ       0 ][−pʸ]    [               cₑ                ]
    if (solver.compute(lhs).info() != Eigen::Success) [[unlikely]] {
      return ExitStatus::FACTORIZATION_FAILED;
    }
 
    kkt_matrix_decomp_profiler.stop();
    ScopedProfiler kkt_system_solve_profiler{kkt_system_solve_prof};
 
    auto compute_step = [&](Step& step, const DenseVector& c_i_minus_s) {
      // p = [ pˣ]
      //     [−pʸ]
      DenseVector p = solver.solve(rhs);
      step.p_x = p.segment(0, x.rows());
      step.p_y = -p.segment(x.rows(), y.rows());
 
      // pˢ = cᵢ − s + Aᵢpˣ
      // pᶻ = μS⁻¹e − z − Σpˢ
      step.p_s = c_i_minus_s + A_i * step.p_x;
      step.p_z = μ * s.cwiseInverse() - z - Σ * step.p_s;
    };
    compute_step(step, c_i - s);
 
    kkt_system_solve_profiler.stop();
    ScopedProfiler line_search_profiler{line_search_prof};
 
    // αᵐᵃˣ = max(α ∈ (0, 1] : sₖ + αpₖˢ ≥ (1−τⱼ)sₖ)
    α_max = fraction_to_the_boundary_rule<Scalar>(s, step.p_s, τ);
    α = α_max;
 
    // If maximum step size is below minimum, invoke feasibility restoration
    if (α < α_min) {
      call_feasibility_restoration = true;
    }
 
    // αₖᶻ = max(α ∈ (0, 1] : zₖ + αpₖᶻ ≥ (1−τⱼ)zₖ)
    α_z = fraction_to_the_boundary_rule<Scalar>(z, step.p_z, τ);
 
    const FilterEntry<Scalar> current_entry{f, s, c_e, c_i, μ};
 
    // Compute the directional derivative of the log-barrier function along the
    // search direction.
    //
    //   ϕ_μ(x, s) = f(x) − μ∑ᵢ ln(sᵢ)
    //
    //   D_ϕ = ∇ϕ_μ(x, s)ᵀ[pˣ pˢ]
    //       = ∇f(x)ᵀpˣ − μ∑ᵢ pᵢˢ/sᵢ
    const Scalar D_ϕ =
        g.transpose() * step.p_x - μ * s.cwiseInverse().dot(step.p_s);
 
    // Loop until a step is accepted
    while (1) {
      trial_x = x + α * step.p_x;
      trial_c_i = matrices.c_i(trial_x);
      if (options.feasible_ipm && c_i.cwiseGreater(Scalar(0)).all()) {
        // If the inequality constraints are all feasible, prevent them from
        // becoming infeasible again.
        //
        // See equation (19.30) in [1].
        trial_s = trial_c_i;
      } else {
        trial_s = s + α * step.p_s;
      }
      trial_y = y + α_z * step.p_y;
      trial_z = z + α_z * step.p_z;
 
      trial_f = matrices.f(trial_x);
      trial_c_e = matrices.c_e(trial_x);
 
      // If f(xₖ + αpₖˣ), cₑ(xₖ + αpₖˣ), or cᵢ(xₖ + αpₖˣ) aren't finite, reduce
      // step size immediately
      if (!isfinite(trial_f) || !trial_c_e.allFinite() ||
          !trial_c_i.allFinite()) {
        // Reduce step size
        α *= α_reduction_factor;
 
        if (α < α_min) {
          call_feasibility_restoration = true;
          break;
        }
        continue;
      }
 
      // Check whether filter accepts trial iterate
      FilterEntry trial_entry{trial_f, trial_s, trial_c_e, trial_c_i, μ};
      if (filter.try_add(current_entry, trial_entry, D_ϕ, α)) {
        // Accept step
        break;
      }
 
      Scalar prev_constraint_violation =
          c_e.template lpNorm<1>() + (c_i - s).template lpNorm<1>();
      Scalar next_constraint_violation =
          trial_c_e.template lpNorm<1>() +
          (trial_c_i - trial_s).template lpNorm<1>();
 
      // Second-order corrections
      //
      // If first trial point was rejected and constraint violation stayed the
      // same or went up, apply second-order corrections
      if (α == α_max &&
          next_constraint_violation >= prev_constraint_violation) {
        // Apply second-order corrections. See section 2.4 of [2].
        auto soc_step = step;
 
        Scalar α_soc = α;
        Scalar α_z_soc = α_z;
        DenseVector c_e_soc = c_e;
        DenseVector c_i_minus_s_soc = c_i - s;
 
        Scalar soc_constraint_violation = next_constraint_violation;
 
        bool step_acceptable = false;
        for (int soc_iteration = 0; soc_iteration < 5 && !step_acceptable;
             ++soc_iteration) {
          ScopedProfiler soc_profiler{soc_prof};
 
          scope_exit soc_exit{[&] {
            soc_profiler.stop();
 
            if (options.diagnostics && step_acceptable) {
              print_iteration_diagnostics(
                  iterations, IterationType::SECOND_ORDER_CORRECTION,
                  soc_profiler.current_duration(),
                  unscaled_kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
                      matrices.scaling, g, A_e, trial_c_e, A_i, trial_c_i,
                      trial_s, trial_y, trial_z, Scalar(0)),
                  trial_f,
                  trial_c_e.template lpNorm<1>() +
                      (trial_c_i - trial_s).template lpNorm<1>(),
                  trial_s.dot(trial_z), μ, solver.hessian_regularization(),
                  solver.constraint_jacobian_regularization(),
                  std::max(soc_step.p_x.template lpNorm<Eigen::Infinity>(),
                           soc_step.p_s.template lpNorm<Eigen::Infinity>()),
                  std::max(soc_step.p_y.template lpNorm<Eigen::Infinity>(),
                           soc_step.p_z.template lpNorm<Eigen::Infinity>()),
                  α_soc, Scalar(1), α_reduction_factor, α_z_soc);
            }
          }};
 
          // Rebuild Newton-KKT rhs with updated constraint values.
          //
          // rhs = −[∇f − Aₑᵀy − Aᵢᵀ(μS⁻¹e − Σ(cᵢ − s)ˢᵒᶜ)]
          //        [               cₑˢᵒᶜ                 ]
          //
          // where
          //
          //   cₑˢᵒᶜ = αˢᵒᶜcₑ(xₖ) + cₑ(xₖ + αˢᵒᶜpˣˢᵒᶜ)
          //   (cᵢ − s)ˢᵒᶜ =
          //     αˢᵒᶜ(cᵢ(xₖ) − sₖ) + cᵢ(xₖ + αˢᵒᶜpˣˢᵒᶜ) − (sₖ + αˢᵒᶜpˢˢᵒᶜ)
          c_e_soc = α_soc * c_e_soc + trial_c_e;
          c_i_minus_s_soc = α_soc * c_i_minus_s_soc + trial_c_i - trial_s;
          rhs.segment(0, x.rows()) =
              -g + A_e.transpose() * y +
              A_i.transpose() * (μ * s.cwiseInverse() - Σ * c_i_minus_s_soc);
          rhs.segment(x.rows(), y.rows()) = -c_e_soc;
 
          // Solve the Newton-KKT system
          compute_step(soc_step, c_i_minus_s_soc);
 
          // αˢᵒᶜ = max(α ∈ (0, 1] : sₖ + αpₖˢ ≥ (1−τⱼ)sₖ)
          // αₖᶻˢᵒᶜ = max(α ∈ (0, 1] : zₖ + αpₖᶻ ≥ (1−τⱼ)zₖ)
          α_soc = fraction_to_the_boundary_rule<Scalar>(s, soc_step.p_s, τ);
          α_z_soc = fraction_to_the_boundary_rule<Scalar>(z, soc_step.p_z, τ);
 
          trial_x = x + α_soc * soc_step.p_x;
          trial_s = s + α_soc * soc_step.p_s;
          trial_y = y + α_z_soc * soc_step.p_y;
          trial_z = z + α_z_soc * soc_step.p_z;
 
          trial_f = matrices.f(trial_x);
          trial_c_e = matrices.c_e(trial_x);
          trial_c_i = matrices.c_i(trial_x);
 
          // Check whether filter accepts trial iterate
          FilterEntry trial_entry{trial_f, trial_s, trial_c_e, trial_c_i, μ};
          if (filter.try_add(current_entry, trial_entry, D_ϕ, α)) {
            step = soc_step;
            α = α_soc;
            α_z = α_z_soc;
            step_acceptable = true;
            break;
          }
 
          // Constraint violation scale factor for second-order corrections
          constexpr Scalar κ_soc(0.99);
 
          // If constraint violation hasn't been sufficiently reduced, stop
          // making second-order corrections
          next_constraint_violation =
              trial_c_e.template lpNorm<1>() +
              (trial_c_i - trial_s).template lpNorm<1>();
          if (next_constraint_violation > κ_soc * soc_constraint_violation) {
            break;
          }
 
          soc_constraint_violation = next_constraint_violation;
        }
 
        if (step_acceptable) {
          // Accept step
          break;
        }
      }
 
      // If we got here and α is the full step, the full step was rejected.
      // Increment the full-step rejected counter to keep track of how many full
      // steps have been rejected in a row.
      if (α == α_max) {
        ++full_step_rejected_counter;
      }
 
      // If the full step was rejected enough times in a row, reset the filter
      // because it may be impeding progress.
      //
      // See section 3.2 case I of [2].
      if (full_step_rejected_counter >= 4 &&
          filter.max_constraint_violation >
              current_entry.constraint_violation / Scalar(10) &&
          filter.last_rejection_due_to_filter()) {
        filter.max_constraint_violation *= Scalar(0.1);
        filter.reset();
        continue;
      }
 
      // Reduce step size
      α *= α_reduction_factor;
 
      // If step size hit a minimum, check if the KKT error was reduced. If it
      // wasn't, invoke feasibility restoration.
      if (α < α_min) {
        Scalar current_kkt_error = kkt_error<Scalar, KKTErrorType::ONE_NORM>(
            g, A_e, c_e, A_i, c_i, s, y, z, μ);
 
        trial_x = x + α_max * step.p_x;
        trial_s = s + α_max * step.p_s;
        trial_y = y + α_z * step.p_y;
        trial_z = z + α_z * step.p_z;
 
        trial_f = matrices.f(trial_x);
        trial_c_e = matrices.c_e(trial_x);
        trial_c_i = matrices.c_i(trial_x);
 
        Scalar next_kkt_error = kkt_error<Scalar, KKTErrorType::ONE_NORM>(
            matrices.g(trial_x), matrices.A_e(trial_x), trial_c_e,
            matrices.A_i(trial_x), trial_c_i, trial_s, trial_y, trial_z, μ);
 
        // If the step using αᵐᵃˣ reduced the KKT error, accept it anyway
        if (next_kkt_error <= Scalar(0.999) * current_kkt_error) {
          // Accept step
          break;
        }
 
        call_feasibility_restoration = true;
        break;
      }
    }
 
    line_search_profiler.stop();
 
    if (call_feasibility_restoration) {
      ScopedProfiler feasibility_restoration_profiler{
          feasibility_restoration_prof};
 
      // If already in feasibility restoration mode, running it again won't help
      if (in_feasibility_restoration) {
        return ExitStatus::FEASIBILITY_RESTORATION_FAILED;
      }
 
      FilterEntry initial_entry{matrices.f(x), s, c_e, c_i, μ};
 
      // Feasibility restoration phase
      gch::small_vector<std::function<bool(const IterationInfo<Scalar>& info)>>
          callbacks;
      for (auto& callback : iteration_callbacks) {
        callbacks.emplace_back(callback);
      }
      callbacks.emplace_back([&](const IterationInfo<Scalar>& info) {
        DenseVector trial_x =
            info.x.segment(0, matrices.num_decision_variables);
        DenseVector trial_s =
            info.s.segment(0, matrices.num_inequality_constraints);
 
        DenseVector trial_c_e = matrices.c_e(trial_x);
        DenseVector trial_c_i = matrices.c_i(trial_x);
 
        // If the current iterate sufficiently reduces constraint violation and
        // is accepted by the normal filter, stop feasibility restoration
        FilterEntry trial_entry{matrices.f(trial_x), trial_s, trial_c_e,
                                trial_c_i, μ};
        const Scalar D_ϕ_restoration = g.transpose() * (trial_x - x) -
                                       μ * s.cwiseInverse().dot(trial_s - s);
        return trial_entry.constraint_violation <
                   Scalar(0.9) * initial_entry.constraint_violation &&
               filter.try_add(initial_entry, trial_entry, D_ϕ_restoration, α);
      });
      auto status =
          feasibility_restoration<Scalar>(matrices, callbacks, options,
#ifdef SLEIPNIR_ENABLE_BOUND_PROJECTION
                                          bound_constraint_mask,
#endif
                                          x, s, y, z, μ, iterations);
 
      if (status != ExitStatus::SUCCESS) {
        // Report failure
        return status;
      }
 
      f = matrices.f(x);
      c_e = matrices.c_e(x);
      c_i = matrices.c_i(x);
    } else {
      // If full step was accepted, reset full-step rejected counter
      if (α == α_max) {
        full_step_rejected_counter = 0;
      }
 
      // Update iterates
      x = trial_x;
      s = trial_s;
      y = trial_y;
      z = trial_z;
 
      // A requirement for the convergence proof is that the primal-dual barrier
      // term Hessian Σₖ₊₁ does not deviate arbitrarily much from the primal
      // barrier term Hessian μSₖ₊₁⁻².
      //
      //   Σₖ₊₁ = μSₖ₊₁⁻²
      //   Sₖ₊₁⁻¹Zₖ₊₁ = μSₖ₊₁⁻²
      //   Zₖ₊₁ = μSₖ₊₁⁻¹
      //
      // We ensure this by resetting
      //
      //   zₖ₊₁ = clamp(zₖ₊₁, 1/κ_Σ μ/sₖ₊₁, κ_Σ μ/sₖ₊₁)
      //
      // for some fixed κ_Σ ≥ 1 after each step. See equation (16) of [2].
      for (int row = 0; row < z.rows(); ++row) {
        constexpr Scalar κ_Σ(1e10);
        z[row] =
            std::clamp(z[row], Scalar(1) / κ_Σ * μ / s[row], κ_Σ * μ / s[row]);
      }
 
      f = trial_f;
      c_e = trial_c_e;
      c_i = trial_c_i;
    }
 
    // Update autodiff for Jacobians and Hessian
    A_e = matrices.A_e(x);
    A_i = matrices.A_i(x);
    g = matrices.g(x);
    H = matrices.H(x, y, z);
 
    // Update the error
    E_0 = unscaled_kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
        matrices.scaling, g, A_e, c_e, A_i, c_i, s, y, z, Scalar(0));
 
    // Update the barrier parameter if necessary
    if (E_0 > Scalar(options.tolerance)) {
      // Barrier parameter scale factor for tolerance checks
      constexpr Scalar κ_ε(10);
 
      // While the error is below the desired threshold for this barrier
      // parameter value, decrease the barrier parameter further
      Scalar E_μ = kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
          g, A_e, c_e, A_i, c_i, s, y, z, μ);
      while (μ > μ_min && E_μ <= κ_ε * μ) {
        update_barrier_parameter_and_reset_filter();
        E_μ = kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(g, A_e, c_e, A_i,
                                                               c_i, s, y, z, μ);
      }
    }
 
    inner_iter_profiler.stop();
 
    if (options.diagnostics) {
      print_iteration_diagnostics(
          iterations,
          in_feasibility_restoration ? IterationType::FEASIBILITY_RESTORATION
                                     : IterationType::NORMAL,
          inner_iter_profiler.current_duration(), E_0, f,
          c_e.template lpNorm<1>() + (c_i - s).template lpNorm<1>(), s.dot(z),
          μ, solver.hessian_regularization(),
          solver.constraint_jacobian_regularization(),
          std::max(step.p_x.template lpNorm<Eigen::Infinity>(),
                   step.p_s.template lpNorm<Eigen::Infinity>()),
          std::max(step.p_y.template lpNorm<Eigen::Infinity>(),
                   step.p_z.template lpNorm<Eigen::Infinity>()),
          α, α_max, α_reduction_factor, α_z);
    }
 
    ++iterations;
 
    // Check for max iterations
    if (iterations >= options.max_iterations) {
      return ExitStatus::MAX_ITERATIONS_EXCEEDED;
    }
 
    // Check for max wall clock time
    if (std::chrono::steady_clock::now() - solve_start_time > options.timeout) {
      return ExitStatus::TIMEOUT;
    }
  }
 
  return ExitStatus::SUCCESS;
}
 
extern template SLEIPNIR_DLLEXPORT ExitStatus
interior_point(const InteriorPointMatrixCallbacks<double>& matrix_callbacks,
               std::span<std::function<bool(const IterationInfo<double>& info)>>
                   iteration_callbacks,
               const Options& options,
#ifdef SLEIPNIR_ENABLE_BOUND_PROJECTION
               const Eigen::ArrayX<bool>& bound_constraint_mask,
#endif
               Eigen::Vector<double, Eigen::Dynamic>& x);
 
}  // namespace slp