sqp.hpp

// Copyright (c) Sleipnir contributors
 
#pragma once
 
#include <algorithm>
#include <chrono>
#include <cmath>
#include <functional>
#include <limits>
#include <optional>
#include <span>
 
#include <Eigen/Core>
#include <Eigen/SparseCore>
#include <gch/small_vector.hpp>
 
#include "sleipnir/optimization/solver/exit_status.hpp"
#include "sleipnir/optimization/solver/iteration_info.hpp"
#include "sleipnir/optimization/solver/options.hpp"
#include "sleipnir/optimization/solver/sqp_matrix_callbacks.hpp"
#include "sleipnir/optimization/solver/util/all_finite.hpp"
#include "sleipnir/optimization/solver/util/append_as_triplets.hpp"
#include "sleipnir/optimization/solver/util/constraint_violation.hpp"
#include "sleipnir/optimization/solver/util/feasibility_restoration.hpp"
#include "sleipnir/optimization/solver/util/feasibility_restoration_cycle_detector.hpp"
#include "sleipnir/optimization/solver/util/filter.hpp"
#include "sleipnir/optimization/solver/util/kkt_error.hpp"
#include "sleipnir/optimization/solver/util/kkt_solver.hpp"
#include "sleipnir/optimization/solver/util/time_trace.hpp"
#include "sleipnir/util/assert.hpp"
#include "sleipnir/util/print_diagnostics.hpp"
#include "sleipnir/util/scope_exit.hpp"
#include "sleipnir/util/symbol_exports.hpp"
#include "sleipnir/util/time_tracer.hpp"
 
// See docs/algorithms.md#Works_cited for citation definitions.
 
namespace slp {
 
template <typename Scalar>
ExitStatus sqp(const SQPMatrixCallbacks<Scalar>& matrix_callbacks,
               std::span<std::function<bool(const IterationInfo<Scalar>& info)>>
                   iteration_callbacks,
               const Options<Scalar>& options, Eigen::VectorX<Scalar>& x) {
  using DenseVector = Eigen::VectorX<Scalar>;
 
  DenseVector y = DenseVector::Zero(matrix_callbacks.num_equality_constraints);
 
  return sqp(matrix_callbacks, iteration_callbacks, options, x, y);
}
 
template <typename Scalar>
ExitStatus sqp(const SQPMatrixCallbacks<Scalar>& matrix_callbacks,
               std::span<std::function<bool(const IterationInfo<Scalar>& info)>>
                   iteration_callbacks,
               const Options<Scalar>& options, Eigen::VectorX<Scalar>& x,
               Eigen::VectorX<Scalar>& y) {
  using DenseVector = Eigen::VectorX<Scalar>;
  using SparseMatrix = Eigen::SparseMatrix<Scalar>;
  using SparseVector = Eigen::SparseVector<Scalar>;
 
  struct Step {
    DenseVector p_x;
    DenseVector p_y;
 
    Scalar primal_inf_norm() const {
      return p_x.template lpNorm<Eigen::Infinity>();
    }
 
    Scalar dual_inf_norm() const {
      return p_y.template lpNorm<Eigen::Infinity>();
    }
 
    Scalar tiny_step_size(const DenseVector& x) const {
      // A primal step αpˣ is negligible if
      //
      //   ‖αpˣ/(1 + |x|)‖_∞ < 10ε
      //
      // where the absolute value and division are elementwise. Since α > 0, it
      // factors out of the norm.
      //
      //   α‖pˣ/(1 + |x|)‖_∞ < 10ε
      //
      // Dividing both sides by the norm gives
      //
      //   α < 10ε / ‖pˣ/(1 + |x|)‖_∞
      return Scalar(10) * std::numeric_limits<Scalar>::epsilon() /
             (p_x.array().abs() / (Scalar(1) + x.array().abs()))
                 .matrix()
                 .template lpNorm<Eigen::Infinity>();
    }
  };
 
  using std::isfinite;
 
  const auto solve_start_time = std::chrono::steady_clock::now();
 
  using enum SolveTimeTracerID;
 
  int iterations = 0;
 
  // Prints final solver diagnostics when the solver exits
  SolveDiagnostics<sqp_time_trace> diagnostics{options.diagnostics,
                                               options.diagnostics};
 
  // Set up traced matrix callbacks if diagnostics are enabled
#ifndef SLEIPNIR_DISABLE_DIAGNOSTICS
  std::optional<SQPMatrixCallbacks<Scalar>> traced_matrices;
  if (options.diagnostics) {
    traced_matrices.emplace(
        matrix_callbacks.num_decision_variables,
        matrix_callbacks.num_equality_constraints,
        traced(diagnostics[COST], matrix_callbacks.f),
        traced(diagnostics[GRADIENT], matrix_callbacks.g),
        traced(diagnostics[HESSIAN], matrix_callbacks.H),
        traced(diagnostics[HESSIAN_CONSTRAINTS], matrix_callbacks.H_c),
        traced(diagnostics[EQUALITY_CONSTRAINTS], matrix_callbacks.c_e),
        traced(diagnostics[EQUALITY_JACOBIAN], matrix_callbacks.A_e),
        matrix_callbacks.scaling);
  }
  const auto& matrices = traced_matrices ? *traced_matrices : matrix_callbacks;
#else
  const auto& matrices = matrix_callbacks;
#endif
 
  ScopedTimeTracer setup_tracer{diagnostics[SETUP]};
 
  Scalar f = matrices.f(x);
  SparseVector g = matrices.g(x);
  SparseMatrix H = matrices.H(x, y);
  DenseVector c_e = matrices.c_e(x);
  SparseMatrix A_e = matrices.A_e(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);
 
  DenseVector trial_x;
  DenseVector trial_y;
 
  Scalar trial_f;
  DenseVector trial_c_e;
 
  // 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)) {
    return ExitStatus::NONFINITE_INITIAL_GUESS;
  }
 
  Filter<Scalar> filter{constraint_violation(c_e)};
 
  // 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;
  KKTSolver<Scalar> solver{
      // Use sparse solver if lower triangle fills < 25% of system
      H.nonZeros() + A_e.nonZeros() < 0.25 * lhs_rows * lhs_rows,
      matrices.num_decision_variables, matrices.num_equality_constraints};
 
  // Step size reduction factor
  constexpr Scalar α_reduction_factor(0.5);
 
  int full_step_rejected_counter = 0;
 
  // The smallest KKT error of a step accepted by the KKT error reduction
  // fallback. Each fallback step must improve on it, so finitely many fallback
  // steps are taken before the solver converges.
  Scalar fallback_kkt_error = std::numeric_limits<Scalar>::infinity();
 
  StallDetector<Scalar> stall_detector;
 
  FeasibilityRestorationCycleDetector<Scalar>
      feasibility_restoration_cycle_detector;
 
  // Error
  Scalar E_0 = unscaled_kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
      matrices.scaling, g, A_e, c_e, y, options.tolerance);
 
  setup_tracer.stop();
 
  // Prints initial iterate diagnostics now and closes the iterate diagnostics
  // table when the solver exits
  IterateDiagnosticsTable table{
      options.diagnostics,       E_0,       f,
      constraint_violation(c_e), Scalar(0), Scalar(0)};
 
  while (E_0 > options.tolerance) {
    ScopedTimeTracer inner_iter_tracer{diagnostics[ITERATION]};
 
    // Check for diverging iterates
    if (x.template lpNorm<Eigen::Infinity>() > Scalar(1e10) || !x.allFinite()) {
      return ExitStatus::DIVERGING_ITERATES;
    }
 
    PhaseTimeTracer phase_tracer{diagnostics[CALLBACKS]};
 
    // Call iteration callbacks
    for (const auto& callback : iteration_callbacks) {
      if (callback({iterations, x, {}, y, {}, g, H, A_e, {}})) {
        return ExitStatus::CALLBACK_REQUESTED_STOP;
      }
    }
 
    phase_tracer.next(diagnostics[KKT_MATRIX_BUILD]);
 
    // lhs = [H   Aₑᵀ]
    //       [Aₑ   0 ]
    //
    // Don't assign upper triangle because solver only uses lower triangle.
    triplets.clear();
    triplets.reserve(H.nonZeros() + A_e.nonZeros());
    append_as_triplets(triplets, 0, 0, {H, 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]
    //        [   cₑ    ]
    DenseVector rhs{x.rows() + y.rows()};
    rhs.segment(0, x.rows()) = -g + A_e.transpose() * y;
    rhs.segment(x.rows(), y.rows()) = -c_e;
 
    phase_tracer.next(diagnostics[KKT_MATRIX_DECOMP]);
 
    Step step;
    constexpr Scalar α_max(1);
    Scalar α(1);
    bool call_feasibility_restoration = false;
 
    // Solve the Newton-KKT system
    //
    // [H   Aₑᵀ][ pˣ] = −[∇f − Aₑᵀy]
    // [Aₑ   0 ][−pʸ]    [   cₑ    ]
    if (solver.compute(lhs).info() != Eigen::Success) [[unlikely]] {
      return ExitStatus::FACTORIZATION_FAILED;
    }
 
    phase_tracer.next(diagnostics[KKT_SYSTEM_SOLVE]);
 
    auto compute_step = [&](Step& step) {
      // 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());
    };
    compute_step(step);
 
    phase_tracer.next(diagnostics[LINE_SEARCH]);
 
    α = α_max;
 
    const FilterEntry<Scalar> current_entry{f, c_e};
    const Scalar D_ϕ = g.transpose() * step.p_x;
 
    const Scalar α_min = filter.min_step_size(current_entry, D_ϕ);
 
    // α_min can be zero (e.g., when the constraint violation is zero), so also
    // end the line search when the step becomes negligible relative to the
    // iterate. This bounds the number of step size reductions.
    const Scalar α_tiny = step.tiny_step_size(x);
 
    // Loop until a step is accepted
    while (1) {
      trial_x = x + α * step.p_x;
      trial_y = y + α * step.p_y;
 
      trial_f = matrices.f(trial_x);
      trial_c_e = matrices.c_e(trial_x);
 
      // If f(xₖ + αpₖˣ) or cₑ(xₖ + αpₖˣ) aren't finite, reduce step size
      // immediately
      if (!isfinite(trial_f) || !trial_c_e.allFinite()) {
        // Reduce step size
        α *= α_reduction_factor;
 
        if (α < α_min || α < α_tiny) {
          call_feasibility_restoration = true;
          break;
        }
        continue;
      }
 
      // Check whether filter accepts trial iterate
      FilterEntry trial_entry{trial_f, trial_c_e};
      if (filter.try_add(current_entry, trial_entry, D_ϕ, α)) {
        // Accept step
        break;
      }
 
      Scalar prev_constraint_violation = constraint_violation(c_e);
      Scalar next_constraint_violation = constraint_violation(trial_c_e);
 
      // 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;
        DenseVector c_e_soc = c_e;
 
        Scalar soc_constraint_violation = next_constraint_violation;
 
        bool step_acceptable = false;
        for (int soc_iteration = 0; soc_iteration < 5 && !step_acceptable;
             ++soc_iteration) {
          ScopedTimeTracer soc_tracer{diagnostics[SOC]};
 
          scope_exit soc_exit{[&] {
            soc_tracer.stop();
 
            if (options.diagnostics && step_acceptable) {
              print_iterate_diagnostics(
                  iterations, IterationType::SECOND_ORDER_CORRECTION,
                  soc_tracer.current_duration(),
                  kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
                      g, A_e, trial_c_e, trial_y, options.tolerance),
                  trial_f, constraint_violation(trial_c_e), Scalar(0),
                  Scalar(0), solver.hessian_regularization(),
                  solver.constraint_jacobian_regularization(),
                  soc_step.primal_inf_norm(), soc_step.dual_inf_norm(), α_soc,
                  Scalar(1), α_reduction_factor, Scalar(1));
            }
          }};
 
          // Rebuild Newton-KKT rhs with updated constraint values.
          //
          // rhs = −[∇f − Aₑᵀy]
          //        [  cₑˢᵒᶜ  ]
          //
          // where cₑˢᵒᶜ = αc(xₖ) + c(xₖ + αpₖˣ)
          c_e_soc = α_soc * c_e_soc + trial_c_e;
          rhs.bottomRows(y.rows()) = -c_e_soc;
 
          // Solve the Newton-KKT system
          compute_step(soc_step);
 
          trial_x = x + α_soc * soc_step.p_x;
          trial_y = y + α_soc * soc_step.p_y;
 
          trial_f = matrices.f(trial_x);
          trial_c_e = matrices.c_e(trial_x);
 
          // Check whether the filter accepts trial iterate
          FilterEntry trial_entry{trial_f, trial_c_e};
          if (filter.try_add(current_entry, trial_entry, D_ϕ, α)) {
            step = soc_step;
            α = α_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 = constraint_violation(trial_c_e);
          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
      // if it may be impeding progress
      if (full_step_rejected_counter >= 4 && filter.try_reset(current_entry)) {
        continue;
      }
 
      // Reduce step size
      α *= α_reduction_factor;
 
      // If step size hit a minimum or the step became negligible, check if the
      // KKT error was reduced. If it wasn't, invoke feasibility restoration.
      if (α < α_min || α < α_tiny) {
        Scalar current_kkt_error = kkt_error<Scalar, KKTErrorType::ONE_NORM>(
            g, A_e, c_e, y, options.tolerance);
 
        trial_x = x + α_max * step.p_x;
        trial_y = y + α_max * step.p_y;
 
        trial_f = matrices.f(trial_x);
        trial_c_e = matrices.c_e(trial_x);
 
        Scalar next_kkt_error = kkt_error<Scalar, KKTErrorType::ONE_NORM>(
            matrices.g(trial_x), matrices.A_e(trial_x), trial_c_e, trial_y,
            options.tolerance);
 
        // If the step using αᵐᵃˣ reduced the KKT error below the current
        // iterate's and every previous fallback step's, accept it anyway
        if (next_kkt_error <= Scalar(0.999) * std::min(current_kkt_error,
                                                       fallback_kkt_error) &&
            constraint_violation(trial_c_e) <=
                filter.max_constraint_violation) {
          // Accept step
          fallback_kkt_error = next_kkt_error;
          α = α_max;
          break;
        }
 
        call_feasibility_restoration = true;
        break;
      }
    }
 
    phase_tracer.stop();
 
    // If solver stalled, invoke feasibility restoration
    if (!call_feasibility_restoration) {
      FilterEntry trial_entry{trial_f, trial_c_e};
      if (stall_detector.update(current_entry, trial_entry, α,
                                filter.min_constraint_violation)) {
        call_feasibility_restoration = true;
      }
    }
 
    if (call_feasibility_restoration) {
      ScopedTimeTracer feasibility_restoration_tracer{
          diagnostics[FEASIBILITY_RESTORATION]};
 
      FilterEntry initial_entry{matrices.f(x), c_e};
 
      // Augment the filter with the current iterate so the normal phase can't
      // return to it
      filter.add(initial_entry);
 
      feasibility_restoration_cycle_detector.record_entry(
          initial_entry.constraint_violation);
      const bool run_feasibility_restoration_to_convergence =
          feasibility_restoration_cycle_detector.is_cycling();
 
      // 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) {
        if (run_feasibility_restoration_to_convergence) {
          return false;
        }
 
        DenseVector trial_x =
            info.x.segment(0, matrices.num_decision_variables);
 
        DenseVector trial_c_e = matrices.c_e(trial_x);
 
        FilterEntry trial_entry{matrices.f(trial_x), trial_c_e};
 
        // If the current iterate sufficiently reduces constraint violation and
        // is accepted by the normal filter, stop feasibility restoration
        return trial_entry.constraint_violation <
                   Scalar(0.9) * initial_entry.constraint_violation &&
               filter.is_acceptable(trial_entry);
      });
      auto status = feasibility_restoration<Scalar>(matrices, callbacks,
                                                    options, x, y, iterations);
 
      if (status != ExitStatus::SUCCESS) {
        // Report failure
        return status;
      }
 
      f = matrices.f(x);
      c_e = matrices.c_e(x);
 
      // Resuming from an iterate the filter rejects would break the filter's
      // convergence guarantee, and continuing feasibility restoration from its
      // converged point would return that point again, so report failure.
      //
      // The returned iterate can be rejected because feasibility restoration
      // stops at its own convergence tolerance instead of driving constraint
      // violation to zero, so the filter can still hold entries with lower cost
      // and constraint violation.
      if (!filter.is_acceptable(FilterEntry{f, c_e})) {
        return ExitStatus::FEASIBILITY_RESTORATION_FAILED;
      }
    } 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;
      y = trial_y;
 
      f = trial_f;
      c_e = trial_c_e;
    }
 
    // Update autodiff for Jacobians and Hessian
    A_e = matrices.A_e(x);
    g = matrices.g(x);
    H = matrices.H(x, y);
 
    // Update the error
    E_0 = unscaled_kkt_error<Scalar, KKTErrorType::INF_NORM_SCALED>(
        matrices.scaling, g, A_e, c_e, y, options.tolerance);
 
    inner_iter_tracer.stop();
 
    if (options.diagnostics) {
      print_iterate_diagnostics(iterations, IterationType::NORMAL,
                                inner_iter_tracer.current_duration(), E_0, f,
                                constraint_violation(c_e), Scalar(0), Scalar(0),
                                solver.hessian_regularization(),
                                solver.constraint_jacobian_regularization(),
                                step.primal_inf_norm(), step.dual_inf_norm(), α,
                                α_max, α_reduction_factor, Scalar(1));
    }
 
    ++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;
    }
  }
 
  if (!isfinite(E_0)) {
    return ExitStatus::DIVERGING_ITERATES;
  } else {
    return ExitStatus::SUCCESS;
  }
}
 
extern template SLEIPNIR_DLLEXPORT ExitStatus
sqp(const SQPMatrixCallbacks<double>& matrix_callbacks,
    std::span<std::function<bool(const IterationInfo<double>& info)>>
        iteration_callbacks,
    const Options<double>& options, Eigen::VectorX<double>& x);
 
}  // namespace slp