panoc-helpers.tpp

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#pragma once
 
#include <alpaqa/config/config.hpp>
#include <alpaqa/export.hpp>
#include <alpaqa/inner/inner-solve-options.hpp>
#include <alpaqa/inner/internal/panoc-stop-crit.hpp>
#include <alpaqa/inner/internal/solverstatus.hpp>
#include <alpaqa/problem/type-erased-problem.hpp>
#include <alpaqa/util/atomic-stop-signal.hpp>
#include <alpaqa/util/not-implemented.hpp>
 
#include <stdexcept>
 
namespace alpaqa::detail {
 
template <Config Conf>
struct PANOCHelpers {
    USING_ALPAQA_CONFIG(Conf);
    using Problem = alpaqa::TypeErasedProblem<config_t>;
    using Box     = alpaqa::Box<config_t>;
 
    /// Calculate the error between ẑ and g(x).
    /// @f[ \hat{z}^k = \Pi_D\left(g(x^k) + \Sigma^{-1}y\right) @f]
    static void calc_err_z(const Problem &p, ///< [in]  Problem description
                           crvec x̂, ///< [in]  Decision variable @f$ \hat{x} @f$
                           crvec y, ///< [in]  Lagrange multipliers @f$ y @f$
                           crvec Σ, ///< [in]  Penalty weights @f$ \Sigma @f$
                           rvec err_z ///< [out] @f$ g(\hat{x}) - \hat{z} @f$
    ) {
        if (p.get_m() == 0) /* [[unlikely]] */
            return;
 
        // g(x̂)
        p.eval_g(x̂, err_z);
        // ζ = g(x̂) + Σ⁻¹y
        err_z += Σ.asDiagonal().inverse() * y;
        // ẑ = Π(ζ, D)
        p.eval_proj_diff_g(err_z, err_z);
        // g(x) - ẑ
        err_z -= Σ.asDiagonal().inverse() * y;
        // TODO: catastrophic cancellation?
    }
 
    static bool stop_crit_requires_grad_ψx̂(PANOCStopCrit crit) {
        switch (crit) {
            case PANOCStopCrit::ApproxKKT: [[fallthrough]];
            case PANOCStopCrit::ApproxKKT2: return true;
            case PANOCStopCrit::ProjGradNorm: [[fallthrough]];
            case PANOCStopCrit::ProjGradNorm2: [[fallthrough]];
            case PANOCStopCrit::ProjGradUnitNorm: [[fallthrough]];
            case PANOCStopCrit::ProjGradUnitNorm2: [[fallthrough]];
            case PANOCStopCrit::FPRNorm: [[fallthrough]];
            case PANOCStopCrit::FPRNorm2: return false;
            case PANOCStopCrit::Ipopt: return true;
            case PANOCStopCrit::LBFGSBpp: return false;
            default:;
        }
        throw std::out_of_range("Invalid PANOCStopCrit");
    }
 
    /// Compute the ε from the stopping criterion, see @ref PANOCStopCrit.
    static real_t calc_error_stop_crit(
        const Problem &problem, ///< [in]  Problem description
        PANOCStopCrit crit,     ///< [in]  What stoppint criterion to use
        crvec pₖ,      ///< [in]  Projected gradient step @f$ \hat x^k - x^k @f$
        real_t γ,      ///< [in]  Step size
        crvec xₖ,      ///< [in]  Current iterate
        crvec x̂ₖ,      ///< [in]  Current iterate after projected gradient step
        crvec ŷₖ,      ///< [in]  Candidate Lagrange multipliers
        crvec grad_ψₖ, ///< [in]  Gradient in @f$ x^k @f$
        crvec grad_̂ψₖ, ///< [in]  Gradient in @f$ \hat x^k @f$
        rvec work_n1,  ///<       Workspace of dimension n
        rvec work_n2   ///<       Workspace of dimension n
    ) {
        switch (crit) {
            case PANOCStopCrit::ApproxKKT: {
                auto err = (1 / γ) * pₖ + (grad_ψₖ - grad_̂ψₖ);
                // These parentheses     ^^^               ^^^     are important
                // to prevent catastrophic cancellation when the step is small
                return vec_util::norm_inf(err);
            }
            case PANOCStopCrit::ApproxKKT2: {
                auto err = (1 / γ) * pₖ + (grad_ψₖ - grad_̂ψₖ);
                // These parentheses     ^^^               ^^^     are important
                // to prevent catastrophic cancellation when the step is small
                return err.norm();
            }
            case PANOCStopCrit::ProjGradNorm: {
                return vec_util::norm_inf(pₖ);
            }
            case PANOCStopCrit::ProjGradNorm2: {
                return pₖ.norm();
            }
            case PANOCStopCrit::ProjGradUnitNorm: {
                problem.eval_prox_grad_step(real_t(1), xₖ, grad_ψₖ, work_n1,
                                            work_n2);
                return vec_util::norm_inf(work_n2);
            }
            case PANOCStopCrit::ProjGradUnitNorm2: {
                problem.eval_prox_grad_step(real_t(1), xₖ, grad_ψₖ, work_n1,
                                            work_n2);
                return work_n2.norm();
            }
            case PANOCStopCrit::FPRNorm: {
                return vec_util::norm_inf(pₖ) / γ;
            }
            case PANOCStopCrit::FPRNorm2: {
                return pₖ.norm() / γ;
            }
            case PANOCStopCrit::Ipopt: {
                // work_n2 ← x̂ₖ - Π_C(x̂ₖ - ∇ψ(x̂ₖ))
                problem.eval_prox_grad_step(real_t(1), x̂ₖ, grad_̂ψₖ, work_n1,
                                            work_n2);
                auto err = vec_util::norm_inf(work_n2);
                auto n   = 2 * (ŷₖ.size() + x̂ₖ.size());
                if (n == 0)
                    return err;
                // work_n2 ← x̂ₖ - ∇ψ(x̂ₖ) - Π_C(x̂ₖ - ∇ψ(x̂ₖ))
                work_n2 -= grad_̂ψₖ;
                auto C_lagr_mult   = vec_util::norm_1(work_n2);
                auto D_lagr_mult   = vec_util::norm_1(ŷₖ);
                const real_t s_max = 100;
                const real_t s_n   = static_cast<real_t>(n);
                real_t s_d =
                    std::max(s_max, (C_lagr_mult + D_lagr_mult) / s_n) / s_max;
                return err / s_d;
            }
            case PANOCStopCrit::LBFGSBpp: {
                problem.eval_prox_grad_step(real_t(1), xₖ, grad_ψₖ, work_n1,
                                            work_n2);
                return vec_util::norm_inf(work_n2) /
                       std::fmax(real_t(1), xₖ.norm());
            }
            default:;
        }
        throw std::out_of_range("Invalid PANOCStopCrit");
    }
 
    /// Increase the estimate of the Lipschitz constant of the objective gradient
    /// and decrease the step size until quadratic upper bound or descent lemma is
    /// satisfied:
    /// @f[ \psi(x + p) \le \psi(x) + \nabla\psi(x)^\top p + \frac{L}{2} \|p\|^2 @f]
    /// The projected gradient iterate @f$ \hat x^k @f$ and step @f$ p^k @f$ are
    /// updated with the new step size @f$ \gamma^k @f$, and so are other quantities
    /// that depend on them, such as @f$ \nabla\psi(x^k)^\top p^k @f$ and
    /// @f$ \|p\|^2 @f$. The intermediate vector @f$ \hat y(x^k) @f$ can be used to
    /// compute the gradient @f$ \nabla\psi(\hat x^k) @f$ or to update the Lagrange
    /// multipliers.
    ///
    /// @return The original step size, before it was reduced by this function.
    static real_t descent_lemma(
        /// [in]  Problem description
        const Problem &problem,
        /// [in]    Tolerance used to ignore rounding errors when the function
        ///         @f$ \psi(x) @f$ is relatively flat or the step size is very
        ///         small, which could cause @f$ \psi(x^k) < \psi(\hat x^k) @f$,
        ///         which is mathematically impossible but could occur in finite
        ///         precision floating point arithmetic.
        real_t rounding_tolerance,
        /// [in]    Maximum allowed Lipschitz constant estimate (prevents infinite
        ///         loop if function or derivatives are discontinuous, and keeps
        ///         step size bounded away from zero).
        real_t L_max,
        /// [in]    Current iterate @f$ x^k @f$
        crvec xₖ,
        /// [in]    Objective function @f$ \psi(x^k) @f$
        real_t ψₖ,
        /// [in]    Gradient of objective @f$ \nabla\psi(x^k) @f$
        crvec grad_ψₖ,
        /// [in]    Lagrange multipliers @f$ y @f$
        crvec y,
        /// [in]    Penalty weights @f$ \Sigma @f$
        crvec Σ,
        /// [out]   Projected gradient iterate @f$ \hat x^k @f$
        rvec x̂ₖ,
        /// [out]   Projected gradient step @f$ p^k @f$
        rvec pₖ,
        /// [out]   Intermediate vector @f$ \hat y(\hat x^k) @f$
        rvec ŷx̂ₖ,
        /// [inout] Objective function @f$ \psi(\hat x^k) @f$
        real_t &ψx̂ₖ,
        /// [inout] Squared norm of the step @f$ \left\| p^k \right\|^2 @f$
        real_t &norm_sq_pₖ,
        /// [inout] Gradient of objective times step @f$ \nabla\psi(x^k)^\top p^k@f$
        real_t &grad_ψₖᵀpₖ,
        /// [inout] Lipschitz constant estimate @f$ L_{\nabla\psi}^k @f$
        real_t &Lₖ,
        /// [inout] Step size @f$ \gamma^k @f$
        real_t &γₖ) {
 
        real_t old_γₖ = γₖ;
        real_t margin = (1 + std::abs(ψₖ)) * rounding_tolerance;
        while (ψx̂ₖ - ψₖ > grad_ψₖᵀpₖ + real_t(0.5) * Lₖ * norm_sq_pₖ + margin) {
            if (not(Lₖ * 2 <= L_max))
                break;
 
            Lₖ *= 2;
            γₖ /= 2;
 
            // Calculate x̂ₖ and pₖ (with new step size)
            problem.eval_prox_grad_step(γₖ, xₖ, grad_ψₖ, /* in ⟹ out */ x̂ₖ, pₖ);
            // Calculate ∇ψ(xₖ)ᵀpₖ and ‖pₖ‖²
            grad_ψₖᵀpₖ = grad_ψₖ.dot(pₖ);
            norm_sq_pₖ = pₖ.squaredNorm();
 
            // Calculate ψ(x̂ₖ) and ŷ(x̂ₖ)
            ψx̂ₖ = problem.eval_ψ(x̂ₖ, y, Σ, /* in ⟹ out */ ŷx̂ₖ);
        }
        return old_γₖ;
    }
 
    /// Check all stop conditions (required tolerance reached, out of time,
    /// maximum number of iterations exceeded, interrupted by user,
    /// infinite iterate, no progress made)
    template <class ParamsT, class DurationT>
    static SolverStatus check_all_stop_conditions(
        /// [in]    Parameters including `max_iter`, `max_time` and `max_no_progress`
        const ParamsT &params,
        /// [in]    Options for the current solve
        const InnerSolveOptions<config_t> &opts,
        /// [in]    Time elapsed since the start of the algorithm
        DurationT time_elapsed,
        /// [in]    The current iteration number
        unsigned iteration,
        /// [in]    A stop signal for the user to interrupt the algorithm
        const AtomicStopSignal &stop_signal,
        /// [in]    Tolerance of the current iterate
        real_t εₖ,
        /// [in]    The number of successive iterations no progress was made
        unsigned no_progress) {
 
        auto max_time = params.max_time;
        if (opts.max_time)
            max_time = std::min(max_time, *opts.max_time);
        auto tolerance   = opts.tolerance > 0 ? opts.tolerance : real_t(1e-8);
        bool out_of_time = time_elapsed > max_time;
        bool out_of_iter = iteration == params.max_iter;
        bool interrupted = stop_signal.stop_requested();
        bool not_finite  = not std::isfinite(εₖ);
        bool converged   = εₖ <= tolerance;
        bool max_no_progress = no_progress > params.max_no_progress;
        return converged         ? SolverStatus::Converged
               : out_of_time     ? SolverStatus::MaxTime
               : out_of_iter     ? SolverStatus::MaxIter
               : not_finite      ? SolverStatus::NotFinite
               : max_no_progress ? SolverStatus::NoProgress
               : interrupted     ? SolverStatus::Interrupted
                                 : SolverStatus::Busy;
    }
 
    /// Compute the Hessian matrix of the augmented Lagrangian function multiplied
    /// by the given vector, using finite differences.
    /// @f[ \nabla^2_{xx} L_\Sigma(x, y)\, v \approx
    ///     \frac{\nabla_x L_\Sigma(x+hv, y) - \nabla_x L_\Sigma(x, y)}{h} @f]
    static void calc_augmented_lagrangian_hessian_prod_fd(
        /// [in]    Problem description
        const Problem &problem,
        /// [in]    Current iterate @f$ x^k @f$
        crvec xₖ,
        /// [in]    Lagrange multipliers @f$ y @f$
        crvec y,
        /// [in]    Penalty weights @f$ \Sigma @f$
        crvec Σ,
        /// [in]    Gradient @f$ \nabla \psi(x^k) @f$
        crvec grad_ψ,
        /// [in]    Vector to multiply with the Hessian
        crvec v,
        /// [out]   Hessian-vector product
        rvec Hv,
        ///         Dimension n
        rvec work_n1,
        ///         Dimension n
        rvec work_n2,
        ///         Dimension m
        rvec work_m) {
 
        real_t cbrt_ε = std::cbrt(std::numeric_limits<real_t>::epsilon());
        real_t h      = cbrt_ε * (1 + xₖ.norm());
        rvec xₖh      = work_n1;
        xₖh           = xₖ + h * v;
        problem.eval_grad_ψ(xₖh, y, Σ, Hv, work_n2, work_m);
        Hv -= grad_ψ;
        Hv /= h;
    }
 
    /// Estimate the Lipschitz constant of the gradient @f$ \nabla \psi @f$ using
    /// finite differences.
    static real_t initial_lipschitz_estimate(
        /// [in]    Problem description
        const Problem &problem,
        /// [in]    Current iterate @f$ x^k @f$
        crvec x,
        /// [in]    Lagrange multipliers @f$ y @f$
        crvec y,
        /// [in]    Penalty weights @f$ \Sigma @f$
        crvec Σ,
        /// [in]    Finite difference step size relative to x
        real_t ε,
        /// [in]    Minimum absolute finite difference step size
        real_t δ,
        /// [in]    Minimum allowed Lipschitz estimate.
        real_t L_min,
        /// [in]    Maximum allowed Lipschitz estimate.
        real_t L_max,
        /// [out]   @f$ \psi(x^k) @f$
        real_t &ψ,
        /// [out]   Gradient @f$ \nabla \psi(x^k) @f$
        rvec grad_ψ,
        ///         Dimension n
        rvec work_x,
        ///         Dimension n
        rvec work_grad_ψ,
        ///         Dimension n
        rvec work_n,
        ///         Dimension m
        rvec work_m) {
 
        // Calculate ψ(x₀), ∇ψ(x₀)
        ψ = problem.eval_ψ_grad_ψ(x, y, Σ, /* in ⟹ out */ grad_ψ, work_n,
                                  work_m);
        // Select a small step h for finite differences
        auto h        = grad_ψ.unaryExpr([&](real_t g) {
            return g > 0 ? std::max(g * ε, δ) : std::min(g * ε, -δ);
        });
        work_x        = x - h;
        real_t norm_h = h.norm();
        // Calculate ∇ψ(x₀ - h)
        problem.eval_grad_ψ(work_x, y, Σ, /* in ⟹ out */ work_grad_ψ, work_n,
                            work_m);
 
        // Estimate Lipschitz constant using finite differences
        real_t L = (work_grad_ψ - grad_ψ).norm() / norm_h;
        return std::clamp(L, L_min, L_max);
    }
 
    /// Estimate the Lipschitz constant of the gradient @f$ \nabla \psi @f$ using
    /// finite differences.
    static real_t initial_lipschitz_estimate(
        /// [in]    Problem description
        const Problem &problem,
        /// [in]    Current iterate @f$ x^k @f$
        crvec xₖ,
        /// [in]    Lagrange multipliers @f$ y @f$
        crvec y,
        /// [in]    Penalty weights @f$ \Sigma @f$
        crvec Σ,
        /// [in]    Finite difference step size relative to xₖ
        real_t ε,
        /// [in]    Minimum absolute finite difference step size
        real_t δ,
        /// [in]    Minimum allowed Lipschitz estimate.
        real_t L_min,
        /// [in]    Maximum allowed Lipschitz estimate.
        real_t L_max,
        /// [out]   Gradient @f$ \nabla \psi(x^k) @f$
        rvec grad_ψ,
        ///         Dimension n
        rvec work_n1,
        ///         Dimension n
        rvec work_n2,
        ///         Dimension n
        rvec work_n3,
        ///         Dimension m
        rvec work_m) {
 
        auto h        = (xₖ * ε).cwiseAbs().cwiseMax(δ);
        work_n1       = xₖ + h;
        real_t norm_h = h.norm();
        // Calculate ∇ψ(x_0 + h)
        problem.eval_grad_ψ(work_n1, y, Σ, /* in ⟹ out */ work_n2, work_n3,
                            work_m);
        // Calculate ∇ψ(x_0)
        problem.eval_grad_ψ(xₖ, y, Σ, /* in ⟹ out */ grad_ψ, work_n1, work_m);
 
        // Estimate Lipschitz constant using finite differences
        real_t L = (work_n2 - grad_ψ).norm() / norm_h;
        return std::clamp(L, L_min, L_max);
    }
};
 
// clang-format off
ALPAQA_EXPORT_EXTERN_TEMPLATE(struct, PANOCHelpers, EigenConfigd);
ALPAQA_IF_FLOAT(ALPAQA_EXPORT_EXTERN_TEMPLATE(struct, PANOCHelpers, EigenConfigf);)
ALPAQA_IF_LONGD(ALPAQA_EXPORT_EXTERN_TEMPLATE(struct, PANOCHelpers, EigenConfigl);)
ALPAQA_IF_QUADF(ALPAQA_EXPORT_EXTERN_TEMPLATE(struct, PANOCHelpers, EigenConfigq);)
// clang-format on
 
} // namespace alpaqa::detail