Functions and operators

group grp_Functions

(Proximable) functions and operators.

Variables

struct alpaqa::prox_fn prox

Compute the proximal mapping.

$$\begin{array}{rl}\mathrm{out}& \leftarrow {\mathbf{prox}}_{\gamma \mathrm{func}}\left(\mathrm{in}\right).\end{array}$$

Param func:

The proximable function $h:{\mathrm{I}\mathrm{R}}^n\to {\mathrm{I}\mathrm{R}}^n$ to apply the proximal mapping of.

Param in:

[in] Input vector or matrix $x$, e.g. current iterate.

Param out:

[out] Proximal mapping of $(\gamma h)$ at $x$. $\hat{x}\leftarrow {\mathbf{prox}}_{\gamma h}\left(x\right)$

Param γ:

[in] Proximal step size $\gamma$.

Return:

The value of the function evaluated in the output, $h(\hat{x})$.

struct alpaqa::prox_step_fn prox_step

Compute a generalized forward-backward step.

$$\begin{array}{r}\begin{array}{rl}\mathrm{out}& \leftarrow {\mathbf{prox}}_{\gamma \mathrm{func}}(\mathrm{in}+{\gamma }_{\mathrm{fwd}}\mathrm{fwd}\mathrm{\_}\mathrm{step})\\ \mathrm{fb}\mathrm{\_}\mathrm{step}& \leftarrow \mathrm{out}-\mathrm{in}.\end{array}\end{array}$$

This function can be used to implement the TypeErasedProblem::eval_prox_grad_step function:

struct Problem {
    alpaqa::functions::NuclearNorm<config_t> h{λ, rows, cols};
    real_t eval_prox_grad_step(real_t γ, crvec x, crvec grad_ψ, rvec x̂, rvec p) const {
        return alpaqa::prox_step(h, x, grad_ψ, x̂, p, γ, -γ);
    }
};

Note the negative sign for the forward step size.

Param func:

The proximable function $h:{\mathrm{I}\mathrm{R}}^n\to {\mathrm{I}\mathrm{R}}^n$ to apply the proximal mapping of.

Param in:

[in] Input vector or matrix $x$, e.g. current iterate.

Param fwd_step:

[in] Step $d$ to add to $x$ before computing the proximal mapping. Scaled by ${\gamma }_{\text{fwd}}$.

Param out:

[out] Proximal mapping of $(\gamma h)$ at $x+{\gamma }_{\text{fwd}}d$. $\hat{x}\leftarrow {\mathbf{prox}}_{\gamma h}(x+{\gamma }_{\text{fwd}}d)$

Param fb_step:

[out] Forward-backward step $p$. $p=\hat{x}-\hat{x}$

Param γ:

[in] Proximal step size $\gamma$.

Param γ_fwd:

[in] Forward step size ${\gamma }_{\mathrm{fwd}}$.

Return:

The value of the function evaluated in the output, $h(\hat{x})$.

template<Config Conf, class Weight = typename Conf::real_t>
struct L1Norm

#include <alpaqa/functions/l1-norm.hpp>

ℓ₁-norm.

Template Parameters:

Weight – Type of weighting factors. Either scalar or vector.

Public Types

using weight_t = Weight

Public Functions

inline L1Norm(weight_t λ)

inline L1Norm()

L1Norm() = default

inline real_t prox(crmat in, rmat out, real_t γ = 1)

Public Members

weight_t λ

Public Static Attributes

static constexpr bool scalar_weight = std::is_same_v<weight_t, real_t>

Friends

inline friend real_t alpaqa_tag_invoke(tag_t<alpaqa::prox>, L1Norm &self, crmat in, rmat out, real_t γ)

template<Config Conf, class Weight = typename Conf::real_t>
struct L1NormComplex

#include <alpaqa/functions/l1-norm.hpp>

ℓ₁-norm of complex numbers.

Template Parameters:

Weight – Type of weighting factors. Either scalar or vector.

Public Types

using weight_t = Weight

Public Functions

inline L1NormComplex(weight_t λ)

inline L1NormComplex()

L1NormComplex() = default

inline real_t prox(crcmat in, rcmat out, real_t γ = 1)

inline real_t prox(crmat in, rmat out, real_t γ = 1)

Note: a complex vector in ℂⁿ is represented by a real vector in ℝ²ⁿ.

Public Members

weight_t λ

Public Static Attributes

static constexpr bool scalar_weight = std::is_same_v<weight_t, real_t>

Friends

inline friend real_t alpaqa_tag_invoke(tag_t<alpaqa::prox>, L1NormComplex &self, crmat in, rmat out, real_t γ)

template<Config Conf, class SVD = DefaultSVD<Conf>>
struct NuclearNorm

#include <alpaqa/functions/nuclear-norm.hpp>

Nuclear norm (ℓ₁-norm of singular values).

Public Functions

inline NuclearNorm(real_t λ = 1)

Construct without pre-allocation.

inline NuclearNorm(real_t λ, length_t rows, length_t cols)

Construct with pre-allocation.

inline real_t prox(crmat in, rmat out, real_t γ = 1)

Public Members

real_t λ

length_t rows = 0

length_t cols = 0

SVD svd

vec singular_values

Friends

inline friend real_t alpaqa_tag_invoke(tag_t<alpaqa::prox>, NuclearNorm &self, crmat in, rmat out, real_t γ)