Functions and operators

Detailed Description

(Proximable) functions and operators.

Classes
struct	L1Norm< Conf, Weight >
	ℓ₁-norm. More...
struct	L1NormComplex< Conf, Weight >
	ℓ₁-norm of complex numbers. More...
struct	NuclearNorm< Conf, SVD >
	Nuclear norm (ℓ₁-norm of singular values). More...

Variables
struct alpaqa::prox_fn	prox
	Compute the proximal mapping.
struct alpaqa::prox_step_fn	prox_step
	Compute a generalized forward-backward step.

Variable Documentation

prox

struct alpaqa::prox_fn prox

Compute the proximal mapping.

\[ \begin{aligned} \mathrm{out} &\leftarrow \prox_{\gamma\, \mathrm{func}} \left( \mathrm{in} \right). \end{aligned} \]

Parameters

	func	The proximable function \( h : \Rn \to \Rn \) to apply the proximal mapping of.
[in]	in	Input vector or matrix \( x \), e.g. current iterate.
[out]	out	Proximal mapping of \( (\gamma\, h) \) at \( x \). \( \hat x \leftarrow \prox_{\gamma\, h}\left( x \right) \)
[in]	γ	Proximal step size \( \gamma \).

Returns

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

prox_step

struct alpaqa::prox_step_fn prox_step

Compute a generalized forward-backward step.

\[ \begin{aligned} \mathrm{out} &\leftarrow \prox_{\gamma\, \mathrm{func}} \left( \mathrm{in} + \gamma_\mathrm{fwd}\, \mathrm{fwd\_step} \right) \\ \mathrm{fb\_step} &\leftarrow \mathrm{out} - \mathrm{in}. \end{aligned} \]

Parameters

	func	The proximable function \( h : \Rn \to \Rn \) to apply the proximal mapping of.
[in]	in	Input vector or matrix \( x \), e.g. current iterate.
[in]	fwd_step	Step \( d \) to add to \( x \) before computing the proximal mapping. Scaled by \( \gamma_\text{fwd} \).
[out]	out	Proximal mapping of \( (\gamma\, h) \) at \( x + \gamma_\text{fwd}\, d \). \( \hat x \leftarrow \prox_{\gamma\, h}\left( x + \gamma_\text{fwd}\, d \right) \)
[out]	fb_step	Forward-backward step \( p \). \( p = \hat x - \hat x \)
[in]	γ	Proximal step size \( \gamma \).
[in]	γ_fwd	Forward step size \( \gamma_\mathrm{fwd} \).

Returns

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

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.

Examples

C++/Advanced/lasso-fbs.cpp.