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.

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

Parameters

	func	The proximable function $h:{\mathrm{I}\mathrm{R}}^n\to {\mathrm{I}\mathrm{R}}^n$ 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 $x$. $\hat{x}\leftarrow {\mathbf{prox}}_{\gamma h}\left(x\right)$
[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.

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

Parameters

	func	The proximable function $h:{\mathrm{I}\mathrm{R}}^n\to {\mathrm{I}\mathrm{R}}^n$ 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 $x+{\gamma }_{\text{fwd}}d$. $\hat{x}\leftarrow {\mathbf{prox}}_{\gamma h}(x+{\gamma }_{\text{fwd}}d)$
[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.