ShiftedProximalOperators.jl

Introduction

ShiftedProximalOperators is a library of proximal operators associated with proper lower-semicontinuous functions such as those implemented in ProximalOperators.jl for use in the algorithms implemented in RegularizedOptimization.jl.

The main differences with the proximal operators implemented in ProximalOperators.jl are that those implemented here involve a shift of the nonsmooth term and may include an extra indicator function. We also implement new proximal operators.

Proximal operators

The operators from ProximalOperators.jl can be written as

\[\mathrm{prox}_{\nu h}(q) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(t) \right\}\]

We consider a proximal operator from ProximalOperators.jl, for example ProximalOperators.NormL1.

using ProximalOperators, ShiftedProximalOperators
h = NormL1(1.0) # proximal operator

This package provides the following shifted proximal operators, where x and s are fixed shifts, h is the nonsmooth term with respect to which we are computing the proximal operator.

Basic shifted proximal operator

ψ = shifted(h, x)

models

\[\mathrm{prox}_{\nu h}(q; x) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(x + t) \right\}\]

ψ can be shifted again using

ψs = shifted(ψ, sj)

which models

\[\mathrm{prox}_{\nu h}(q; x, s) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(x + s + t) \right\}\]

Ball shifted proximal operator

Let χ(.; ΔB) be the indicator of a ball of radius Δ defined by a certain norm.

χ = NormL2(1.0) # choose the 2-norm for χ
ψ = shifted(h, x, Δ, χ) # Δ is the radius of the ball ΔB

models

\[\mathrm{prox}_{\nu h + \chi}(q; x) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(x + t) + \chi (t; \Delta \mathcal{B}) \right\}\]

ψ can be shifted again using

ψs = shifted(ψ, sj)

which models

\[\mathrm{prox}_{\nu h + \chi}(q; x, s) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(x + s + t) + \chi (s + t; \Delta \mathcal{B}) \right\}\]

Box shifted proximal operator

ψ = shifted(h, x, l, u)

models

\[\mathrm{prox}_{\nu h + \chi}(q; x) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(x + t) + \chi (t; [\ell, u]) \right\}\]

where χ(t; [ℓ, u]) is 0 if tᵢ ∈ [ℓᵢ, uᵢ] for all i ∈ [1, length(t)] and +∞ otherwise. ψ can be shifted again using

ψs = shifted(ψ, sj)

which models

\[\mathrm{prox}_{\nu h + \chi}(q; x, s) = \arg\min_t \left\{ \tfrac{1}{2}\|t-q\|^2 + \nu h(x + s + t) + \chi (s + t; [\ell, u]) \right\}\]