From perspective maps to epigraphical projections

TL;DR

This paper develops a variational analysis framework for projecting onto epigraphs and level sets of convex functions using root-finding of scalar equations involving proximal operators, with applications to optimization.

Contribution

It introduces a unified variational analysis approach for epigraphical projections, extending properties of the proximal map to a broader class of convex optimization problems.

Findings

Derived properties like Lipschitz continuity, differentiability, and semismoothness of the solution map.

Established an SC^1 optimization framework for projections.

Numerical experiments demonstrate the approach's effectiveness.

Abstract

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map -- including local Lipschitz continuity, directional differentiability, and semismoothness -- are derived. An SC$^1$ optimization framework for computing epigraphical and level-set projections is thus established. Numerical experiments on 1-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms