Polar Convolution

TL;DR

This paper introduces the polar envelope, an operation analogous to the Moreau envelope, specifically designed for gauge functions, and explores its properties to aid in developing algorithms for gauge optimization.

Contribution

It develops the theory of the polar envelope for gauge functions, mirroring key properties of the Moreau envelope to facilitate optimization algorithms.

Findings

Polar envelope shares properties with the Moreau envelope, such as smoothness and proximal map continuity.

The paper establishes tools for algorithm development in gauge optimization.

Properties of the polar envelope support duality and algorithm construction.

Abstract

The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an analogous convolution operation, called the polar envelope, specialized to gauge functions. Many important properties of the Moreau envelope and the proximal map are mirrored by the polar envelope and its corresponding proximal map. These properties include smoothness of the envelope function, uniqueness and continuity of the proximal map, which play important roles in duality and in the construction of algorithms for gauge optimization. A suite of tools with which to build algorithms for this family of optimization problems is thus established.