Constant Along Primal Rays Conjugacies and Generalized Convexity for Functions of the Support

TL;DR

This paper introduces a new generalized convexity framework for functions of the support of vectors, using a novel coupling called Capra, which overcomes limitations of Fenchel conjugacy for 0-homogeneous functions.

Contribution

It defines the Capra coupling and proves that nondecreasing functions of support are Capra-convex, revealing a hidden convexity and variational formulation for these functions.

Findings

Capra coupling generalizes Fenchel conjugacy for support functions.

Nondecreasing support functions are Capra-convex and can be expressed via convex functions on the sphere.

Normalized support functions admit a variational formulation involving generalized dual norms.

Abstract

The support of a vector in R d is the set of indices with nonzero entries. Functions of the support have the property to be 0-homogeneous and, because of that, the Fenchel conjugacy fails to provide relevant analysis. In this paper, we define the coupling Capra between R d and itself by dividing the classic Fenchel scalar product coupling by a given (source) norm on R d. Our main result is that, when both the source norm and its dual norm are orthant-strictly monotonic, any nondecreasing finite-valued function of the support mapping is Capra-convex, that is, is equal to its Capra-biconjugate (generalized convexity). We also establish that any such function is the composition of a proper convex lower semi continuous function on R d with the normalization mapping on the unit sphere (hidden convexity), and that, when normalized, it admits a variational formulation, which involves a family of generalized local-K-support dual norms.