Perspective Functions with Nonlinear Scaling

TL;DR

This paper extends the classical perspective function by replacing the linear scaling variable with a nonlinear term, broadening its applicability in convex analysis and related fields.

Contribution

It introduces a generalized perspective function with nonlinear scaling, providing new theoretical insights and closed-form expressions in locally convex spaces.

Findings

Generates lower semicontinuous convex functions under broad conditions

Establishes various convex-analytical properties

Provides multiple applications of the generalized perspective

Abstract

The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented.