Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness

TL;DR

This paper investigates the conditions under which the threshold of prox-boundedness exists, how to identify it, and how to compute it for various classes of functions, enhancing the theoretical foundation for proximal algorithms.

Contribution

It provides explicit conditions, bounds, and calculus rules for the threshold of prox-boundedness, including for sums and compositions of functions, advancing the theoretical understanding.

Findings

Explicit thresholds determined for certain functions

Bounds established for general prox-bounded functions

Calculus rules for sums and compositions of functions

Abstract

This work advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite dimensions, we study general prox-bounded functions and then focus on some useful families such as piecewise functions and Lipschitz continuous functions. The thresholds are explicitly determined when possible and bounds are established otherwise. Some calculus rules are constructed; we consider functions with known thresholds and find the thresholds of their sum and composition.