Optimality conditions and subdifferential calculus for infinite sums of functions

TL;DR

This paper develops new optimality conditions and subdifferential calculus for infinite sums of functions, removing traditional Lipschitz assumptions and introducing fuzzy subdifferential rules for local minima.

Contribution

It extends decoupling techniques and subdifferential calculus to infinite function collections, providing fuzzy rules and weaker continuity conditions.

Findings

Fuzzy subdifferential necessary conditions for local minima.

Sum rules without Lipschitz continuity assumptions.

Introduction of quasi uniform lower semicontinuity concepts.

Abstract

The paper extends the widely used in optimisation theory decoupling techniques to infinite collections of functions. Extended concepts of uniform lower semicontinuity and firm uniform lower semicontinuity are discussed. The main theorems give fuzzy subdifferential necessary conditions (multiplier rules) for a local minimum of the sum of an infinite collection of functions and fuzzy subdifferential sum rules without the traditional Lipschitz continuity assumptions. More subtle "quasi" versions of the uniform infimum and uniform lower semicontinuity properties are also discussed.