Alternative theorem for sequences of functions and applications to optimisation

TL;DR

This paper introduces a new theorem for sequences of functions, extending optimality conditions to problems with countably many constraints using Dini differentiability, applicable even when functions lack Gâteaux differentiability.

Contribution

It extends first-order optimality conditions to countable constraints using Dini differentiability, broadening applicability beyond Gâteaux differentiability.

Findings

Extended optimality conditions to countable constraints

Demonstrated applicability with non-Gâteaux differentiable functions

Provided examples with finite and countable constraints

Abstract

We present a new alternative theorems for sequences of functions. As applications, we extend recent results in the literature related to first-order necessary conditions for optimality problems. Our contributions involve extending well-known results, previously established for a finite number of inequality constraints to a countable number of inequality constraints. This extension is achieved using the Dini differentiability concept which is more general than Fr\'echet or G\^ateaux differentiability. We will illustrate our results by giving examples of optimisation problems with a finite or countable number of inequality constraints where the functions are not Gateaux-differentiable but only upper Dini-differentiable.