Interval-Valued Optimization Problems for Strongly LU-E-Invex and Strongly LU-E-Preinvex Functions

TL;DR

This paper introduces new classes of interval-valued functions, explores their properties, and applies them to nonlinear programming, establishing conditions under which local minima are global and linking to KKT optimality.

Contribution

It defines and analyzes strongly LU-E-preinvex and related functions, extending invexity concepts to interval-valued functions and applying these to optimization problems.

Findings

Established relationships between SLUEP and PSLUEP functions

Proved conditions under which local minima are global in SLUEP-based problems

Validated theoretical results with examples and counterexamples

Abstract

In this paper, we introduce and explore the concepts of strongly LU-E-preinvex (SLUEP), pseudo strongly LU-E-preinvex (PSLUEP) and strongly LU-E-invex (SLUEI) functions. To illustrate and validate these definitions, we provide several non-trivial examples. Additionally, we extend the idea of strongly-G invex sets to the context of interval-valued functions. The epigraph of a SLUEP function is derived, and a relationship between SLUEP and PSLUEP functions have been explored. A key contribution of this work is the identification of a significant connection between weakly-strongly E-invex functions and SLUEP functions. As an application, we consider a nonlinear programming problem involving SLUEP functions. Under certain conditions, we prove that a local minimum of the problem is also a global minimum. Moreover, the sufficiency of Karush-Kuhn-Tucker (KKT) optimality conditions by considering the objective and constraint functions are SLUEI and SEI respectively. The theoretical results are validated through illustrative examples and counterexamples.