Optimality Conditions for Interval-Valued Optimization Problems on Riemannian Manifolds Under a Total Order Relation

TL;DR

This paper establishes optimality conditions for interval-valued optimization problems on Riemannian manifolds using generalized Hukuhara differentiability, covering various cases of objective functions and constraints, with theoretical justification and examples.

Contribution

It introduces KKT-type optimality conditions for interval-valued problems on Riemannian manifolds considering different function types, expanding the theoretical framework in this area.

Findings

Derived KKT-type optimality conditions for interval-valued functions.

Analyzed three cases based on the nature of objective and constraint functions.

Validated the theory with illustrative examples.

Abstract

This article explores fundamental properties of convex interval-valued functions defined on Riemannian manifolds. The study employs generalized Hukuhara directional differentiability to derive KKT-type optimality conditions for an interval-valued optimization problem on Riemannian manifolds. Based on type of functions involved in optimization problems, we consider the following cases: 1. objective function as well as constraints are real-valued; 2. objective function is interval-valued, and constraints are real-valued; 3. objective function as well as constraints are interval-valued. The whole theory is justified with the help of examples. The order relation that we use throughout the paper is a total order relation defined on the collection of all closed and bounded intervals in $\mathbb{R}$.