Interval Optimization Problems on Hadamard manifolds

TL;DR

This paper introduces interval optimization problems on Hadamard manifolds, establishing theoretical foundations and relationships with variational inequalities, and developing new concepts for interval-valued functions on these manifolds.

Contribution

It develops new concepts of $gH$-directional derivatives and $gH$-Gâteaux differentiability for interval functions on Hadamard manifolds, advancing the theory of Riemannian interval optimization.

Findings

Established relationships between IOPs and interval variational inequalities.

Developed new concepts of derivatives for interval functions on Hadamard manifolds.

Laid groundwork for further research on Riemannian interval optimization problems.

Abstract

In this article, we introduce the interval optimization problems (IOPs) on Hadamard manifolds as well as study the relationship between them and the interval variational inequalities. To achieve the theoretical results, we build up some new concepts about $gH$-directional derivative and $gH$-G\^ateaux differentiability of interval valued functions and their properties on the Hadamard manifolds. The obtained results pave a way to further study on Riemannian interval optimization problems (RIOPs).