Optimality Conditions for Interval-Valued Optimization Problems on Riemannian Manifolds Under a Total Order Relation
Hilal Ahmad Bhat, Akhlad Iqbal, Mahwash Aftab
TL;DR
The paper develops KKT-type optimality conditions for interval-valued optimization problems on Riemannian manifolds by leveraging a total interval order and generalized Hukuhara directional differentiability. It introduces cw-convexity for interval-valued functions and proves that geodesic-based convexity along manifolds is equivalent to IVF convexity, including epigraph-based characterizations. It derives unconstrained and constrained optimality results for real-valued and interval-valued objectives and constraints, with multiplier-based conditions that generalize Euclidean-space results to nonlinear manifolds. The framework is demonstrated through examples and highlights potential applications in machine learning and AI, while outlining future directions in duality, saddle-point criteria, and constraint qualifications on nonlinear spaces.
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 $\mathbb{R}$.