Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

Version 1 : Received: 5 December 2020 / Approved: 7 December 2020 / Online: 7 December 2020 (07:09:52 CET)

How to cite: Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A.; Hernández-Jiménez, B. Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions. Preprints 2020, 2020120124 (doi: 10.20944/preprints202012.0124.v1). Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A.; Hernández-Jiménez, B. Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions. Preprints 2020, 2020120124 (doi: 10.20944/preprints202012.0124.v1).

Abstract

This article has two objectives. Firstly, we will use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of Vector Optimization Problem within the novel field of the Hadamard manifolds. Previously, we will introduce the concepts of generalized approximate geodesic convex functions and illustrate them with examples. We will see the minimum requirements under which critical points, solutions of Stampacchia and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we will show an economical application, using again solutions of the variational problems to identify with Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

Subject Areas

Generalized convexity; Hadamard manifold; Approximate efficient solution; Stackelberg equilibrium point

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