Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

Gabriel Ruiz-Garzón,Rafaela Osuna-Gómez,Beatriz Hernández-Jiménez,Antonio Rufián-Lizana

doi:10.3390/math8122196

Gabriel Ruiz-Garzón, Rafaela Osuna-Gómez + Show 2 more

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https://doi.org/10.3390/math8122196

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Abstract

This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We 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 show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

Highlights

We have extended Theorem 4.2 given by Wang et al [18], Theorem 3.5 given by Mishra and Upadhyay [17], and Theorem 4.5 given by Ruiz-Garzón et al [16] for Euclidean spaces to Hadamard manifolds with local approximate weakly efficient (AWE) points and approximate geodesic pseudoconvex (AGPCX) functions
Inspired by the work of Ngai, Luc and Thera [5], we have introduced the concepts of generalized approximate geodesic convex functions on Hadamard manifolds, and we have illustrated them with examples
Under conditions of approximate geodesic convexity functions, the solutions of the Stampacchia vector variational-like inequality problem are local approximate efficient points ( AE)

Summary

Introduction

This article is about Pareto’s approximate solutions for the unconstrained vector optimization problem on Hadamard manifolds. In practice, the best idea is not to get solutions to vector optimization problems by solving them directly but through other problems that are related to the first ones. Sometimes we have to be satisfied with finding approximate solutions to the vector optimization problem instead of the exact solutions. Those feasible points whose objective values are at a small e distance from the optimal objective vector values are considered an approximate Pareto solution. Many computer algorithms that exactly search for efficient points after finite numbers of steps only reach an approximate solution.

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