Multiobjective Convex Optimization in Real Banach Space

Kin Keung Lai,Jitendra Kumar Maurya,Shashi Kant Mishra,Mohd Hassan,Sanjeev Kumar Singh

doi:10.3390/sym13112148

Kin Keung Lai, Jitendra Kumar Maurya + Show 3 more

Open Access

https://doi.org/10.3390/sym13112148

Copy DOI

Journal: Symmetry	Publication Date: Nov 10, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Shaanxi Normal University, Banaras Hindu University

Abstract

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn–Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019.

Highlights

Barbu and Precupanu [18] studied the saddle point optimality conditions of convex optimization problem for real Banach space
Consider the general multiobjective optimization problem (MOP) min f (x) = ( f1(x), · · ·, fp(x)), subject to g(x) 0, h(x) = 0, (1)where the functions f : X → Rp, g : X → Rq, and h : X → Rr are real vector valued functions and X is real Banach space.Multiobjective optimization problem (MOP) arises when two or more objective functions are simultaneously optimized over a feasible region
Motivated by the work of Barbu and Precupanu [18], Rooyen et al [14] and Wendell and Lee [12], we extend the results related to saddle point optimality conditions and Karush–Kuhn–Tucker optimality conditions from single objective function to multiobjective function with the help of Slater’s constraint qualifications [13]

Summary

Introduction

Barbu and Precupanu [18] studied the saddle point optimality conditions of convex optimization problem for real Banach space. Motivated by the work of Barbu and Precupanu [18], Rooyen et al [14] and Wendell and Lee [12], we extend the results related to saddle point optimality conditions and Karush–Kuhn–Tucker optimality conditions from single objective function to multiobjective function with the help of Slater’s constraint qualifications [13]. We establish the relationship between the Pareto solution and the saddle point for the Lagrange function using Slater’s constraint qualification.

Results

Conclusion