Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

Jutamas Kerdkaew; Rabian Wangkeeree; Rattanaporn Wangkeeree

doi:10.3934/naco.2021053

Article Contents

2022, Volume 12, Issue 1: 93-107. Doi: 10.3934/naco.2021053

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Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

1.
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2.
Department of Mathematics, Faculty of Science, Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand

^* Corresponding author
^* Corresponding author

Received: March 2020

Revised: October 2021

Early access: November 2021

Published: March 2022

This research was supported by Naresuan University, Grant No. R2563C024

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Abstract

In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [1,2] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.

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Mathematics Subject Classification: Primary: 90C26, 90C30, 90C46.

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