Optimality conditions and duality results for a robust bi-level programming problem

¹ School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, Punjab, India
² Department of Mathematics, Interdisciplinary Research Center for Intelligent Secure Systems, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

^* Corresponding author: nkailey@thapar.edu

Received: 8 September 2022
Accepted: 22 February 2023

Abstract

Robust bi-level programming problems are a newborn branch of optimization theory. In this study, we have considered a bi-level model with constraint-wise uncertainty at the upper-level, and the lower-level problem is fully convex. We use the optimal value reformulation to transform the given bi-level problem into a single-level mathematical problem and the concept of robust counterpart optimization to deal with uncertainty in the upper-level problem. Necessary optimality conditions are beneficial because any local minimum must satisfy these conditions. As a result, one can only look for local (or global) minima among points that hold the necessary optimality conditions. Here we have introduced an extended non-smooth robust constraint qualification (RCQ) and developed the KKT type necessary optimality conditions in terms of convexifactors and subdifferentials for the considered uncertain two-level problem. Further, we establish as an application the robust bi-level Mond-Weir dual (MWD) for the considered problem and produce the duality results. Moreover, an example is proposed to show the applicability of necessary optimality conditions.