Mond-Weir Duality and Optimality Conditions for Nonsmooth Bilevel Optimization Under Uncertainty

Abstract

In this paper, we study a class of nondifferentiable bilevel optimization problems in which uncertainty is incorporated through both the upper- and lower-level constraints. By utilizing an optimal value reformulation, we reduce the original hierarchical model to an equivalent single-level nonsmooth optimization problem. Under the assumptions that the objective function is ∂_c-pseudoconvex and the constraints are ∂_c-quasiconvex, both characterized using Clarke subdifferentials, we derive sufficient optimality conditions for the reformulated problem. Moreover, we develop a Mond-Weir-type dual corresponding to the original bilevel model and derive several duality results under the same generalized convexity framework. To demonstrate the practical relevance of our theoretical contributions, we provide numerical examples of nonsmooth bilevel optimization problems in which uncertainty affects both the upper-level and lower-level constraints.