Global efficiency for multiobjective bilevel programming problems under generalized invexity

Abstract

In this paper, we consider a nonlinear optimistic bilevel programming problem \(({\textit{NMBP}})\) where the upper-level is a vector optimization problem and the lower-level is a scalar optimization problem. By using the Karush–Kuhn–Tucker conditions associated to the lower-level problem, we reformulate the bilevel programming problem into a nonlinear multiobjective single-level programming problem with equality and inequality constraints \(({\textit{MP}})\). Similarly to Dempe and Dutta (2012), we establish relationships between the problems \(({\textit{NMBP}})\) and \(({\textit{MP}})\). We prove that under appropriate constraint qualification and convexity assumptions, global (weakly or properly) efficient solutions of \(({\textit{MP}})\) correspond to global (weakly or properly) efficient solutions of \(({\textit{NMBP}})\). We establish Fritz John type necessary efficiency conditions for \(({\textit{NMBP}})\) without using any constraint qualification. Furthermore, we obtain (Fritz John type) sufficient efficiency conditions for a feasible point of \(({\textit{MP}})\) corresponds to a (weakly or properly) efficient solution for the bilevel problem \(({\textit{NMBP}})\) under various forms of generalized invexity and infineness. Moreover, a linear multiobjective bilevel programming problem is studied and sufficient efficiency conditions are derived. To illustrate the obtained results some examples are given.