Necessary Optimality Conditions for Semi-vectorial Bi-level Optimization with Convex Lower Level: Theoretical Results and Applications to the Quadratic Case

Abstract

This paper explores related aspects to post-Pareto analysis arising from the multi-criteria optimization problem. It consists of two main parts. In the first one, we give first-order necessary optimality conditions for a semi-vectorial bi-level optimization problem: the upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multi-objective optimization problem to be solved by several followers acting in a cooperative way (greatest coalition multi-players game). For the lower level, we deal with weakly or properly Pareto (efficient) solutions and we consider the so-called optimistic problem, i.e. when followers choose amongst Pareto solutions one which is the most favourable for the leader. In order to handle real-life applications, in the second part of the paper, we consider the case where each follower objective is expressed in a quadratic form. In this setting, we give explicit first-order necessary optimality conditions. Finally, some computational results are given to illustrate the paper.