Approximate Karush-Kuhn-Tucker condition for multi-objective optimistic bilevel programming problems
Jinman Lv, Zhenhua Peng, Zhongping Wan · Journal of Industrial and Management Optimization · 2023
The approximate Karush-Kuhn-Tucker (AKKT) condition is introduced being a necessary condition of the local weak efficient solution for optimistic bilevel optimization problems with multiple objectives in upper-level problems. We transform the multi-objective bilevel optimization problem into single-level multi-objective optimization problem by means of the value function transformation or the KKT transformation. We then prove that the AKKT condition is necessary for the point to be a local weak efficient solution without any constraint qualification for the transformed one-level problem. Besides, we give examples to show that the bilevel problem has no KKT point or the lower-level problem violates the Slater CQ, but may have an AKKT point, and we introduce some suitable constraint qualifications that can ensure that the AKKT condition implies the KKT condition. Finally, numerical results are given to show the AKKT conditions' necessity.