Optimality Conditions and Duality for Multiobjective Fractional Bilevel Optimization Problems
Felipe Lara, Rishabh Pandey, Vinay Singh
TL;DR
This work tackles multiobjective bilevel optimization with fractional objectives in a nondifferentiable setting. It reformulates the bilevel problem $(A)$ into a single-level problem $(A^*)$ using an auxiliary function $\Psi$ and derives refined necessary and sufficient first-order conditions based on directional convexificators $\partial^{u}_D$ under a $\partial_D$-nonsmooth Abadie-type constraint qualification. A Mond-Weir dual is developed and shown to satisfy weak and strong duality under directional generalized convexity assumptions (pseudoconvexity and quasiconvexity); the theory is supported by illustrative examples. The framework accommodates discontinuities and non-Lipschitz data, extending existing results for fractional bilevel programs. Overall, the results provide practical optimality tests and dual formulations for complex hierarchical problems with ratio-based objectives.
Abstract
This paper studies a multiobjective bilevel optimization problem where each objective is a fractional function. By reformulating the problem into a single-level one, we establish refined necessary and sufficient optimality conditions. These results are derived using ${\partial}_D$-nonsmooth Abadie-type constraint qualifications and generalized convexity concepts (quasiconvexity and pseudoconvexity) based on directional convexificators. We also prove weak and strong duality theorems for a Mond-Weir dual problem formulated with directional convexificators. Finally, several examples are provided to illustrate the advantages of our approach.