Rachid El Idrissi, El Mostafa Kalmoun, Lahoussine Lafhim, Approximate KKT conditions for generalized bilevel optimization with a variational-inequality lower level

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DOI: 10.23952/jnva.10.2026.4.07

Volume 10, Issue 4, 1 August 2026, Pages 829-844

Abstract. We study necessary optimality conditions for nonsmooth generalized bilevel optimization problems in which the lower level is a variational inequality. Using a gap-function reformulation and partial calmness, we derive Karush–Kuhn–Tucker (KKT) type conditions via the basic (limiting) subdifferential. We then introduce an approximate KKT (AKKT) framework tailored to this nonsmooth, hierarchical setting and prove that every KKT point is an AKKT point, while the converse may fail in general, as demonstrated by a counterexample. Under natural regularity assumptions, including local Lipschitz continuity of the gap function and a nondegeneracy condition on the argmax set, we recover the equivalence AKKT $\Rightarrow$ KKT. A model example illustrates the applicability of the approach and the sharpness of the assumptions. Our results position AKKT conditions as a robust tool for bilevel analysis and computation when smoothness or classical constraint qualifications are unavailable.

How to Cite this Article:
R. El Idrissi, E.M. Kalmoun, L. Lafhim, Approximate KKT conditions for generalized bilevel optimization with a variational-inequality lower level, J. Nonlinear Var. Anal. 10 (2026), 829-844.