Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.8.2026.2.07
Volume 8, Issue 2, 1 August 2026, Pages 271-293

 

Abstract. This paper studies directional optimality conditions for mathematical programs with equilibrium constraints (MPECs), a class of optimization problems with constraints of equilibrium type such as complementarity conditions. Due to the non-smooth and non-convex nature of MPECs, classical optimality conditions are usually not applicable. To overcome these difficulties, we introduce a directional analysis framework that provides a more accurate characterization of stationarity and optimality. We first present the basic building blocks of variational analysis, which are needed for directional analysis. We next derive general directional necessary optimality conditions for a non-linear optimization problem. Building on this foundation, we develop and compare several notions of directional stationarity, namely DW-, DM-, DC-, and DS-stationarity, emphasizing their theoretical distinctions and practical implications. Finally, we provide directional sufficient conditions that ensure global optimality, thus finishing a full optimality framework for MPECs. The findings present an insightful view of the solution structure of MPECs and provide a theoretical basis for future algorithmic developments.

 

How to Cite this Article:
L. Lahoussine, R. El Idrissi, Y. El-Yahyaoui, A theoretical study on directional necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints, J. Appl. Numer. Optim. 8 (2026), 271-293.