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Facial Approach for Constructing Stationary Points for Mathematical Programs with Cone Complementarity Constraints

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  • Javier I. Madariaga

    (North Carolina State University)

  • Héctor Ramírez

    (Universidad de Chile)

Abstract

This paper studies stationary points in mathematical programs with cone complementarity constraints (CMPCC). We begin by reviewing various formulations of CMPCC and revisiting definitions for Bouligand, proximal strong, regular strong, Wachsmuth’s strong, L-strong, weak, as well as Mordukhovich and Clarke stationary points, establishing a comprehensive framework for CMPCC. Building on key principles related to cone faces and their properties, we introduce a novel stationarity concept, facial stationarity, which naturally extends the weak stationarity condition in the CMPCC context. Finally, we analyze the hierarchical relations between these different types of stationary points.

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  • Javier I. Madariaga & Héctor Ramírez, 2025. "Facial Approach for Constructing Stationary Points for Mathematical Programs with Cone Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-27, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02562-8
    DOI: 10.1007/s10957-024-02562-8
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    References listed on IDEAS

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    1. Gerd Wachsmuth, 2015. "Mathematical Programs with Complementarity Constraints in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 480-507, August.
    2. Didier Aussel & Anton Svensson, 2019. "Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 504-520, May.
    3. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    4. Yi Zhang & Jia Wu & Liwei Zhang, 2015. "First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints," Journal of Global Optimization, Springer, vol. 63(2), pages 253-279, October.
    5. Xide Zhu & Jin Zhang & Jinchuan Zhou & Xinmin Yang, 2019. "Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 521-540, May.
    6. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
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