On Stability of the MPCC Feasible Set

Jongen, Hubertus Th.;Shikhman, Vladimir;Ruckmann, Jan-J.
(2009) SIAM Journal on Optimization — Vol. 20, n° 3, p. 1171-1184 (2009)

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  • Jongen, Hubertus Th.RWTH Aachen University
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  • Shikhman, VladimirUCLouvain
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  • Ruckmann, Jan-J.The University of Birmingham
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Abstract
The feasible set of mathematical programs with complementarity constraints (MPCC) is considered. We discuss local stability of the feasible set with respect to perturbations (up to first order) of the defining functions. Here, stability refers to homeomorphy invariance under small perturbations. For stability we propose a kind of Mangasarian-Fromovitz Condition (MFC) and its stronger version (SMFC). MFC is a natural Constraint Qualification for C-stationarity and SMFC is a generalization of the well-known Clarke's maximal rank condition. It turns out that SMFC implies local stability. MFC and SMFC coincide in case that the number of complementarity constraints (k) equals to the dimension of the state space (n). Moreover, the equivalence of MFC and SMFC is also proven for the cases k=2 as well as under Linear Independence Constraint Qualification (LICQ) for MPCC.
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Jongen, H. Th., Shikhman, V., & Ruckmann, J.-J. (2009). On Stability of the MPCC Feasible Set. SIAM Journal on Optimization, 20(3), 1171-1184. https://doi.org/10.1137/08072694X (Original work published 2009)