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Complementarity constraints as nonlinear equations: Theory and numerical experience

In: Optimization with Multivalued Mappings

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  • Sven Leyffer

    (Argonne National Laboratory)

Abstract

Summary Recently, it has been shown that mathematical programs with complementarity constraints (MPCCs) can be solved efficiently and reliably as nonlinear programs. This paper examines various nonlinear formulations of the complementarity constraints. Several nonlinear complementarity functions are considered for use in MPCC. Unlike standard smoothing techniques, however, the reformulations do not require the control of a smoothing parameter. Thus they have the advantage that the smoothing is exact in the sense that Karush-Kuhn-Tucker points of the reformulation correspond to strongly stationary points of the MPCC. A new exact smoothing of the well-known min function is also introduced and shown to possess desirable theoretical properties. It is shown how the new formulations can be integrated into a sequential quadratic programming solver, and their practical performance is compared on a range of test problems.

Suggested Citation

  • Sven Leyffer, 2006. "Complementarity constraints as nonlinear equations: Theory and numerical experience," Springer Optimization and Its Applications, in: Stephan Dempe & Vyacheslav Kalashnikov (ed.), Optimization with Multivalued Mappings, pages 169-208, Springer.
    • Handle: RePEc:spr:spochp:978-0-387-34221-4_9
    DOI: 10.1007/0-387-34221-4_9
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    Citations

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    Cited by:

    1. Sven Leyffer, 2009. "A Complementarity Constraint Formulation of Convex Multiobjective Optimization Problems," INFORMS Journal on Computing, INFORMS, vol. 21(2), pages 257-267, May.
    2. Patrick Mehlitz, 2020. "A comparison of solution approaches for the numerical treatment of or-constrained optimization problems," Computational Optimization and Applications, Springer, vol. 76(1), pages 233-275, May.
    3. Jacquet, Quentin & van Ackooij, Wim & Alasseur, Clémence & Gaubert, Stéphane, 2024. "Quadratic regularization of bilevel pricing problems and application to electricity retail markets," European Journal of Operational Research, Elsevier, vol. 313(3), pages 841-857.

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