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Optimization problems with equilibrium constraints and their numerical solution

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Abstract.

We consider a class of optimization problems with a generalized equation among the constraints. This class covers several problem types like MPEC (Mathematical Programs with Equilibrium Constraints) and MPCC (Mathematical Programs with Complementarity Constraints). We briefly review techniques used for numerical solution of these problems: penalty methods, nonlinear programming (NLP) techniques and Implicit Programming approach (ImP). We further present a new theoretical framework for the ImP technique that is particularly useful in case of difficult equilibria. Finally, three numerical examples are presented: an MPEC that can be solved by ImP but can hardly be formulated as a nonlinear program, an MPCC that cannot be solved by ImP and finally an MPEC solvable by both, ImP and NLP techniques. In the last example we compare the efficiency of the two approaches.

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Authors and Affiliations

  1. Institute of Applied Mathematics, University of Erlangen, Martensstr. 3, 91058, Erlangen, Germany

    Michal Kočvara

  2. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 18208, Praha 8, Czech Republic

    Jiří V. Outrata

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  1. Michal Kočvara
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  2. Jiří V. Outrata
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Correspondence to Michal Kočvara.

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On leave from the Academy of Sciences of the Czech Republic.

Mathematics Subject Classification (2000):49J40, 49J52, 90C30, 90C33

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Kočvara, M., Outrata, J. Optimization problems with equilibrium constraints and their numerical solution. Math. Program., Ser. A 101, 119–149 (2004). https://doi.org/10.1007/s10107-004-0539-2

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