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On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory

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Abstract

We consider a stochastic mathematical program with equilibrium constraints (SMPEC) and show that, under certain assumptions, global optima and stationary solutions are robust with respect to changes in the underlying probability distribution. In particular, the discretization scheme sample average approximation (SAA), which is convergent for both global optima and stationary solutions, can be combined with the robustness results to motivate the use of SMPECs in practice. We then study two new and natural extensions of the SMPEC model. First, we establish the robustness of global optima and stationary solutions to an SMPEC model where the upper-level objective is the risk measure known as conditional value-at-risk (CVaR). Second, we analyze a multiobjective SMPEC model, establishing the robustness of weakly Pareto optimal and weakly Pareto stationary solutions. In the accompanying paper (Cromvik and Patriksson, Part 2, J. Optim. Theory Appl., 2010, to appear) we present applications of these results to robust traffic network design and robust intensity modulated radiation therapy.

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Authors and Affiliations

  1. Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, University of Gothenburg, 412 96, Gothenburg, Sweden

    C. Cromvik & M. Patriksson

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  1. C. Cromvik
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  2. M. Patriksson
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Correspondence to M. Patriksson.

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Communicated by F. Giannessi.

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Cromvik, C., Patriksson, M. On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory. J Optim Theory Appl 144, 461–478 (2010). https://doi.org/10.1007/s10957-009-9639-8

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