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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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  • C. Cromvik

    (Chalmers University of Technology and Mathematical Sciences, University of Gothenburg)

  • M. Patriksson

    (Chalmers University of Technology and Mathematical Sciences, University of Gothenburg)

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.

Suggested Citation

  • C. Cromvik & M. Patriksson, 2010. "On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 461-478, March.
    • Handle: RePEc:spr:joptap:v:144:y:2010:i:3:d:10.1007_s10957-009-9639-8
    DOI: 10.1007/s10957-009-9639-8
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    1. C. Cromvik & M. Patriksson, 2010. "On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 2: Applications," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 479-500, March.
    2. A. Shapiro, 2006. "Stochastic Programming with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 221-243, January.
    3. Patriksson, Michael, 2008. "On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs," Transportation Research Part B: Methodological, Elsevier, vol. 42(10), pages 843-860, December.
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    8. A. Evgrafov & M. Patriksson, 2004. "On the Existence of Solutions to Stochastic Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 65-76, April.
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    10. Ş. İlker Birbil & Gül Gürkan & Ovidiu Listeş, 2006. "Solving Stochastic Mathematical Programs with Complementarity Constraints Using Simulation," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 739-760, November.
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