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A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

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  • M. Diehl
  • B. Houska
  • O. Stein
  • P. Steuermann

Abstract

This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent finite minimization problems by exploiting the concept of Wolfe duality for convex lower level problems. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate reformulations of the original GSIP allow us to apply standard nonlinear optimization algorithms. Copyright Springer Science+Business Media, LLC 2013

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  • M. Diehl & B. Houska & O. Stein & P. Steuermann, 2013. "A lifting method for generalized semi-infinite programs based on lower level Wolfe duality," Computational Optimization and Applications, Springer, vol. 54(1), pages 189-210, January.
    • Handle: RePEc:spr:coopap:v:54:y:2013:i:1:p:189-210
    DOI: 10.1007/s10589-012-9489-4
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
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    3. Harald Günzel & Hubertus Jongen & Oliver Stein, 2007. "On the closure of the feasible set in generalized semi-infinite programming," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 15(3), pages 271-280, September.
    4. Oliver Stein, 2001. "First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Optimization," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 565-582, August.
    5. Gerhard-Wilhelm Weber & Aysun Tezel, 2007. "On generalized semi-infinite optimization of genetic networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(1), pages 65-77, July.
    6. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    7. O. Stein & A. Winterfeld, 2010. "Feasible Method for Generalized Semi-Infinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 419-443, August.
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    Cited by:

    1. Stuart M. Harwood & Paul I. Barton, 2017. "How to solve a design centering problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 215-254, August.
    2. Peter Kirst & Oliver Stein, 2016. "Solving Disjunctive Optimization Problems by Generalized Semi-infinite Optimization Techniques," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1079-1109, June.
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    4. Peter Kirst & Oliver Stein, 2019. "Global optimization of generalized semi-infinite programs using disjunctive programming," Journal of Global Optimization, Springer, vol. 73(1), pages 1-25, January.

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