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

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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.

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Acknowledgements

We thank two anonymous referees and the associated editor for their precise and substantial remarks which significantly improved this manuscript.

The research was supported by the Research Council KUL via GOA/11/05 Ambiorics, GOA/10/09 MaNet, CoE EF/05/006 Optimization in Engineering (OPTEC), IOF-SCORES4CHEM and PhD/postdoc/fellow grants, the Flemish Government via FWO (PhD/postdoc grants, projects G0226.06, G0321.06, G.0302.07, G.0320.08, G.0558.08, G.0557.08, G.0588.09, G.0377.09, research communities ICCoS, ANMMM, MLDM) and via IWT (PhD Grants, Eureka-Flite+, SBO LeCoPro, SBO Climaqs, SBO POM, O&O-Dsquare), the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, Dynamical systems, control and optimization, 2007–2011), the IBBT, the EU (ERNSI; FP7-HD-MPC (INFSO-ICT-223854), COST intelliCIS, FP7-EMBOCON (ICT-248940), FP7-SADCO (MC ITN-264735), ERC HIGHWIND (259 166)), the Contract Research (AMINAL), the Helmholtz Gemeinschaft via viCERP and the ACCM.

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Authors and Affiliations

  1. Electrical Engineering Department (ESAT) and Optimization in Engineering Center (OPTEC), K.U. Leuven, Leuven, Belgium

    M. Diehl & B. Houska

  2. Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    O. Stein & P. Steuermann

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  1. M. Diehl
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  2. B. Houska
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  3. O. Stein
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  4. P. Steuermann
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Correspondence to O. Stein.

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Diehl, M., Houska, B., Stein, O. et al. A lifting method for generalized semi-infinite programs based on lower level Wolfe duality. Comput Optim Appl 54, 189–210 (2013). https://doi.org/10.1007/s10589-012-9489-4

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