Computational Optimization and Applications

, Volume 54, Issue 1, pp 189–210 | Cite as

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



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.


Semi-infinite optimization Lower level duality Lifting approach Adaptive convexification Mathematical program with complementarity constraints 



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.


  1. 1.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009) MATHGoogle Scholar
  3. 3.
    Bhattacharjee, B., Green, W.H., Barton, P.I.: Interval methods for semi-infinite programs. Comput. Optim. Appl. 30, 63–93 (2005) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bhattacharjee, B., Lemonidis, P., Green, W.H., Barton, P.I.: Global solution of semi-infinite programs. Math. Program. 103, 283–307 (2005) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dantzig, G.B., Thapa, M.N.: Linear Programming. Springer, Berlin (2003) MATHGoogle Scholar
  6. 6.
    Flegel, M.L., Kanzow, C.: A Fritz John approach to first order optimality conditions for mathematical programs with equilibrium constraints. Optimization 52, 277–286 (2003) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM J. Optim. 18, 1187–1208 (2007) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Günzel, H., Jongen, H.Th., Stein, O.: On the closure of the feasible set in generalized semi-infinite programming. Cent. Eur. J. Oper. Res. 15, 271–280 (2007) MATHCrossRefGoogle Scholar
  9. 9.
    Guerra-Vázquez, F., Jongen, H.Th., Shikhman, V.: General semi-infinite programming: symmetric Mangasarian-Fromovitz constraint qualification and the closure of the feasible set. SIAM J. Optim. 20, 2487–2503 (2010) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hettich, R., Jongen, H.Th.: Semi-infinite programming: conditions of optimality and applications. In: Stoer, J. (ed.) Optimization Techniques, Part 2. Lecture Notes in Control and Information Sciences, vol. 7, pp. 1–11. Springer, Berlin (1978) Google Scholar
  11. 11.
    Hettich, R., Kortanek, K.: Semi infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hettich, R., Zencke, P.: Numerische Methoden der Approximation und Semi-Infiniten Optimierung. Teubner, Stuttgart (1982) MATHGoogle Scholar
  13. 13.
    Jongen, H.Th., Rückmann, J.J., Stein, O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83, 145–158 (1998) MATHGoogle Scholar
  14. 14.
    Kropat, E., Weber, G.W.: Robust regression analysis for gene-environment and eco-finance networks under polyhedral and ellipsoidal uncertainty. Preprint at IAM, METU Google Scholar
  15. 15.
    Levitin, E., Tichatschke, R.: A branch-and-bound approach for solving a class of generalized semi-infinite programming problems. J. Glob. Optim. 13, 299–315 (1998) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Mitsos, A., Lemonidis, P., Lee, C.K., Barton, P.I.: Relaxation-based bounds for semi-infinite programs. SIAM J. Optim. 19, 77–113 (2007) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mitsos, A., Lemonidis, P., Barton, P.I.: Global solution of bilevel programs with a nonconvex inner program. J. Glob. Optim. 42, 475–513 (2008) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Polak, E.: On the mathematical foundation of nondifferentiable optimization in engineering design. SIAM Rev. 29, 21–89 (1987) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Polak, E.: Optimization. Algorithms and Consistent Approximations. Springer, Berlin (1997) MATHGoogle Scholar
  20. 20.
    Polak, E., Royset, J.O.: On the use of augmented Lagrangians in the solution of generalized semi-infinite min-max problems. Comput. Optim. Appl. 31, 173–192 (2005) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Reemtsen, R., Görner, S.: Numerical methods for semi-infinite programming: a survey. In: Reemtsen, R., Rückmann, J.J. (eds.) Semi-Infinite Programming, pp. 195–275. Kluwer, Boston (1998) Google Scholar
  22. 22.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Stein, O.: First order optimality conditions for degenerate index sets in generalized semi-infinite programming. Math. Oper. Res. 26, 565–582 (2001) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Stein, O.: Bi-level Strategies in Semi-infinite Programming. Kluwer Academic, Boston (2003) MATHGoogle Scholar
  25. 25.
    Stein, O.: How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. (2012). doi:10.1016/j.ejor.2012.06.009 Google Scholar
  26. 26.
    Stein, O., Steuermann, P.: The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets. Math. Program. B (2012). doi:10.1007/s10107-012-0556-5 MathSciNetGoogle Scholar
  27. 27.
    Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42, 769–788 (2003) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Stein, O., Winterfeld, A.: A feasible method for generalized semi-infinite programming. J. Optim. Theory Appl. 146, 419–443 (2010) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Still, G.: Discretization in semi-infinite programming: the rate of convergence. Math. Program. 91, 53–69 (2001) MathSciNetMATHGoogle Scholar
  30. 30.
    Still, G.: Generalized semi-infinite programming: numerical aspects. Optimization 49, 223–242 (2001) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Weber, G.W., Tezel, A.: On generalized semi-infinite optimization of genetic networks. Top 15, 65–77 (2007) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Yakubovich, V.A.: S-procedure in nonlinear control theory. Vestn. Leningr. Univ. 4, 73–93 (1977) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Electrical Engineering Department (ESAT) and Optimization in Engineering Center (OPTEC)K.U. LeuvenLeuvenBelgium
  2. 2.Institute of Operations ResearchKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations