Journal of Global Optimization

, Volume 44, Issue 2, pp 235–250 | Cite as

A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems

  • Angelos Tsoukalas
  • Berç Rustem
  • Efstratios N. Pistikopoulos
Article

Abstract

We propose an algorithm for the global optimization of three problem classes: generalized semi-infinite, continuous coupled minimax and bi-level problems. We make no convexity assumptions. For each problem class, we construct an oracle that decides whether a given objective value is achievable or not. If a given value is achievable, the oracle returns a point with a value better than or equal to the target. A binary search is then performed until the global optimum is obtained with the desired accuracy. This is achieved by solving a series of appropriate finite minimax and min-max-min problems to global optimality. We use Laplace’s smoothing technique and a simulated annealing approach for the solution of these problems. We present computational examples for all three problem classes.

Keywords

Generalized semi-infinite Minimax Bi-level Globaloptimization Min-max-min 

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References

  1. 1.
    Winterfeld, A.: Application of general semi infinite programming to lapidary cutting problems. Technical Report ISSN 1434-9973, Fraunhofer-Institut Techno-and Wirtschaftsmathematik (2006)Google Scholar
  2. 2.
    Hoffman, A., Reinhardt, R.: On reverse chebyshev approximation problems. Technical Report M08/94, Technical University of Illmenau (1994)Google Scholar
  3. 3.
    Kaplan, A., Tichatschke, R.: On a class of terminal variational problems. In: Parametric Optimization and Related Topics, IV (Enschede, 1995) (vol. 9 of Approx. Optim., pp. 185–199). Lang, Frankfurt am Main (1997)Google Scholar
  4. 4.
    Still G.: Generalized semi-infinite programming: theory and methods. Eur. J. Oper. Res. 119, 301–313 (1999)CrossRefGoogle Scholar
  5. 5.
    Hettich R., Kortanek K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)CrossRefGoogle Scholar
  6. 6.
    Bhattacharjee B., Lemonidis P., Green W.H. Jr, Barton P.I.: Global solution of semi-infinite programs. Math. Program 103(2, Ser. B), 283–307 (2005)CrossRefGoogle Scholar
  7. 7.
    Bansal V., Perkins J.D., Pistikopoulos E.N.: Flexibility analysis and design of linear systems by parametric programming. AIChE J. 46(2), 335–354 (2000)CrossRefGoogle Scholar
  8. 8.
    Polak, E., Wets, R.J.-B., Der Kiureghian, A.: On an approach to optimization problems with a probabilistic cost and or constraints. In: Nonlinear Optimization and Related Topics (Erice, 1998) (vol. 36 of Appl. Optim., pp. 299–315). Kluwer Academic Publisher, Dordrecht (2000)Google Scholar
  9. 9.
    Rustem B., Howe M.: Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton, NJ (2002)Google Scholar
  10. 10.
    Kawaguchi T., Maruyama Y.: Generalized constrained games in farm planning. Am. J. Agric. Econ. 54, 591–602 (1972)CrossRefGoogle Scholar
  11. 11.
    Lanckriet G., Ghaoui L., Bhattacharyya C., Jordan M.: A robust minimax approach to classification. J. Mach. Learn. Res. 3, 555–582 (2002)CrossRefGoogle Scholar
  12. 12.
    Hurtado F., Sacristán V., Toussaint G.: Some constrained minimax and maximin location problems. Stud. Locat. Anal. 15, 17–35 (2000)Google Scholar
  13. 13.
    Kiwiel K.C.: A direct method of linearization for continuous minimax problems. J. Optim. Theory Appl. 55(2), 271–287 (1987)CrossRefGoogle Scholar
  14. 14.
    Monahan G.E.: Finding saddle points on polyhedra: solving certain continuous minimax problems. Naval Res. Logist. 43(6), 821–837 (1996)CrossRefGoogle Scholar
  15. 15.
    Rustem B., Zakovic S., Parpas P.: Convergence of an interior point algorithm for continuous minimax. J. Optim. Theory Appl. 136, 87–103 (2008)CrossRefGoogle Scholar
  16. 16.
    Sasai H.: An interior penalty method for minimax problems with constraints. SIAM J. Control 12, 643–649 (1974)CrossRefGoogle Scholar
  17. 17.
    Parpas, P., Rustem, B.: An algorithm for the global optimization of a class of continuous minimax problems. To appear in J. Optim. Theory Appl.Google Scholar
  18. 18.
    Shimizu K., Aiyoshi E.: Necessary conditions for min-max problems and algorithms by a relaxation procedure. IEEE Trans. Automat. Control 25(1), 62–66 (1980)CrossRefGoogle Scholar
  19. 19.
    Žaković S., Rustem B.: Semi-infinite programming and applications to minimax problems. Ann. Oper. Res. 124, 81–110 (2003) (Applied mathematical programming and modelling)CrossRefGoogle Scholar
  20. 20.
    Royset J.O., Polak E., Der Kiureghian A.: Adaptive approximations and exact penalization for the solution of generalized semi-infinite min-max problems. SIAM J. Optim. 14(1), 1–33 (2003) (electronic)CrossRefGoogle Scholar
  21. 21.
    Visweswaran, V., Floudas, C.A., Ierapetritou, M.G., Pistikopoulos, E.N.: A decomposition-based global optimization approach for solving bilevel linear and quadratic programs. In: State of the Art in Global Optimization (Princeton, NJ, 1995) (vol. 7 of Nonconvex Optim. Appl., pp. 139–162). Kluwer Academic Publisher, Dordrecht (1996)Google Scholar
  22. 22.
    Dua V., Papalexandri K.P., Pistikopoulos E.N.: Global optimization issues in multiparametric continuous and mixed-integer optimization problems. J. Global Optim. 30(1), 59–89 (2004)CrossRefGoogle Scholar
  23. 23.
    Faisca N., Dua V., Rustem B., Saraiva P., Pistikopoulos E.: Parametric global optimization for bilevel programming. J. Global Optim. 38, 609–623 (2007)CrossRefGoogle Scholar
  24. 24.
    Mitsos, A., Lemonidis, P., Barton, P.: Global solution of bilevel programs with a nonconvex inner program. J. Global Optim. doi:10.1007/s10898-007-9260-z
  25. 25.
    Stein O., Still G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142(3), 444–462 (2002)CrossRefGoogle Scholar
  26. 26.
    Blankenship J.W., Falk J.E.: Infinitely constrained optimization problems. J. Optim. Theory Appl. 19(2), 261–281 (1976)CrossRefGoogle Scholar
  27. 27.
    Polak E., Royset J.O., Womersley R.S.: Algorithms with adaptive smoothing for finite minimax problems. J. Optim. Theory Appl. 119(3), 459–484 (2003)CrossRefGoogle Scholar
  28. 28.
    Polak E., Royset J.O.: Algorithms for finite and semi-infinite min-max-min problems using adaptive smoothing techniques. J. Optim. Theory Appl. 119(3), 421–457 (2003)CrossRefGoogle Scholar
  29. 29.
    Tsoukalas, A., Parpas, P., Rustem, B.: A smooting algorithm for min-max-min problems. To be published in Optim. Lett.Google Scholar
  30. 30.
    Parpas P., Rustem B., Pistikopoulos E.N.: Linearly constrained global optimization and stochastic differential equations. J. Global Optim. 36(2), 191–217 (2006)CrossRefGoogle Scholar
  31. 31.
    Horst R., Tuy H.: Global Optimization, 2nd edn. Springer-Verlag, Berlin (1993) (Deterministic approaches)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Angelos Tsoukalas
    • 1
  • Berç Rustem
    • 1
  • Efstratios N. Pistikopoulos
    • 2
  1. 1.Department of ComputingImperial CollegeLondonUK
  2. 2.Centre for Process Systems EngineeringImperial CollegeLondonUK

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