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

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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.

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References

  1. Winterfeld, A.: Application of general semi infinite programming to lapidary cutting problems. Technical Report ISSN 1434-9973, Fraunhofer-Institut Techno-and Wirtschaftsmathematik (2006)

  2. Hoffman, A., Reinhardt, R.: On reverse chebyshev approximation problems. Technical Report M08/94, Technical University of Illmenau (1994)

  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)

  4. Still G.: Generalized semi-infinite programming: theory and methods. Eur. J. Oper. Res. 119, 301–313 (1999)

    Article  Google Scholar 

  5. Hettich R., Kortanek K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  9. Rustem B., Howe M.: Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton, NJ (2002)

    Google Scholar 

  10. Kawaguchi T., Maruyama Y.: Generalized constrained games in farm planning. Am. J. Agric. Econ. 54, 591–602 (1972)

    Article  Google Scholar 

  11. Lanckriet G., Ghaoui L., Bhattacharyya C., Jordan M.: A robust minimax approach to classification. J. Mach. Learn. Res. 3, 555–582 (2002)

    Article  Google Scholar 

  12. Hurtado F., Sacristán V., Toussaint G.: Some constrained minimax and maximin location problems. Stud. Locat. Anal. 15, 17–35 (2000)

    Google Scholar 

  13. Kiwiel K.C.: A direct method of linearization for continuous minimax problems. J. Optim. Theory Appl. 55(2), 271–287 (1987)

    Article  Google Scholar 

  14. Monahan G.E.: Finding saddle points on polyhedra: solving certain continuous minimax problems. Naval Res. Logist. 43(6), 821–837 (1996)

    Article  Google Scholar 

  15. Rustem B., Zakovic S., Parpas P.: Convergence of an interior point algorithm for continuous minimax. J. Optim. Theory Appl. 136, 87–103 (2008)

    Article  Google Scholar 

  16. Sasai H.: An interior penalty method for minimax problems with constraints. SIAM J. Control 12, 643–649 (1974)

    Article  Google Scholar 

  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.

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  23. Faisca N., Dua V., Rustem B., Saraiva P., Pistikopoulos E.: Parametric global optimization for bilevel programming. J. Global Optim. 38, 609–623 (2007)

    Article  Google Scholar 

  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. Stein O., Still G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142(3), 444–462 (2002)

    Article  Google Scholar 

  26. Blankenship J.W., Falk J.E.: Infinitely constrained optimization problems. J. Optim. Theory Appl. 19(2), 261–281 (1976)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  29. Tsoukalas, A., Parpas, P., Rustem, B.: A smooting algorithm for min-max-min problems. To be published in Optim. Lett.

  30. Parpas P., Rustem B., Pistikopoulos E.N.: Linearly constrained global optimization and stochastic differential equations. J. Global Optim. 36(2), 191–217 (2006)

    Article  Google Scholar 

  31. Horst R., Tuy H.: Global Optimization, 2nd edn. Springer-Verlag, Berlin (1993) (Deterministic approaches)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computing, Imperial College, London, UK

    Angelos Tsoukalas & Berç Rustem

  2. Centre for Process Systems Engineering, Imperial College, London, UK

    Efstratios N. Pistikopoulos

Authors
  1. Angelos Tsoukalas
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  2. Berç Rustem
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  3. Efstratios N. Pistikopoulos
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Correspondence to Angelos Tsoukalas.

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Tsoukalas, A., Rustem, B. & Pistikopoulos, E.N. A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems. J Glob Optim 44, 235–250 (2009). https://doi.org/10.1007/s10898-008-9321-y

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  • DOI: https://doi.org/10.1007/s10898-008-9321-y

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