Advertisement

Survey of Piecewise Convex Maximization and PCMP over Spherical Sets

  • Ider TseveendorjEmail author
  • Dominique Fortin
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 107)

Abstract

The main investigation in this chapter is concerned with a piecewise convex function which can be defined by the pointwise minimum of convex functions, \(F(x) =\min \{ f_{1}(x),\ldots,f_{m}(x)\}\). Such piecewise convex functions closely approximate nonconvex functions, that seems to us as a natural extension of the piecewise affine approximation from convex analysis. Maximizing F(⋅ ) over a convex domain have been investigated during the last decade by carrying tools based mostly on linearization and affine separation. In this chapter, we present a brief overview of optimality conditions, methods, and some attempts to solve this difficult nonconvex optimization problem. We also review how the line search paradigm leads to a radius search paradigm, in the sense that sphere separation which seems to us more appropriate than the affine separation. Some simple, but illustrative, examples showing the issues in searching for a global solution are given.

Keywords

Piecewise convex Nonconvex optimization Nonsmooth optimization 

Notes

Acknowledgements

This research benefited from the support of the FMJH “Program Gaspard Monge for optimization and operations research”, and from the support from EDF. The authors acknowledge use of the IBM ILOG CPLEX under the academic initiative license. The authors would like to thank the anonymous referee for his/her useful comments on this chapter.

References

  1. 1.
    Ehrgott, M.: Multicriteria Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 491. Springer, Berlin (2000)Google Scholar
  2. 2.
    Enkhbat, R.: Quasiconvex Programming. National University of Mongolia, Ulaanbaatar (2004)Google Scholar
  3. 3.
    Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gümüş, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and its Applications, vol. 33. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  4. 4.
    Fortin, D., Tsevendorj, I.: Piecewise-convex maximization problems: algorithm and computational experiments. J. Global Optim. 24 (1), 61–77 (2002). doi: 10.1023/A:1016221931922. http://dx.doi.org/10.1023/A:1016221931922
  5. 5.
    Fortin, D., Tseveendorj, I.: Piecewise convex maximization approach to multiknapsack. Optimization 58 (7), 883–895 (2009). doi: 10.1080/02331930902945033. http://dx.doi.org/10.1080/02331930902945033
  6. 6.
    Fortin, D., Tseveendorj, I.: A trust branching path heuristic for permutation problems. Int. J. Pure Appl. Math. 56 (3), 329–343 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Fortin, D., Tseveendorj, I.: Piece adding technique for convex maximization problems. J. Global Optim. 48 (4), 583–593 (2010). doi:10.1007/s10898-009-9506-z. http://dx.doi.org/10.1007/s10898-009-9506-z MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fortin, D., Tseveendorj, I.: Piecewise convex maximization problems: piece adding technique. J. Optim. Theory Appl. 148 (3), 471–487 (2011). doi:10.1007/s10957-010-9763-5. http://dx.doi.org/10.1007/s10957-010-9763-5 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fortin, D., Tseveendorj, I.: Attractive force search algorithm for piecewise convex maximization problems. Optim. Lett. 6 (7), 1317–1333 (2012). doi:10.1007/s11590-011-0395-y. http://dx.doi.org/10.1007/s11590-011-0395-y MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 2nd edn. Springer, Berlin (1993). doi: 10.1007/978-3-662-02947-3. http://dx.doi.org/10.1007/978-3-662-02947-3
  11. 11.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Nonconvex Optimization and its Applications, vol. 48, 2nd edn. Kluwer Academic Publishers, Dordrecht (2000). doi: 10.1007/978-1-4615-0015-5. http://dx.doi.org/10.1007/978-1-4615-0015-5
  12. 12.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)Google Scholar
  13. 13.
    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. Springer Briefs in Optimization. Springer, New York (2014). doi: 10.1007/978-1-4614-9093-7. http://dx.doi.org/10.1007/978-1-4614-9093-7
  14. 14.
    Penot, J.P.: Duality for anticonvex programs. J. Global Optim. 19 (2), 163–182 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sergeyev, Y.D., Kvasov, D.: Diagonal global optimization methods (in Russian). Fizmatlit, Moscow (2008)zbMATHGoogle Scholar
  16. 16.
    Strekalovskiĭ, A.S.: On the problem of the global extremum. Dokl. Akad. Nauk SSSR 292 (5), 1062–1066 (1987)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Strekalovskii, A.S.: Elements of Nonconvex Optimization (in Russian). NAUKA, Novosibirsk (2003)zbMATHGoogle Scholar
  18. 18.
    Törn, A., Žilinskas, A.: Global Optimization. Lecture Notes in Computer Science, vol. 350. Springer, Berlin (1989). doi:10.1007/3-540-50871-6. http://dx.doi.org/10.1007/3-540-50871-6
  19. 19.
    Tseveendorj, I.: On the conditions for global optimality. J. Mong. Math. Soc. (2), 58–61 (1998)MathSciNetGoogle Scholar
  20. 20.
    Tsevendorj, I.: Piecewise-convex maximization problems: global optimality conditions. J. Global Optim. 21 (1), 1–14 (2001). doi:10.1023/A:1017979506314. http://dx.doi.org/10.1023/A:1017979506314 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tseveendorj, I., Fortin, D.: Pareto optima and local optimality of piecewise convex maximization problems. In: Proceedings of the 15-th Baikal International Conference on Optimization Methods and Their Applications, “Mathematical Programming”, vol. 2, pp. 25–29. ISDCT SB of the Russian Academy of Sciences’ Publisher, Irkutsk (2011)Google Scholar
  22. 22.
    Tuy, H.: Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications, vol. 22. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  23. 23.
    Zălinescu, C.: On the maximization of (not necessarily) convex functions on convex sets. J. Global Optim. 36 (3), 379–389 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhiglyavskiĭ, A.A., Zhilinskas, A.G.: Metody poiska globalnogo ekstremuma. “Nauka”, Moscow (1991). With an English summaryGoogle Scholar
  25. 25.
    Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer Optimization and Its Applications, vol. 9. Springer, New York (2008)Google Scholar
  26. 26.
    Zhilinskas, A.: Globalnaya optimizatsiya. “Mokslas”, Vilnius (1986). Aksiomatika statisticheskikh modelei, algoritmy, primeneniya. [Axiomatics of statistical models, algorithms, and applications.] With Lithuanian and English summariesGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of VersaillesUniversité Paris-SaclayVersailles CedexFrance
  2. 2.INRIA, Domaine de Voluceau, RocquencourtLe Chesnay CedexFrance

Personalised recommendations