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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ehrgott, M.: Multicriteria Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 491. Springer, Berlin (2000)
Enkhbat, R.: Quasiconvex Programming. National University of Mongolia, Ulaanbaatar (2004)
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)
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
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
Fortin, D., Tseveendorj, I.: A trust branching path heuristic for permutation problems. Int. J. Pure Appl. Math. 56 (3), 329–343 (2009)
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
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
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
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
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
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)
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
Penot, J.P.: Duality for anticonvex programs. J. Global Optim. 19 (2), 163–182 (2001)
Sergeyev, Y.D., Kvasov, D.: Diagonal global optimization methods (in Russian). Fizmatlit, Moscow (2008)
Strekalovskiĭ, A.S.: On the problem of the global extremum. Dokl. Akad. Nauk SSSR 292 (5), 1062–1066 (1987)
Strekalovskii, A.S.: Elements of Nonconvex Optimization (in Russian). NAUKA, Novosibirsk (2003)
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
Tseveendorj, I.: On the conditions for global optimality. J. Mong. Math. Soc. (2), 58–61 (1998)
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
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)
Tuy, H.: Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications, vol. 22. Kluwer Academic Publishers, Dordrecht (1998)
Zălinescu, C.: On the maximization of (not necessarily) convex functions on convex sets. J. Global Optim. 36 (3), 379–389 (2006)
Zhiglyavskiĭ, A.A., Zhilinskas, A.G.: Metody poiska globalnogo ekstremuma. “Nauka”, Moscow (1991). With an English summary
Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer Optimization and Its Applications, vol. 9. Springer, New York (2008)
Zhilinskas, A.: Globalnaya optimizatsiya. “Mokslas”, Vilnius (1986). Aksiomatika statisticheskikh modelei, algoritmy, primeneniya. [Axiomatics of statistical models, algorithms, and applications.] With Lithuanian and English summaries
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Tseveendorj, I., Fortin, D. (2016). Survey of Piecewise Convex Maximization and PCMP over Spherical Sets. In: Pardalos, P., Zhigljavsky, A., Žilinskas, J. (eds) Advances in Stochastic and Deterministic Global Optimization. Springer Optimization and Its Applications, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-29975-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-29975-4_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29973-0
Online ISBN: 978-3-319-29975-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)