Abstract
The aim of this chapter is to compare different smoothing techniques for solving finite minimax problems. We consider the local smoothing technique which approximates the function in some neighborhood of a point of nondifferentiability and also global smoothing techniques such as the exponential and hyperbolic smoothing which approximate the function in the whole domain. Computational results on the collection of academic test problems are used to compare different smoothing techniques. Results show the superiority of the local smoothing technique for convex problems and global smoothing techniques for nonconvex problems.
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References
Bagirov, A.M., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization: Theory. Practice and Software. Springer, Cham (2014)
Mäkelä, M.M., Neittaanmaki, P.: Nonsmooth Optimization. World Scientific, Singapore (1992)
Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax. Wiley, New York (1974)
Du, D.Z., Pardalos, P.M.: Minimax and Applications. Kluwer Academic Publishers, Dordrecht (1995)
Bagirov, A.M., Ganjehlou, A.N.: A quasisecant method for minimizing nonsmooth functions. Optim. Methods Softw. 25(1), 3–18 (2010)
Bagirov, A.M., Karasozen, B., Sezer, M.: Discrete gradient method: derivative-free method for nonsmooth optimization. J. Optim. Theory Appl. 137, 317–334 (2008)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer, Berlin (1993)
Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. Springer, Berlin (1985)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)
Li, X.S.: An entropy-based aggregate method for minimax optimization. Eng. Optim. 18, 277–285 (1992)
Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20(3), 267–279 (2001)
Yang, X.Q.: Smoothing approximations to nonsmooth optimization problems. J. Austral. Math. Soc. Ser. B. 36, 274–285 (1994)
Polak, E., Womersley, R.S., Yin, H.X.: An algorithm based on active sets and smoothing for discretized semi-infinite minimax problems. J. Optim. Theory Appl. 138, 311–328 (2008)
Yin, H.-X.: Error bounds of two smoothing approximations for semi-infinite minimax problems. Acta Math. Applicatae Sinica. 25(4), 685–696 (2009)
Chen, X.: Smoothing methods for complementarity problems and their applications: a survey. J. Oper. Res. Soc. Jpn. 43(1), 32–47 (2000)
Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999)
Fukushima, M., Luo, Z.Q., Pang, J.S.: A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints. Comput. Optim. Appl. 10, 5–34 (1998)
Sun, D., Qi, L.: Solving variational inequality problems via smoothing-nonsmooth reformulations. J. Comput. Appl. Math. 129, 37–62 (2001)
Ansari, M.R., Mahdavi-Amiri, N.: A robust combined trust region-line search exact penalty projected structured scheme for constrained nonlinear least squares. Optim. Methods Softw. 30(1), 162–190 (2015)
Bagirov, A.M., Taheri, S.: DC Programming algorithm for clusterwise linear \(L_1\) regression. J. Oper. Res. Soc. China 5(2), 233–256 (2017)
Bagirov, A.M., Mohebi, E.: An algorithm for clustering using \(L_1\)-norm based on hyperbolic smoothing technique. Comput. Intell. 32(3), 439–457 (2016)
Bagirov, A.M., Mohebi, E.: Nonsmooth optimization based algorithms in cluster analysis. In: Celebi, E. (ed.) Partitional Clustering Algorithms, pp. 99–146. Springer, Berlin (2015)
Feng, Z.G., Yiu, K.F.C., Teo, K.L.: A smoothing approach for the optimal parameter selection problem with continuous inequality constraint. Optim. Methods Softw. 28(4), 689–705 (2013)
Ye, F., Liu, H., Zhou, Sh, Liu, S.: A smoothing trust-region Newton-CG method for minimax problem. Appl. Math. Comput. 199, 581–589 (2008)
Zang, I.: A smoothing-out technique for min-max optimization. Math. Program. 19, 61–77 (1980)
Polak, E., Royset, J.O., Womersley, R.S.: Algorithms with adaptive smoothing for finite minimax problems. J. Optim. Theory Appl. 119, 459–484 (2003)
Xiao, Y., Yu, B.: A truncated aggregate smoothing Newton method for minimax problems. Appl. Math. Comput. 216, 1868–1879 (2010)
Bagirov, A.M., Al Nuaimat, A., Sultanova, N.: Hyperbolic smoothing function method for minimax problems. Optimization 62(6), 759–782 (2013)
Xavier, A.E.: The hyperbolic smoothing clustering method. Pattern Recog. 43, 731–737 (2010)
Xavier, A.E., Oliveira, A.A.F.D.: Optimal covering of plane domains by circles via hyperbolic smoothing. J. Glob. Optim. 31(3), 493–504 (2005)
Xavier, A.E.: Penalizaćao hiperbólica. I Congresso Latino-Americano de Pesquisa Operacional e Engenharia de Sistemas. 8 a 11 de Novembro, pp. 468–482. Brasil, Rio de Janeiro (1982)
Vazquez, F.G., Gunzel, H., Jongen, HTh: On logarithmic smoothing of the maximum function. Ann. Oper. Res. 101, 209–220 (2001)
Nesterov, Yu.: Smooth minimization of nonsmooth functions. Math. Program. 103(1), 127–152 (2005)
Ermoliev, Y.M., Norkin, V.I., Wets, R.J-B.: The minimization of semicontinuous functions: Mollifier subgradients. SIAM J. Control Optim. 33, 149–167 (1995)
Ben-Tal, A., Teboulle, M.: A smoothing technique for nondifferentiable optimization problems. In: Dolecki, S. (ed.) Lecture Notes in Mathematics, vol. 1405, pp. 1–11. Springer, Heidelberg (1989)
Peng, J.: A smoothing function and its applications. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 293–316. Kluwer, Dordrecht (1998)
Lukšan, L., Vlček, J.: Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization. In: Technical Report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2000) Available via http://hdl.handle.net/11104/0124190
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)
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Bagirov, A.M., Sultanova, N., Al Nuaimat, A., Taheri, S. (2018). Solving Minimax Problems: Local Smoothing Versus Global Smoothing. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. NAO 2017. Springer Proceedings in Mathematics & Statistics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-319-90026-1_2
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