Diagonal bundle method with convex and concave updates for large-scale nonconvex and nonsmooth optimization

References

• P. Apkarian, D. Noll, and O. Prot, A trust region spectral bundle method for non-convex eigenvalue optimization, SIAM J. Optim. 19(1) (2008), pp. 281–306. doi: 10.1137/060665191
   Web of Science ®Google Scholar
• A. Astorino and A. Fuduli, Nonsmooth optimization techniques for semi-supervised classification, IEEE Trans. Pattern Anal. Mach. Intell. 29(12) (2007), pp. 2135–2142. doi: 10.1109/TPAMI.2007.1102
   PubMed Web of Science ®Google Scholar
• A. Astorino, A. Fuduli, and E. Gorgone, Non-smoothness in classification problems, Optim. Methods Softw. 23(5) (2008), pp. 675–688. doi: 10.1080/10556780802264071
   Web of Science ®Google Scholar
• S. Äyrämö, Knowledge mining using robust clustering, Ph.D. thesis, University of Jyväskylä, Department of Mathematical Information Technology, 2006.
   Google Scholar
• A. Bagirov, N. Karmitsa, and M.M. Mäkelä, Introduction to Nonsmooth Optimization: Theory, Practice and Software, Springer, Cham, 2014.
   Google Scholar
• A.M. Bagirov, S. Taheri, and J. Ugon, Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems, Pattern Recognit. 53 (2016), pp. 12–24. doi: 10.1016/j.patcog.2015.11.011
   Web of Science ®Google Scholar
• C. Bergeron, G. Moore, J. Zaretzki, C.M. Breneman, and K.P. Bennett, Fast bundle algorithm for multiple instance learning, IEEE Trans. Pattern Anal. Mach. Intell. 34(6) (2012), pp. 1068–1079. doi: 10.1109/TPAMI.2011.194
   PubMed Web of Science ®Google Scholar
• A. Bihain, Optimization of upper semidifferentiable functions, J. Optim. Theory Appl. 4 (1984), pp. 545–568. doi: 10.1007/BF00938396
   Web of Science ®Google Scholar
• P.S. Bradley, U.M. Fayyad, and O.L. Mangasarian, Mathematical programming for data mining: Formulations and challenges, INFORMS J. Comput. 11 (1999), pp. 217–238. doi: 10.1287/ijoc.11.3.217
   Web of Science ®Google Scholar
• J.V. Burke, A.S. Lewis, and M.L. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim. 15 (2005), pp. 751–779. doi: 10.1137/030601296
   Web of Science ®Google Scholar
• R.H. Byrd, J. Nocedal, and R.B. Schnabel, Representations of quasi-Newton matrices and their use in limited memory methods, Math. Program. 63 (1994), pp. 129–156. doi: 10.1007/BF01582063
   Web of Science ®Google Scholar
• E. Carrizosa and D. Romero Morales, Supervised classification and mathematical optimization, Comput. Oper. Res. 40(1) (2013), pp. 150–165. doi: 10.1016/j.cor.2012.05.015
   Web of Science ®Google Scholar
• F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.
   Google Scholar
• F.H. Clarke, Y.S. Ledyaev, R.J. Stern, and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
   Google Scholar
• V.F. Demyanov, A. Bagirov, and A. Rubinov, A method of truncated codifferential with application to some problems of cluster analysis, J. Global Optim. 23(1) (2002), pp. 63–80. doi: 10.1023/A:1014075113874
   Web of Science ®Google Scholar
• E.D. Dolan and J.J. Moré, Benchmarking optimization software with performance profiles, Math. Program. 91 (2002), pp. 201–213. doi: 10.1007/s101070100263
   Web of Science ®Google Scholar
• A. Fuduli, M. Gaudioso, and G. Giallombardo, A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization, Optim. Methods Softw. 19(1) (2004), pp. 89–102. doi: 10.1080/10556780410001648112
   Web of Science ®Google Scholar
• A. Fuduli, M. Gaudioso, and G. Giallombardo, Minimizing nonconvex nonsmooth functions via cutting planes and proximity control, SIAM J. Optim. 14(3) (2004), pp. 743–756. doi: 10.1137/S1052623402411459
   Web of Science ®Google Scholar
• A. Fuduli, M. Gaudioso, and E.A. Nurminski, A splitting bundle approach for non-smooth non-convex minimization, Optimization 64(5) (2015), pp. 1131–1151. doi: 10.1080/02331934.2013.840625
   Web of Science ®Google Scholar
• M. Gaudioso and E. Gorgone, Gradient set splitting in nonconvex nonsmooth numerical optimization, Optim. Methods Softw. 25 (2010), pp. 59–74. doi: 10.1080/10556780903236911
   Web of Science ®Google Scholar
• M. Gaudioso, G. Giallombardo, and G. Miglionico, A partially inexact bundle method for convex semi-infinite minmax problems, Commun. Nonlinear Sci. Numer. Simul. 21 (2015), pp. 172–180. doi: 10.1016/j.cnsns.2014.07.033
   Web of Science ®Google Scholar
• M. Haarala, K. Miettinen, and M.M. Mäkelä, New limited memory bundle method for large-scale nonsmooth optimization, Optim. Methods Softw. 19(6) (2004), pp. 673–692. doi: 10.1080/10556780410001689225
   Web of Science ®Google Scholar
• N. Haarala, K. Miettinen, and M.M. Mäkelä, Globally convergent limited memory bundle method for large-scale nonsmooth optimization, Math. Program. 109(1) (2007), pp. 181–205. doi: 10.1007/s10107-006-0728-2
   Web of Science ®Google Scholar
• W. Hare and C. Sagastizábal, A redistributed proximal bundle method for nonconvex optimization, SIAM J. Optim. 20(5) (2010), pp. 2442–2473. doi: 10.1137/090754595
   Web of Science ®Google Scholar
• J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd ed., John Wiley & Sons, Chichester, 1996.
   Google Scholar
• J. Herskovits and E. Goulart, Sparse quasi-Newton matrices for large scale nonlinear optimization, in Proceedings of the 6th Word Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, 2005.
   Google Scholar
• J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms II, Springer-Verlag, Berlin, 1993.
   Google Scholar
• T. Kärkkäinen and E. Heikkola, Robust formulations for training multilayer perceptrons, Neural Comput. 16 (2004), pp. 837–862. doi: 10.1162/089976604322860721
   Web of Science ®Google Scholar
• N. Karmitsa, Diagonal bundle method for nonsmooth sparse optimization, J. Optim. Theory Appl. 166(3) (2015), pp. 889–905. doi:10.1007/s10957-014-0666-8.
   Web of Science ®Google Scholar
• N. Karmitsa, M. Tanaka Filho, and J. Herskovits, Globally convergent cutting plane method for nonconvex nonsmooth minimization, J. Optim. Theory Appl. 148(3) (2011), pp. 528–549. doi: 10.1007/s10957-010-9766-2
   Web of Science ®Google Scholar
• N. Karmitsa, A. Bagirov, and M.M. Mäkelä, Comparing different nonsmooth minimization methods and software, Optim. Methods Softw. 27(1) (2012), pp. 131–153. doi: 10.1080/10556788.2010.526116
   Web of Science ®Google Scholar
• N. Karmitsa, A. Bagirov, and S. Taheri, New diagonal bundle method for clustering problems in large data sets, Eur J Oper Res. 263(2) (2017), pp. 367–379. doi: 10.1016/j.ejor.2017.06.010
   Web of Science ®Google Scholar
• N. Karmitsa, A. Bagirov, and S. Taheri, Mssc clustering of large data using the limited memory bundle method. TUCS Technical Report, No. 1164, Turku Centre for Computer Science, Turku, 2016. The report is available online at http://tucs.fi/publications/view/?pub_id=tKaBaTa16b.
   Google Scholar
• K.C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Vol. 1133, Springer-Verlag, Berlin, 1985.
   Google Scholar
• C. Lemaréchal, J.-J. Strodiot, and A. Bihain, On a bundle algorithm for nonsmooth optimization, in Nonlinear Programming, O.L. Mangasarian, R.R. Mayer, and S.M. Robinson, eds., Academic Press, New York, 1981, pp. 245–281.
   Google Scholar
• L. Lukšan and J. Vlček, Globally convergent variable metric method for convex nonsmooth unconstrained minimization, J. Optim. Theory Appl. 102(3) (1999), pp. 593–613. doi: 10.1023/A:1022650107080
   Web of Science ®Google Scholar
• M.M. Mäkelä and P. Neittaanmäki, Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control, World Scientific Publishing Co., Singapore, 1992.
   Google Scholar
• R. Mifflin, A modification and an extension of Lemaréchal's algorithm for nonsmooth minimization, Math. Program. Study 17 (1982), pp. 77–90. doi: 10.1007/BFb0120960
   Web of Science ®Google Scholar
• E.S. Mistakidis and G.E. Stavroulakis, Nonconvex Optimization in Mechanics. Smooth and Nonsmooth Algorithms, Heuristics and Engineering Applications by the F.E.M., Kluwert Academic Publishers, Dordrecht, 1998.
   Google Scholar
• J. Moreau, P.D. Panagiotopoulos, and G. Strang, Eds., Topics in Nonsmooth Mechanics, Birkhäuser Verlag, Basel, 1988.
   Google Scholar
• D. Noll, O. Prot, and A. Rondepierre, A proximity control algorithm to minimize nonsmooth and nonconvex functions, Pacific J. Optim. 4(3) (2008), pp. 571–604.
   Web of Science ®Google Scholar
• J. Outrata and J. Zowe, Nonsmooth Approach to Optimization Problems With Equilibrium Constraints. Theory, Applications and Numerical Results, Kluwert Academic Publisher, Dordrecht, 1998.
   Google Scholar
• J. Vlček and L. Lukšan, Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization, J. Optim. Theory Appl. 111(2) (2001), pp. 407–430. doi: 10.1023/A:1011990503369
   Web of Science ®Google Scholar