Abstract
We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, L.T.H., Tao, P.D.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. J. Glob. Optim. 133, 23–46 (2005)
Astorino, A., Fuduli, A., Gaudioso, M.: DC models for spherical separation. J. Glob. Optim. 48(4), 657–669 (2010)
Bagirov, A.M.: A method for minimization of quasidifferentiable functions. Optim. Methods Softw. 17(1), 31–60 (2002)
Bagirov, A.M., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Berlin (2014)
Bagirov, A.M., Taheri, S., Ugon, J.: Nonsmooth DC programming approach to the minimum sum-of-squares clustering problems. Pattern Recognit. 53, 12–24 (2016)
Bagirov, A.M., Ugon, J.: Codifferential method for minimizing nonsmooth DC functions. J. Glob. Optim. 50(1), 3–22 (2011)
Bagirov, A.M., Ugon, J.: Nonsmooth DC programming approach to clusterwise linear regression: optimality conditions and algorithms. Optim. Methods Softw. (in press). doi:10.1080/10556788.2017.1371717
Demyanov, A.V., Fuduli, A., Miglionico, G.: A bundle modification strategy for convex minimization. Eur. J. Oper. Res. 180, 38–47 (2007)
Demyanov, V.F., Malozemov, V.N.: On the theory of non-linear minimax problems. Russ. Math. Surv. 26(3) (1971). http://iopscience.iop.org/0036-0279/26/3/R02
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Fuduli, A., Gaudioso, M., Giallombardo, G.: Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14(3), 743–756 (2004)
Fuduli, A., Gaudioso, M., Giallombardo, G., Miglionico, G.: A partially inexact bundle method for convex semi-infinite minmax problems. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 172–180 (2014)
Gaudioso, M., Giallombardo, G., Miglionico, G.: Minimizing piecewise-concave functions over polytopes. Math. Op. Res. (in press). doi:10.1287/moor.2017.0873
Gaudioso, M., Gorgone, E.: Gradient set splitting in nonconvex nonsmooth numerical optimization. Optim. Methods Softw. 25(1), 59–74 (2010)
Gaudioso, M., Monaco, M.F.: Variants to the cutting plane approach for convex nondifferentiable optimization. Optimization 25, 65–75 (1992)
Hiriart-Urruty, J.B.: From Convex Optimization to Nonconvex Optimization: Necessary and Sufficient Conditions for Global Optimality, in Nonsmooth Optimization and Related Topics, pp. 219–240. Plenum, New York/London (1989)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I-II. Springer, Berlin (1993)
Holmberg, K., Tuy, H.: A production-transportation problem with stochastic demand and concave production costs. Math. Program. 85, 157–179 (1999)
Horst, R., Thoai, N.V.: DC programming: overview. J. Optim. Theory Appl. 103, 1–43 (1999)
Ibm Ilog Cplex Optimizer: http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/
Joki, K., Bagirov, A.M., Karmitsa, N., Mäkelä, M.M.: A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes. J. Global Optim. 68(3), 501–535 (2017). doi:10.1007/s10898-016-0488-3
Ordin, B., Bagirov, A.M.: A heuristic algorithm for solving the minimum sum-of-squares clustering problems. J. Glob. Optim. 61(2), 341–361 (2015)
Pey-Chun, C., Hansen, P., Jaumard, B., Tuy, H.: Solution of the multisource Weber and conditional Weber problems by DC programming. Oper. Res. 46(4), 548–562 (1998)
Souza, J.C.O., Oliveira, P.R., Soubeyran, A.: Global convergence of a proximal linearized algorithm for difference of convex functions. Optim. Lett. 10, 1529–1539 (2016)
Strekalovsky, A.S.: On Global Optimality Conditions for D.C. Programming Problems. Irkutsk State University, Russia (1997)
Strekalovsky, A.S.: Global optimality conditions for nonconvex optimization. J. Glob. Optim. 12, 415–434 (1998)
Tuy, H.: Convex Analysis and Global Optimization. Kluwer Academic Publishers, Dordrescht (1998)
Acknowledgements
The authors are grateful to two anonymous reviewers for many helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gaudioso, M., Giallombardo, G., Miglionico, G. et al. Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations. J Glob Optim 71, 37–55 (2018). https://doi.org/10.1007/s10898-017-0568-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-017-0568-z