Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints

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Abstract

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.

Keywords

Multiobjective optimization Nonsmooth optimization Nonconvex optimization DC programming Bundle methods 

Mathematics Subject Classification

90C29 90C26 65K05 

References

  1. 1.
    Astorino, A., Miglionico, G.: Optimizing sensor cover energy via DC programming. Optim. Lett. 10(2), 355–368 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bagirov, A., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Cham (2014)CrossRefMATHGoogle Scholar
  3. 3.
    Bagirov, A., Yearwood, J.: A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems. Eur. J. Oper. Res. 170(2), 578–596 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent method for nonsmooth convex minimization in Hilbert spaces. Numer. Funct. Anal. Optim. 32(10), 1009–1018 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Carrizosa, E., Guerrero, V., Romero Morales, D.: Visualizing data as objects by DC (difference of convex) optimization. Math. Program. (2017).  https://doi.org/10.1007/s10107-017-1156-1 Google Scholar
  7. 7.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  8. 8.
    Gadhi, N., Metrane, A.: Sufficient optimality condition for vector optimization problems under DC data. J. Global Optim. 28(1), 55–66 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gaudioso, M., Giallombardo, G., Miglionico, G., Bagirov, A.: Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations. J. Global Optim. (2017).  https://doi.org/10.1007/s10898-017-0568-z Google Scholar
  10. 10.
    Gaudioso, M., Gruzdeva, T.V., Strekalovsky, A.S.: On numerical solving the spherical separability problem. J. Global Optim. 66(1), 21–34 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math. 9(3), 707–713 (1959)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hiriart-Urruty, J.B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. Lect. Note Econ. Math. Syst. 256, 37–70 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Holmberg, K., Tuy, H.: A production-transportation problem with stochastic demand and concave production costs. Math. Program. 85(1), 157–179 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Horst, R., Thoai, N.V.: DC programming: overview. J. Optim. Theory Appl. 103(1), 1–43 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ji, Y., Goh, M., de Souza, R.: Proximal point algorithms for multi-criteria optimization with the difference of convex objective functions. J. Optim. Theory Appl. 169(1), 280–289 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Joki, K., Bagirov, A., 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Joki, K., Bagirov, A., Karmitsa, N., Mäkelä, M.M., Taheri, S.: Double bundle method for finding Clarke stationary points in nonsmooth DC programming. SIAM J. Optim. (2018) (to appear)Google Scholar
  18. 18.
    Kiwiel, K.C.: An aggregate subgradient method for nonsmooth convex minimization. Math. Program. 27(3), 320–341 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kiwiel, K.C.: A descent method for nonsmooth convex multiobjective minimization. Large Scale Syst. 8(2), 119–129 (1985)MathSciNetMATHGoogle Scholar
  20. 20.
    Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable optimization. Math. Program. 46(1), 105–122 (1990)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J. Global Optim. 11(3), 253–285 (1997)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Annals Oper. Res. 133(1), 23–46 (2005)MathSciNetMATHGoogle Scholar
  23. 23.
    Lukšan, L.: Dual method for solving a special problem of quadratic programming as a subproblem at linearly constrained nonlinear minimax approximation. Kybernetika 20(6), 445–457 (1984)MathSciNetMATHGoogle Scholar
  24. 24.
    Mäkelä, M.M.: Multiobjective Proximal Bundle Method for Nonconvex Nonsmooth Optimization: Fortran Subroutine MPBNGC 2.0. Technical Representative B 13/2003, Reports of the Department of Mathematical Information Technology, Series B, Scientific computing, University of Jyväskylä, Jyväskylä (2003)Google Scholar
  25. 25.
    Mäkelä, M.M., Eronen, V.P., Karmitsa, N.: On Nonsmooth Multiobjective Optimality Conditions with Generalized Convexities. Tech. Rep. 1056, TUCS Technical Reports, Turku Centre for Computer Science, Turku (2012)Google Scholar
  26. 26.
    Mäkelä, M.M., Eronen, V.P., Karmitsa, N.: On nonsmooth multiobjective optimality conditions with generalized convexities. In: Rassias, T.M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 333–357. Springer, Berlin (2014)CrossRefGoogle Scholar
  27. 27.
    Mäkelä, M.M., Karmitsa, N., Wilppu, O.: Proximal bundle method for nonsmooth and nonconvex multiobjective optimization. In: Tuovinen, T., Repin, S., Neittaanmäki, P. (eds.) Mathematical Modeling and Optimization of Complex Structures, Computational Methods in Applied Sciences, vol. 40, pp. 191–204. Springer, Berlin (2016)CrossRefGoogle Scholar
  28. 28.
    Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific Publishing Co., Singapore (1992)CrossRefMATHGoogle Scholar
  29. 29.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)MATHGoogle Scholar
  30. 30.
    Miettinen, K., Mäkelä, M.M.: Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization 34(3), 231–246 (1995)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mistakidis, E.S., Stavroulakis, G.E.: Nonconvex Optimization in Mechanics. Smooth and Nonsmooth Algorithms, Heuristics and Engineering Applications by the F.E.M. Kluwer Academic Publisher, Dordrecht (1998)MATHGoogle Scholar
  32. 32.
    Moreau, J.J., Panagiotopoulos, P.D., Strang, G. (eds.): Topics in Nonsmooth Mechanics. Birkhäuser, Basel (1988)MATHGoogle Scholar
  33. 33.
    Mukai, H.: Algorithms for multicriterion optimization. IEEE Trans. Autom. Control ac–25(2), 177–186 (1979)MathSciNetMATHGoogle Scholar
  34. 34.
    Outrata, J., Koĉvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Theory, Applications and Numerical Results. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  35. 35.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to DC programming: Theory, algorithms and applications. Acta Math. Vietnam. 22(1), 289–355 (1997)MathSciNetMATHGoogle Scholar
  36. 36.
    Qu, S., Goh, M., Wu, S.Y., De Souza, R.: Multiobjective DC programs with infinite convex constraints. J. Global Optim. 59(1), 41–58 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Qu, S., Liu, C., Goh, M., Li, Y., Ji, Y.: Nonsmooth multiobjective programming with quasi-Newton methods. Eur. J. Oper. Res. 235(3), 503–510 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–152 (1992)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sun, W.Y., Sampaio, R.J.B., Candido, M.A.B.: Proximal point algorithm for minimization of DC functions. J. Comput. Math. 21(4), 451–462 (2003)MathSciNetMATHGoogle Scholar
  40. 40.
    Taa, A.: Optimality conditions for vector optimization problems of a difference of convex mappings. J. Global Optim. 31(3), 421–436 (2005)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Toland, J.F.: On subdifferential calculus and duality in nonconvex optimization. Mémoires de la Société Mathématique de France 60, 177–183 (1979)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Wang, S.: Algorithms for multiobjective and nonsmooth optimization. In: Kleinschmidt, P., Radermacher, F., Sweitzer, W., Wildermann, H. (eds.) Methods of Operations Research, vol. 58, pp. 131–142. Athenaum Verlag, Kronberg im Taunus (1989)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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