Multiobjective Double Bundle Method for DC Optimization

  • Outi MontonenEmail author
  • Kaisa Joki


We discuss about the multiobjective double bundle method for nonsmooth multiobjective optimization where objective and constraint functions are presented as a difference of two convex (DC) functions. By utilizing a special technique called the improvement function, we are able to handle several objectives and constraints simultaneously. The method improves every objective at each iteration and the improvement function preserves the DC property of the objectives and constraints. Once the improvement function is formed, we can approximate it by using a special cutting plane model capturing the convex and concave behaviour of a DC function. We solve the problem with a modified version of the single-objective double bundle method using the cutting plane model as an objective. The multiobjective double bundle method is proved to be finitely convergent to a weakly Pareto stationary solution under mild assumptions. Moreover, the applicability of the method is considered.



This work was financially supported by the University of Turku and the Academy of Finland (Project No. 294002).


  1. 1.
    Astorino, A., Miglionico, G.: Optimizing sensor cover energy via DC programming. Optim. Lett. 10(2), 355–368 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carrizosa, E., Guerrero, V., Romero Morales, D.: Visualizing data as objects by DC (difference of convex) optimization. Math. Program. 169(1), 119–140 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Craft, D., Halabi, T., Shih, H.A., Bortfeld, T.: An approach for practical multiobjective IMRT treatment planning. Int. J. Radiat. Oncol. Biol. Phys. 69(5), 1600–1607 (2007)CrossRefGoogle Scholar
  7. 7.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  9. 9.
    Gadhi, N., Metrane, A.: Sufficient optimality condition for vector optimization problems under D.C. data. J. Global Optim. 28(1), 55–66 (2004)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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gaudioso, M., Giallombardo, G., Miglionico, G., Bagirov, A.: Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations. J. Global Optim. 71(1), 37–55 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gutjahr, W.J., Nolz, P.C.: Multicriteria optimization in humanitarian aid. Eur. J. Oper. Res. 252(2), 351–366 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math. 9(3), 707–713 (1959)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hiriart-Urruty, J-.B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization, vol. 256, pp. 37–70. Springer, Berlin (1985)Google Scholar
  15. 15.
    Holmberg, K., Tuy, H.: A production-transportation problem with stochastic demand and concave production costs. Math. Program. 85(1), 157–179 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Horst, R., Thoai, N.V.: DC programming: Overview. J. Optim. Theory Appl. 103(1), 1–43 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ji, Y., Qu, S.: Proximal point algorithms for vector DC programming with applications to probabilistic lot sizing with service levels. Discret. Dyn. Nat. Soc. 2017, 5675183 (2017).
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Joki, K., Bagirov, A.M., Karmitsa, N., Mäkelä, M.M., Taheri, S.: Double bundle method for finding Clarke stationary points in nonsmooth DC programming. SIAM J. Optim. 28(2), 1892–1919 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kiwiel, K.C.: A descent method for nonsmooth convex multiobjective minimization. Large Scale Syst. 8(2), 119–129 (1985)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable optimization. Math. Program. 46(1–3), 105–122 (1990)CrossRefGoogle Scholar
  23. 23.
    Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by D.C. algorithms. J. Global Optim. 11(3), 253–285 (1997)Google Scholar
  24. 24.
    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. Ann. Oper. Res. 133(1–4), 23–46 (2005)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: analysis and algorithms with applications to optimal control. World Scientific, Singapore (1992)CrossRefGoogle 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.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  29. 29.
    Miettinen, K., Mäkelä, M.M.: Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization 34(3), 231–246 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Montonen, O., Joki, K.: Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints. J. Global Optim. 72(3), 403–429 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Montonen, O., Karmitsa, N., Mäkelä, M.M.: Multiple subgradient descent bundle method for convex nonsmooth multiobjective optimization. Optimization 67(1), 139–158 (2018)MathSciNetCrossRefGoogle Scholar
  32. 32.
    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)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Poirion, F., Mercier, Q., Désidéri, J.-A.: Descent algorithm for nonsmooth stochastic multiobjective optimization. Comput. Optim. Appl. 68(2), 317–331 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefGoogle Scholar
  35. 35.
    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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rangaiah, G.P.: Multi-Objective Optimization: Techniques and Applications in Chemical Engineering. Advances in Process Systems Engineering. World Scientific, Singapore (2009)Google Scholar
  37. 37.
    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)MathSciNetCrossRefGoogle Scholar
  38. 38.
    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)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Taa, A.: Optimality conditions for vector optimization problems of a difference of convex mappings. J. Global Optim. 31(3), 421–436 (2005)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Toland, J.F.: On subdifferential calculus and duality in nonconvex optimization. Mémoires de la Société Mathématique de France 60, 177–183 (1979)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wang, S.: Algorithms for Multiobjective and Nonsmooth Optimization. In: Kleinschmidt, P., Radermacher, F.J., Sweitzer, W., Wildermann, H. (eds.) Methods of Operations Research, 58, pp. 131–142. Athenaum Verlag, Frankfurt (1989)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

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