Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2020 | OriginalPaper | Buchkapitel

14. Multiobjective Double Bundle Method for DC Optimization

verfasst von : Outi Montonen, Kaisa Joki

Erschienen in: Numerical Nonsmooth Optimization

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel New Multiobjective Proximal Bundle Method with Scaled Improvement Function
Nächstes Kapitel Mixed-Integer Linear Optimization: Primal–Dual Relations and Dual Subgradient and Cutting-Plane Methods
Literatur
1.
Astorino, A., Miglionico, G.: Optimizing sensor cover energy via DC programming. Optim. Lett. 10(2), 355–368 (2016)MathSciNetCrossRef
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)MathSciNetCrossRef
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)MathSciNetCrossRef
4.
Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)MathSciNetCrossRef
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)MathSciNetCrossRef
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)CrossRef
7.
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRef
8.
Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)MATH
9.
Gadhi, N., Metrane, A.: Sufficient optimality condition for vector optimization problems under D.C. data. J. Global Optim. 28(1), 55–66 (2004)
10.
Gaudioso, M., Gruzdeva, T.V., Strekalovsky, A.S.: On numerical solving the spherical separability problem. J. Global Optim. 66(1), 21–34 (2016)MathSciNetCrossRef
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)MathSciNetCrossRef
12.
Gutjahr, W.J., Nolz, P.C.: Multicriteria optimization in humanitarian aid. Eur. J. Oper. Res. 252(2), 351–366 (2016)MathSciNetCrossRef
13.
Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math. 9(3), 707–713 (1959)MathSciNetCrossRef
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)
15.
Holmberg, K., Tuy, H.: A production-transportation problem with stochastic demand and concave production costs. Math. Program. 85(1), 157–179 (1999)MathSciNetCrossRef
16.
17.
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)MathSciNetCrossRef
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)MathSciNetCrossRef
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)MathSciNetCrossRef
21.
Kiwiel, K.C.: A descent method for nonsmooth convex multiobjective minimization. Large Scale Syst. 8(2), 119–129 (1985)MathSciNetMATH
22.
Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable optimization. Math. Program. 46(1–3), 105–122 (1990)CrossRef
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)
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)MathSciNetMATH
25.
Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: analysis and algorithms with applications to optimal control. World Scientific, Singapore (1992)CrossRef
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)CrossRef
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)CrossRef
28.
Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)MATH
29.
Miettinen, K., Mäkelä, M.M.: Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization 34(3), 231–246 (1995)MathSciNetCrossRef
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)MathSciNetCrossRef
31.
Montonen, O., Karmitsa, N., Mäkelä, M.M.: Multiple subgradient descent bundle method for convex nonsmooth multiobjective optimization. Optimization 67(1), 139–158 (2018)MathSciNetCrossRef
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)MathSciNetMATH
33.
Poirion, F., Mercier, Q., Désidéri, J.-A.: Descent algorithm for nonsmooth stochastic multiobjective optimization. Comput. Optim. Appl. 68(2), 317–331 (2017)MathSciNetCrossRef
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)MathSciNetCrossRef
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)MathSciNetCrossRef
36.
Rangaiah, G.P.: Multi-Objective Optimization: Techniques and Applications in Chemical Engineering. Advances in Process Systems Engineering. World Scientific, Singapore (2009)
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)MathSciNetCrossRef
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)MathSciNetMATH
39.
Taa, A.: Optimality conditions for vector optimization problems of a difference of convex mappings. J. Global Optim. 31(3), 421–436 (2005)MathSciNetCrossRef
40.
Toland, J.F.: On subdifferential calculus and duality in nonconvex optimization. Mémoires de la Société Mathématique de France 60, 177–183 (1979)MathSciNetCrossRef
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)
Metadaten
Titel
Multiobjective Double Bundle Method for DC Optimization
verfasst von
Outi Montonen
Kaisa Joki
Copyright-Jahr
2020
Buch
Numerical Nonsmooth Optimization

Print ISBN: 978-3-030-34909-7

Electronic ISBN: 978-3-030-34910-3

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-3-030-34910-3_14