Skip to main content

2020 | OriginalPaper | Buchkapitel

Constructing Mixed Algorithms on the Basis of Some Bundle Method

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

search-config
loading …

Abstract

In the paper, a method is proposed for minimizing a nondifferentiable convex function. This method belongs to a class of bundle methods. In the developed method it is possible to periodically produce discarding all previously constructed cutting planes that form the model of the objective function. These discards are applied when approximation of the epigraph of the objective function is sufficiently good in the a neighborhood of the current iteration point, and the quality of this approximation is estimated by using the model of the objective function. It is proposed an approach for constructing mixed minimization algorithms on the basis of the developed bundle method with involving any relaxation methods. The opportunity to mix the developed bundle method with other methods is provided as follows. In the proposed method during discarding the cutting planes the main iteration points are fixed with the relaxation condition. Any relaxation minimization method can be used to build these points. Moreover, the convergence of all such mixed algorithms will be guaranteed by the convergence of the developed bundle method. It is important to note that the procedures for updating cutting planes introduced in the bundle method will be transferred to mixed algorithms. The convergence of the proposed method is investigated, its properties are discussed, an estimate of the accuracy of the solution and estimation of the complexity of finding an approximate solution are obtained.

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!

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!

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!

Literatur
1.
Zurück zum Zitat Zangwill, W.I.: Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs (1969)MATH Zangwill, W.I.: Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs (1969)MATH
2.
Zurück zum Zitat Zabotin, I.Y., Yarullin, R.S.: A cutting method and construction of mixed minimization algorithms on its basis. Uchen. Zap. Kazansk. Univ. Ser. Fiz.-Matem. Nauki. 156(4), 14–24 (2014). (in Russian)MATH Zabotin, I.Y., Yarullin, R.S.: A cutting method and construction of mixed minimization algorithms on its basis. Uchen. Zap. Kazansk. Univ. Ser. Fiz.-Matem. Nauki. 156(4), 14–24 (2014). (in Russian)MATH
4.
Zurück zum Zitat Polyak, B.T.: Introduction to Optimization. Nauka, Moscow (1983). [in Russian]MATH Polyak, B.T.: Introduction to Optimization. Nauka, Moscow (1983). [in Russian]MATH
5.
Zurück zum Zitat de Oliveira, W., Eckstein, J.: A bundle method for exploiting additive structure in difficult optimization problems. Technical report (2015) de Oliveira, W., Eckstein, J.: A bundle method for exploiting additive structure in difficult optimization problems. Technical report (2015)
8.
Zurück zum Zitat Zabotin, I.Y., Yarullin, R.S.: One approach to constructing cutting algorithms with dropping of cutting planes. Russ. Math. (Iz. VUZ). 57(3), 60–64 (2013)MathSciNetCrossRef Zabotin, I.Y., Yarullin, R.S.: One approach to constructing cutting algorithms with dropping of cutting planes. Russ. Math. (Iz. VUZ). 57(3), 60–64 (2013)MathSciNetCrossRef
Metadaten
Titel
Constructing Mixed Algorithms on the Basis of Some Bundle Method
verfasst von
Rashid Yarullin
Copyright-Jahr
2020
DOI
https://doi.org/10.1007/978-3-030-58657-7_14