Skip to main content

2021 | OriginalPaper | Buchkapitel

6. Self-Concordant Functions and IPM Complexity

verfasst von : Roman A. Polyak

Erschienen in: Introduction to Continuous Optimization

Verlag: Springer International Publishing

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

search-config
loading …

Abstract

The introduction by N. Karmarkar of his projective transformation method for LP in 1984; finding by P. Gill et al. (1986) of a, connection between Karmarkar’s and Newton log-barrier methods; rediscovery of the affine scaling (AS) method.

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
Zurück zum Zitat Adler, I., Monteiro, R.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Program. 50, 29–51 (1991)MathSciNetCrossRef Adler, I., Monteiro, R.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Math. Program. 50, 29–51 (1991)MathSciNetCrossRef
Zurück zum Zitat Adler, I., Karmarkar, N., Resende, M., Veiga, G.: An implementation of Karmarkar’s algorithm for linear programming. Math. Program. 44, 297–335 (1989). Errata in Math. Program. 50, 415 (1991) Adler, I., Karmarkar, N., Resende, M., Veiga, G.: An implementation of Karmarkar’s algorithm for linear programming. Math. Program. 44, 297–335 (1989). Errata in Math. Program. 50, 415 (1991)
Zurück zum Zitat Barnes, E.R.: A variation on Karmarkar’s algorithm for solving linear programming problem. Math. Program. 36, 174–182 (1986)MathSciNetCrossRef Barnes, E.R.: A variation on Karmarkar’s algorithm for solving linear programming problem. Math. Program. 36, 174–182 (1986)MathSciNetCrossRef
Zurück zum Zitat Gill, P., Murray, W., Saunders, J., Tomlin, J., Wright, M.: On projected Newton Barrier methods for linear programming and an equivalence to Karmarkar’s projective method. Math. Program. 36, 183–209 (1986)MathSciNetCrossRef Gill, P., Murray, W., Saunders, J., Tomlin, J., Wright, M.: On projected Newton Barrier methods for linear programming and an equivalence to Karmarkar’s projective method. Math. Program. 36, 183–209 (1986)MathSciNetCrossRef
Zurück zum Zitat Gonzaga, C.: An algorithm for solving linear programming problems in O(n 3 L) operations. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 1–28. Springer, New York (1989) Gonzaga, C.: An algorithm for solving linear programming problems in O(n 3 L) operations. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 1–28. Springer, New York (1989)
Zurück zum Zitat Jensen, B., Roos, C., Terlaky, T., Vial, J.: Primal-dual algorithms for linear programming based on th logarithmic barrier method. J. Optim. Theory Appl. 83, 1–26 (1994)MathSciNetCrossRef Jensen, B., Roos, C., Terlaky, T., Vial, J.: Primal-dual algorithms for linear programming based on th logarithmic barrier method. J. Optim. Theory Appl. 83, 1–26 (1994)MathSciNetCrossRef
Zurück zum Zitat Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming, In: Megiddo, N. (ed.) Progress in mathematical programming: interior point and related methods, pp 29–47. Springer, New York (1989a)CrossRef Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming, In: Megiddo, N. (ed.) Progress in mathematical programming: interior point and related methods, pp 29–47. Springer, New York (1989a)CrossRef
Zurück zum Zitat Lustig, I., Marsten, R., Shanno, D.: Computational experience with a primal-dual interior point method for linear programming. Linear Algebra Appl. 152, 191–222 (1991)MathSciNetCrossRef Lustig, I., Marsten, R., Shanno, D.: Computational experience with a primal-dual interior point method for linear programming. Linear Algebra Appl. 152, 191–222 (1991)MathSciNetCrossRef
Zurück zum Zitat Lustig, I., Marsten, R., Shanno, D.: On implementing Mehrotra’s predictor-corrector interior point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)MathSciNetCrossRef Lustig, I., Marsten, R., Shanno, D.: On implementing Mehrotra’s predictor-corrector interior point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)MathSciNetCrossRef
Zurück zum Zitat Lustig, I., Marsten, R., Shanno, D.: Interior point methods for linear programming: computational state of the art. ORSA J. Comput. 6(1), 1–14 (1994)MathSciNetCrossRef Lustig, I., Marsten, R., Shanno, D.: Interior point methods for linear programming: computational state of the art. ORSA J. Comput. 6(1), 1–14 (1994)MathSciNetCrossRef
Zurück zum Zitat Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming—Interior Point and Related Methods. Springer, Berlin, pp. 131–158 (1989) Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming—Interior Point and Related Methods. Springer, Berlin, pp. 131–158 (1989)
Zurück zum Zitat Megiddo, N., Shub, M.: Boundary behavior of interior point methods in linear programming. Math. Oper. Res. 14(1), 97–146 (1989)MathSciNetCrossRef Megiddo, N., Shub, M.: Boundary behavior of interior point methods in linear programming. Math. Oper. Res. 14(1), 97–146 (1989)MathSciNetCrossRef
Zurück zum Zitat Mehrotra, S.: Higher order methods and their performance. Technical Report 90-16R1, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, IL 60208, USA (1990). Revised July 1991 Mehrotra, S.: Higher order methods and their performance. Technical Report 90-16R1, Department of Industrial Engineering and Management Science, Northwestern University, Evanston, IL 60208, USA (1990). Revised July 1991
Zurück zum Zitat Mehrotra, S.: On the implementation of a (primal-dual) interior point method. SIAM J. Optim. 2(4), 575–601 (1992)MathSciNetCrossRef Mehrotra, S.: On the implementation of a (primal-dual) interior point method. SIAM J. Optim. 2(4), 575–601 (1992)MathSciNetCrossRef
Zurück zum Zitat Monteiro, R., Adler, I., Resende, M.: A polynomial - time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res. 15, 191–214 (1990)MathSciNetCrossRef Monteiro, R., Adler, I., Resende, M.: A polynomial - time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res. 15, 191–214 (1990)MathSciNetCrossRef
Zurück zum Zitat Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, Norwell, (2004)CrossRef Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, Norwell, (2004)CrossRef
Zurück zum Zitat Nesterov, Y., Nemirovski, A.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRef Nesterov, Y., Nemirovski, A.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRef
Zurück zum Zitat Pardalos, P., Ye, Y., Han, G.: Computational Aspects of an Interior Point Algorithm for Quadratic Problems with box Constraints. In: Large-Scale Numerical Optimization. SIAM, Philadelphia, pp. 92–112 (1990) Pardalos, P., Ye, Y., Han, G.: Computational Aspects of an Interior Point Algorithm for Quadratic Problems with box Constraints. In: Large-Scale Numerical Optimization. SIAM, Philadelphia, pp. 92–112 (1990)
Zurück zum Zitat Polyak, R.: The Legendre Transformation in Modern Optimization in “Optimization and its Applications in Control and Data Sciences”. Springer, Berlin (2016) Polyak, R.: The Legendre Transformation in Modern Optimization in “Optimization and its Applications in Control and Data Sciences”. Springer, Berlin (2016)
Zurück zum Zitat Potra, F.: A quadratically convergent predictor - corrector method for solving linear programs from infeasible starting points. Math. Program. 67(3), 383–406 (1994)MathSciNetCrossRef Potra, F.: A quadratically convergent predictor - corrector method for solving linear programs from infeasible starting points. Math. Program. 67(3), 383–406 (1994)MathSciNetCrossRef
Zurück zum Zitat Renegar, J.: A polynomial -time algorithm, based on Newton’s method, for linear programming. Math. Program. 40, 59–93 (1988)MathSciNetCrossRef Renegar, J.: A polynomial -time algorithm, based on Newton’s method, for linear programming. Math. Program. 40, 59–93 (1988)MathSciNetCrossRef
Zurück zum Zitat Renegar, J.: A mathematical view of interior - point methods in convex optimization. In: MPS-SIAM Series on Optimization. SIAM, New York (2001) Renegar, J.: A mathematical view of interior - point methods in convex optimization. In: MPS-SIAM Series on Optimization. SIAM, New York (2001)
Zurück zum Zitat Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Wiley, Chichester (1997)MATH Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Wiley, Chichester (1997)MATH
Zurück zum Zitat Vanderbei, R., Meketon, M., Freedman, B.: A modification of Karmarkar’s linear programming algorithm. Algorithmica 1(4), 395–407 (1986)MathSciNetCrossRef Vanderbei, R., Meketon, M., Freedman, B.: A modification of Karmarkar’s linear programming algorithm. Algorithmica 1(4), 395–407 (1986)MathSciNetCrossRef
Zurück zum Zitat Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, vol. 27. Kluwer Academic Publisher, Dordrecht (2000) Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, vol. 27. Kluwer Academic Publisher, Dordrecht (2000)
Zurück zum Zitat Wright, S.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)CrossRef Wright, S.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)CrossRef
Zurück zum Zitat Ye, Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)CrossRef Ye, Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)CrossRef
Metadaten
Titel
Self-Concordant Functions and IPM Complexity
verfasst von
Roman A. Polyak
Copyright-Jahr
2021
DOI
https://doi.org/10.1007/978-3-030-68713-7_6