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2021 | OriginalPaper | Chapter

9. Lagrangian Transformation and Interior Ellipsoid Methods

Author : Roman A. Polyak

Published in: Introduction to Continuous Optimization

Publisher: Springer International Publishing

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Abstract

The NR approach produced a number of multipliers methods, which are primal exterior and dual interior.

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previous chapter Realizations of the NR Principle
next chapter Finding Nonlinear Equilibrium
Literature
Bregman, L.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)MathSciNetCrossRef
Bregman, L., Censor, Y., Reich, S.: Dykstra algorithm as the nonlinear extention of Bregman’s optimization method. J. Convex Anal. 6(2), 319–333 (1999)MathSciNetMATH
Censor, Y., Zenios, S.: The Proximal minimization algorithm with d–functions. J. Optim. Theory Appl. 73, 451–464 (1992)MathSciNetCrossRef
Chen, G., Teboulle, M.: Convergence analysis of a proximal–like minimization algorithm using Bregman functions. SIAM J. Optim. 3(4), 538–543 (1993)MathSciNetCrossRef
Dikin, I.: Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR 174, 747–748 (1967). Translated in Soviet Mathematics Doklady 8, 674–675 (1967)
Eckstein, J.: Nonlinear proximal point algorithms using bregman functions with applications to convex programming. Y. Math. Oper. Res. 18(1), 202–226 (1993)MathSciNetCrossRef
Matioli, L., Gonzaga, C.: A new family of penalties for augmented lagrangian methods. Numer. Linear Algebra Appl. 15, 925–944 (2008)MathSciNetCrossRef
Polyak, R.: Lagrangian transformation in convex optimization, Research Report-072004. Department of SEOR and Mathematical Science Department, GMU, Fairfax, pp 1–23 (2004)
Polyak, R.: Lagrangian transformation and interior ellipsoid methods. J. Optim. Theory Appl. (JOTA) 164(3), 966–992 (2015)
Metadata
Title
Lagrangian Transformation and Interior Ellipsoid Methods
Author
Roman A. Polyak
Copyright Year
2021
Book
Introduction to Continuous Optimization

Print ISBN: 978-3-030-68711-3

Electronic ISBN: 978-3-030-68713-7

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-68713-7_9

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