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Log-Sigmoid Multipliers Method in Constrained Optimization

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  • Roman Polyak

Abstract

In this paper we introduced and analyzed the Log-Sigmoid (LS) multipliers method for constrained optimization. The LS method is to the recently developed smoothing technique as augmented Lagrangian to the penalty method or modified barrier to classical barrier methods. At the same time the LS method has some specific properties, which make it substantially different from other nonquadratic augmented Lagrangian techniques. We established convergence of the LS type penalty method under very mild assumptions on the input data and estimated the rate of convergence of the LS multipliers method under the standard second order optimality condition for both exact and nonexact minimization. Some important properties of the dual function and the dual problem, which are based on the LS Lagrangian, were discovered and the primal–dual LS method was introduced. Copyright Kluwer Academic Publishers 2001

Suggested Citation

  • Roman Polyak, 2001. "Log-Sigmoid Multipliers Method in Constrained Optimization," Annals of Operations Research, Springer, vol. 101(1), pages 427-460, January.
    • Handle: RePEc:spr:annopr:v:101:y:2001:i:1:p:427-460:10.1023/a:1010938423538
    DOI: 10.1023/A:1010938423538
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    Cited by:

    1. M. V. Dolgopolik, 2018. "Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property," Journal of Global Optimization, Springer, vol. 71(2), pages 237-296, June.
    2. R. Polyak & I. Griva, 2004. "Primal-Dual Nonlinear Rescaling Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 111-156, July.
    3. Liwei Zhang & Jian Gu & Xiantao Xiao, 2011. "A class of nonlinear Lagrangians for nonconvex second order cone programming," Computational Optimization and Applications, Springer, vol. 49(1), pages 61-99, May.
    4. H. Luo & X. Sun & Y. Xu & H. Wu, 2010. "On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints," Journal of Global Optimization, Springer, vol. 46(2), pages 217-232, February.

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