Annals of Operations Research

, Volume 101, Issue 1, pp 427–460

Log-Sigmoid Multipliers Method in Constrained Optimization

  • Roman A. Polyak

DOI: 10.1023/A:1010938423538

Cite this article as:
Polyak, R.A. Annals of Operations Research (2001) 101: 427. doi:10.1023/A:1010938423538


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.

log-sigmoidmultipliers methoddualitysmoothing technique

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Roman A. Polyak
    • 1
  1. 1.Systems Engineering and Operations Research Department and Mathematical Sciences DepartmentGeorge Mason UniversityFairfaxUSA