Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems

Liu, Yong-Jin; Zhang, Liwei

doi:10.1016/j.na.2006.07.022

Cited by 33 publications

(17 citation statements)

References 10 publications

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“…(see [51,52,97]). The function Φ(y, λ, c) defined above is a particular case of the function Φ(y, λ, c) for the general cone constrained optimization problem from Example 1 with σ(y) = y 2 /2.…”

Section: Nonlinear Second Order Cone Programmingmentioning

confidence: 99%

Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

Долгополик

2018

J Glob Optim

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In this article we present a general theory of augmented Lagrangian functions for cone constrained optimization problems that allows one to study almost all known augmented Lagrangians for these problems within a unified framework. We develop a new general method for proving the existence of global saddle points of augmented Lagrangian functions, called the localization principle. The localization principle unifies, generalizes and sharpens most of the known results on the existence of global saddle points, and, in essence, reduces the problem of the existence of global saddle points to a local analysis of optimality conditions. With the use of the localization principle we obtain first necessary and sufficient conditions for the existence of a global saddle point of an augmented Lagrangian for cone constrained minimax problems via both second and first order optimality conditions. In the second part of the paper, we present a general approach to the construction of globally exact augmented Lagrangian functions. The general approach developed in this paper allowed us not only to sharpen most of the existing results on globally exact augmented Lagrangians, but also to construct first globally exact augmented Lagrangian functions for equality constrained optimization problems, for nonlinear second order cone programs and for nonlinear semidefinite programs. These globally exact augmented Lagrangians can be utilized in order to design new superlinearly (or even quadratically) convergent optimization methods for cone constrained optimization problems.

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Section: Nonlinear Second Order Cone Programmingmentioning

confidence: 99%

Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

Долгополик

2018

J Glob Optim

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“…The main idea was given by Di Pillo and Grippo [11,12] for nonlinear programming and it consists in incorporating a multipliers estimate in an augmented Lagrangian function. In the case of the nonlinear SOCP, the augmented Lagrangian [26,37] is given by…”

Section: Construction Of Exact Penalty Functionmentioning

confidence: 99%

Differentiable Exact Penalty Functions for Nonlinear Second-Order Cone Programs

Fukuda¹,

Silva²,

Fukushima³

2012

SIAM J. Optim.

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We propose a method to solve nonlinear second-order cone programs (SOCPs), based on a continuously dierentiable exact penalty function. The construction of the penalty function is given by incorporating a multipliers estimate in the augmented Lagrangian for SOCPs. Under the nondegeneracy assumption and the strong second-order sucient condition, we show that a generalized Newton method has global and superlinear convergence. We also present some preliminary numerical experiments.

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“…Liu and Zhang [16,17] studied the rate of the convergence for the proximal augmented Lagrangian method for problem (1.1) for this case. But we have not seen other published works about nonlinear Lagrangians for second order cone optimization.…”

Section: Introductionmentioning

confidence: 99%