On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming

Luo, Hezhi; Wu, Han‐Qing; Chen, G. T.

doi:10.1007/s10898-011-9779-x

Cited by 23 publications

(16 citation statements)

References 49 publications

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“…From (57) and the fact that (x n , λ n ) ∈ S e (c n ) it follows that f (x * ) ≤ f * , which implies that x * is a globally optimal solution of problem (55). Consequently, LICQ holds at this point.…”

Section: Mathematical Programmingmentioning

confidence: 94%

“…Arguing as above, one can easily verify that lim inf n→∞ u n /p(x n , λ n ) ≥ 0 and lim inf n→∞ w n /q(x n , λ n ) ≥ 0. From (57) and the fact that (x n , λ n ) ∈ S e (c n ) it follows that the sequences {f (x n )} and {η 1 (x n , λ n )} are bounded. Moreover, by the definition of {(x n , λ n )} one has τ ≤ L e (x n , λ n , c n ) < f * .…”

Section: Mathematical Programmingmentioning

confidence: 99%

“…The axiomatic approach that we use was inspired by [50]. It provides one with a simple and unified framework for the study of various augmented Lagrangian functions, such as the Hestenes-Powell-Rockafellar augmented Lagrangian [2,66,67], the cubic augmented Lagrangian [42], Mangasarian's augmented Lagrangian [59,89], the exponential penalty function [1,80,79,50,83], the Log-Sigmoid Lagrangian [64,65], the penalized exponential-type augmented Lagrangians [1,79,50,83], the modified barrier functions [63,79,50,83], the p-th power augmented Lagrangian [43,44,91,46,45,88,50], He-Wu-Meng's augmented Lagrangian [36], extensions of the Hestenes-Powell-Rockafellar augmented Lagrangian to the case of nonlinear second order cone programs [51,52,97], nonlinear semidefinite programs [40,78,76,96,77,86,57,87,90,92], and semi-infinite programs [70,…”

Section: Introductionmentioning

confidence: 99%

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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: Mathematical Programmingmentioning

confidence: 94%

Section: Mathematical Programmingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…if x ∈ Ω α , and F (x, c) = +∞ otherwise. Let us point out that F (x, c) is, in essence, a direct modification of the Hestenes-Powell-Rockafellar augmented Lagrangian function to the case of nonlinear semidefinite programming problems [103,101,123,102,111,81,112,114,117] with Lagrange multipliers λ and µ replaced by their estimates λ(x) and µ(x). One can verify that the function F (·, c) is l.s.c.…”

Section: Example Iii: Continuously Differentiable Exact Penalty Functmentioning

confidence: 99%

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Долгополик¹

2018

J Optim Theory Appl

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In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications.In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

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“…However, the conventional SRC algorithm usually has a "sparser" solution than the generalized SRC algorithm. Typical conventional sparse representation algorithms include l1-regularized least squares (L1LS) [29], fast iterative shrinkage and thresholding algorithm (FISTA) [30], augmented Lagrangian [31], orthogonal matching pursuit (OMP) [32] etc. Typical generalized sparse representation algorithms include linear regression classification (LRC) [33], collaborative representation (CRC) [34], two phase sparse representation etc.…”

Section: Introductionmentioning

confidence: 99%

Multiple representations and sparse representation for image classification

Zhang

Zhong

2015

Pattern Recognition Letters

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On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming

Cited by 23 publications

References 49 publications

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

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

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Multiple representations and sparse representation for image classification

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