Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming

Chan, Zi Xian; Sun, Defeng

doi:10.1137/070681235

Cited by 60 publications

(84 citation statements)

References 50 publications

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“…In order to analyze the rate of convergence of the Newton-CG augmented Lagrangian method to be presented in Section 4, we need the following result which characterizes the Lipschitz continuity of T −1 g at the origin. The result we establish here is stronger than that appeared in Proposition 15 of [8].…”

Section: Assumption 1 Problem (P ) Satisfies the Slater Conditioncontrasting

confidence: 66%

“…In spite of that, under mild conditions, a linear rate of convergence analysis is available (superlinear convergence is also possible when σ k goes to infinity, which should be avoided in numerical implementations) [33]. However, recent studies conducted by Sun, Sun, and Zhang [37] and Chan and Sun [8] revealed that under the constraint nondegenerate conditions for (D) and (P ) (i.e., the dual nondegeneracy and primal nondegeneracy in the IPMs literature, e.g., [1]), respectively, the augmented Lagrangian method can be locally regarded as an approximate generalized Newton method applied to a semismooth equation. It is this connection that inspired us to investigate the augmented Lagrangian method for SDPs.…”

Section: Introductionmentioning

confidence: 99%

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A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

Zhao¹,

Sun²,

Toh³

2010

SIAM J. Optim.

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366

387

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Abstract. We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth Newton-CG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts.

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Section: Assumption 1 Problem (P ) Satisfies the Slater Conditioncontrasting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

Zhao¹,

Sun²,

Toh³

2010

SIAM J. Optim.

Self Cite

366

387

View full text Add to dashboard Cite

show abstract

“…Constraint nondegeneracy plays a very important role in optimization, see [3, Def. 5], [10,Def. 9], and [37, Sect.…”

Section: Existence Of Optimal Dualmentioning

confidence: 99%

“…Since y is an optimal solution of (8), A := Π K n + (D + Diag(y)) is the optimal solution of (3) by (10). Obviously, A is feasible with respect to the constraints of (3).…”

Section: Generalized Jacobian Ofmentioning

confidence: 99%

A Semismooth Newton Method for the Nearest Euclidean Distance Matrix Problem

Qi¹

2013

SIAM J. Matrix Anal. & Appl.

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Abstract. The Nearest Euclidean distance matrix problem (NEDM) is a fundamental computational problem in applications such as multidimensional scaling and molecular conformation from nuclear magnetic resonance data in computational chemistry. Especially in the latter application, the problem is often large scale with the number of atoms ranging from a few hundreds to a few thousands. In this paper, we introduce a semismooth Newton method that solves the dual problem of (NEDM). We prove that the method is quadratically convergent. We then present an application of the Newton method to NEDM with H-weights. We demonstrate the superior performance of the Newton method over existing methods including the latest quadratic semi-definite programming solver. This research also opens a new avenue towards efficient solution methods for the molecular embedding problem.

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“…We refer to [8] for more references, results, and comments on its application to SDP. We describe it below.…”

Section: Nonsingularity Of ∂F (Y)mentioning

confidence: 99%

Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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Abstract:The conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem.For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy.We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP. Response to Reviewers:The author would like to thank the referees and the associate editor for their efforts in their speedy review. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Conditional Quadratic Semidefinite Programming: Examples and MethodsHou-Duo Qi * April 14, 2014; Revised May 3, 2014 AbstractThe conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem. For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy. We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP.

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Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming

Cited by 60 publications

References 50 publications

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

A Semismooth Newton Method for the Nearest Euclidean Distance Matrix Problem

Conditional Quadratic Semidefinite Programming: Examples and Methods

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