A semismooth Newton-CG based dual PPA for matrix spectral norm approximation problems

Chen, Caihua; Liu, Yong-Jin; Sun, Defeng; Toh, Kim-Chuan

doi:10.1007/s10107-014-0853-2

Cited by 21 publications

(30 citation statements)

References 45 publications

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“…Denote σ = σ (X). Let G S (X) and G R (X) be defined by (12). Therefore, by (13), we know that for any H → 0,…”

Section: Lipschitz Continuitymentioning

confidence: 99%

“…Denote σ = σ (X). Recall the mappings G S and G R defined in (12). We know from [20,Proposition 8] that there exists an open neighborhood B ⊆ N of X such that G S twice continuously differentiable on B and…”

Section: S(h a L A Lmentioning

confidence: 99%

“…Therefore, the study of the general spectral operators will provide the necessary foundations for both computational and theoretical study of the general MOPs. In particular, the first-and second-order properties of spectral operators obtained in [19] including the well-definedness, continuity, directional differentiability, and Fréchet-differentiability are of fundamental importance in the study of MOPs [18,43,12,17].…”

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confidence: 99%

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Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized Jacobian

Ding

Sun²,

Sun

et al. 2020

SIAM J. Optim.

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Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. 168, 509-531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector function to all eigenvalues/singular values of the underlying matrices. Spectral operators play a crucial role in the study of various applications involving matrices such as matrix optimization problems (MOPs) that include semidefinite programming as one of the most important example classes. In this paper, we will study more fundamental first-and second-order properties of spectral operators, including the Lipschitz continuity, ρ-order B(ouligand)-differentiability (0 < ρ ≤ 1), ρ-order G-semismoothness (0 < ρ ≤ 1), and characterization of generalized Jacobians.

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“…Denote σ = σ (X). Let G S (X) and G R (X) be defined by (12). Therefore, by (13), we know that for any H → 0,…”

Section: Lipschitz Continuitymentioning

confidence: 99%

Section: S(h a L A Lmentioning

confidence: 99%

mentioning

confidence: 99%

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Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized Jacobian

Ding

Sun²,

Sun

et al. 2020

SIAM J. Optim.

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“…To illustrate the usefulness of our derived results, we investigate the fast convergence rates of the augmented Lagrangian method (ALM) for solving problem (2) under the quadratic growth conditions. Our motivation of this part stemmed from the highly promising numerical results of the ALM incorporated with the semismooth Newton-CG algorithm for solving large scale convex matrix problems [63,32,10,62,38]. We extend the results in the current literatures on the rates of the ALM for solving convex optimization problems [49,40] and show that the (super)linear convergence rates of the ALM may still be valid even if problem (1) admits multiple solutions.…”

mentioning

confidence: 90%

Quadratic Growth Conditions for Convex Matrix Optimization Problems Associated with Spectral Functions

Cui¹,

Ding²,

Zhao³

2017

SIAM J. Optim.

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Abstract. In this paper, we provide two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex symmetric and non-symmetric matrix optimization problems regularized by nonsmooth spectral functions. These sufficient conditions are derived via the study of the C 2 -cone reducibility of spectral functions and the metric subregularity of their subdifferentials, respectively. As an application, we demonstrate how quadratic growth conditions are used to guarantee the desirable fast convergence rates of the augmented Lagrangian methods (ALM) for solving convex matrix optimization problems. Numerical experiments on an easy-toimplement ALM applied to the fastest mixing Markov chain problem are also presented to illustrate the significance of the obtained results.Key words. matrix optimization, spectral functions, quadratic growth conditions, metric subregularity, augmented Lagrangian function, fast convergence rates AMS subject classifications. 65K05, 90C25, 90C311. Introduction. The quadratic growth condition is an important concept in optimization. It is closely related to the metric subregularity and calmness of setvalued mappings (see Section 2 for definitions), and the existence of error bounds. From different perspectives, the study of the metric subregularity and the calmness of set-valued mappings plays a central role in variational analysis, such as nonsmooth calculus and perturbation analysis of variational problems. We refer the reader to the monograph by Dontchev and Rockafellar [17] for a comprehensive study on both theory and applications of related subjects. See also [31,18,24,37,25,26,43] and references therein for recent advances.Instead of considering general set-valued mappings, in this paper, we mainly focus on the solution mappings of convex matrix optimization problems. It is known from [1] that for convex problems, the metric subregularity of the subdifferentials of the essential objective functions (or the calmness of the solution mappings) can be equivalently characterized by the corresponding quadratic growth conditions. This connection motivates us to study the sufficient conditions for ensuring the latter properties. Beyond their own interests in second order variational analysis, the quadratic growth conditions can be employed for deriving the convergence rates of various first order and second order algorithms, including the proximal gradient methods [39,58], the proximal point algorithms [50,40,36,37] and the generalized Newton-type methods [22,19,42].The convex matrix optimization problems concerned in our paper take the fol-

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“…The motivation of studying MOSN arises from its wide applications such as the spectral matrix norm approximation (MNA) problem studied in [2] and references therein. The MNA problem has the following form,…”

Section: Introductionmentioning

confidence: 99%

On Some Aspects of Perturbation Analysis for Matrix Cone Optimization Induced by Spectral Norm

Guo

Zhang

Hao

2015

J. Oper. Res. Soc. China

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In this paper, we consider a cone problem of matrix optimization induced by spectral norm (MOSN). By Schur complement, MOSN can be reformulated as a nonlinear semidefinite programming (NLSDP) problem. Then we discuss the constraint nondegeneracy conditions and strong second-order sufficient conditions of MOSN and its SDP reformulation, and obtain that the constraint nondegeneracy condition of MOSN is not always equivalent to that of NLSDP. However, the strong second-order sufficient conditions of these two problems are equivalent without any assumption. Finally, a sufficient condition is given to ensure the nonsingularity of the Clarke's generalized Jacobian of the KKT system for MOSN.

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A semismooth Newton-CG based dual PPA for matrix spectral norm approximation problems

Cited by 21 publications

References 45 publications

Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized Jacobian

Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized Jacobian

Quadratic Growth Conditions for Convex Matrix Optimization Problems Associated with Spectral Functions

On Some Aspects of Perturbation Analysis for Matrix Cone Optimization Induced by Spectral Norm

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