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The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(·, εk), and the derivative(More)
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(More)
In this paper, we aim at finding a nearest correlation matrix to a given symmetric matrix, measured by the componentwise weighted Frobenius norm, with a prescribed rank and bound constraints on its correlations. This is in general a non-convex and difficult problem due to the presence of the rank constraint. To deal with this difficulty, we first consider a(More)
Matrix-valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized differential properties of such functions related to nonsmooth-smoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher’s theorem,(More)
In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as(More)
The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods like BFGS have been used directly to(More)
For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the(More)
The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in(More)
We study analyticity, differentiability, and semismoothness of Löwner’s operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any(More)
It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of(More)