Lingchen Kong

A Regularized Smoothing Newton Method for Symmetric Cone Complementarity Problems

This paper extends the regularized smoothing Newton method in vector complementarity problems to symmetric cone complementarity problems (SCCP), which includes the nonlinear complementarity problem, the second-order cone complementarity problem, and the semidefinite complementarity problem as special cases. In particular, we study strong semismoothness and Jacobian nonsingularity of the total natural residual function for SCCP. We also derive the uniform approximation property and the Jacobian consistency of the Chen-Mangasarian smoothing function of the natural residual. Based on these properties, global and quadratical convergence of the proposed algorithm is established.

VECTOR-VALUED IMPLICIT LAGRANGIAN FOR SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.

New smooth C-functions for symmetric cone complementarity problems

In this paper, with the help of the Jordan-algebraic technique we introduce two new complementarity functions (C-functions) for symmetric cone complementary problems, and show that they are continuously differentiable and strongly semismooth everywhere.

EXACT LOW-RANK MATRIX RECOVERY VIA NONCONVEX SCHATTEN p-MINIMIZATION

The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, quantum state tomography, magnetic resonance imaging, system identification and control, and it is generally NP-hard. Recently, Majumdar and Ward [Majumdar, A and RK Ward (2011). An algorithm for sparse MRI reconstruction by Schatten p-norm minimization. Magnetic Resonance Imaging, 29, 408–417]. had successfully applied nonconvex Schatten p-minimization relaxation of LMR in magnetic resonance imaging. In this paper, our main aim is to establish RIP theoretical result for exact LMR via nonconvex Schatten p-minimization. Carefully speaking, letting

Equivalent Conditions for Jacobian Nonsingularity in Linear Symmetric Cone Programming

\mathcal{A}

Sufficiency of linear transformations on Euclidean Jordan algebras

be a linear transformation from ℝm×n into ℝs and r be the rank of recovered matrix X ∈ ℝm×n, and if

Improved Complexity Analysis of Full Nesterov---Todd Step Feasible Interior-Point Method for Symmetric Optimization

\mathcal{A}

Monotonicity of Löwner operators and its applications to symmetric cone complementarity problems

satisfies the RIP condition

The solution set structure of monotone linear complementarity problems over second-order cone

\sqrt{2}\delta_{\max\{r+\lceil\frac{3}{2}k\rceil, 2k\}}+{(\frac{k}{2r})}^{\frac{1}{p}-\frac{1}{2}}\delta_{2r+k}

Sparse and Low-Rank Covariance Matrix Estimation

In this paper we consider the linear symmetric cone programming (SCP). At a Karush-Kuhn-Tucker (KKT) point of SCP, we present the important conditions equivalent to the nonsingularity of Clarke’s generalized Jacobian of the KKT nonsmooth system, such as primal and dual constraint nondegeneracy, the strong regularity, and the nonsingularity of the B-subdifferential of the KKT system. This affirmatively answers an open question by Chan and Sun (SIAM J. Optim. 19:370–396, 2008).