Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Quadratic and Semi-Definite Programming

In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength … Read more

On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization

In this note, we prove that the KKT mapping for nonlinear semidefinite optimization problem is upper Lipschitz continuous at the KKT point, under the second-order sufficient optimality conditions and the strict Robinson constraint qualification. Article Download View On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization

Quantitative Stability Analysis of Stochastic Quasi-Variational Inequality Problems and Applications

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is de ned over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation … Read more


We discuss the $l_2$-$l_p$ (with $p\in (0,1)$ matrix minimization for recovering low rank matrix. A smoothing approach is developed for solving this non-smooth, non-Lipschitz and non-convex optimization problem, in which the smoothing parameter is used as a variable and a majorization method is adopted to solve the smoothing problem. The convergence theorem shows that any … Read more

The Second Order Directional Derivative of Symmetric Matrix-valued Functions

This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form $F(X)=P\mbox{diag}[f(\lambda_1(X)),\cdots,f(\lambda_n(X))]P^T$. For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki \cite{Tor01}; Second, we establish a formula for the (parabolic) … Read more

The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming

We analyze the rate of local convergence of the augmented Lagrangian method for nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and certain variational analysis on the projection operator in the symmetric-matrix space. Without … Read more