On efficiently solving the subproblems of a level-set method for fused lasso problems

In applying the level-set method developed in [Van den Berg and Friedlander, SIAM J. on Scientific Computing, 31 (2008), pp.~890–912 and SIAM J. on Optimization, 21 (2011), pp.~1201–1229] to solve the fused lasso problems, one needs to solve a sequence of regularized least squares subproblems. In order to make the level-set method practical, we develop … Read more

A block symmetric Gauss-Seidel decomposition theorem for convex composite quadratic programming and its applications

For a symmetric positive semidefinite linear system of equations $\mathcal{Q} {\bf x} = {\bf b}$, where ${\bf x} = (x_1,\ldots,x_s)$ is partitioned into $s$ blocks, with $s \geq 2$, we show that each cycle of the classical block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an … Read more

A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems

We develop a fast and robust algorithm for solving large scale convex composite optimization models with an emphasis on the $\ell_1$-regularized least squares regression (Lasso) problems. Despite the fact that there exist a large number of solvers in the literature for the Lasso problems, we found that no solver can efficiently handle difficult large scale … Read more

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

Spectral Operators of Matrices

The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important applications involving structured low rank matrices. This trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing. The … Read more

Hankel Matrix Rank Minimization with Applications to System Identification and Realization

We introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz and moment structures, and catalog applications from diverse fields under this framework. We discuss various first-order methods for solving the resulting optimization problem, including alternating direction methods of multipliers, proximal point algorithms and gradient projection methods. We … Read more

An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP

The accelerated proximal gradient (APG) method, first proposed by Nesterov, and later refined by Beck and Teboulle, and studied in a unifying manner by Tseng has proven to be highly efficient in solving some classes of large scale structured convex optimization (possibly nonsmooth) problems, including nuclear norm minimization problems in matrix completion and $l_1$ minimization … Read more

Preferences for Travel Time under Risk and Ambiguity: Implications in Path Selection and Network Equilibrium

In this paper, we study the preferences for uncertain travel time in which the probability distribution may not be fully characterized. In evaluating an uncertain travel time, we explicitly distinguish between risk, where probability distribution is precisely known, and ambiguity, where it is not. In particular, we propose a new criterion called ambiguity-aware CARA travel … Read more

On the Moreau-Yosida regularization of the vector k-norm related functions

In this paper, we conduct a thorough study on the first and second order properties of the Moreau-Yosida regularization of the vector $k$-norm function, the indicator function of its epigraph, and the indicator function of the vector $k$-norm ball. We start with settling the vector $k$-norm case via applying the existing breakpoint searching algorithms to … Read more

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions and give constraint … Read more