The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems. Despite of its success in practice, the convergence of the standard ADMM for minimizing the sum of $N$ $(N\geq 3)$ convex functions whose variables are linked by linear constraints, has remained unclear for a very long time. Recently, Chen … Read more
The alternating direction method of multipliers (ADMM) has been widely used for solving structured convex optimization problems. In particular, the ADMM can solve convex programs that minimize the sum of $N$ convex functions with $N$-block variables linked by some linear constraints. While the convergence of the ADMM for $N=2$ was well established in the literature, … Read more
A new algorithm is proposed to project, exactly and in finite time, a vector of arbitrary size onto a simplex or a l1-norm ball. The algorithm is demonstrated to be faster than existing methods. In addition, a wrong statement in a paper by Duchi et al. is corrected and an adversary sequence for Michelot’s algorithm … Read more
We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. SA spectral theory on semi-simple complex Jordan algebras is used as atool to prove the convexity results. Possible applications to optimization and statistics are indicated CitationPreprint, University of Notre Dame, August 2014ArticleDownload View PDF
The stochastic gradient (SG) method can minimize an objective function composed of a large number of differentiable functions, or solve a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, handles problems with multiple blocks of variables by updating them one at a time; when the blocks … Read more
The alternating direction method of multipliers (ADMM) is a benchmark for solving a linearly constrained convex minimization model with a two-block separable objective function; and it has been shown that its direct extension to a multiple-block case where the objective function is the sum of more than two functions is not necessarily convergent. For the … Read more
The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems. To apply ADM, we first formulate a given problem into the “ADM-ready” form, so the final algorithm depends … Read more
In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the L2−Lp minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an epsilon-KKT point within O(log(1/epsilon)) iterations. The same result is also … Read more
In this paper, we first propose a general inertial proximal point method for the mixed variational inequality (VI) problem. Based on our knowledge, without stronger assumptions, convergence rate result is not known in the literature for inertial type proximal point methods. Under certain conditions, we are able to establish the global convergence and a $o(1/k)$ … Read more
The composite $L_q$ (0