Xiao-Ming Yuan In this paper, we focus on the convergence analysis for the application of the Peaceman-Rachford splitting method to a convex minimization model whose objective function is the sum of a smooth and nonsmooth convex functions. The sublinear convergence rate in term of the worst-case O(1/t) iteration complexity is established if the gradient of the smooth … Read more
Recently, in [17] we have showed the first possibility of combining the Douglas-Rachford alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving a convex minimization model with a general separable structure. This paper is a further study on theoretical aspects of this theme. We first derive a general algorithmic framework … Read more
Recently, a worst-case O(1/t) convergence rate was established for the Douglas-Rachford alternating direction method of multipliers in an ergodic sense. This note proposes a novel approach to derive the same convergence rate while in a non-ergodic sense. Article Download View On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers
This note provides a simple proof on a O(1/k) convergence rate for the Douglas-Rachford operator splitting method where $k$ denotes the iteration counter. Article Download View On convergence rate of the Douglas-Rachford operator splitting method
This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddle-point problem. We model these two classes of problems uniformly by a mixed variational inequality, and show how PPA with … Read more
Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show … Read more
The old alternating direction method (ADM) has found many new applications recently, and its empirical efficiency has been well illustrated in various fields. However, the estimate of ADM’s convergence rate remains a theoretical challenge for a few decades. In this note, we provide a uniform proof to show the O(1/t) convergence rate for both the … Read more
The alternating direction method (ADM) is classical for solving a linearly constrained separable convex programming problem (primal problem), and it is well known that ADM is essentially the application of a concrete form of the proximal point algorithm (PPA) (more precisely, the Douglas-Rachford splitting method) to the corresponding dual problem. This paper shows that an … Read more
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
We consider the linearly constrained separable convex programming whose objective function is separable into m individual convex functions without crossed variables. The alternating direction method (ADM) has been well studied in the literature for the special case m=2. But the convergence of extending ADM to the general case m>=3 is still open. In this paper, … Read more