Discrete Approximation Scheme in Distributionally Robust Optimization

Discrete approximation which is the prevailing scheme in stochastic programming in the past decade has been extended to distributionally robust optimization (DRO) recently. In this paper we conduct rigorous quantitative stability analysis of discrete approximation schemes for DRO, which measures the approximation error in terms of discretization sample size. For the ambiguity set defined through … Read more

Policies for Inventory Models with Product Returns Forecast from Past Demands and Past Sales

Finite horizon periodic review backlog models are considered in this paper for an inventory system that remanufactures two types of cores: buyback cores and normal cores. Returns of used products as buyback cores are modelled to depend on past demands and past sales. We obtain an optimal inventory policy for the model in which returns … Read more

Variational analysis perspective on linear convergence of some first order methods for nonsmooth convex optimization problems

We understand linear convergence of some first-order methods such as the proximal gradient method (PGM), the proximal alternating linearized minimization (PALM) algorithm and the randomized block coordinate proximal gradient method (R-BCPGM) for minimizing the sum of a smooth convex function and a nonsmooth convex function from a variational analysis perspective. We introduce a new analytic … Read more

Discerning the linear convergence of ADMM for structured convex optimization through the lens of variational analysis

Despite the rich literature, the linear convergence of alternating direction method of multipliers (ADMM) has not been fully understood even for the convex case. For example, the linear convergence of ADMM can be empirically observed in a wide range of applications, while existing theoretical results seem to be too stringent to be satisfied or too … Read more

Block Coordinate Proximal Gradient Method for Nonconvex Optimization Problems: Convergence Analysis

We propose a block coordinate proximal gradient method for a composite minimization problem with two nonconvex function components in the objective while only one of them is assumed to be differentiable. Under some per-block Lipschitz-like conditions based on Bregman distance, but without the global Lipschitz continuity of the gradient of the differentiable function, we prove … Read more

Primal-Dual Hybrid Gradient Method for Distributionally Robust Optimization Problems

We focus on the discretization approach to distributionally robust optimization (DRO) problems and propose a numerical scheme originated from the primal-dual hybrid gradient (PDHG) method that recently has been well studied in convex optimization area. Specifically, we consider the cases where the ambiguity set of the discretized DRO model is defined through the moment condition … Read more

On the Optimal Proximal Parameter of an ADMM-like Splitting Method for Separable Convex Programming

An ADMM-based splitting method is proposed in [11] for solving convex minimization problems with linear constraints and multi-block separable objective functions; while a relatively large proximal parameter is required for theoretically ensuring the convergence. In this paper, we further study this method and find its optimal (smallest) proximal parameter. For succinctness, we focus on the … Read more

Optimal Linearized Alternating Direction Method of Multipliers for Convex Programming

The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention from many areas because of its efficiency and easy implementation. To theoretically guarantee … Read more

Implementing the ADMM to Big Datasets: A Case Study of LASSO

The alternating direction method of multipliers (ADMM) has been popularly used for a wide range of applications in the literature. When big datasets with high-dimensional variables are considered, subproblems arising from the ADMM must be solved inexactly even though theoretically they may have closed-form solutions. Such a scenario immediately poses mathematical ambiguities such as how … Read more

On Glowinski’s Open Question of Alternating Direction Method of Multipliers

The alternating direction method of multipliers (ADMM) was proposed by Glowinski and Marrocco in 1975; and it has been widely used in a broad spectrum of areas, especially in some sparsity-driven application domains. In 1982, Fortin and Glowinski suggested to enlarge the range of the step size for updating the dual variable in ADMM from … Read more