Solving Two Stage Stochastic Programming Problems Using ADMM

In this paper, we have dealt with the solution of a two-stage stochastic programming problem using ADMM. We have formulated the problem into a deterministic three-block separable optimization problem, and then we applied ADMM to solve it. We have established the theoretical convergence of ADMM to the optimal solution based on the concept of lower semicontinuity and the Kurdyka-Lojasiewicz property. We have compared ADMM with Progressive Hedging in terms of performance criteria using five benchmark problems from the literature. The comparison shows that ADMM outperforms Progressive Hedging.

A Dynamical Alternating Direction Multiplier Method for Two-Block Optimization Problems

In this paper, we propose a dynamic alternating direction method of multipliers for two-block separable optimization problems. The well-known classical ADMM can be obtained after the time discretization of the dynamical system. Under suitable condition, we prove that the trajectory asymptotically converges to a saddle point of the Lagrangian function of the problems. When the coefficient matrices in the constraint are identiy matrices, we prove the worst-case O(1/t) convergence rate in ergodic sense.

On the Convergence Rate of Inexact Majorized sGS ADMM with Indefinite Proximal Terms for Convex Composite Programming

In this paper, we propose an inexact majorized symmetric Gauss–Seidel (sGS) alternating direction method of multipliers (ADMM) with indefinite proximal terms for multi-block convex composite programming. This method is a specific form of the inexact majorized ADMM which is further proposed to solve a general two-block separable optimization problem. The new methods adopt certain relative error criteria to solve the involving subproblems approximately, and the step-sizes allow to choose in the scope [Formula: see text]. Under more general conditions, we establish the global convergence and Q-linear convergence rate of the proposed methods.