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We design inexact proximal augmented Lagrangian based decomposition methodsfor convex composite programming problems with dual block-angular structures.Our methods are particularly well suited for convex quadratic programmingproblems arising from stochastic programming models. The algorithmic frameworkis based on the application of the abstract inexact proximal ADMM frameworkdeveloped in [Chen, Sun, Toh, Math. Prog. 161:237--270] to the dual of thetarget problem, as well as the application of the recently developed symmetricGauss-Seidel decomposition theorem for solving a proximal multi-block convexcomposite quadratic programming problem. The key issues in our algorithmicdesign are firstly in designing appropriate proximal terms to decompose thecomputation of the dual variable blocks of the target problem to make thesubproblems in each iteration easier to solve, and secondly to develop novelnumerical schemes to solve the decomposed subproblems efficiently. Our inexactaugmented Lagrangian based decomposition methods have guaranteed convergence.We present an application of the proposed algorithms to the doubly nonnegativerelaxations of uncapacitated facility location problems, as well as totwo-stage stochastic optimization problems. We conduct numerous numericalexperiments to evaluate the performance of our method against state-of-the-artsolvers such as Gurobi and MOSEK. Moreover, our proposed algorithms alsocompare favourably to the well-known progressive hedging algorithm ofRockafellar and Wets.

On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems