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

Kuang-Yu Ding (), Xin-Yee Lam () and Kim-Chuan Toh ()
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Kuang-Yu Ding: National University of Singapore
Xin-Yee Lam: National University of Singapore
Kim-Chuan Toh: National University of Singapore

Computational Optimization and Applications, 2023, vol. 86, issue 1, No 4, 117-161

Abstract: Abstract We design inexact proximal augmented Lagrangian based decomposition methods for convex composite programming problems with dual block-angular structures. Our methods are particularly well suited for convex quadratic programming problems arising from stochastic programming models. The algorithmic framework is based on the application of the abstract inexact proximal ADMM framework developed in [Chen, Sun, Toh, Math. Prog. 161:237–270] to the dual of the target problem, as well as the application of the recently developed symmetric Gauss-Seidel decomposition theorem for solving a proximal multi-block convex composite quadratic programming problem. The key issues in our algorithmic design are firstly in designing appropriate proximal terms to decompose the computation of the dual variable blocks of the target problem to make the subproblems in each iteration easier to solve, and secondly to develop novel numerical schemes to solve the decomposed subproblems efficiently. Our inexact augmented Lagrangian based decomposition methods have guaranteed convergence. We present an application of the proposed algorithms to the doubly nonnegative relaxations of uncapacitated facility location problems, as well as to two-stage stochastic optimization problems. We conduct numerous numerical experiments to evaluate the performance of our method against state-of-the-art solvers such as Gurobi and MOSEK. Moreover, our proposed algorithms also compare favourably to the well-known progressive hedging algorithm of Rockafellar and Wets.

Keywords: Dual block angular structure; Progressive hedging algorithm; Symmetric Gauss-Seidel decomposition; ADMM; Proximal augmented Lagrangian method; Convex composite programming (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00493-0

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