A semi-proximal augmented Lagrangian based decomposition method for primal block angular convex composite quadratic conic programming problems

TL;DR

This paper introduces a semi-proximal augmented Lagrangian decomposition method tailored for large-scale primal block angular convex quadratic conic programming, demonstrating significant efficiency improvements over Gurobi in solving massive instances.

Contribution

It develops a novel semi-proximal augmented Lagrangian framework that generalizes existing methods and introduces a semi-proximal symmetric Gauss-Seidel based ADMM for dual problems.

Findings

Efficiently solves large-scale problems with over 300,000 constraints in minutes.

Outperforms Gurobi significantly on large instances.

Provides new algorithmic insights for convex quadratic conic programming.

Abstract

We propose a semi-proximal augmented Lagrangian based decomposition method for convex composite quadratic conic programming problems with primal block angular structures. Using our algorithmic framework, we are able to naturally derive several well known augmented Lagrangian based decomposition methods for stochastic programming such as the diagonal quadratic approximation method of Mulvey and Ruszczy\'{n}ski. Moreover, we are able to derive novel enhancements and generalizations of these well known methods. We also propose a semi-proximal symmetric Gauss-Seidel based alternating direction method of multipliers for solving the corresponding dual problem. Numerical results show that our algorithms can perform well even for very large instances of primal block angular convex QP problems. For example, one instance with more than $300,000$ linear constraints and $12,500,000$ nonnegative variables is solved in less than a minute whereas Gurobi took more than 3 hours, and another instance {\tt qp-gridgen1} with more than $331,000$ linear constraints and $986,000$ nonnegative variables is solved in about 5 minutes whereas Gurobi took more than 35 minutes.