A Lagrangian Dual Based Approach to Sparse Linear Programming

TL;DR

This paper introduces a novel Lagrangian dual approach and a semi-proximal ADMM algorithm to efficiently solve large-scale sparse linear programming problems with nonconvex sparsity constraints.

Contribution

It develops an explicit dual formulation for sparse linear programming and proposes an efficient dual-primal algorithm leveraging the proximal mapping of the Ky-Fan norm.

Findings

The proposed algorithm effectively solves large-scale SLP problems.

Strong duality holds for the reformulated dual problem.

Numerical results demonstrate promising performance on large-scale instances.

Abstract

A sparse linear programming (SLP) problem is a linear programming problem equipped with a sparsity (or cardinality) constraint, which is nonconvex and discontinuous theoretically and generally NP-hard computationally due to the combinatorial property involved. By rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of its Lagrangian dual in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality. A semi-proximal alternating direction method of multipliers (sPADMM) is then proposed to solve this dual problem by taking advantage of the efficient computation of the proximal mapping of the vector Ky-Fan norm function. Based on the optimal solution of the dual problem, we design a dual-primal algorithm for pursuing a global solution of the original SLP problem. Numerical results illustrate that our proposed algorithm is promising especially for large-scale problems.