Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization

TL;DR

This paper introduces two new ADMM algorithms employing dual variable factorization to efficiently solve doubly nonnegative programs, improving convergence speed and enabling high-quality bounds with moderate precision solutions.

Contribution

The paper proposes novel ADMM methods with dual factorization for DNNs, enhancing convergence and bound quality, and demonstrates their effectiveness through numerical experiments.

Findings

Reduced iteration count and CPU time for DNNs

High-quality bounds achievable from moderate precision solutions

Effective post-processing for primal objective bounds

Abstract

Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs. Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.