A low-rank augmented Lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs

TL;DR

This paper introduces RNNAL, a globally convergent low-rank augmented Lagrangian method for efficiently solving large-scale doubly nonnegative relaxations of mixed-binary quadratic programs by leveraging low-rank structures and algebraic varieties.

Contribution

The paper proposes a novel RNNAL method that combines low-rank decomposition, a modified Burer-Monteiro approach, and convex reformulations to efficiently solve large-scale DNN relaxations.

Findings

RNNAL effectively handles large-scale DNN problems.

The method demonstrates superior computational efficiency.

Numerical experiments validate the approach's effectiveness.

Abstract

Doubly nonnegative (DNN) programming problems are known to be challenging to solve because of their huge number of $\Omega(n^2)$ constraints and $\Omega(n^2)$ variables. In this work, we introduce RNNAL, a method for solving DNN relaxations of large-scale mixed-binary quadratic programs by leveraging their solutions' possible low-rank property. RNNAL is a globally convergent Riemannian augmented Lagrangian method (ALM) that penalizes the nonnegativity and complementarity constraints while preserving all other constraints as an algebraic variety. After applying the low-rank decomposition to the ALM subproblem, its feasible region becomes an algebraic variety with favorable geometric properties. Our low-rank decomposition model is different from the standard Burer-Monteiro (BM) decomposition model in that we make the key improvement to equivalently reformulate most of the quadratic constraints after the BM decomposition into fewer and more manageable affine constraints. This modification is also important in helping us to alleviate the violation of Slater's condition for the primal DNN problem. Moreover, we make the crucial step to show that the metric projection onto the algebraic variety, although non-convex, can be transformed into a solvable convex optimization problem under certain regularity conditions, which can be ensured by a constraint-relaxation strategy. RNNAL is able to handle general semidefinite programming (SDP) with additional polyhedral cone constraints, thus serving as a prototype algorithm for solving general DNN problems. Numerous numerical experiments are conducted to validate the efficiency of the proposed RNNAL method.