On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope

TL;DR

This paper presents an efficient method for computing a generalized Jacobian of the projection onto the Birkhoff polytope, enabling faster solutions for large-scale convex quadratic programming problems.

Contribution

We derive an explicit formula and an efficient procedure for the generalized Jacobian of the Birkhoff polytope projection, and develop a semismooth Newton method with superior performance.

Findings

Successfully solved projection problems with over one billion variables in under 15 minutes.

Our method outperforms Gurobi and PPROJ in speed and accuracy.

The approach significantly accelerates solving convex QP problems constrained by the Birkhoff polytope.

Abstract

We derive an explicit formula, as well as an efficient procedure, for constructing a generalized Jacobian for the projector of a given square matrix onto the Birkhoff polytope, i.e., the set of doubly stochastic matrices. To guarantee the high efficiency of our procedure, a semismooth Newton method for solving the dual of the projection problem is proposed and efficiently implemented. Extensive numerical experiments are presented to demonstrate the merits and effectiveness of our method by comparing its performance against other powerful solvers such as the commercial software Gurobi and the academic code PPROJ [{\sc Hager and Zhang}, SIAM Journal on Optimization, 26 (2016), pp.~1773--1798]. In particular, our algorithm is able to solve the projection problem with over one billion variables and nonnegative constraints to a very high accuracy in less than 15 minutes on a modest desktop computer. More importantly, based on our efficient computation of the projections and their generalized Jacobians, we can design a highly efficient augmented Lagrangian method (ALM) for solving a class of convex quadratic programming (QP) problems constrained by the Birkhoff polytope. The resulted ALM is demonstrated to be much more efficient than Gurobi in solving a collection of QP problems arising from the relaxation of quadratic assignment problems.