An Update-and-Stabilize Framework for the Minimum-Norm-Point Problem

TL;DR

This paper introduces a unified framework combining active set and first order methods to efficiently solve the minimum-norm-point problem, with polynomial iteration bounds and practical performance improvements.

Contribution

The paper proposes a new algorithmic framework that unifies active set and first order methods for the MNP problem, providing polynomial bounds and extending classical algorithms.

Findings

Polynomial iteration bounds for the algorithm.

Strongly polynomial for network flow instances.

Preliminary experiments show promising results.

Abstract

We consider the minimum-norm-point (MNP) problem over polyhedra, a well-studied problem that encompasses linear programming. We present a general algorithmic framework that combines two fundamental approaches for this problem: active set methods and first order methods. Our algorithm performs first order update steps, followed by iterations that aim to `stabilize' the current iterate with additional projections, i.e., find a locally optimal solution whilst keeping the current tight inequalities. Such steps have been previously used in active set methods for the nonnegative least squares (NNLS) problem. We bound on the number of iterations polynomially in the dimension and in the associated circuit imbalance measure. In particular, the algorithm is strongly polynomial for network flow instances. Classical NNLS algorithms such as the Lawson-Hanson algorithm are special instantiations of our framework; as a consequence, we obtain convergence bounds for these algorithms. Our preliminary computational experiments show promising practical performance.