A Semismooth Newton-Type Method for the Nearest Doubly Stochastic Matrix Problem

TL;DR

This paper introduces a novel semismooth Newton-type method for efficiently finding the nearest doubly stochastic matrix, achieving quadratic convergence without relying on the local error bound condition.

Contribution

The paper develops a modified two-step semismooth Newton method that guarantees quadratic convergence and nonsingular Jacobians, even when traditional conditions fail.

Findings

Achieves Q-quadratic convergence for the problem.

Ensures nonsingular Jacobian matrices at each iteration.

First Newton-type method with quadratic convergence without local error bound.

Abstract

We study a semismooth Newton-type method for the nearest doubly stochastic matrix problem where both differentiability and nonsingularity of the Jacobian can fail. The optimality conditions for this problem are formulated as a system of strongly semismooth functions. We show that the so-called local error bound condition does not hold for this system. Thus the guaranteed convergence rate of Newton-type methods is at most superlinear. By exploiting the problem structure, we construct a modified two step semismooth Newton method that guarantees a nonsingular Jacobian matrix at each iteration, and that converges to the nearest doubly stochastic matrix quadratically. To the best of our knowledge, this is the first Newton-type method which converges $Q$-quadratically in the absence of the local error bound condition.