Computing the nearest doubly stochastic matrix with a prescribed entry

Abstract

In this paper a nearest doubly stochastic matrix problem is studied. This problem is to find the closest doubly stochastic matrix with the prescribed (1,1) entry to a given matrix. According to the well-established dual theory in optimization, the dual of the underlying problem is an unconstrained differentiable, but not twice differentiable, convex optimization problem. A Newton-type method is used for solving the associated dual problem, and then the desired nearest doubly stochastic matrix is obtained. Under some mild assumptions, the quadratic convergence of the proposed Newton method is proved. The numerical performance of the method is also demonstrated by numerical examples. © 2007 Society for Industrial and Applied Mathematics.