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

Abstract

We study a semismooth Newton-type method for the nearest doubly stochastic matrix problem where the 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 nonsingularity of the Jacobian does not hold for this system. 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. Funding: This work was supported by Canadian Network for Research and Innovation in Machining Technology. The research of H. Hu, H. Im, and H. Wolkowocz was supported by The Natural Sciences and Engineering Research Council of Canada. The research of X. Li was supported by the National Natural Science Foundation of China [No. 11601183] and Natural Science Foundation for Young Scientist of Jilin Province [No. 20180520212JH].