Computing The Best Approximation Over the Intersection of a Polyhedral Set and the Doubly Nonnegative Cone

This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities, and the doubly nonnegative cone (the cone of all positive semidefinite matrices whose elements are nonnegative). In contrast to directly applying the block coordinate descent type methods, we propose an inexact accelerated (two-) block coordinate descent algorithm to tackle the four-block unconstrained nonsmooth dual program. The proposed algorithm hinges on the superlinearly convergent semismooth Newton method to solve each of the two subproblems generated from the (two-)block coordinate descent, which have no closed form solutions due to the merger of the original four blocks of variables. The O(1/k²) iteration complexity of the proposed algorithm is established. Extensive numerical results over various large scale semidefinite programming instances from relaxations of combinatorial problems demonstrate the effectiveness of the proposed algorithm.