  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
efficient semismooth Newton method to solve the subproblems, which have no
closed form solutions since the original four blocks are merged into two larger
blocks. The $O(1/k^2)$ 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.
