  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.
