Permanents of doubly stochastic matrices with fixed zero pattern

The doubly stochastic matrices with a given zero pattern which are closest in Euclidean norm to J_nn, the matrix with each entry equal to 1/n, are identified. If the permanent is restricted to matrices having a given zero pattern confined to one row or to one column, the permanent achieves a local minimum at those matrices with that zero pattern which are closest to J_nn. This need no longer be true if the zeros lie in more than one row or column.