Entropy Minimization, DAD Problems, and Doubly Stochastic Kernels

The classical DAD problem asks, for a square matrix A with nonnegative entries, when it is possible to find positive diagonal matrices D1 and D2 with D1AD2 doubly stochastic. We consider various continuous and measurable generalizations of this problem. Through a fusion of variational and fixed point techniques we obtain strong analogues of the classical results. Our extensions appear inaccessible by either technique separately.