Maximization of the information divergence from an exponential family and criticality

The problem to maximize the information divergence from an exponential family is compared to the maximization of an entropy-like quantity over the boundary of a polytope. First-order conditions on directional derivatives define critical sets for the two problems. The bijection between the sets of global maximizers in the two problems found earlier is extended here to bijections between the sets of local maximizers and the critical sets. This is based on new inequalities relating the maximized quantities and a reformulation of the first order criticality conditions for the second problem.