DUAL METHODS IN ENTROPY MAXIMIZATION. APPLICATION TO SOME PROBLEMS IN CRYSTALLOGRAPHY

This paper is devoted to some infinite-dimensional optimization problems with finitely many constraints. These deal with entropy maximization, and this paper is particularly concerned with those originating from Fourier analysis. These problems have a structure that makes them amenable to dual methods, for theoretical as well as numerical solutions. Existence results are recalled, and the use of duality to construct suitable and efficient optimization algorithms is demonstrated. Finally, the so-called phase-problem of crystallographers, which is of crucial importance in biology and pharmacology, is studied. Although it is a nonconvex optimization problem, a solution algorithm also based on duality is proposed and some numerical illustrations are given.