Convex Optimization over Probability Measures

Abstract

The thesis studies convex optimization over the Banach space of regular Borel measures on a compact set. The focus is on problems where the variables are constrained to be probability measures. Applications include non-parametric maximum likelihood estimation of mixture densities, optimal experimental design, and distributionally robust stochastic programming. The theoretical study begins by developing the duality theory for optimization problems having non-finite-valued convex objectives over the set of probability measures. It is then shown that the infinite-dimensional problems can be posed as non-convex optimization problems in finite dimensions. The duality theory and constraint qualifications for these finite dimensional problems are derived and applied in each of the applications studied. It is then shown that the non-convex finite-dimensional problems can be decomposed by first optimizing over a subset of the variables, called the "weights", for which the associated optimization problem is convex. Optimization over the remaining variables, called the "support points", is then considered. The objective function in the reduced problem is neither convex nor finite-valued. For these reasons, a smoothing of this function is introduced. The epi-continuity of this smoothing and the convergence of critical points is established. It is shown that all known constraint qualifications fail for this problem, requiring a detailed analysis of the behavior of optimal solutions as the smoothing converges to the original problem. Finally, proof of concept numerical results are given.