Geometry, Moments, and Semidefinite Optimization

Optimization formulations seek to take advantage of structure and information in a particular problem, and investigate how successfully this information constrains the performance measures of interest. In this paper, we apply some recent results of algebraic geometry, to show how the underlying geometry of the problem may be incorporated in a natural way, in a semidefinite optimization formulation. In particular, we discuss two examples; the first, an application to partial differential equations, and the second, an application to deriving optimal inequalities in probability theory. In both cases, the additional “geometry–constraints” significantly improve the quality of the bounds. Furthermore, we believe this framework to be a general one, with many potential applications.