Algebraic subfamily optimization

Algebraic methods for optimization have been used for maximum likelihood estimation and optimizing Euclidean distance. A complexity measure of these problems is given by the algebraic degree of the problem. This is a good measure because it counts the number of trials needed to solve the problem. These optimization problems are described by a correspondence between critical points, Lagrange-multipliers, and functions to be optimized. In this talk we present a study of subfamilies of these problems. Our main contribution is to formulate a unified duality theory encompassing maximum likelihood degree and Euclidean distance degree duality. We use this duality theory to develop algorithms and describe special loci. Moreover, we do a case analysis for problems in statistics, kinematics, and engineering.