Optimality conditions and algorithms for conic optimization and applications in economics

Abstract

This research project deals with the development of tools for optimization problems over conic constraints; in particular, we are interested in so-called optimization problems over symmetric cones. Symmetric cones have a rich underlying algebraic structure that does not allow to use the approach similar to the classic approach of nonlinear programming. Symmetric cones encompass the nonnegative ortant, the semidefinite matrix cone and the second order cone (Lorentz cone), thus extending as the most important classes of nonlinear optimization problems. Our focus will be on treating the problem in all its generality, extending and unifying the existing theory, mainly in terms of optimality conditions under minimal assumptions and their relations with a convergence of algorithms. Connected to the practical appeal of the developed theory, we propose to explore relevant applications of problems that can be modeled with this type of technique, such as the problem of identifying financial risk factors. (AU)