Conic geometric programming

This invited submission summarizes recent work by the authors on conic geometric programs (CGPs), which are convex optimization problems obtained by blending geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). GPs and SDPs are two prominent families of structured convex programs that each generalize linear programs (LPs) in different ways, and that are both employed in a broad range of applications. This submission provides a summary of a unified mathematical and algorithmic treatment of GPs and SDPs under the framework of CGPs. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications - permanent maximization, hitting-time estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs - that fall outside the scope of SDPs and GPs alone.