Conic Geometric Programming

TL;DR

This paper introduces conic geometric programs (CGPs), a new convex optimization framework that unifies geometric and conic programs, enabling solutions to complex problems like quantum channel capacity and robust optimization.

Contribution

The paper presents CGPs as a novel class of convex programs that generalize GPs and SDPs, with efficient solvability and broad applicability.

Findings

CGPs unify GPs and SDPs in a single framework.

CGPs can solve problems like permanent maximization and quantum channel capacity.

CGPs are computationally comparable to GPs and SDPs.

Abstract

We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as SDPs, while upper bounds on combined exponential/affine functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. 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 that fall outside the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hitting-time estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.