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Published January 2017 | public
Journal Article

Relative entropy optimization and its applications


In this expository article, we study optimization problems specified via linear and relative entropy inequalities. Such relative entropy programs (REPs) are convex optimization problems as the relative entropy function is jointly convex with respect to both its arguments. Prominent families of convex programs such as geometric programs (GPs), second-order cone programs, and entropy maximization problems are special cases of REPs, although REPs are more general than these classes of problems. We provide solutions based on REPs to a range of problems such as permanent maximization, robust optimization formulations of GPs, and hitting-time estimation in dynamical systems. We survey previous approaches to some of these problems and the limitations of those methods, and we highlight the more powerful generalizations afforded by REPs. We conclude with a discussion of quantum analogs of the relative entropy function, including a review of the similarities and distinctions with respect to the classical case. We also describe a stylized application of quantum relative entropy optimization that exploits the joint convexity of the quantum relative entropy function.

Additional Information

© 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society. Received: 14 January 2015; Accepted: 22 February 2016; First online: 01 April 2016. The authors would like to thank Pablo Parrilo and Yong-Sheng Soh for helpful conversations, and Leonard Schulman for pointers to the literature on Von-Neumann entropy. Venkat Chandrasekaran was supported in part by National Science Foundation Career award CCF-1350590 and Air Force Office of Scientific Research grant FA9550-14-1-0098.

Additional details

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October 24, 2023