Applications of convex analysis to signomial and polynomial nonnegativity problems Thesis by Riley John Murray In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2021 Defended May 13, 2021 ii © 2021 Riley John Murray ORCID: 0000-0003-1461-6458 All rights reserved iii ACKNOWLEDGEMENTS So many people, in so many areas of my life, have gone into making this thesis possible. It feels natural to first acknowledge my two incredible advisors, Adam Wierman and Venkat Chandrasekaran. Over the years Adam and Venkat have challenged me in every way necessary to help me grow into an independent researcher. They provided up-close guidance when I needed it, and they afforded me complete freedom to define and pursue my own projects once I felt I was ready. Adam in particular taught me the essential skill of communicating the importance of my work to the larger scientific community, while Venkat taught me what it means to hold my work to the highest of standards. I am also grateful to my committee members, Steven Low and Joel Tropp. For this industrial-engineer-turned-applied-mathematician, they serve as role models in the synthesis of mathematics and engineering, and the development of mathematical theories with practical importance. Greg Blekherman, Didier Henrion, Pablo Parrilo, Rekha Thomas, Cynthia Vinzant, Ankur Moitra, Levent Tunçel, and Daniel Plaumann graciously invited me to give presentations at various polynomial optimization workshops in the past few years. One chapter of this thesis is the result of a collaboration with Mareike Dressler that was first proposed at such a workshop in Banff, Canada. I am indebted to Timo de Wolff for suggesting to Bernd Sturmfels that I be invited to spend time at MPI Leipzig, and to Bernd for agreeing to Timo’s suggestion. While in Leipzig I had the great pleasure of beginning a collaboration with Thorsten Theobald and Helen Naumann, which became a major chapter of this thesis. Beyond my immediate research area in signomial and polynomial optimization, I am extremely grateful for the mentorship provided by Parikshit Shah during my time at Facebook Core Data Science. Time in grad school is about more than just research. I would be remiss to not mention the amazing friends I have made while at Caltech. In my first year I was fortunate to meet my housemates Elliott Williams and Doug Ober, fellow CMS (and CMS-adjacent) first years Natalie Bernat, Eric Zhan, Chen Liang, Sara Beery, Nikola Kovachki, Fengyu Zhou, Zach Lee, and Fang Xiao, my research group-mates Gautam Goel, Palma London, Yong-Sheng Soh, Armeen Taeb, and John Peng, as well as postdocs James Anderson and Shai Vardi. As the years went on I would iv only add to this list, getting to know wonderful people like Dima Burov, SooJean Han, Andrew Taylor, Victor Dorobantu, Jennifer Sun, Ameera Abdelaziz, Yu Su, Jiajie Chen, Dim Ho, Newton Nguyen, and Angela Gui. I was also fortunate to build friendships outside Caltech with Steven Diamond, Akshay Agrawal, and Berk Ozturk. Thank you, all of you, for being a part of this journey with me. Rhonda Righter, Dorit Hochbaum, and Samir Khuller each played transformative roles in my undergraduate career. Rhonda and Dorit resoundingly dispelled the myth that professors at a large school like Berkeley are out of touch with their students; I would never have discovered my love for mathematics if not for their graduate courses in the department of IEOR. Samir helped me discover my passion for research over a summer at the University of Maryland. Through his brilliant mentorship and some wizardry in letter writing, I somehow found myself accepted to Caltech with that single summer of research under my belt. Over the course of undergrad I also had the privilege of meeting Lousia Avellar (who I would follow to Caltech), Duy Vo (the best Industrial Engineer I know), and Bill Ruth. Bill isn’t around to read these acknowledgements, but the world should know, he changed my life and so many others over the decades that he ran Barrows House. Which brings me to the people who’ve been in it for the long haul: my brothers John and Patrick, my uncle Tom, my excellent-forever-person Sumita, my mother Nancy, father Paul, and step-dad Scott, as well as my best friend Derek and my high school student government teacher Kim. Thank you so much for all the love you’ve given to me over the years. I’m sure you understand that it would be a fool’s errand for me to try and summarize the profound ways each of you has affected my life. So I will close these acknowledgements by offering that this thesis belongs to all of you as much as it belongs to me. And that should make you proud. v ABSTRACT Here is a question that is easy to state, but often hard to answer: Is this function nonnegative on this set? When faced with such a question, one often makes appeals to known inequalities. One crafts arguments that are sufficient to establish the nonnegativity of the function, rather than determining the function’s precise range of values. This thesis studies sufficient conditions for nonnegativity of signomials and polynomials. Conceptually, signomials may be viewed as generalized polynomials that feature arbitrary real exponents, but with variables restricted to the positive orthant. Our methods leverage efficient algorithms for a type of convex optimization known as relative entropy programming (REP). By virtue of this integration with REP, our methods can help answer questions like the following: Is there some function, in this particular space of functions, that is nonnegative on this set? The ability to answer such questions is extremely useful in applied mathematics. Alternative approaches in this same vein (e.g., methods for polynomials based on semidefinite programming) have been used successfully as convex relaxation frameworks for nonconvex optimization, as mechanisms for analyzing dynamical systems, and even as tools for solving nonlinear partial differential equations. This thesis builds from the sums of arithmetic-geometric exponentials or SAGE approach to signomial nonnegativity. The term “exponential” appears in the SAGE acronym because SAGE parameterizes signomials in terms of exponential functions. Our first round of contributions concern the original SAGE approach. We employ basic techniques in convex analysis and convex geometry to derive structural re- sults for spaces of SAGE signomials and exactness results for SAGE-based REP relaxations of nonconvex signomial optimization problems. We frame our analysis primarily in terms of the coefficients of a signomial’s basis expansion rather than in terms of signomials themselves. The effect of this framing is that our results for signomials readily transfer to polynomials. In particular, we are led to define a new concept of SAGE polynomials. For sparse polynomials, this method offers an vi exponential efficiency improvement relative to certificates of nonnegativity obtained through semidefinite programming. We go on to create the conditional SAGE methodology for exploiting convex sub- structure in constrained signomial nonnegativity problems. The basic insight here is that since the standard relative entropy representation of SAGE signomials is obtained by a suitable application of convex duality, we are free to add additional convex constraints into the duality argument. In the course of explaining this idea we provide some illustrative examples in signomial optimization and analysis of chemical dynamics. The majority of this thesis is dedicated to exploring fundamental questions surround- ing conditional SAGE signomials. We approach these questions through analysis frameworks of sublinear circuits and signomial rings. These sublinear circuits generalize simplicial circuits of affine-linear matroids, and lead to rich modes of analysis for sets that are simultaneously convex in the usual sense and convex under a logarithmic transformation. The concept of signomial rings lets us develop a powerful signomial Positivstellensatz and an elementary signomial moment theory. The Positivstellensatz provides for an effective hierarchy of REP relaxations for approaching the value of a nonconvex signomial minimization problem from below, as well as a first-of-its-kind hierarchy for approaching the same value from above. In parallel with our mathematical work, we have developed the sageopt python package. Sageopt drives all the examples and experiments used throughout this thesis, and has been used by engineers to solve high-degree polynomial optimization problems at scales unattainable by alternative methods. We conclude this thesis with an explanation of how our theoretical results affected sageopt’s design. vii PUBLISHED CONTENT AND CONTRIBUTIONS Murray, R., Chandrasekaran, V., Wierman, A. (2021). “Newton polytopes and rel- ative entropy optimization”. In: Foundations of Computational Mathematics. doi: 10.1007/s10208-021-09497-w. arXiv pre-print (2018). R.M. partici- pated in the conception of the project, performed the analysis, and participated in the writing of the manuscript. Murray, R., Chandrasekaran, V., Wierman, A. (2020). “Signomial and polynomial optimization via relative entropy and partial dualization”. In: Mathematical Programming Computation. doi: 10.1007/s12532-020-00193-4. arXiv pre-print (2019). R.M. participated in the conception of the project, performed the analysis, developed the software, undertook the experiments, and partici- pated in the writing of the manuscript. Murray, R., Naumann, H., Theobald, T. “Sublinear circuits and the constrained signomial nonnegativity problem.” arXiv pre-print (2020). R.M. participated in the conception of the project, the analysis, and the writing of the manuscript. Dressler, M., Murray, R. “An algebraic approach to signomial optimization.” In preparation (2021). R.M. conceived the project, created the analysis frame- work, performed the analysis of the lower bounds, performed the experiments, participated in the analysis of the upper bounds, and participated in the writing of the manuscript. viii TABLE OF CONTENTS Acknowledgements .