Research

My area of research is mathematical optimization. For the last three years I’ve been focusing on convex relaxations for continuous nonconvex problems. The principle for obtaining these relaxations is to exploit a certain equivalence between optimization and certifying function nonnegativity. The mathematical techniques I have developed are implemented in the comprehensive sageopt python package. I have developing interests in applied algebraic geometry and second-order methods for constrained convex optimization. In parallel with these efforts I’ve regularly contributed to CVXPY, and I currently serve as one of three members of CVXPY’s core development team.

PhD Thesis

Applications of convex analysis to signomial and polynomial nonnegativity problems. [CaltechTHESIS record, PDF].

Received the Amori Doctoral Prize in Computing and Mathematical Sciences.

This thesis combines (and slightly expands upon) four papers I wrote in graduate school. The new content includes applications in chemical reaction networks and more discussion of the sageopt python package. There is also a chapter on preliminaries that covers generic ideas like nonnegativity cones, moment relaxations, and convex cone programming.  This is an excellent on-boarding resource to learn about sums of arithmetic-geometric exponentials (SAGE) and related ideas such as sums of nonnegative circuits (SONC).

Preprints

Sublinear Circuits and the Constrained Signomial Nonnegativity Problem

With Helen Naumann and Thorsten Theobald. Currently available on arXiv. I mentioned this work towards the end of my MIT vsOPT talk. I also have slides from a 30 minute presentation focused on the paper (notation in slides is slightly different from the paper).

Publications

Signomial and Polynomial Optimization via Relative Entropy and Partial Dualization

With Venkat Chandrasekaran and Adam Wierman. Available at Mathematical Programming Computation (2020) , with an earlier version on arXiv (2019). The mathematics described in this paper are implemented by the sageopt python package. I have two sets of slides which address the whole paper (MPI MiS 2019ICCOPT 2019). On April 17 2020 I gave a recorded talk through MIT, which addressed the signomial parts of the paper in detail, along with other recent results in this area. Errata.

Newton Polytopes and Relative Entropy Optimization.

With Venkat Chandrasekaran and Adam Wierman. Available at Foundations of Computational Mathematics (2021). You can find a freely available read-only copy from Springer at this link (see also arXiv). In June 2019 I gave a recorded talk on this paper at the Banff International Research Station. Venkat gave a talk based on this work at ICERM in November 2018.

Robust Market Equilibria with Uncertain Preferences. With Christian Kroer, Alex Peysakhovich, and Parikshit Shah. [arXiv, AAAI 2020, blog post].

Structured State Feedback for Metzler Dynamics. With James Anderson. [CDC 2018].

Scheduling Distributed Clusters of Parallel Machines : Primal-Dual and LP-Based Algorithms. With Samir Khuller and Megan Chao. [arXiv, ESA 2016, extended version in Algorithmica].