Tuesday, February 6, 2018 at 4:15pm
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others. A number of breakthrough papers in the early 2000s showed that this problem, long thought to be intractable, could be solved by using sum of squares programming. This technique however has proved to be expensive for large-scale problems, as it involves solving large semidefinite programs (SDPs).
In the first part of this talk, we present two methods for approximately solving large-scale sum of squares programs that dispense altogether with semidefinite programming and only involve solving a sequence of linear or second order cone programs generated in an adaptive fashion. In the second part of the talk, we focus on the problem of finding tight lower bounds on polynomial optimization problems (POPs), a fundamental task in this area that is most commonly handled through the use of SDP-based sum of squares hierarchies (e.g., due to Lasserre and Parrilo). In contrast to previous approaches, we provide the first theoretical framework for efficiently constructing converging hierarchies of lower bounds on POPs whose computation does not require any optimization, but simply the ability to multiply certain fixed polynomials together and check nonnegativity of the coefficients of the product.
Georgina Hall is a final-year graduate student and a Gordon Wu fellow in the Department of Operations Research and Financial Engineering at Princeton University, where she is advised by Professor Amir Ali Ahmadi. She was the valedictorian of Ecole Centrale, Paris, where she obtained a B.S. and an M.S., in 2011 and 2013 respectively. Her interests lie in convex relaxations of NP-hard problems, particularly those arising in polynomial optimization. Georgina is the recipient of the Médaille de l’Ecole Centrale from the French Académie des Sciences and the Princeton School of Engineering and Applied Sciences Award for Excellence. She was also chosen for the 2017 Rising Stars in EECS workshop at Stanford and the 2017 Young Researchers Workshop at Cornell University. Her paper “DC decomposition of nonconvex polynomials using algebraic techniques” is the recent recipient of the 2016 Informs Computing Society Prize for Best Student Paper. She has also been the recipient of a number of teaching awards, including the Princeton University's Engineering Council Teaching Award, the university-wide Excellence in Teaching Award of the Princeton Graduate School, and the 2017 Excellence in Teaching of Operations Research Award of the Institute for Industrial and Systems Engineers.