Vijay Bhattiprolu

Member, School of Mathematics

April 28, 2020

We investigate the approximability of the following optimization problem, whose input is an

n-by-n matrix A and an origin symmetric convex set C that is given by a membership oracle:

"Maximize the quadratic form as x ranges over C."

n-by-n matrix A and an origin symmetric convex set C that is given by a membership oracle:

"Maximize the quadratic form

This is a rich and expressive family of optimization problems; for different choices of forms A

and convex bodies C it includes a diverse range of interesting combinatorial and continuous

optimization problems. To name some examples, max-cut, Grothendieck's inequality, the

non-commutative Grothendieck inequality, certifying hypercontractivity, small set expansion,

vertex expansion, and the spread constant of a metric, are all captured by this class. While the

literature studied these special cases using case-specific reasoning, here we develop a general

methodology for treatment of the approximability and inapproximability aspects of these questions.

Based on joint work with Euiwoong Lee and Assaf Naor.