A Framework for Quadratic Form Maximization over Convex Sets

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."

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.



Member, School of Mathematics