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.