We introduce a method for proving lower bounds on the efficacy of semidefinite
programming (SDP) relaxations for combinatorial problems. In particular, we show that the …

We prove super-polynomial lower bounds on the size of linear programming relaxations for
approximation versions of constraint satisfaction problems. We show that for these problems …

We study the convex hull of SO(n), the set of n*n orthogonal matrices with unit determinant,
from the point of view of semidefinite programming. We show that the convex hull of SO(n) is …

The abbreviations LMI and SOS stand for “linear matrix inequality" and “sum of squares,"
respectively. The cone n,2d of SOS polynomials in n variables of degree at most 2d is known …

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G
that are sparse with respect to the Fourier basis. We establish combinatorial conditions on …

Convex sets arising in a variety of applications are well-defined for every relevant
dimension. Examples include the simplex and the spectraplex that correspond to probability …

This paper presents a selected tour through the theory and applications of lifts of convex
sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original …

The dictionary learning problem concerns the task of representing data as sparse linear
sums drawn from a smaller collection of basic building blocks. In application domains where …

Sums-of-squares techniques have played an important role in optimization and control. One
question that has attracted a lot of attention is to exploit sparsity in order to reduce the size of …

We study the computational power of general symmetric relaxations for combinatorial
optimization problems, both in the linear programming (LP) and semidefinite programming …