In this review article we discuss connections between the physics of disordered systems,
phase transitions in inference problems, and computational hardness. We introduce two …

Abstract Let M ∈ R^ p * q M∈ R p× q be a nonnegative matrix. The positive semidefinite
rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite …

Consider the problem of minimizing a polynomial f over a compact semialgebraic set
X⊆R^n. Lasserre introduces hierarchies of semidefinite programs to approximate this hard …

In this survey we consider polynomial optimization problems, asking to minimize a
polynomial function over a compact semialgebraic set, defined by polynomial inequalities …

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a
polynomial f over the boolean hypercube B^ n={0, 1\}^ n B n= 0, 1 n. This hierarchy provides …

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 …

The aim of this paper is to introduce a new code for the solution of large-and-sparse linear
semidefinite programs (SDPs) with low-rank solutions or solutions with few outlying …

This paper develops a correspondence relating convex hulls of fractional functions with
those of polynomial functions over the same domain. Using this result, we develop a number …

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 …

We show that, if WW is an N * NN× N matrix drawn from the gaussian orthogonal ensemble,
then with high probability the degree 4 sum-of-squares relaxation cannot certify an upper …