## The Burer-Monteiro SDP method can fail even above the Barvinok-Pataki bound

### Liam O'Carroll · Vaidehi Srinivas · Aravindan Vijayaraghavan

##### Hall J #837

Keywords: [ low-rank SDP ] [ manifold optimization ] [ Semidefinite programming ] [ Burer-Monteiro ] [ Non-Convex Optimization ]

[ Abstract ]
[ [ [
Thu 1 Dec 2 p.m. PST — 4 p.m. PST

Abstract: The most widely used technique for solving large-scale semidefinite programs (SDPs) in practice is the non-convex Burer-Monteiro method, which explicitly maintains a low-rank SDP solution for memory efficiency. There has been much recent interest in obtaining a better theoretical understanding of the Burer-Monteiro method. When the maximum allowed rank $p$ of the SDP solution is above the Barvinok-Pataki bound (where a globally optimal solution of rank at most $$p$$ is guaranteed to exist), a recent line of work established convergence to a global optimum for generic or smoothed instances of the problem. However, it was open whether there even exists an instance in this regime where the Burer-Monteiro method fails. We prove that the Burer-Monteiro method can fail for the Max-Cut SDP on $n$ vertices when the rank is above the Barvinok-Pataki bound ($p \ge \sqrt{2n}$). We provide a family of instances that have spurious local minima even when the rank $p = n/2$. Combined with existing guarantees, this settles the question of the existence of spurious local minima for the Max-Cut formulation in all ranges of the rank and justifies the use of beyond worst-case paradigms like smoothed analysis to obtain guarantees for the Burer-Monteiro method.

Chat is not available.