### Abstract

We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer-Monteiro approach, we optimize a factor Y of size n × p instead, such that X = YY^{T}. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a nonconvex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite nonconvexity, first- and second-order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YY^{T} of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust-region subproblem, and quadratic optimization over several spheres, as well as for the Max-Cut and Orthogonal-Cut SDPs, which are common relaxations in stochastic block modeling and synchronization of rotations.

Original language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21830

**Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs.** / Boumal, Nicolas; Voroninski, Vladislav; Bandeira, Afonso.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21830

}

TY - JOUR

T1 - Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs

AU - Boumal, Nicolas

AU - Voroninski, Vladislav

AU - Bandeira, Afonso

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer-Monteiro approach, we optimize a factor Y of size n × p instead, such that X = YYT. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a nonconvex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite nonconvexity, first- and second-order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YYT of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust-region subproblem, and quadratic optimization over several spheres, as well as for the Max-Cut and Orthogonal-Cut SDPs, which are common relaxations in stochastic block modeling and synchronization of rotations.

AB - We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer-Monteiro approach, we optimize a factor Y of size n × p instead, such that X = YYT. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a nonconvex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite nonconvexity, first- and second-order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YYT of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust-region subproblem, and quadratic optimization over several spheres, as well as for the Max-Cut and Orthogonal-Cut SDPs, which are common relaxations in stochastic block modeling and synchronization of rotations.

UR - http://www.scopus.com/inward/record.url?scp=85066896631&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066896631&partnerID=8YFLogxK

U2 - 10.1002/cpa.21830

DO - 10.1002/cpa.21830

M3 - Article

AN - SCOPUS:85066896631

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -