### 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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Pages (from-to) | 581-608 |

Number of pages | 28 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 73 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2020 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*73*(3), 581-608. https://doi.org/10.1002/cpa.21830