Sampling Algebraic Varieties for Sum of Squares Programs
Author(s)Cifuentes, Diego Fernando; Parrilo, Pablo A.
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We study sum of squares (SOS) relaxations to optimize polynomial functions over a set V ∩ Rn, where V is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents V with a generic set of complex samples. This approach depends only on the geometry of V, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties, we can obtain the required samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.
DepartmentMassachusetts Institute of Technology. Department of Mathematics; Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
SIAM journal on optimization
Society for Industrial & Applied Mathematics (SIAM)
Cifuentes, Diego and Pablo A. Parrilo. "Sampling Algebraic Varieties for Sum of Squares Programs." SIAM journal on optimization 27, no. 4 (2017): pp. 2381-2404.
Final published version