A note on the computational complexity of the moment-SOS hierarchy for polynomial optimization
Gribling,Sander Polak,Sven C. Slot,Lucas
Gribling,Sander
Polak,Sven C.
Slot,Lucas
Abstract
The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O’Donnell [16] and later Raghavendra & Weitz [20] show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz [20]. In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.
Description
Publisher Copyright: © 2023 Owner/Author.
Date
2023-07-24
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Publisher
ACM Digital Library
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Keywords
computational complexity, moment-SOS hierarchy, moments, polynomial optimization, semidefinite programming, sums of squares
Citation
Gribling, S, Polak, S C & Slot, L 2023, A note on the computational complexity of the moment-SOS hierarchy for polynomial optimization. in G Jeronimo (ed.), ISSAC 2023 - Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation. ACM International Conference Proceeding Series, ACM Digital Library, Tromso, pp. 280-288, The 2023 International Symposium on Symbolic and Algebraic Computation , Tromso, Norway, 24/07/23. https://doi.org/10.1145/3597066.3597075