182. Conjectures in Real Algebra and Polynomial Optimization through High Precision Semidefinite Programming
Invited abstract in session TC-11: Advances in Manifold and Conic Optimization, stream Riemannian Manifold and Conic Optimization.
Tuesday, 14:00-16:00Room: B100/5017
Authors (first author is the speaker)
| 1. | Michal Kocvara
|
| School of Mathematics, University of Birmingham | |
| 2. | Lorenzo Baldi
|
| Max Planck Institute for Mathematics in the Sciences |
Abstract
We study degree bounds for the denominator-free Positivstellens\"atze in real algebra, based on sums of squares (SOS), or equivalently the convergence rate for the moment-sums of squares hierarchy in polynomial optimization, from a numerical point of view. As standard semidefinite programming (SDP) solvers do not provide reliable answers in many important instances, we use a new high-precision SDP solver, Loraine.jl, to support our investigation.
We study small instances (low-degree, small number of variables) of one-parameter families of examples, and propose several conjectures for the asymptotic behavior of the degree bounds. Our objective is twofold: first, to raise awareness on the bad performance of standard SDP solvers in such examples, and then to guide future research on the Effective Positivstellens\"atze.
Keywords
- Conic and semidefinite optimization
- Global optimization
Status: accepted
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