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Positivity certificates and polynomial optimization on non-compact semialgebraic sets
- Source :
- Mathematical Programming, Series A, Mathematical Programming, Series A, 2021, ⟨10.1007/s10107-021-01634-1⟩, Mathematical Programming, Series A, Springer, 2021, ⟨10.1007/s10107-021-01634-1⟩
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu [Comptes Rendus de l'Acad\'emie des Sciences-Series I-Mathematics, 328(6) (1999) pp. 495-499]. We use Jacobi's technique from [Mathematische Zeitschrift, 237(2) (2001) pp. 259-273] to provide an alternative proof with an effective degree bound on the sums of squares multipliers in such certificates. As a consequence, it allows one to define a hierarchy of semidefinite relaxations for a general polynomial optimization problem. Convergence of this hierarchy to a neighborhood of the optimal value as well as strong duality and analysis are guaranteed. In a second contribution, we introduce a new numerical method for solving systems of polynomial inequalities and equalities with possibly uncountably many solutions. As a bonus, one may apply this method to obtain approximate global optimizers in polynomial optimization.<br />Comment: 33 pages, 2 figures, 5 tables
- Subjects :
- Discrete mathematics
021103 operations research
Optimization problem
Degree (graph theory)
Hierarchy (mathematics)
General Mathematics
Numerical analysis
0211 other engineering and technologies
Value (computer science)
010103 numerical & computational mathematics
02 engineering and technology
01 natural sciences
Optimization and Control (math.OC)
Convergence (routing)
FOS: Mathematics
Strong duality
Polynomial optimization
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
[MATH]Mathematics [math]
0101 mathematics
Mathematics - Optimization and Control
Software
Mathematics
Subjects
Details
- ISSN :
- 14364646 and 00255610
- Volume :
- 194
- Database :
- OpenAIRE
- Journal :
- Mathematical Programming
- Accession number :
- edsair.doi.dedup.....73b3202770b36fc24eca8e8af01c4f9a