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Positivity certificates and polynomial optimization on non-compact semialgebraic sets

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.
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https://hal.laas.fr/hal-02382059
Contributor : Ngoc Hoang Anh Mai <>
Submitted on : Wednesday, November 27, 2019 - 7:25:26 AM
Last modification on : Thursday, June 10, 2021 - 3:02:26 AM

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Ngoc Hoang Anh Mai, Jean-Bernard Lasserre, Victor Magron. Positivity certificates and polynomial optimization on non-compact semialgebraic sets. Mathematical Programming, Series A, Springer, 2021, ⟨10.1007/s10107-021-01634-1⟩. ⟨hal-02382059⟩

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