In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu (C R Acad Sci Ser I Math 328(6):495–499, 1999). We use Jacobi’s technique from (Math Z 237(2):259–273, 2001) to provide an alternative proof with an effective degree bound on the sums of squares weights 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 can apply this method to obtain approximate global optimizers in polynomial optimization.

在第一个贡献中，我们将重新研究基于普蒂纳尔和瓦西莱斯库的（可能是非紧实的）基本半代数集的两个正定性证明（CR Acad Sci Ser I Math 328（6）：495-499，1999）。我们使用（Math Z 237（2）：259-273，2001）中的Jacobi技术提供具有有效度界的替代证明此类证明中的平方和的总和。结果，它允许为一般的多项式优化问题定义半定松弛的层次。保证了该层次结构收敛到最佳值的邻域以及强大的对偶性和分析能力。在第二个贡献中，我们介绍了一种新的数值方法，用于求解多项式不等式和等式的系统，其中可能包含无数个解。作为奖励，可以将这种方法应用于多项式优化中的近似全局优化器。