This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre’s hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence.

中文翻译：

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多项式的鞍点问题

本文研究多项式的鞍点问题。我们给出了计算鞍点的算法。它基于求解 Lasserre 的半定松弛层次。在定义多项式的一些通用假设下，我们证明： (i) 如果存在鞍点，我们的算法可以通过求解 Lasserre 型半定松弛的有限层次来得到一个；(ii) 如果没有鞍点，我们的算法可以检测到它的不存在。