局部极小值的复杂性及相关概念
Complexity aspects of local minima and related notions
摘要：
我们考虑的概念（i）临界点，（ii）二阶点，（iii）局部极小值，（iv）严格局部极小值的多元多项式。对于每种类型的点，作为多项式次数的函数，我们研究了决定（1）给定点是否属于该类型以及（2）多项式是否具有该类型的点的复杂性。我们的研究结果的特点，这两个问题的复杂性，为所有程度的开放先前的文献。我们的主要贡献表明，这些问题中的许多原来是易处理的三次多项式。特别地，我们给出了三次多项式点的局部极小的一个可有效检验的充要条件。我们还表明，一个局部极小值的三次多项式可以有效地解决半定规划的大小线性的变量的数量。相反，我们证明了判定一个三次多项式是否有临界点是强NP-难的。我们还证明了任何三次多项式的二阶点集是一个谱面体，反之，任何谱面体是一个三次多项式的二阶点集的投影。在我们的最后一节中，我们简要地介绍了一个潜在的应用程序，找到局部极小的三次多项式的设计一个三阶牛顿法。
英文摘要：
We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if a polynomial has a point of that type. Our results characterize the complexity of these two questions for all degrees left open by prior literature. Our main contributions reveal that many of these questions turn out to be tractable for cubic polynomials. In particular, we present an efficiently-checkable necessary and sufficient condition for local minimality of a point for a cubic polynomial. We also show that a local minimum of a cubic polynomial can be efficiently found by solving semidefinite programs of size linear in the number of variables. By contrast, we show that it is strongly NP-hard to decide if a cubic polynomial has a critical point. We also prove that the set of second-order points of any cubic polynomial is a spectrahedron, and conversely that any spectrahedron is the projection of the set of second-order points of a cubic polynomial. In our final section, we briefly present a potential application of finding local minima of cubic polynomials to the design of a third-order Newton method.
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