关于一元极大函数在局部极大点处的性质
On Properties of Univariate Max Functions at Local Maximizers
摘要：
30多年前，Boyd和Balakrishnan建立了传递函数在最大化点的两范数的正则性结果。他们的结果很容易扩展到声明的最大特征值的一个单变量的实解析埃尔米特矩阵家庭是两次连续可微的，与Lipschitz二阶导数，在所有的本地最大化，一个属性，这是有用的几个应用程序，我们描述。我们还研究了这种光滑性是否更普遍地扩展到最大函数。我们证明了一有限集的$q$ -倍连续可微单变量函数的点态最大值在$q=1$的最大值处必须有零导数，但任意接近最大值，导数可能不被定义，即使当$q=3$和最大值是孤立的。
英文摘要：
More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set of $q$-times continuously differentiable univariate functions must have zero derivative at a maximizer for $q=1$, but arbitrarily close to the maximizer, the derivative may not be defined, even when $q=3$ and the maximizer is isolated.
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