单变量整体Optimization的Steklov正则化和轨迹方法
Steklov Regularization and Trajectory Methods for Univariate Global Optimization
摘要：
我们引入了一种新的正则化技术，使用我们所称的Steklov正则化函数，并应用这种技术来设计一种算法，计算一个全局极小的单变量强制函数。首先，我们证明了Steklov正则化对一个给定的单变量强制函数进行了凸化。然后，通过使用正则化参数作为自变量，在Steklov函数生成的表面上构造轨迹。对于一元四次多项式，我们证明了这条轨迹确实产生一个全局极小。在这个过程中，我们得到了四次多项式的一些性质。与以前的方法，使用二次正则化函数进行比较。通过数值实验说明了新方法对不同次数多项式以及非多项式函数的有效性。
英文摘要：
We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions. First, we show that the Steklov regularization convexifies a given univariate coercive function. Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function. For monic quartic polynomials, we prove that this trajectory does generate a global minimizer. In the process, we derive some properties of quartic polynomials. Comparisons are made with a previous approach which uses a quadratic regularization function. We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function.
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