The Steklov function $\mu_f(\cdot,t)$ is defined to average a continuous function $f$ at each point of its domain by using a window of size given by $t>0$. It has traditionally been used to approximate $f$ smoothly with small values of $t$. In this paper, we first find a concise and useful expression for $\mu_f$ for the case when $f$ is a multivariate quartic polynomial. Then we show that, for large enough $t$, $\mu_f(\cdot,t)$ is convex; in other words, $\mu_f(\cdot,t)$ convexifies $f$. We provide an easy-to-compute formula for $t$ with which $\mu_f$ convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving $\mu_f$, a trajectory $x(t)$ emanating from the minimizer of the convexified $f$ and ending at $x(0)$, an estimate of the global minimizer of $f$. For a family of quartic polynomials, we provide an estimate for the size of a ball that contains all its global minimizers. Finally, we illustrate the working of our method by means of numerous computational examples.

中文翻译：

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Steklov凸化和多元四次多项式全局优化的轨迹方法

Steklov 函数 $\mu_f(\cdot,t)$ 被定义为使用 $t>0$ 给出的大小窗口在其域的每个点对连续函数 $f$ 进行平均。传统上，它被用来用 $t$ 的小值平滑地近似 $f$。在本文中，我们首先针对 $f$ 是多元四次多项式的情况，找到了 $\mu_f$ 的简洁而有用的表达式。然后我们证明，对于足够大的 $t$，$\mu_f(\cdot,t)$ 是凸的；换句话说，$\mu_f(\cdot,t)$ 凸化了 $f$。我们为 $t$ 提供了一个易于计算的公式，其中 $\mu_f$ 凸化了某些类别的多项式。我们提出了一种算法，该算法通过涉及 $\mu_f$ 的 ODE 构造轨迹 $x(t)$，该轨迹 $x(t)$ 从凸化 $f$ 的极小值发出并以 $x(0)$ 结束，这是全局极小值的估计$f$。对于四次多项式族，我们提供了一个包含所有全局极小值的球大小的估计值。最后，我们通过大量计算示例来说明我们方法的工作原理。