We identify a class of root-searching methods that surprisingly outperform the bisection method on the average performance while retaining minmax optimality. The improvement on the average applies for any continuous distributional hypothesis. We also pinpoint one specific method within the class and show that under mild initial conditions it can attain an order of convergence of up to 1.618, i.e., the same as the secant method. Hence, we attain both an improved average performance and an improved order of convergence with no cost on the minmax optimality of the bisection method. Numerical experiments show that, on regular functions, the proposed method requires a number of function evaluations similar to current state-of-the-art methods, about 24% to 37% of the evaluations required by the bisection procedure. In problems with non-regular functions, the proposed method performs significantly better than the state-of-the-art, requiring on average 82% of the total evaluations required for the bisection method, while the other methods were outperformed by bisection. In the worst case, while current state-of-the-art commercial solvers required two to three times the number of function evaluations of bisection, our proposed method remained within the minmax bounds of the bisection method.

中文翻译：

---

保持 Minmax 最优性的二分法平均性能的增强

我们确定了一类根搜索方法，它们在平均性能上出人意料地优于二分法，同时保持了 minmax 最优性。平均值的改进适用于任何连续分布假设。我们还确定了该类中的一种特定方法，并表明在温和的初始条件下，它可以获得高达 1.618 的收敛阶数，即与割线法相同。因此，我们获得了改进的平均性能和改进的收敛顺序，而对二分法的最小最大最优性没有成本。数值实验表明，在常规函数上，所提出的方法需要许多类似于当前最先进方法的函数评估，大约是二分法所需评估的 24% 到 37%。在非常规函数的问题中，所提出的方法的性能明显优于最先进的方法，平均需要二分法所需总评估的 82%，而其他方法的性能则优于二分法。在最坏的情况下，虽然当前最先进的商业求解器需要两到三倍的二分函数评估次数，但我们提出的方法仍然在二分法的最小最大值范围内。