论文标题
通过优化的三个立方体的总和
Sum of Three Cubes via Optimisation
论文作者
论文摘要
首先求解等式$ x^3+y^3+z^3 = k $，固定$ k $ for $ z $，然后考虑到结果的最接近整数函数的距离，我们将三个立方体问题的总和变成了优化。然后，我们在此功能的情况下对此功能应用三种随机优化算法，其中有许多已知的解决方案。目的是测试该方法在寻找整数解决方案方面的有效性。算法是粒子群优化的修改和两个模拟退火的实现。我们想比较它们的有效性，如算法的运行时间所衡量。为此，我们通过假设两个潜在的概率分布（指数和对数正态）来对时间数据进行建模，并为它们计算一些数值特征。最后，我们通过相应的Fisher Information指标评估了模型相对于多种地质距离的统计区分性。
By first solving the equation $x^3+y^3+z^3=k$ with fixed $k$ for $z$ and then considering the distance to the nearest integer function of the result, we turn the sum of three cubes problem into an optimisation one. We then apply three stochastic optimisation algorithms to this function in the case with $k=2$, where there are many known solutions. The goal is to test the effectiveness of the method in searching for integer solutions. The algorithms are a modification of particle swarm optimisation and two implementations of simulated annealing. We want to compare their effectiveness as measured by the running times of the algorithms. To this end, we model the time data by assuming two underlying probability distributions -- exponential and log-normal, and calculate some numerical characteristics for them. Finally, we evaluate the statistical distinguishability of our models with respect to the geodesic distance in the manifold with the corresponding Fisher information metric.
