希尔伯特空间中的优胜劣汰全局优化算法
Global Optimisation in Hilbert Spaces using the Survival of the Fittest Algorithm
摘要：
在高维和无限维空间中的全局优化问题出现在各种现实世界的应用，如工程，经济学，物理学，生物学，机器学习，最优控制等随机方法，全局优化，生物启发的方法是目前流行的文献中，模仿自然生态和进化过程，并报告是有效的，在许多实际的研究案例。然而，许多生物启发的方法有一些重要的缺点。由于它们的半经验性质，收敛到全局最优解并不总是能得到保证，并且与高维空间作斗争，显示出缓慢的收敛。在这里，我们提出了一种生物启发的全局随机优化方法，适用于希尔伯特函数空间，灵感来自达尔文的著名的适者生存的思想，因此，被称为“适者生存算法”（SoFA）。在数学上，SoFA的收敛性是概率测度在希尔伯特空间中局部化的一个基本性质的结果，我们严格地证明了所引入的算法对于一般泛函类的收敛性.作为一个有见地的现实世界的问题，我们应用SoFA来寻找海洋和湖泊中浮游动物每日垂直迁移的全球最佳轨迹，这被认为是地球上生物量最大的同步运动。我们最大限度地提高健身函数空间来自冯福斯特阶段结构的人口模型与生物现实的参数。我们表明，在高维空间中的适应度最大化的问题，SoFA比其他一些随机全局优化算法表现更好。我们强调了新的优化算法和自然选择过程之间的联系，在生态系统中发生的人口通过逐步排除竞争的con-specific菌株。
英文摘要：
Global optimisation problems in high-dimensional and infinite dimensional spaces arise in various real-world applications such as engineering, economics, geophysics, biology, machine learning, optimal control, etc. Among stochastic approaches to global optimisation, biology-inspired methods are currently popular in the literature, imitating natural ecological and evolutionary processes and reported to be efficient in many practical study cases. However, many of bio-inspired methods have some vital drawbacks. Due to their semi-empirical nature, convergence to the globally optimal solution cannot always be guaranteed and struggles with the high dimensionality of space, showing a slow convergence. Here, we present a bio-inspired global stochastic optimisation method, applicable in Hilbert function spaces, inspired by Darwin's' famous idea of the survival of the fittest, therefore, referred to as the `Survival of the Fittest Algorithm' (SoFA). Mathematically, the convergence of SoFA is a consequence of a fundamental property of localisation of probabilistic measure in a Hilbert space, we rigorously prove the convergence of the introduced algorithm for a generic class of functionals. As an insightful real-world problem, we apply SoFA to find globally optimal trajectories for daily vertical migrations of zooplankton in ocean and lakes, considered to be the largest synchronised movement of biomass on Earth. We maximise fitness in a function space derived from a von-Foerster stage-structured population model with biologically realistic parameters. We show that for problems of fitness maximisation in high-dimensional spaces, SoFA performs better than some other stochastic global optimisation algorithms. We highlight the links between the new optimisation algorithm and natural selection process in ecosystems occurring within a population via gradual exclusion of competitive con-specific strains.
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