Global Optimisation in Hilbert Spaces using the Survival of the Fittest Algorithm

Abstract

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.