Revisiting ‘survival of the fittest’ principle in global stochastic optimisation: Incorporating anisotropic mutations

Abstract

Evolutionary methods of global stochastic optimisation are very popular in the literature with various practical applications, including modelling of natural disasters. To enhance efficiency of optimisation, advanced algorithms seek to incorporate the heterogeneity of population (trial points) and consider more complex scenarios of mutations and recombination. On the other hand, most of the existing evolutionary algorithms have the major drawback that they cannot guarantee the convergence to the global extremum, in other words, such algorithms are purely heuristic. Here we proposed a new approach to global stochastic optimisation using evolutionary algorithms, where we can rigorously show convergence to the optimal solution with probability of one. This was a challenge for previous heuristic algorithms of optimisation. We provide the proof of convergence of the novel algorithm both for the case, where the parameter space has a finite dimensionality, and for optimisation in function spaces, with infinite number of dimensions. The crux of the proposed algorithms is to describe optimisation as a long-term natural selection process, which is a result of interplay between competition for limited resources and mutation, which is known as survival of the fittest principle. This is a modification of Monte-Carlo optimisation, where at each iteration, the search is refined by generating mutations to eventually attain the neighbourhood with the maximal population fitness. Unlike the previous algorithms, using survival of the fittest philosophy, we incorporate the anisotropic mutation for different spatial coordinates. We also suggest implementation of combination of the local and the global search. Using various multi-modal test functions in high dimensional spaces, we show that the new optimisation methods can demonstrate a better performance than some other well-known bio-inspired optimisation algorithms. As an important practical application, we implement our method to find the optimal control strategy in a multi-agent model of spread of infectious diseases. We argue that incorporating anisotropy in mutations is a promising direction in constructing more efficient evolutionary methods of optimisation, which has not been properly exploited in previous algorithms yet.