Quantum modification of the freshwater hydra algorithm for solving the optimization problem

Abstract

One of the stages of solving a number of mathematical modelling problems is finding a point where some function of several variables reaches the largest or the least value. This function is usually high-dimensional and has many local extrema. Such a problem can be successfully solved using population optimisation algorithms, for example, the particle swarm algorithm, the freshwater hydra algorithm, and others. The aim of this work was to improve the fresh-water hydra algorithm (H-algorithm) taking into account the quantum mechanics approach. The developed quantum modifications of this algorithm are based on the Analytic Hierarchy Process (QH-AHP) and the Bayesian approach (QH-B). The convergence rate of quantum modifications is higher than the convergence rate of the original H-algorithm due to the fact that the position of individuals is determined directly without manipulating their speeds (according to the concepts of quantum mechanics). As a result, during one iteration, individuals can move over a considerable distance, which increases the range of the search space. In addition, this approach allows the individuals to overcome the domains of attraction of local extrema, which prevents the premature algorithm convergence. Optimum values of the parameters of the developed algorithms were found as a result of solving the problem of meta-optimisation. The conversion rates of the suggested algorithms of optimisation were compared through the example of various multiextremal test functions, such as the Rosenbrock, Davis, Ackley, and Rastrigin functions. The suggested quantum modifications of the H-algorithm (QH-AHP and QH-B) for various test functions showed different, but on average similar, convergence rates. The convergence rate of quantum modifications of algorithms for various test functions is higher than the convergence rate of the original freshwater hydra algorithm and particle swarm optimisation algorithm. The developed algorithms for solving the optimisation problem can be used in training neural networks, in mathematical modelling of processes and systems in various subject areas at the stage of identifying model parameters, optimisation of characteristics, and the creation of optimal control for such systems, etc..