Particle swarm method for solving the problems of nonlinear optimization

The optimal design of buildings and structures requires solving a number of important tasks related to determining the best topology and geometric shape of structures, physical properties of elements, connections of elements among themselves, etc. In this case, it is necessary to take into account the influence of many factors: the distribution of static and dynamic loads, corrosive effects, the nature of the operating conditions, the requirements for the reliability and durability of the unit. The complexity of solving such problems is connected with the fact that, as a rule, the objective functions of such problems are nonlinear. Methods. Various approaches are used to solve nonlinear optimization problems: stochastic search (Monte Carlo method); search methods in which the step is successively reduced, following to the given relation (halving, golden ratio, inverse Fibonacci numbers); gradient descent method; evolutionary algorithms; penalty function method and others. In recent years, a new class of methods of numerical optimization has been intensively developed, in various works it is called social-behavioral, population, or swarm. For practical verification of the particle swarm method, the finding of the extrema of the test functions of Rosenbrock and Rastrigin is considered. Results. A new software implementation of one of the methods of artificial collective intelligence, the particle swarm method, is proposed for solving nonlinear optimization problems in the Maxima open-source computer algebra system. The high computational efficiency of this method for finding global extrema of “ravine” and multimodal functions in those cases where the application of many classical algorithms can be difficult is shown. Scientific novelty. Compared to classical methods, swarm intelligence methods are especially effective for finding extrema of nonlinear multimodal functions, as well as for solving high-dimensional problems. Practical significance. The effect of the method parameters (particle number and weight coefficients) on the rate of practical convergence is investigated. The developed method can be used, inter alia, to solve the problems of optimal design of building structures, buildings and structures.