On reqularization and vector optimization of machine design variables

Abstract

Abstract The correct determination of the feasible solutions set and the set of Pareto-optimal solutions are the most important in engineering optimization problems. To create a machine with optimal parameters, you need to know exactly the boundaries of the feasible set on which to search for these parameters. And to claim that the resulting machine design is optimal, we need to approximate the Pareto set with the necessary accuracy. To create a feasible set the method of obtain his constrains was proposed. Earlier, the results of our research on determining the rate of convergence of this method and approximation of feasible set of solutions were solved. In this paper, the problems of approximation and regularization Pareto optimal set are solved. Due to the fact that the set of Pareto-optimal solutions is not stable, even small errors in calculating the values of the system performance criteria can significantly change this set. It follows from this that, approximating a feasible solutions set with a given accuracy, we cannot guarantee approximations of the Pareto set. Such problems are called ill-posedaccording to Tikhonov. In these case reqularization of the Pareto-optimal set is a solution of these ill-posed problem.. To get a complete solution to this problem, acceptable for most practical tasks, is quite difficult. In this paper, this problem is solved for criteria that satisfy Lipschitz condition. The results obtained here are not only theoretical in nature, but are already used in the design and identification of parameters of mathematical models of machines.