On the choice of the initial approximation in the numerical solution of parametric optimization problems

Abstract

Background. In recent years, the complexity of the applied problems of parametric optimization has increased significantly. Since computational possibilities are finite, it is necessary to look for ways to increase the efficiency of the optimization search process. One of the very significant factors for the effectiveness of optimization studies is a good choice of initial approximations. To make such a choice, it is proposed to probe the parameter space before starting the optimization. The Sobol generator and the generator based on random numbers are considered as sounding algorithms. Materials and methods. The comparative analysis of the efficiency of such sounding depending on the number of variables and the number of sounded points is carried out in the work. The deformable polyhedron method (Nelder-Mead) was used as an optimization algorithm. To increase the reliability of the research, a statistical analysis of the results of solving different problems on the same topology was carried out - the search ranges were randomly varied. The criterion for the success of a particular solution tactic was the probability of finding an acceptable extremum. Results and conclusions. As a result of the research, it turned out that space probing effectively reduces the required number of calls to the mathematical model in multiextremal issues. If the optimization method makes it possible to consistently find a global extremum for the objective function of the topology under study, the choice of generator is not critical. In complex problems where the global extremum is not reached using the optimization method used, the use of the Sobol generator gives a higher probability of obtaining an acceptable solution than random generation. An increase in the number of points probed with the Sobol generator leads to an increase in the optimization efficiency in terms of the probability of finding an acceptable solution.

Keywords

• parametric optimization
• sobol generator
• monte carlo method
• deformable polyhedron method
• space probing
• irregular grids