Об одном методе нелинейного программирования с произвольными ограничениями

Abstract

Introduction: Modeling and identification of nonlinear and non-stationary systems, as well as the analysis and filtration of experimental data often require the use of optimization methods, in particular, nonlinear programming. These problems belong to the class of problems of multidimensional optimization with restrictions. Unfortunately, at the moment there are no efficient numerical methods to solve such problems for the case when the number of restrictions т exceeds manifold the number of the varied parameters n. Purpose: The goal is to develop new methods and algorithms for multiple-parameter optimization with restrictions. Results: A new nonlinear programming method is developed which is called "Multidimensional Patchwork Shell Method" (MPSM). It allows you to solve problems of conditional multiple-parameter optimization for the case of n << т. The method uses both necessary and sufficient optimality conditions for metric-like criterion functions. It has been shown that if the only minimum (maximum) of a metric is out of the area of admissible values formed by the restrictions, the solution of the problem will be found on the border of the area of admissible decisions. In this case, the problem can have several solutions which will correspond to identical minimum (maximum) values of the criterion function, and all these solutions can be obtained by MPSM. The basis of MPSM is creating an area of admissible decisions by analyzing restrictions of any type and approximating them with piecewise and polynomial restrictions or with rational fractions. This reduces the initial problem to a set of interconnected subtasks with the linear restrictions forming multidimensional shells, significantly narrowing the search area. Practical relevance: The proposed method of nonlinear programming has been tested. It will find application in conditional optimization in such fields as physical metrology, optimal control systems and logical-interval systems. This algorithm can be improved by developing more efficient algorithms for new types of restrictions.