Using feedback functions to solve parametric programming problems

Q4 Computer Science
Aleksandr Evgen'evich Umnov, Egor Alexandrovich Umnov
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引用次数: 0

Abstract

We consider a finite-dimensional optimization problem, the formulation of which in addition to the required variables contains parameters. The solution to this problem is a dependence of optimal values of variables on parameters. In general, these dependencies are not functions because they can have ambiguous meanings and in the functional case be non-differentiable. In addition, their domain of definition may be narrower than the domains of definition of functions in the condition of the original problem. All these properties make it difficult to solve both the original parametric problem and other tasks, the statement of which includes these dependencies. To overcome these difficulties, usually methods such as non-differentiable optimization are used. This article proposes an alternative approach that makes it possible to obtain solutions to parametric problems in a form devoid of the specified properties. It is shown that such representations can be explored using standard algorithms, based on the Taylor formula. This form is a function smoothly approximating the solution of the original problem for any parameter values, specified in its statement. In this case, the value of the approximation error is controlled by a special parameter. Construction of proposed approximations is performed using special functions that establish feedback (within optimality conditions for the original problem) between variables and Lagrange multipliers. This method is described for linear problems with subsequent generalization to the nonlinear case. From a computational point of view the construction of the approximation consists in finding the saddle point of the modified Lagrange function of the original problem. Moreover, this modification is performed in a special way using feedback functions. It is shown that the necessary conditions for the existence of such a saddle point are similar to the conditions of the Karush –Kuhn –Tucker theorem, but do not contain constraints such as inequalities and conditions of complementary slackness. Necessary conditions for the existence of a saddle point determine this approximation implicitly. Therefore, to calculate its differential characteristics, the implicit function theorem is used. The same theorem is used to reduce the approximation error to an acceptable level. Features of the practical implementation feedback function method, including estimates of the rate of convergence to the exact solution are demonstrated for several specific classes of parametric optimization problems. Specifically, tasks searching for the global extremum of functions of many variables and the problem of multiple extremum (maximin-minimax) are considered. Optimization problems that arise when using multicriteria mathematical models are also considered. For each of these classes, there are demo examples.
利用反馈函数解决参数化规划问题
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来源期刊
Computer Research and Modeling
Computer Research and Modeling Computer Science-Computational Theory and Mathematics
CiteScore
0.80
自引率
0.00%
发文量
82
审稿时长
15 weeks
期刊介绍: The journal publishes original research papers and review articles in the field of computer research and mathematical modeling in physics, engineering, biology, ecology, economics, psychology etc. The journal covers research on computer methods and simulation of systems of various nature in the leading scientific schools of Russia and other countries. Of particular interest are papers devoted to simulation in thriving fields of science such as nanotechnology, bioinformatics, and econophysics. The main goal of the journal is to cover the development of computer and mathematical methods for the study of processes in complex structured and developing systems. The primary criterion for publication of papers in the journal is their scientific level. The journal does not charge a publication fee. The decision made on publication is based on the results of an independent review. The journal is oriented towards a wide readership – specialists in mathematical modeling in various areas of science and engineering. The scope of the journal includes: — mathematical modeling and numerical simulation; — numerical methods and the basics of their application; — models in physics and technology; — analysis and modeling of complex living systems; — models of economic and social systems. New sections and headings may be included in the next volumes.
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