Abstract
Typically, the search for solutions in discrete optimization problems is associated with fundamental computational difficulties. The known methods of accurate or approximated solution of such problems are studied talking into consideration their belonging to so-called problems from P and NP class (algorithms for polynomial and exponential implementation of solution). Modern combinatorial methods for practical solution of discrete optimization problems are focused on the development of algorithms which allow obtaining an approximated solution with guaranteed evaluation of deviations from the optimum. Simplification algorithms are an effective technique of the search for solutions to an optimization problem. If we make a projection of a multi-dimensional process onto a two-dimensional plane, this technique will make it possible to clearly display a set of all solutions to the problem in graphical form. The method for simplification of the combinatorial solution to the discrete optimization problem was proposed in the framework of this research. It is based on performance of decomposition of a system that reflects the system of constraints of the original five-dimensional original problem on a two-dimensional plane. This method enables obtaining a simple system of graphic solutions to a complex problem of linear discrete optimization. From the practical point of view, the proposed method enables us to simplify computational complexity of optimization problems of such a class. The applied aspect of the proposed approach is the use of obtained scientific results in order to provide a possibility to improve the typical technological processes, described by systems of linear equations with existence of systems of linear constraints. This is a prerequisite for subsequent development and improvement of similar systems. In this study, the technique for decomposition of a discrete optimization system through projection of an original problem on two-dimensional coordinate planes was proposed. In this case, the original problem is transformed to a combinatorial family of subsystems, which makes it possible to obtain a system of graphic solutions to a complex problem of linear discrete optimization.
Highlights
Discrete optimization, which has already been defined as a separate section of the optimization theory, in most cases operates the combinatorial methods of solution [1]
The aim of present research is to develop the algorithm of simplification of the solution of multidimensional discrete optimization problems using the standard computational procedures of linear algebra and some techniques of linear optimization
It is shown that the solution to a linear optimization problem is possible by simplification with the use of decomposition of a system due to construction of projections of a multi-dimensional system of the original problem onto two-dimensional coordinate planes
Summary
Discrete optimization, which has already been defined as a separate section of the optimization theory, in most cases operates the combinatorial methods of solution [1]. Are solved using the methods of linear programming [2]. The typical tasks of the combinatorial methods are to obtain the original reference plan, optimality assessment, improvement of the plan and boundary estimation of objective function. Since most discrete optimization problems belong to the class of NP problems, the use of algorithms for a problem simplification without losing control of the solution accuracy is relevant [3]. Direct computing simplification technique was implemented with the use of the Jordan-Gauss method [5].
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