Визуальное представление многомерных задач линейного программирования

Abstract

The article proposes an n-dimensional mathematical model of the visual representation of a linear programming problem. This model makes it possible to use artificial neural networks to solve multidimensional linear optimization problems, the feasible region of which is a bounded non-empty set. To visualize the linear programming problem, an objective hyperplane is introduced, the orientation of which is determined by the gradient of the linear objective function: the gradient is the normal to the objective hyperplane. In case of searching a maximum, the objective hyperplane is positioned in such a way that the value of the objective function at all its points exceeds the value of the objective function at all points of the feasible region, which is a bounded convex polytope. For an arbitrary point of the objective hyperplane, the objective projection onto the polytope is determined: the closer the objective projection point is to the objective hyperplane, the greater the value of the objective function at this point. Based on the objective hyperplane, a finite regular set of points is constructed, called the retina. Using objective projections, an image of a polytope is constructed. This image includes the points of the retina and the distances to the corresponding points of the polytope surface. Based on the proposed model, parallel algorithms for visualizing a linear programming problem are constructed. An analytical estimation of its scalability is performed. Information about the software implementation and the results of largescale computational experiments confirming the efficiency of the proposed approaches are presented.