  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 the 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 receptive field. Using
objective projections, an image of a polytope is constructed. This image
includes the distances from the receptive points 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 large-scale computational experiments
confirming the efficiency of the proposed approaches are presented.
