Surface Movement Method for Linear Programming

TL;DR

This paper introduces the surface movement method for linear programming, which constructs an optimal path on the feasible polytope surface using a neural network to determine the direction, with proven convergence.

Contribution

It presents a novel surface movement approach for linear programming that integrates deep neural networks to guide the optimal path construction.

Findings

The method guarantees convergence to the optimal solution.

Neural network effectively determines the optimal movement direction.

The approach can be implemented efficiently with deep learning techniques.

Abstract

The article presents a new method of linear programming, called the surface movement method. This method constructs an optimal objective path on the surface of the feasible polytope from the initial boundary point to the point at which the optimal value of the objective function is achieved. The optimality of the path means moving in the direction of maximum increase/decrease in the value of the objective function. A formal description of the algorithm implementing the surface movement method is described. The convergence theorem of this algorithm is proved. The presented method can be effectively implemented using a feed forward deep neural network to determine the optimal direction of movement along the faces of the feasible polytope. To do this, a multidimensional local image of the linear programming problem is constructed at the point of the current approximation. This image is fed to the input of the deep neural network, which returns a vector determining the direction of the optimal objective path on the polytope surface.