Numerical implementation of surface movement method in linear programming

The article is devoted to the numerical implementation of a new linear programming method called the surface movement method. The implementation is based on the AlFaMove algorithm, which builds, on the surface of the feasible polytope, the optimal objective path from an arbitrary boundary point to a point that is a solution to the linear programming problem. The optimality of the path lies in the fact that the direction of movement along the face of the polytope corresponds to the maximum increase in the value of the objective function. To calculate the optimal direction of movement, a method based on the operation of pseudoprojection onto a linear manifold is used. The pseudoprojection operation is a generalization of the notion of orthogonal projection. Pseudo projection is implemented using an iterative projection-type algorithm. The proposition is proved that the pseudoprojection coincides with the orthogonal projection in the case of a linear manifold that is the intersection of hyperplanes. It is also proved that, in the case of a linear manifold, the pseudoprojection method calculates a movement vector that coincides with the direction of maximum increase of the objective function. A parallel implementation of the AlFaMove algorithm is performed. The results of computational experiments on a cluster computing system are presented, which demonstrate the high scalability of the proposed numerical implementation.