Increasing the efficiency of solving linear programming problems when solving systems of partial derivative differential equations using the control point method

Abstract

The paper discusses the basic principles of the control point method, which reduces the solution of a linear system of partial differential equations to the solution of a linear programming (LP) problem. An analysis of various methods for solving LP problems is given and it is shown that the most suitable for the problems under consideration is the interior point method with unconditional sequential minimizations of the logarithmic penalty function. Moreover, with a large dimension of the LP problem, the main costs of computer time are associated with solving systems of linear algebraic equations (SLAE) that determine the direction of descent in Newton’s method, used for unconditional minimization. A structural approach is proposed to the formation of optimized parameters of the LP problem and the organization of the solution of SLAE, in which the solution of SLAE of large dimension is reduced to the inversion of square matrices and the solution of SLAE of much smaller dimensions, which radically reduces the cost of computer time both for solving the SLAE and the LP problem.