Optimization Algorithm for Finding Solutions in Linear Programming Problems

When speaking about linear programming problems of big dimensions with rare matrix of the system, solved through simplex method, it is necessary, at each iteration, to calculate the inverse of the base matrix, which leads to the loss of the rarity character of the matrix. The paper proposes the replacement of the calculus of the inverse of the base matrix with solving through iterative parallel methods a linear system with rare matrix of the system. General Presentation Linear programs for big real systems are characterized by rare matrices, having a low percentage of non-zero elements. The rare character appears at each base matrix, but disappears at the inverse of this matrix. In its classical form, the simplex method uses a square matrix, the inverse of the base matrix, whose value is putting up-to-date at each iteration. The number of non-zero elements of the inverse matrix increases rapidly and depends on the number of iterations. Because of this, in the place of the calculus of the rare matrix, one can solve the linear systems with a rare matrix of the system through iterative parallel methods. Let’s take the linear programming problem in the standard form: