About matrix correction of a dual pair of improper linear programming problems with respectto the minimum weighted euclidean norm

Abstract

The paper focused on problem of matrix correction of a dual pair of improper linear programming problems with respect to the minimum weighted Euclidean norm. Weighted matrix is achieved by multiplying the extended left and right correction matrix by non-degenerate matrices. The main purpose of weighing is to include in the linear programming problem information about the complexity of the correction of the expanded matrix of the constraint system. Matrix correction of linear programming problems is a change (correction) of any coefficients of both left and right parts of equations and inequalities of constraints of primal and dual linear programming problems. The indicated problem is reduced to the auxiliary problem of unconditional differentiable minimization. The theorem about the optimal matrix correction of a dual pair of improper linear programming problems with respect to the minimum weighted Euclidean norm is presented in the article. This theorem is a consequence of the theorem about the existence of a solution to the matrix correction problem of an extended constraint matrix of a dual pair of improper linear programming problems with respect to the minimum weighted Euclidean norm. In turn, the last theorem is based on the theorem about matrix solution for the inverse linear programming problem. The statements of the that theorems are also given in the article. As a possible tool for the numerical solution of this problem, the quasinewton Broyden-Fletcher-Goldfarb-Channo algorithm is considered. The problem of searching for an expanded correction matrix that is minimal in terms of a weighted Euclidean norm is considered. This problem is determined by the following parameters: the expanded matrix of the constraint system, non-degenerate weight matrices, and the initial approximation. The solution is represented by the argument of the objective function and its value. The results of computational experiments of algorithm convergence in terms of the objective function and the argument are given.