Matrix correction of a dual pair of improper linear programming problems

Linear-quadratic programming and its application to data correction of improper linear programming problems

“…To formalize this problem in the form of a special case of the problem (1), we introduce augmented vectors c = (0, c) T Corollary of the Theorem 1. If the system of equations Az = e is consistent, the matrix B has linearly independent columns, d > ⟨(AD −1 A T ) −1 e, e⟩, the vector c does not belong to the subspace of vector-rows of the matrix A, then the problem (6) has a solution and it can be represented in the form…”

“…Vector correction (right side) and matrix correction (of all initial data) are applied to inconsistent systems of linear algebraic equations and inequalities and improper problems of LP (regression analysis [2,3], production planning [4,5], etc.). If the obtained correction problem of LP has no solution, then the correction of a pair of dual problems may be considered [6]. However, there is another problem.…”

“…We introduce a restriction on the value of correc-tion ⟨Dz, z⟩ ≤ d. The system A 1 x = b 1 is transformed into Az = e, taking into account the additional condition z 0 = 1 (in the augmented vector z the first component is z 0 ). We obtain the problem of the form(1) ⟨c, z⟩ → max z∈Z , Z = {z|⟨Dz, z⟩ ≤ d, Az = e} (6).…”