The Main Theorems of Linear Programming

Abstract

Recall that the general problem of linear programming can be formulated as follows:

$$ \begin{array}{l}f(x) = \left\langle {c,x} \right\rangle = \left\langle {{c_1},{x_1}} \right\rangle + \left\langle {{c_2},{x_2}} \right\rangle \to \inf ,x = \left( {{x_1},{x_2}} \right) \in X, \\X = {\left\{ {x = \left( {{x_1},{x_2}} \right):x} \right._1} \in {E^{n1}},{x_2} \in {E^{n2}},{x_1} \ge 0, \\{A_{11}}{x_1} + {A_{12}}{x_2} \le b,{A_{21}}{x_1} + {A_{22}}{x_2} = \left. {{b_2}} \right\} \\\end{array} $$

((2.1.1))

where A _ij are m _i × n _j matrices, c _j ∈ E^nj, b_i ∈E ^mi, i,j = 1,2. As before, we denote f _* = inf _x∈X f(x) assuming that X ∈ Ø. For the case where f _* > -∞ we introduce a set ${X_*} = \left\{ {x \in X:f(x) = {F_*}} \right\}$. Recall that problem (2.1.1) is solvable if X_* ≠ Ø; every point x_* ∈ X_* is a solution of this problem.