Duality Theory Revisited

Abstract

Starting with the primal problem, let us

$$ \begin{gathered} maxf(X) = C'X\quad s.t. \hfill \\ AX \leqq b, X \geqq 0, X\varepsilon {{R}^{n}}, \hfill \\ \end{gathered} $$

where A is of order (mxn) with rows α_1,..., αm and b is an (m x 1) vector with components b₁,…,b _m . Alternatively, if

$$\bar A = \left[ {\begin{array}{*{20}{c}}A\\\ldots \\{ - {I_n}}\end{array}} \right],\bar b = \left[ {\begin{array}{*{20}{c}}b\\\ldots \\0\end{array}} \right]$$

are respectively of order (m + n x n) and (m + n x 1), then we may

$$\max f\left( x \right) = C'Xs.t.$$

$$\bar b - \bar AX0,$$

where X€Rⁿ is now unrestricted. In this formulation Xj ≥ 0, j = l,…,n,$\bar b - \bar AX0,$ is treated as a structural constraint so that defines a region of feasibleor admissible solutions K ⊂ Rⁿ.