Duality in Linear Programming

1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our primal linear programming problem, so x ∈ P , i.e., Ax ≤ b and x ≥ 0. Also suppose we can choose y1, . . . , ym defining an m-covector y such that (yA)i ≥ ci and yi ≥ 0 for i = 1, . . . , n. Then c x ≤ yAx, and we have yAx ≤ yb, so altogether we have cx ≤ yAx ≤ yb. We see that yAx and yb are both upper-bounds of our primal objective function ζ(x) = cx for admissible choices of x and y, and yb is independent of x, so for any admissible choice of y, yb is an upper-bound of cx for all feasible points x ∈ P .