Dual Theorems - Introduction to Operations Research - Lecture Slides, Slides of Operational Research

Download Dual Theorems - Introduction to Operations Research - Lecture Slides and more Slides Operational Research in PDF only on Docsity! Dual Theorems z Z cx s t Ax b x x *: max . . = = ≤ ≥ 0 w* := min x w = yb s.t. yA ≥ c y ≥ 0 Primal Problem Dual Problem b is not assumed to be non-negative Docsity.com 7.4.1 Weak Duality Theorem • If x is a feasible solution to the primal problem from (7.13) and y is a feasible solution to the dual problem, then cx ≤ by. • Proof: If x is a feasible solution to the Primal Problem, then we must have: Ax ≤ b So, if y is a feasible solution to the Dual Problem, we must have (why?) yAx ≤ yb Docsity.com 7.4.2 Lemma • If a linear programming problem has an unbounded objective, then its dual problem has no feasible solution. Observation: • It is possible that both the primal and the dual problem are infeasible. Docsity.com 7.4.3 Lemma • Let x* be a feasible solution to the primal problem and y* a feasible solution to the dual. Then cx* = by* implies that x* is an optimal solution for the primal and y* is optimal for the dual. Proof: • From the Weak Duality theorem we have cx ≤ by hence we also have cx ≤ by* Docsity.com for any feasible solution x of the primal and any optimal solution y* of the dual. Thus, for any feasible solution x* of the primal for which cx*=by* we have (why?) cx ≤ cx* for any feasible solution, x, of the primal. This implis that x* is an optimal solution for the primal. Docsity.com Eq. # x1 --- xn s1 --- sm RHS --- --- --- --- --- --- --- (7.57) Z a y ci jim 1 11 −∑ = --- a y cimim i n=∑ −1 y1 --- ym yb y = cBB-1 for any simplex tableau not necessarily the final one This is implied by our beloved recipe: r = cBB-1D - c Docsity.com Important Observations • IN ANY TABLEAU, THE DUAL VARIABLES CORRESPONDING TO THE PRIMAL BASIC FEASIBLE SOLUTION ARE THE REDUCED COSTS FOR THE PRIMAL SLACK VARIABLES. • IN APPLYING THE SIMPLEX METHOD TO THE PRIMAL PROBLEM, WE ALSO SOLVE ITS DUAL. Docsity.com 7.4.6 Example BV Eq. # x1 x2 x3 s1 s2 RHS x1 1 1 4/5 4/5 0 1/5 12 s1 2 0 - 2/5 - 12/5 1 - 8/5 4 Z Z 0 4 14 0 16 960 BV Eq. # y1 y2 t1 t2 t3 RHS y2 1 8/5 1 - 1/5 0 0 16 t2 2 2/5 0 - 4/5 1 0 4 t3 3 12/5 0 - 4/5 0 1 14 w w - 4 0 - 12 0 0 960 Docsity.com