Linear Programming: Solving Primal and Dual Problems with Duality Theorems, Assignments of Optimization Techniques in Engineering

Download Linear Programming: Solving Primal and Dual Problems with Duality Theorems and more Assignments Optimization Techniques in Engineering in PDF only on Docsity! MAT 168 – Spring 2009 HW Set 6, Part 1 of 3 (Due Wed. May 13) (H6P1.1) Consider the following linear program min z = 2x1 + 9x2 + 3x3 s.t. −2x1 + 2x2 + x3 ≥ 1 x1 + 4x2 − x3 ≥ 1 xj ≥ 0 (i) Find the dual to this problem and solve it graphically. (ii) Use complementary slackness to obtain the solution to the primal problem. (H6P1.2) Consider the following pair of primal/dual linear programs in canonical form min z = cT x max w = bT y s.t. s.t. Ax ≥ b x ≥ 0 AT y ≤ c y ≥ 0 Suppose x and y are feasible for the primal and dual problems, respectively. Prove the following (i) (Weak Duality Theorem) cT x ≥ bT y. (ii) (Corollary to Weak Duality Theorem) If cT x = bT y, then x and y are optimal for their respective problems. (iii) (Complementary Slackness) x and y are optimal for their respective problems if and only if xT (c−AT y) = 0 and yT (Ax− b) = 0. 1