II. Consider the feasible region F = {x € R" | Ax 0}, where A E Rmxn and be Rm. Let JC {1,2,...,m+n} with |J| = m. Define the dual of J by Jd Je + m (mod m + n), = where Je is the complement of J from {1,2,...,m+n}, and X + m = {x+m|x€ X}. A Define B = = [A Im] and Ba If B is nonsingular, define B = B-¹b, and [n] jd: b if Bd is nonsingular, define v = (134)-1 [01] ₁ (Bd Finally, define x Rmxn such that IJTB and x je = 0. 1 1 1 Set m2, n = 3, A= 1₁ and b = = 2. 2 1 0 2 1. Fill out the following table. Three rows have been filled up for you. J Jd Bd V B TJ and [₁ 2 3] 1 1 1 2 0 0 {1,2} {1,2,5} (Bd)-1 2 = 2 2 1 0 00-1 1 ; 2 2 0 0 0 {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} 2 1 0 {3,4} {2,3,4} -1 0 0 singular Bd 3 singular B 0 -1 0 1 1 1 2 0 0 {3,5} {1,3,4} -1 0 0 (B)-10 = 0 61 ; 0 2 0 -1 0 0 2 2 {4,5} 2. Draw the feasible region F in R³ and label its vertices. 3. Determine which vertices are degenerate or nondegenerate. Then, correspond to each vertex u of F all Js which produce feasible basic solutions corresponding to v. 2
