Midterm Exam with Solution for Optimization | MAT 168, Exams of Optimization Techniques in Engineering

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MAT 168 Spring 2009 Midterm Instructor: L. Lien Instructions: Write legibly. Show work for cradit. (Total 100 points) (1} (30pts) Tn cach of the tableaus below, identify if ¢ it is optimal, then find all solutions ® it is not optimal, then compute the next tableau ¢ the linear program is unbounded, then identify a direction of unboundedness fp Lo 23 ry 25 rhs “= 7 + aan “ 7 —F 0 0 0 0 13 (A) Toca eT af 2? 00 7 7 a bt 1. 8 20 12101 4 id * t 00 3 01 1 5 (2) (25pts) Determine whether the linear prograin below is feasible. If it is, identify a basic feasible point and the corresponding basis. If not, explain. mins = 2+ 22+ 24 s.t. Zz) > —1, =2 BZ +29 +43 >0 Zz >0 (3) (20pts) Let A be an m x n matrix and b an m x 1 vector. Prove that the set S={xeR": Av 0} is convex. (4) (25pts) Consider a linear program mins = chr 29 where the feasible region S = {Ar ~ b.r > 0} is bounded. Suppose the problem has optimal extreme points v,--- ey. Prove that if a point x is optimal then it can be expressed as a convex combination of the optimal extreme points v1, +++ , vy.