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Fina. Exam Math 168 : Optimization Spring 09, Lieu Instructions us exam is closed book and NO caleutators. © Carofilly read and provide answers to all questions. @ Write legibly and show work for credit. e There are 190 total possible points. (1) (20pts total) Let S = {xs Aw = bn > O}. where A is an m x 2 matrix and 6€ 2”, Show that de®", d > 0, is a direction of wboundedness in the region S if and only if d is in the null space of A. (2) (30pts total) Graphically find the solution to the following finear program and identify all linearly independent directions of unboundedness in the feasible region (note: there are at most. two). (3) (S0pts total) The following linear program mins = —32, — Lax, - ty sit. 21 + Xe T ky kt Sry + 2rz 15 < 2a. + 32x, <9 £ 2 has optimal basis {27 ,, 22,23} with 3/2 -3/2 Bta( -3/2 3/2 =1 1-1 ia) (LOpts) What is the optimal basic feasible solution auc the optimal objective value? What are the optimal dual variables? ib) (20pts) By how much can the objective coctticiont of az increase or decrease without chang- ing the optimal basis? 20pts) What is the solution of the linear program obtained by decreasing the right-land side of the second constraint by 5? 
