sol11 - C O250 I NTRODUCTION TO O PTIMIZATION S OLUTIONS 11...

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CO250 INTRODUCTION TO OPTIMIZATION - SOLUTIONS 11 Solution 1. (a) The feasible region is the convex region in the middle of the following diagram, with extreme points (0 , 2) , (1 , 3) , (2 , 3) and (2 , 0) . The cone of tight constraints at one of these points consists of the set of vectors pointing from the extreme point into the shaded region which touches the feasible region at that point (these regions extend to infinity in the appropriate directions). (b) Solving ( - 2 , 1) = y 1 ( - 1 , - 1) + y 2 ( - 1 , 1) for y 1 and y 2 , we find y 1 = 1 2 , y 2 = 3 2 . (c) The dual (D) is min ( - 2 , 2 , 2 , 3) y subject to - y 1 - y 2 + y 3 = - 2 - y 1 + y 2 + y 4 = 1 y 0 So we may set y 1 and y 2 to the solution of (b), and y i = 0 for y i corresponding to the non-tight constraints at this extreme point. In this way, we arrive at a solution of (D) to be y = ( 1 2 , 3 2 , 0 , 0) . One easily checks that this really is feasible for (D), and and the objective function has value 2 at this point. This is equal to the objective function of (P) at the extreme point in question,
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sol11 - C O250 I NTRODUCTION TO O PTIMIZATION S OLUTIONS 11...

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