homework-13-solutions

homework-13-solutions - MthSc810 Mathematical Programming...

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MthSc810 – Mathematical Programming Fall 2011, Homework #13. Due Thursday, December 8, 2011, 6PM EDT. From the syllabus: homework will be penalized 50% for each day they are late. After two days, they will not be accepted. No exception. Problem 1. Compute G ( c ) for the following problem, where c = ( c 1 ,c 2 ): min c 1 x 1 + c 2 x 2 x 1 - x 2 1 x 1 , x 2 0 . Solution. Since the feasible set is nonempty and unbounded, the problem is al- ways feasible and, depending on c 1 and c 2 , admits a finite or an infinite optimum. Observe that there is only one extreme point, (1 , 0) (the other basic solu- tions are (0 , - 1) and (0 , 0), both infeasible) and two extreme rays, given by the recession cone C = { ( d 1 ,d 2 ) R 2 : d 1 0 ,d 2 0 ,d 1 - d 2 0 } = { ( d 1 ,d 2 ) R 2 : d 2 0 ,d 1 d 2 } , whose extreme rays are (1 , 0) and (1 , 1). For the problem to be unbounded, c d < 0 for at least one of the two extreme rays, hence for c 1 < 0 or c 1 + c 2 < 0. Hence for c 1 0 and c 1 + c 2 0 the problem is bounded and admits (1 , 0) as its optimal solution. When the problem
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