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hw3sol - MIT 2.098/6.255/15.093 Optimization Methods Prof J...

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MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 3. Solution Problem 1: BT, Exercise 6.4 Solution. (a) The dual problem is: maximize b 0 1 p 1 + b 0 2 p 2 s.t. F 0 1 p 1 + F 0 2 p 2 = c 0 (1) D 0 1 p 1 c 1 D 0 2 p 2 c 2 p 1 , p 2 0 . Constraint (1) is the coupling constraint and the subproblems are: maximize ( b 0 i - q 0 F 0 i ) p i s.t. D 0 i p i c i p i 0 , where q is the dual variable associated with the coupling constraint. For a maximization problem, the reduced costs at optimality are non-positive. So the subproblem is to maximize the reduced cost. If you change the original problem to a minimization problem, then you will get a minus sign before q . But it is equivalent, because for the minimization problem, the cost vector is - b i then the dual variable q is the negative of the q in the maximization problem. (b) Rewrite the problem as follows: minimize c 0 1 x 1 + c 0 2 x 2 + 0 . 5 c 0 0 y 1 + 0 . 5 c 0 0 y 2 s.t. y 1 - y 2 = 0 D 1 x 1 + F 1 y 1 b 1 D 2 x 2 + F 2 y 2 b 2 x 1 , x 2 0 .
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