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Unformatted text preview: MthSc810 – Mathematical Programming Fall 2011, Solutions to Homework #12. Problem 1. For each of the following problems, represent on a Cartesian plane the set of points ( b 1 , b 2 ) such that the problem is bounded, infeasible, or un bounded. P ′ : min x 1 + x 2 x 1 x 2 = b 1 x 1 + x 2 = b 2 P ′′ : min x 1 x 1 ≥ b 1 x 1 ≤ b 2 P ′′′ : min x 1 x 2 2 x 1 + x 2 ≤ b 1 x 1 + 2 x 2 ≤ b 2 x 1 , x 2 ≥ Solutions. 1. The feasible set of P ′ is the set of solutions of a system of two equations in two variables with invertible coefficient matrix. Hence it admits one solution regardless of b 1 , b 2 , which is also the optimal solution. Hence the problem is bounded for ( b 1 , b 2 ) ∈ R 2 . 2. Since the feasibility set is { x 1 ∈ R : b 1 ≤ x 1 ≤ b 2 } , this is a bounded nonempty interval for b 1 ≤ b 2 and the empty set for b 1 > b 2 . 3. It is easy to verify that the feasible set is never unbounded, for finite b 1 , b 2 ....
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 Fall '08
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 Math, Optimization, Empty set, Category theory

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