homework-1-solutions

# homework-1-solutions - MthSc810 – Mathematical...

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Unformatted text preview: MthSc810 – Mathematical Programming Fall 2011, Solutions of Homework #1. 1 Relaxations and restrictions Provide two relaxations for each of the following problems: min 2 x + 3 y min x + y + z s.t. ≤ x ≤ 2 s.t. x- y + z = 0 1 ≤ y ≤ 4 x + 2 y + 3 z ≥ 4 x, y, z ≥ Solution. min 2 x + 2 y min x + y s.t. ≤ x ≤ 5 s.t. x- y + z ≤ ≤ y ≤ 4 x + 2 y + 3 z ≥ x, y, z ≥ min 2 x + 3 y min x + y + z s.t. ≤ x ≤ 2 s.t. x- y + z = 0 y ≥ x + 3 y + 3 z ≥ 4 x, y, z ≥ 2 Lower and upper bounds Given the following problem: min 2 x + 3 y s.t. x + y ≥ 3 1 ≤ x ≤ 4 1 ≤ y ≤ 3 – Is the solution ( x, y ) = (4 , 3) feasible? A.: Yes, because it satisfies the constraint and the variable bounds. – What is the corresponding value of the objective function? A.: 2 · 4 + 3 · 3 = 17. – Is it a lower or an upper bound on the optimum? A.: A feasible solution always gives an upper bound on the global optimum....
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homework-1-solutions - MthSc810 – Mathematical...

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