homework-1 - Is it an upper bound or a lower bound on the...

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MthSc810 – Mathematical Programming Fall 2011, Homework #1. Due August 30, 2011. 1 Relaxations and restrictions Provide two relaxations for each of the following problems: min 2 x + 3 y min x + y + z s.t. 0 x 2 s.t. x - y + z = 0 1 y 4 x + 2 y + 3 z 4 x, y, z 0 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? What is the corresponding value of the objective function? Is it a lower or an upper bound on the optimum? Write the problem obtained by relaxing the ±rst constraint, x + y 3. What is the optimal solution of this relaxation (i.e., what are the optimal values of x and y ), and what is its objective function value? Note: no need to use the simplex method, this relaxation is trivially solved.
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Unformatted text preview: Is it an upper bound or a lower bound on the optimal value of the original problem? The optimal solution of this problem is ( x, y ) = (2 , 1). Are the lower and upper bound you got really a lower and an upper bound? If so, you did this exercise correctly. 3 Min-max problems Consider the following nonlinear optimization problem: P : max { f ( x, y ) : ( x, y ) F } where f ( x, y ) = min { x + 4 y-2 , 5 x + 2 y-1 } and F = { ( x, y ) R 2 :-3 x 1 , 1 y 7 } . Formulate a linear programming problem equivalent to P ....
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