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MATH 407 QUIZ August 7, 2008 Name (Please print): Problem 1 40 points
Problem: We are given the optimization problem
maximize x 1 + x 2 w.r.t x ∈ R^2 + subject to x 1 + 2x 2 ≤ 2
- Write down the two linear equations corresponding to each vertex of the feasible region cor- responding to the problem above.
- Write down the objective value corresponding to each of the vertices and which of the vertices correspond to the optimal value for the objective function.
Solution:
- The only possible linear equations for this problem are
x 1 = 0 , x 2 = 0 , and x 1 + 2x 2 = 2
Each of the possible pairs of these equations, that has a unique feasible solution, is
First vertex x 1 = 0 , x 2 = 0 Second vertex x 2 = 0 , x 1 + 2x 2 = 2 Third vertex x 1 + 2x 2 = 2 , x 1 = 0
- The vertex points and corresponding value of the objective x 1 + x 2 are Vertex Objective First Vertex (0, 0) 0 Second Vertex (2, 0) 2 Third Vertex (0, 1) 1 Since the optimial value for the objective must occur at a vertex, the point (2, 0) is optimal and it is the only vertex that corresponds to the optimal value for the objective.
MATH 407 QUIZ August 7, 2008 Name (Please print): Problem 2 40 points
Problem: We are given the optimization problem
maximize x 1 + x 2 /2 + x 3 w.r.t x ∈ R^3 + subject to 2 x 1 + x 2 + 2x 3 ≤ 2
- Write down the three linear equations corresponding to each vertex of the feasible region corresponding to the problem above.
- Write down the objective value corresponding to each of the vertices and which of the vertices correspond to the optimal value for the objective function.
Solution:
- The only possible linear equations for this problem are
x 1 = 0 , x 2 = 0 , x 3 = 0 , and 2x 1 + x 2 + 2x 3 = 2
Each of the possible triples of these equations, that has a unique feasible solution, is
First vertex x 1 = 0 , x 2 = 0 , x 3 = 0 Second vertex x 2 = 0 , x 3 = 0 , 2 x 1 + x 2 + 2x 3 = 2 Third vertex x 3 = 0 , 2 x 1 + x 2 + 2x 3 = 2 , x 1 = 0 Fourth vertex 2 x 1 + x 2 + 2x 3 = 2 , x 1 = 0 , x 2 = 0
- The vertex points and corresponding value of the objective x 1 + x 2 /2 + x 3 are Vertex Objective First Vertex (0, 0, 0) 0 Second Vertex (1, 0, 0) 1 Third Vertex (0, 2, 0) 1 Fourth Vertex (0, 0, 1) 1 Since the optimial value for the objective must occur at a vertex, the vertices (1, 0, 0), (0, 2, 0), and (0, 0, 1) are optimal and these are all the vertices that correspond to the optimal value for the objective.
