Consider the following problem:

Maximize Z 4x1  x1 2 10x2  x2 2

, subject to x1 2 4x2 2 16 and x1  0, x2  0.

(a) Is this a convex programming problem? Answer yes or no, and then justify your answer.

(b) Can the modified simplex method be used to solve this problem? Answer yes or no, and then justify your answer (but do not actually solve.)

(c) Can the Frank-Wolfe algorithm be used to solve this problem?

Answer yes or no, and then justify your answer (but do not actually solve).

(d) What are the KKT conditions for this problem? Use these conditions to determine whether (x1, x2) (1, 1) can be optimal.

(e) Use the separable programming technique to formulate an approximate linear programming model for this problem. Use the feasible integers as the breakpoints for each piecewise linear function.

C

(f) Use the simplex method to solve the problem as formulated in part (e).

(g) Give the function P(x; r) to be maximized at each iteration when applying SUMT to this problem. (Do not actually solve.)

D,C (h) Use SUMT (the automatic routine in your OR Courseware) to solve the problem as formulated in part (g). Begin with the initial trial solution (x1, x2) (2, 1) and use r 1, 102

, 104

, 106

.