# Question: Consider the following problem Maximize Z 4x1 x12

Consider the following problem:

Maximize Z = 4x1 – x12 + 10x2 – x22,

subject to

x12 + 4x22 ≤ 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.

(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.)

Maximize Z = 4x1 – x12 + 10x2 – x22,

subject to

x12 + 4x22 ≤ 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.

(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.)

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