Jim Matthews, Vice President for Marketing of the J.R. Nickel Company, is planning advertising campaigns for two unrelated products. These two campaigns need to use some of the same resources. Therefore, Jim knows that his decisions on the levels of the two campaigns need to be made jointly after considering these resource constraints. In particular, letting x1 and x2 denote the levels of campaigns 1 and 2, respectively, these constraints are 4x1 + x2 <= 20 and x1 + 4x2 <= 20.
In facing these decisions, Jim is well aware that there is a point of diminishing returns when raising the level of an advertising campaign too far. At that point,
the cost of additional advertising becomes larger than the increase in net revenue (excluding advertising costs) generated by the advertising. After careful analysis, he and his staff estimate that the net profit from the first product (including advertising costs) when conducting the first campaign at level x1 would be 3x1 - (x1 - 1)2 in millions of dollars. The corresponding estimate for the second product is 3x2 -(x2 - 2)^2. This analysis led to the following quadratic programming model for determining the levels of the two advertising campaigns:
max f(x1; x2) = 3x1 - (x1 - 1)2 + 3x2 - (x2 - 2)^2,
s.t. 4x1 + x2 <= 20
x1 + 4x2 <= 20
x1 >= 0; x2 >= 0:
(a) Obtain the KKT conditions for this problem.
(b) You are given the information that the optimal solution does not lie on the boundary of the feasible region. Use this information to derive the optimal solution from the KKT conditions.