3 This problem will give you an idea why the restricted entry rule is unnecessary when (for a maximization problem)

each fj(xj) is concave and each gij(xj) is convex. Consider the Oilco example. When we solve the approximating problem by the simplex, show that a solution that violates the adjacency assumption cannot be obtained. For example, why can the simplex not yield a solution (x*) of 11  0.4 and 15  0.6? To show that this cannot occur, find a feasible solution to the approximating problem that has a larger

ˆz-value than x*. [Hint: Show that the solution that is identical to x* with the exception that 11  0, 15  0, 13  0.6, and 14  0.4 is feasible for the approximating problem

[use the convexity of gij(xj) for this part] and has a larger

ˆz-value than x* [use concavity of fj(xj) for this part].]