For each of the following cases, prove that the key property of separable programming given in Sec. 13.8 must hold. (Hint:

Assume that there exists an optimal solution that violates this property, and then contradict this assumption by showing that there exists a better feasible solution.)

(a) The special case of separable programming where all the gi(x)

are linear functions.

(b) The general case of separable programming where all the functions are nonlinear functions of the designated form. [Hint:

Think of the functional constraints as constraints on resources, where gij(xj) represents the amount of resource i used by running activity j at level xj, and then use what the convexity assumption implies about the slopes of the approximating piecewise linear function.]