In Section 4 of Chapter 1, we formulated the following nonlinear-programming version of the custom older example:

Do not solve the separable program. 10. In the network of Fig. we wish to ship 10 units from node 1 to node 4 as cheaply as possible. The flow on each arc is incapacitated. The costs on arcs 1–2, 2–3, and 3–4 are linear, with costs per unit of 8, 2, and 4, respectively; the costs on arcs 1–3 and 2–4 are quadratic and are given by x 2 13 and x 2 24.

a) Suppose that we apply the separable-programming δ-technique discussed in Chapter 9, using the breakpoints 

Interpret the resulting linear-programming approximation as a network-flow problem with parallel arcs joining nodes 1 and 3 and nodes 2 and 4, as in Fig. Specify the per-unit cost and the arc capacity for each arc in the linear approximation.

b) Solve the separable-programming approximation from part (a), comparing the solution with the optimal solution 

and minimum cost = 110.25, to the original nonlinear problem formulation.