A young R & D engineer at Carron Chemical Company has synthesized a sensational new fertilizer made of just two interchangeable basic raw materials. The company wants to take advantage of this opportunity and produce as much as possible of the new fertilizer. The company currently has $40, 000 to buy raw materials at a unit price of $8000 and $5000 per unit, respectively. When amounts x₁ and x₂ of the basic raw materials are combined, a quantity q of fertilizer results given by:

a) Formulate as a nonlinear program.

b) Apply four iterations of the Frank-Wolfe algorithm; graphically identify the optimal point, using the property that even- and odd-numbered points lie on lines directly toward the optimum. Start from x ⁰ = (0, 0) and use three decimal places in your computations.

c) Solve the problem using the algorithm for quadratic programming discussed in Section 13.7.