Consider the nonlinear program min 1x1 - 822 + 21x2 - 422 s.t. 2x1 + 8x2 16 x1 7 x1, x2 0

Consider the nonlinear program min 1x1 - 822 + 21x2 - 422 s.t. 2x1 + 8x2 … 16 x1 … 7 x1, x2 Ú 0

(a) Introduce slack variables x3 and x4 to place the model in standard form for reduced gradient Algorithm 17C.

(b) Show that x2 and x4 form a basic set of variables for your standard form.

(c) Assuming the basis of part (b), classify variables as basic, nonbasic, or superbasic at initial solution x102 = 10, 1, 8, 72.

(d) Compute the reduced gradient corresponding to the basis and x102 of part (c).

(e) Construct the move direction that would be pursued by Algorithm 17C at the basis and x102 of part (c).

(f) Compute the maximum feasible step l in the direction of part (e). Then, assuming (correctly) that the direction of part (e)
remains improving all the way to the maximum l, compute the resulting new solution x112.
(g) Explain why a basis change would by required by Algorithm 17C at the x112 of part (f), and choose an appropriate new basis.