Question 1:
For each of the following objective functions and current solutions, determine whether the given directions are improving or not.
(i) maximize 3x1 + 4x2 - 6x3 at the point x = (2, 3, 1).
(ii) minimize 5x1 - 2x2 + 3x3 at the point x = (-1, 2, 1).
(a) d = (1, 2, 3).
(b) d = (1, 0, -2).
(c) d = (0, -1, -1).
Question 2:
For each of the following objective functions and current solutions x, use the gradient (or negative gradient for minimization problems) as our search direction to determine the (positive) step size λmax that will yield the largest improvement in our objective value or show that λmax → ∞
(a) min 3x2 - 2xy + y2 - 10; x = (1, 3).
(b) min x2 - 3xy - 2y2; x = (2, -1).
(c) max x2 + 12xy - y2; x = (-1, 1).
(d) max x2 - 8xy - 10xz - yz - y2 - 3z2; x = (1, 1, 1).
Question 3:
Consider a mathematical program with constraints
x1 + 3x2 + 2x3 ≤ 15
2x1 - x2 + x3 ≥ 5
x1,x3 ≥ 0.
Determine the maximum step size (possibly +∞) that preserves feasibility in the direction indicated from the solution specified. Also, indicate whether that step indicates that the model is unbounded, assuming that directions improve everywhere.
(a) d = (1, 2, 1) from x = (2, 2, 3).
(b) d = (4, 1, 2) from x = (10, 0, 2).
(c) d = (-2, 1, 1) from x = (2, 1, 5).
(d) d = (3, 4, 0) from x = (5, 1, 1).
Question 4:
Determine which of the constraints
[i] 3y1 - 2y2 ≤ 9
[ii] 2y12 - Y1Y2 ≥ 5
[iii] Y1 + Y2 = 3
[iv] Y1 ≥ 0
[v] Y2 ≥ 0
are active at each of the following solutions.
(a) y = (3, 0).
(b) y = (2, 1).
Question 5:
Consider the following linear program
max 6x1 + 5x2
s.t.
5x1 + 2x2 ≤ 34
x1 - x2 ≥ -3
x1, x2 ≥ O.
(a) Show that the directions d(1) =7 (3, 1) and d(2) = (-2, 5) are both improving directions at every feasible solution.
(b) Beginning at x(0) = (0, 0), execute Algorithm 6.2 using only these two directions. Continue until neither direction is both improving and feasible.
(c) Show in a two-dimensional plot the feasible region of this problem. Then plot the path of your search in part (b).