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 λ_max that will yield the largest improvement in our objective value or show that λ_max → ∞

(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).