Assignment

1. Given f(x,y) = x³ - 3xy² + y⁴, identify the stationary points of f. Determine whether they are local/global max, min or saddle points.

2. Use Newton's method to solve the following problem:

minimize f(x) = e^x - 2x

Use an error tolerance ε = 0.001 and start with initial solution x₀ = 1. (two iterations are enough)

3. Use the bisection method to solve following with an error tolerance ε = 0.001

minimize f(x) = x³ - 3x² + 2x -1

s.t. 0 ≤ x ≤ 3

(two iterations are enough)

4. Find the optimal solution to the following problem:

minimize f(x) = x³ - 3x² + 2x -1

s.t. 2 ≤ x ≤ 4

5. Consider the following unconstrained optimization problem:

maximize  f(x₁, x₂)= -(x₁ -2)² - x₁ - x₂²

Starting from the initial trial solution (x₁, x₂) = (2.5, 1.5), apply the gradient search procedure and perform two iterations to obtain an approximate solution. (Do not solve it by setting the first order partial derivatives to zero.)

6. Use the KKT conditions to derive an optimal solution for the following problem:

max f (x₁, x₂ ) = 8x₁ - x₁ +2x₂ + x₃²

s.t.

x₁ + 3x₂ + 2x3 ≤ 12
x₁, x_2, x3 ≥ 0

7. Consider the following quadratic programming problem:

max f (x₁ , x₂ ) = 8x₁ - x₁ +4x₂- x₂²

s.t.

x₁ + x₂ ≤ 2

x₁, x₂ ≥ 0

a. Write down the KKT conditions and explain why KKT conditions are necessary and sufficient.

b. Apply the modified simplex method to find the optimal solution.