Reference no: EM133715628

Question 1 Consider the system of linear equations

x - y = 2,

kx - y = k.

a. Reduce the augmented matrix for this system to row-echelon (or upper-triangular) form. (You do not need to make the leading nonzero entries 1.)
b. Find the values of k (if any) when the system has
(i) No solution.
(ii) Exactly one solution (if this is possible, find the solution in terms of k).
(iii) Infinitely many solutions (if this is possible, find the solutions).

Question 2

Consider the following constrained optimization problem. We will solve it using analytical methods, explicitly employing the Karush-Kuhn-Tucker(KKT) approach.

minimize f(x₁, x₂) = x₁ + 2x₂² - x₁x₂ - x₁ - x₂,

subject to

2x₁ + x₂ ≤ 1.

(i) State the KKT conditions for this problem. Use KKT conditions:  ∇f (x) + λ∇g(x) = 0, λg(x) = 0 and λ ≥ 0.
(ii) Determine KKT points, if they exist.

Question 3

Consider the problem introduced in Question 2.

minimize f(x₁, x₂) = x₁ + 2x₂ - x₁x₂ - x₁ - x₂,

subject to

2x₁ + x₂ ≤ 1.

(i) Express the Lagrangian function of the problem in the form

L(x. λ) = f(x) + λg(x), where λ = 3/11.

(ii) Identify the critical points by setting the gradient equal to zero.
(iii) Apply the second derivative test using the Hessian of L(x, λ) to classify the critical points.
(iv) Compare the optimal solution obtained in Question 2 and comment on the comparison.

Question 4

We will now apply a numerical technique to solve the problem

minimize f (x₁, x₂) = x₁² +2x₂² -x₁x² -x₁ - x₂,
subject to

2X₁ + X₂ ≤ 1.

Specifically, we will utilize the gradient descent method to optimize the function

L(x, λ) = x₁² + 2x₂² - x₁x₂ - x₁ - x₂ + 3/11(2x₁ + x₂ - 1).

Perform two iterations of the gradient descent algorithm, where the parameters of the model are updated iteratively in a defined direction with starting at x₀ = (0.2636,0.1727) and t = 0.1. This iterative process helps refine the parameters and aims to converge towards an optimal solution.

Hints: update the point using gradient descent steps x_k+1 = x_k - t∇f (x_k).

Question 5 A logistics company is tasked with distributing goods from three warehouses to four retail stores. Each warehouse has a specific supply of goods, and each store has a specific demand. The company aims to minimize the total transportation cost. Below is the cost matrix, representing the cost of transporting one unit of goods from each warehouse to each store. Determine minimum transportation cost using Vogel's approximation method.

Table 1: Cost table.

[You are not required to write/convert to the mathematical model. Follow the VAM steps to get the solution. Here are the shorter forms of the steps.

Steps: identify the smallest and next-to-smallest costs and find the difference between them for each row and column. Identify the largest difference among all rows and columns and allocate the maximum feasible amount.]

Question 6
Consider the problem of asigning four jobs to four persons. The assignment costs are given as follows. Determine the optimum assignment schedule and costs using the Hungarian Method.

Table 2: Cost table.

Note: You are not required to write/convert to the mathematical model. Follow the steps of the Hungarian method and find the optimal solution.