Assessment 2. Sem2.2017
Statistical and Optimization Methods for Engineers
- Your submission should be entirely reproducible by using R
- Two documents should be submitted: one is the R Markdown file
and one is the final product (which can be in html, word, or PDF)
generated by using your R Markdown file.
Q1: a) Find the maxima and minima, if any, of the function
𝑓𝑓(𝑥𝑥) = 4𝑥𝑥3 − 18𝑥𝑥2 + 27𝑥𝑥 − 7 (5 marks)
- b) Determine the convexity of the following function using the second derivative test.
𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 + 𝑥𝑥2 (10 marks)
(Note: You are encouraged to use functions provided in R to check whether your solution is correct or not)
Q2: Minimize 𝑓𝑓(𝑥𝑥) = −𝑥𝑥3 − 2𝑥𝑥 + 2𝑥𝑥2 + 0.25𝑥𝑥4
- Apply the bisection method with initial bounds 𝑥𝑥 = 0 and 𝑥𝑥 = 2.4 and with an error tolerance 𝜖𝜖 = 0.04. Please present your search procedure in a tabular form.
- Apply Newton’s method with an error tolerance 𝜖𝜖 = 0.001 and 𝑥𝑥1 = 1.2. Please present your search procedures in a tabular form.
Q3: The attached dataset (please download it separately from the blackboard)
𝑥𝑥 was generated using the model: 𝑦𝑦 = 𝑎𝑎1exp (𝑎𝑎2). You are asked to use the dataset to estimate the two parameters 𝑎𝑎1 and 𝑎𝑎2 in the model, by treating it as a leastsquares problem. Hint: please do not attempt to solve it manually; you need to use optimization functions available in RStudio. (20 marks)
Q4: Solve the following linear programming problem using the two methods below:
Minimize 𝑓𝑓 = −18𝑥𝑥1 − 15𝑥𝑥2+20 Subject to 𝑥𝑥1 + 𝑥𝑥2 ≤ 5
3𝑥𝑥1 + 2𝑥𝑥2 ≤ 12
𝑥𝑥1 ≥ 0
𝑥𝑥2 ≥ 0
- The graphical solution; (5 marks)
- The Simplex method; (10 marks)
Queensland University of Technology Assessment 2.Sem2.2017
ENN542 Statistical and Optimization Methods for Engineers
Q5: Find the minimum value of the following function using Lagrange Multipliers:
𝑓𝑓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) = 2𝑥𝑥2 + 𝑦𝑦2 + 3𝑧𝑧2
subject to 2𝑥𝑥 − 3𝑦𝑦 − 4𝑧𝑧 = 49