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Assessment 2. Sem2.2017 Statistical and Optimization Methods for Engineers IMPORTANT: β€’ Your submission should be entirely reproducible by using R Markdown. β€’ 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(π‘₯π‘₯) = 4π‘₯π‘₯3 βˆ’ 18π‘₯π‘₯2 + 27π‘₯π‘₯ βˆ’ 7 (5 marks) b) Determine the convexity of the following function using the second derivative test. 𝑓𝑓(π‘₯π‘₯) = π‘₯π‘₯4 + π‘₯π‘₯2 (10 marks) (Note: You are encouraged to use functions provided in R to check whether your solution is correct or not) Q2: Minimize 𝑓𝑓(π‘₯π‘₯) = βˆ’π‘₯π‘₯3 βˆ’ 2π‘₯π‘₯ + 2π‘₯π‘₯2 + 0.25π‘₯π‘₯4 (1) Apply the bisection method with initial bounds π‘₯π‘₯ = 0 and π‘₯π‘₯ = 2.4 and with an error tolerance πœ–πœ– = 0.04. Please present your search procedure in a tabular form. (10 marks) (2) Apply Newton’s method with an error tolerance πœ–πœ– = 0.001 and π‘₯π‘₯1 = 1.2. Please present your search procedures in a tabular form. (10 marks) Q3: The attached dataset (please download it separately from the blackboard) π‘₯π‘₯ was generated using the model: 𝑦𝑦 = π‘Žπ‘Ž1exp (π‘Žπ‘Ž2). You are asked to use the dataset to estimate the two parameters π‘Žπ‘Ž1 and π‘Žπ‘Ž2 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 = βˆ’18π‘₯π‘₯1 βˆ’ 15π‘₯π‘₯2+20 Subject to π‘₯π‘₯1 + π‘₯π‘₯2 ≀ 5 3π‘₯π‘₯1 + 2π‘₯π‘₯2 ≀ 12 π‘₯π‘₯1 β‰₯ 0 π‘₯π‘₯2 β‰₯ 0 (1) The graphical solution; (5 marks) (2) 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(π‘₯π‘₯, 𝑦𝑦, 𝑧𝑧) = 2π‘₯π‘₯2 + 𝑦𝑦2 + 3𝑧𝑧2 subject to 2π‘₯π‘₯ βˆ’ 3𝑦𝑦 βˆ’ 4𝑧𝑧 = 49 (10 marks) The End

Assessment 2. Sem2.2017

Statistical and Optimization Methods for Engineers

 

IMPORTANT:Β 

Β 

  • Your submission should be entirely reproducible by using R

Markdown.

  • Two documents should be submitted: one is the R Markdown file

Β Β Β Β Β Β Β  and one is the final product (which can be in html, word, or PDF)

Β Β Β Β Β Β Β  generated by using your R Markdown file.Β 

 

 

Q1:Β  a) Find the maxima and minima, if any, of the function

𝑓𝑓(π‘₯π‘₯) = 4π‘₯π‘₯3 βˆ’ 18π‘₯π‘₯2 + 27π‘₯π‘₯ βˆ’ 7Β  (5 marks)

 

  1. b) Determine the convexity of the following function using the second derivative test.

𝑓𝑓(π‘₯π‘₯) = π‘₯π‘₯4 + π‘₯π‘₯2Β Β  (10 marks)

(Note: You are encouraged to use functions provided in R to check whether your solution is correct or not)

 

Q2:Β  MinimizeΒ  𝑓𝑓(π‘₯π‘₯) = βˆ’π‘₯π‘₯3 βˆ’ 2π‘₯π‘₯ + 2π‘₯π‘₯2 + 0.25π‘₯π‘₯4

  • Apply the bisection method with initial bounds π‘₯π‘₯ = 0 and π‘₯π‘₯ = 2.4 and with an error tolerance πœ–πœ– = 0.04. Please present your search procedure in a tabular form.
    • marks)

 

  • Apply Newton’s method with an error tolerance πœ–πœ– = 0.001 and π‘₯π‘₯1 = 1.2. Please present your search procedures in a tabular form.
    • marks)

 

Q3: The attached dataset (please download it separately from the blackboard)

π‘₯π‘₯ was generated using the model: 𝑦𝑦 = π‘Žπ‘Ž1exp (π‘Žπ‘Ž2). You are asked to use the dataset to estimate the two parameters π‘Žπ‘Ž1 and π‘Žπ‘Ž2 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.Β Β  (20 marks)

 

 

Q4:Β  Solve the following linear programming problem using the two methods below:

MinimizeΒ Β Β Β Β Β Β Β  𝑓𝑓 = βˆ’18π‘₯π‘₯1 βˆ’ 15π‘₯π‘₯2+20 Subject toΒ Β Β Β Β Β Β Β  π‘₯π‘₯1 + π‘₯π‘₯2 ≀ 5

3π‘₯π‘₯1 + 2π‘₯π‘₯2 ≀ 12

π‘₯π‘₯1 β‰₯ 0

π‘₯π‘₯2 β‰₯ 0

 

  • The graphical solution; (5 marks)

 

  • The Simplex method; (10 marks)

Queensland University of Technology Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β  Assessment 2.Sem2.2017

ENN542 Statistical and Optimization Methods for Engineers

Q5:Β  Find the minimum value of the following function using Lagrange Multipliers:

𝑓𝑓(π‘₯π‘₯, 𝑦𝑦, 𝑧𝑧) = 2π‘₯π‘₯2 + 𝑦𝑦2 + 3𝑧𝑧2

subject to 2π‘₯π‘₯ βˆ’ 3𝑦𝑦 βˆ’ 4𝑧𝑧 = 49

(10 marks)

 

 

 

The End

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