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# Give the description of Simulated Annealing function optimisation method for minimising continuous functions.

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For the function 𝑓(𝑥𝑖) = ∑𝑛 −𝑥𝑖 sin √𝑥𝑖 defined in the region 𝑥𝑖 < 1000, 𝑖 = 1. . 𝑛 :

1. Give the description of Simulated Annealing function optimisation method for minimising continuous functions. Include its concept, mathematical formulation, and algorithmic implementation. (4 marks)

2. In what aspect(s) the method can be varied? How do variations influence its performance? What are the advantages and deficiencies of the variants of the method? (4 marks)

3. Minimise the function for 𝑛 = 2 using your implementation of the method.  Analyse the results for various values of the parameters of the method. (4 marks)

4. Quantitatively estimate the performance of the method. Explain the differences in performance depending on the parameters of the method. (4 marks)

5. Plot the optimisation trajectory, analyse its dependence on the parameter of the method. (4 marks)

Total: 20 marks.

For the function defined in Coursework 1 and 𝑛 = 2 :

1. Construct a new function as the intersection of 𝑓(𝑥𝑖) with the plane 𝑥0 = 𝑥1. Provide mathematical expression for this function and plot it. Analyse the function for its roots, extreme points and its behaviour at the limits (5 marks)

2. Find the root(s) of the function using a numerical method of your choice. Compare to the analytical result. (5 marks)

3. Integrate the function using a numerical method of your choice and analytically. Compare the results. (5 marks)

4. How will the function change if constructed as the intersection with the plain containing a different line on the 𝑥0 − 𝑥1 plane? (5 marks)

Total: 20 marks.

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