AM41AC: Give the description of the Simulated Annealing function optimization method for minimizing continuous functions: Algorithmic and Computational Mathematics Assignment, UoL, UK

Coursework 1

The coursework aims at assessing understanding of the material taught at the lectures and practical skills training at the tutorials. The tasks are specific numerical problems that should be solved by providing necessary explanations. The specific requirements on what is expected in the report describing the solutions are given in the accompanied “Assignment Brief” document.

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

• Give the description of the Simulated Annealing function optimization method for minimizing continuous functions. Include its concept, mathematical formulation, and algorithmic implementation.
• 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?
• Minimize the function for 𝑛 = 2 using your implementation of the method. Analyze the results for various values of the parameters of the method.
• Quantitatively estimate the performance of the method. Explain the differences in performance depending on the parameters of the method.
• Plot the optimization trajectory, and analyze its dependence on the parameter of the method.

Coursework 2

The coursework aims at assessing understanding of the material taught at the lectures and practical skills training at the tutorials. The tasks are specific numerical problems that should be solved by providing necessary explanations. The specific requirements on what is expected in the report describing the solutions are given in the accompanied “Assignment Brief” document.

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

• Construct a new function as the intersection of 𝑓(𝑥𝑖) with the plane 𝑥0 = 𝑥1. Provide a mathematical expression for this function and plot it. Analyze the function of its roots, extreme points, and its behavior at the limits.
• Find the root(s) of the function using a numerical method of your choice. Compare to the analytical result.
• Integrate the function using a numerical method of your choice and analytically. Compare the results.
• How will the function change if constructed as the intersection with the plane containing a different line on the 𝑥0 − 𝑥1 plane?