1. Hirophysics.com The Genetic Algorithm vs. Simulated Annealing Charles Barnes PHY 327

2. Hirophysics.com The genetic algorithm and simulated annealing processes are used for determining the global minimum and maximum of a selected range for any given function. Finding the min/max of a function Global minimum The function E(x) is equivalent to the internal energy of the system.

3. Hirophysics.com The genetic algorithm is a computer-performed optimization method that mimics the process of natural evolution. An initial population is generated [A(0):=(A 1 (0), A 2 (0), … A n (0)] Each individual in a population is assigned a fitness function. There are three types of operators for genetic algorithm: 1.Reproduction (Selection) 2.Crossover 3.Mutation Genetic Algorithm

4. Hirophysics.com Reproduction

5. Hirophysics.com The crossover operator randomly recombines pairs through mating. Crossover Parents P 1 P 2 1001 | 0111 | 0111 1110 | 0100 | 1111 Child 1110 | 0111 | 1111 Parents P 1,2 P 3 1110 | 0111 | 1111 1010 | 1101 | 0001 Child 1010 | 0111 | 0001 etc. This is a genetic operator and evolutionary algorithm known as crossover.

6. Hirophysics.com The mutation operator is a sudden change of chromosome. Mutation 001100101001011101 001100101101011101 001100101101001101 A number is randomly switched in the code.

7. Hirophysics.com There are three main parameters that can be changed in the function of the genetic algorithm: M: the population of a specific farm N: the length of an individual’s binary string k: the temperature interval In addition to these, the function can also itself be changed. The research done on the genetic algorithm was to find which parameters, if any, influenced the production of accurate results. Genetic Algorithm - Parameters

8. Hirophysics.com Functions Used in Genetic Algorithm

9. Hirophysics.com The genetic algorithm proved quite accurate on each experiment, producing exceptional results entirely independent of the function. When the variables M, N, and k were changed, little to no effect on the global min/max was observed. The genetic algorithm had no problem finding the minimum and maximum on any type of function. Results – Genetic Algorithm

10. Hirophysics.com Simulated Annealing

11. Hirophysics.com The Simulated Annealing Process

12. Hirophysics.com Finding the global maximum of a function typically produced a graph as follows: Results – Finding the Global Max The graph displays how the solutions converge.

13. Hirophysics.com Finding the global minimum of a function typically produced a graph as follows: Results – Finding the Global Min The graph displays how the solutions converge.

14. Hirophysics.com The algorithm used for finding the maximum function was generally more accurate than the minimum algorithm, producing similar graphs with similar maximums each time. e.g. Results – Changing Parameters It is interesting to note that the more calculations done by the algorithm, the faster the convergence is to the solution. 100 Calculations10,000 Calculations

15. Hirophysics.com Results – Accuracy Max 1 = Min 2 Results of the Algorithms: Graph 1 –(x-2) 2 +3 Graph 2 (x-2) 2 +3 Max 1 =3 at x=2Min 2 =5.04 at x=0.57 

16. Hirophysics.com Accurate results for both the maximum and minimum simulated annealing algorithms were dependent on functions, as more complicated functions (typically those with powers or exponentials) had trouble producing accurate results In general, the more iterations the algorithm would undergo, the more accurate the final data would be for simpler functions. Finding the maximum via simulated annealing is much more accurate than finding the minimum, though not usually typically accurate. Findings – Simulated Annealing

17. Hirophysics.com As observed, the genetic algorithm seems to be the most accurate method of the two to find both the maximum and minimum of any function. The simulated annealing process seems to have trouble finding the maximum and minimum of more complicated functions. Conclusion

18. Hirophysics.com To understand why the simulated annealing algorithm was not accurate, especially at finding the global minimum of a function Future Research