This document discusses using ANOVA analysis to determine the most significant parameters of the simulated annealing algorithm. It analyzes the influence of parameters like cooling scheme, population size, and initial temperature on optimization problems like Griewangk, Rastrigin, normalized Schwefel, and Shekel functions. The analysis found that cooling scheme, population size, and initial temperature most influence results, with larger population sizes and initial temperatures based on initial solutions yielding better fitness. The number of changes and iterations were not found to be significant parameters.

1. Statistical analysis of the parameters of
the simulated annealing algorithm
Pedro A. Castillo Valdivieso
University of Granada

2. Introduction
 We propose using the ANOVA method to carry
out an exhaustive analysis of the Simulated
Annealing (Sim-Ann) parameters
neighbourhood
cooling scheme
initial temperature
etc.

3. Introduction
 If we know the most significant parameters, we
can control and guide our method searchs
Searching
for a cat
 Significance and relative importance of the
parameters have been obtained using ANOVA

4. State of the art
 When using search heuristics, several
parameters must first be chosen
 Obtaining suitable values for the parameters is
a time-consuming and laborious task
 Nobody knows the optimal parameter settings

5. State of the art
 Several ways of setting these parameters:
 Values given in the bibliography
 Trial and error
 Intensive experimentation
 Using meta-algorithms
 Solid tuning methods are needed:
 Setting parameter values during the run instead of
testing
 Self-adaptation of parameters (coding them in the
genome)

6. ANalysis Of the VAriance
 It is very important to know which parameter
have the greatest influence on the optimization
method
 ANOVA allows to determine whether a change
in the results is due to a change in a parameter
 It is possible to determine the variables that
have the greatest effect on the method

7. The Sim-Ann algorithm
 A cost function to be minimized is defined
 From an initial random solution, different
solutions are derived
 The better solution is kept.
 Retaining a worse solution is allowed with a
certain probability
 Our implementation uses several states at the
same time instead of an isolated state.

8. The Sim-Ann algorithm
 Parameters:
 Cooling scheme (CS): how the temperature is
reduced as te simulation proceeds
 Number of changes (NC): number of times to apply
the cooling schedule
 Population size (PS): several solutions are
evaluated
 Number of iterations (NI): how many times the
algorithm generates new neighbours
 Initial temperature (IT): either fixed or initialized
using information from the first random solution

9. Experimental setup (I)
 Using R, ANOVA statistical tool is applied
....... . ...... .......

10. Experimental setup (II)
 For each problem, obtain the fitness for each
combination of parameters
 The set of values for each parameter was
chosen taking into account those found in the
bibliography

11. Experimental setup (III)
 Four function approximation problems:
Griewangk
Rastrigin
Normalized Schwefel
Shekel

12. Griewangk and Rastrigin funct.
 ANOVA shows that changes in CS, PS and IT parameters
influence the results significantly.
 Increasing PS improves fitness.
 Exponential cooling scheme leads to good results faster.
 Initialising the temperature depending on the first random
solution yields better results.
 NC and NI are not significant; using the higher values tested
leads to better fitness

13. Norm. Schwefel and Shekel funct.
 Results are similar in both problems:
 ANOVA shows that changes in CS, PS and IT parameters
influence the results significantly.
 Cauchy cooling scheme leads to better results.
 Initialising the temperature as a fixed value yields better results.
 NC and NI are not significant; using the higher values tested
leads to better fitness

14. Conclusions (I)
 Methodology to analyse and adjust
parameters of any optimization method
 Tested using a Simulated Annealing algorithm
 Determined which parameter have the higher
influence on obtained fitness
 Obtained accurate values for those parameters

15. Conclusions (II)
 High population sizes yield to better results
(increases number of evaluations and time)
 If initial solution is not good enough, a cooling
scheme that allows accepting worse solutions
is more accurate (Cauchy). In other case,
Exponential cooling scheme is better.
 NC and NI are not reported as significant.
High values yied to better fitness.

16. Work in progress
 Applying the methodology proposed to:
- solve and analyse complex problems
- analyse modified Sim-Ann algorithms
(cooling schedules, operators)
- analyse other meta-heuristics
 Implementation of a parameter control method
[Eiben et al.]

17. Thank you!
Pedro A. Castillo Valdivieso
pedro@geneura.ugr.es