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Modern optimization methods 1
Comparison of optimization algorithms
Modern optimization methods 2
Overview of single-objective algorithms
• With one solution in given time:– Gradient methods– Hill-climbing– Simulated annealing– TABU search– (1+1)-ES
• Advantages: small number of evaluations, fast convergence
Modern optimization methods 3
• With a set of solutions:– Binary Genetic Algorithms– Evolution Strategies– Differential Evolution– SADE/GRADE + CERAF
• Advantage: Robustness
Overview of single-objective algorithms
Modern optimization methods 4
Example of comparison
Modern optimization methods 5
Benchmarks• 20 mathematical functions from
1 to 20 variables• In Matlab and C++
F1
http://klobouk.fsv.cvut.cz/~anicka/testfunc/testfunc.html
Quartic
PShubert1Goldprice
Branin
F3
Modern optimization methods 6
One run comparison:“Progress plots”
Modern optimization methods 7
One hundred runs comparison
Modern optimization methods 8
One hundred runs comparison
Modern optimization methods 9
Traditional measures
Mean best fitness
Success rate
Modern optimization methods 10
Comparison on reliabilityFunction Dim Fmincon GRADE GRADE+CERAFF1 1 100 100 100F3 1 100 100 100Branin 2 100 100 100Camelback 2 100 100 100Goldprice 2 100 100 100PShubert1 2 100 100 100PShubert2 2 100 100 100Quartic 2 100 100 100Shubert 2 100 100 100Hartman1 3 100 100 100Shekel1 4 100 100 100Shekel2 4 100 100 100Shekel3 4 100 100 100Hartman2 6 100 59 100Hosc45 10 100 100 100Brown1 20 100 100 100Brown3 20 100 100 100F5n 20 100 100 100F10n 20 0 78 100F15n 20 1 100 100
Modern optimization methods 11
Comparison on convergence speedFunction Dim Fmincon GRADE GRADE+CERAFF1 1 27 55 55F3 1 57 95 95Branin 2 24 348 348Camelback 2 40 198 198Goldprice 2 63 337 337PShubert1 2 2097 3879 1402PShubert2 2 1615 2333 896Quartic 2 56 320 331Shubert 2 375 606 603Hartman1 3 63 284 292Shekel1 4 335 47577 4078Shekel2 4 255 15356 2686Shekel3 4 284 7310 2496Hartman2 6 200 123727 9881Hosc45 10 264 2147 2096Brown1 20 286979 176628 182390Brown3 20 5660 36568 36090F5n 20 15838 6734 7284F10n 20 ------ 89715 226374F15n 20 374110 22378 25528
Modern optimization methods 12
Summary
• Disadvantages of traditional measures:– Unpractical setting of functions calls limit for
MBF type of measures– Need of optimum value knowledge for SR based
measures
• Result:– Whole progress plot is of importance
• Disadvantage:– Too much data need to be stored
Modern optimization methods 13
Proposed solution for two methods
• Store only 10 Dim results (like generations)• Use statistical test to judge the result of
comparison (Mann-Whitney-Wilcoxon test)
Modern optimization methods 14
Proposed solution cont.
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Proposed solution for more methods
• How to graphically compare more than two methods?– From multi-objective domain: Pair-vise comparison table
Modern optimization methods 16
Proposed solution for more methods
• How to graphically compare more than two methods?– From single-objective domain: Partial ordering?
[Carrano et. al.: GECCO’08]
Modern optimization methods 17
Proposed solution for more methods
• Relative Winning Score
scoresofwinnerisicasesNo
RWSi
.
11,0 ii RWSRWS
Modern optimization methods 18
Modern optimization methods 19
Modern optimization methods 20
Modern optimization methods 21
Traditional sizing problems
Modern optimization methods 22
Proposed solution for more methods
Modern optimization methods 23
References on Traditional measures
• A.E. Eiben, J.E. Smith: Introduction to Evolutionary Computing, Springer (2008).
• A.E. Eiben, M. Jelasity: A critical note on experimental research methodology in EC, Proceedings of the 2002 Congress on Evolutionary Computation, 2002. CEC '02.
• Thomas Bartz-Beielstein: Experimental Research in Evolutionary Computation - The New Experimentalism. Springer, Berlin, 2006.
• Thomas Bartz-Beielstein web-page: http://ls11-www.informatik.uni-
dortmund.de/people/tom/
Modern optimization methods 24
A humble plea. Please feel free to e-mail any suggestions, errors andtypos to [email protected].
Date of the last version: 23.11.2011Version: 001