12. Lecture
Stochastic Optimization
Differential Evolution
Soft Control
(AT 3, RMA)
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12. Structure of the lecture
1. Soft control: the definition and limitations, basics of “expert"
systems
2. Knowledge representation and knowledge processing (Symbolic AI)
application: expert systems
3. Fuzzy Systems: Dealing with Fuzzy knowledge application: Fuzzy
Control
4. Connective systems: neural networks application: Identification and
neural controller
5. Genetic Algorithms: Stochastic Optimization
Genetic Algorithms
Simulated Annealing
Differential Evolution
Application: Optimization
6. Summary and Literarture reference
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• Differential Evolution (DE), as well as genetic algorithms, belong to the
population-based optimization methods
• DE has no natural model
• DE was founded and presented in 1996 by PricewaterhouseCoopers
and Storn
R. Storn, R. and K. Price, K. Differential Evolution - A Simple and Efficient
Heuristic for Global Optimization over Continuous Spaces, Journal of
Global Optimization, 11, (1997) pp. 341–359.
• Procedures can be applied directly on minimum and maximum applied
problems (see GA only Maximum-Problems)
• Scope
Optimization in multi search areas with floating
e.g. Controller design
Differential Evolution: Introduction
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• DE is used to search for a optimum in a multi-dimensional continuous
search space
A solution (x, optimum potential) is represented by a vector with the
dimension (D) of the search description
The elements of the vector are floating point numbers:
• The search comes with several solutions (vectors, individuals)
simultaneously searches (population-based)
The quantity of solutions called population (p), with N individuals
• The kindness of a solution is a function described
: The goodness of a solution is a function described
Differential Evolution: Basic idea
Dx
x
x
x2
1
ix
DiN xxxxp ,,,, 21
Dxf :)(
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• Initialising
create Initial Population (such as random solutions)
• Mutation
produce a new random solution by modifying an existing solution of the old
generation
• Recombination
Combine two solutions to a new solution
• Selection
Solution for identifying new generation
Differential Evolution: Basic algorithm 1/2
Initialisingg Mutation Recombination Selection
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Differential Evolution: Basic algorithm 2/2
4 Vectors of old
Generation
Mutation
Recombination
1 Donator-Vector (v)
Selection
3 Vectors (randomly chosen, xr1,xr2,xr3)
1 Vektor (x)
1 Test vector (u)
New Generation
New Vector (x+)
Each vector of
the old
generation is
exactly once this
vector
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• Each vector X of the old generation provides additional three vectors from
the old generation(xr1,xr2,xr3), that holds: x≠xr1≠xr2≠xr3
• Give the donor vector (v) as a linear combination of xr1,xr2,xr3
• Colorful interpretation
Create a new solution based on xr1 from the difference of xr2 and xr3
Enhances heterogeneity of the solutions
• v x, and together are the parents pair for recombination
Differential Evolution: Mutation
xr1 xr2
xr3
xr2-xr3
F*(xr2-xr3)
v
2,0),(* 321 FxxFxv rrr
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• Create a test vector (u) by mixing the elements of x and v
• The mixture of the element of x and v is randomly controlled
x,v,u sind Vectors of Dimension D
CR is the Cross-Over Rate:
y is a random number:
ri is a real random number:
• x and u are competitors in the selection
Differential Evolution: Recombination
DDD u
u
u
v
v
v
x
x
x
2
1
2
1
2
1
,,
1,0CR
Dj ,1
1,0ir
sonst,
oderfalls,
i
ii
ix
j iCR rvu
j sorgt dafür, dass
sich x und u in
mindestens einem
Element
unterscheiden
CR ist ein Parameter des
Optimierungsverfahrens
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• Choose one of the two vectors x, u for the new generation
• Selections are made solely on the basis of goodness (fitness) of an
individual (Vector)
Only the better of the two individuals is included in the new generation over
No dependence of random variables in the selection
f: to optimize Goodness function (fitness function)
By the same goodness through mutation and recombination results individual in the
new generation
Enhances heterogeneity across generations
• Selection in DE has implicit elitism
Only better or equally good individuals form the new generation
Differential Evolution: Selection
sonst ,
falls ,
x
f(x)f(u)ux
sonst ,
falls ,
x
f(x)f(u)ux
Minimization Maximization
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• Ackleys Function
2-dimensonale continuous function with several local minima and a global
minimum for (0.0)
Optimization problem: Minimize f (x1, x2)
Differential Evolution: Application example
))**2cos()**2*(cos(5.0)*(5.0*2,0
2121
22
21*2020),(
xxxxeeexxf
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• Parameter for Optimization
20 Individuals
CR: 50%
F: 0,8
• Initial population
Differential Evolution: Application example (Initializing)
Minimum: 4,355
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Differential Evolution: Application (1 new generation)
Minimum: 4,355
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Differential Evolution: Application (2nd new generation)
Minimum: 4,355
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Differential Evolution: Application (3rd new generation)
Minimum: 3,866
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Differential Evolution: Application (4. new generation)
Minimum: 1,664
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Differential Evolution: Application (5. new generation)
Minimum: 1,664
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Differential Evolution: Application (15. new generation)
Minimum: 0,348
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Differential Evolution: Application (50. new generation)
Minimum: 0,001
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Differential Evolution: Application (50. new generation)
Minimum: 0,001
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Summary and learning from the 12th Lecture
• Genetic Algorithms and Genetic Programming
Optimization through mutation and selection on the model of evolution in
biological systems
Parallel browsing for the search areas
Well suited for new computer structures with multi-core processors
When floats cost high for encoding the solution
• Simulated Annealing
Optimization methods inspired by the emergence of lattice structures in crystals
Only one solution is to use scanning
No speed advantage through multi-core processors
Feature: temporary deterioration is understood as an improvement
• Differential Evolution
Artificial population-based optimization methods
Well suited for new computer structures with multi-core processors
Procedures for the optimization of floating point numbers
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Literature (additional / continuing) 1/2
Chapter 1 or entire lecture: General information on methods of AI Götz, Güntzer (Hrsg.): Handbuch der künstlichen Intelligenz. Oldenbourg Verlag, 2000.
"Umfassendes Nachschlagewerk für Interessierte.„
King R.E.: Computational Intelligence in Control Engineering. Marcel Dekker, 1999
"Sehr schöne Übersicht zu Soft-Control.„
Chapter 2: Expert Systems Polke, M.: Prozeßleittechnik. Oldenbourg Verlag, 1994.
"Einige Ideen für die Anwendung in der Leittechnik in Kapitel 13.„
Ahrens, W.; Scheurlen, H.-J.; Spohr, G.-U.: Informationsorientierte Leittechnik. Oldenbourg Verlag,
1997.
"Einführung in XPS für leittechnische Aufgaben (und etwas Fuzzy) in Kapitel 9.„
Lunze, J.: Künstliche Intelligenz für Ingenieure I und II. Oldenbourg Verlag, 1994/1995.
"Sehr Ausführliche Behandlung von XPS.„
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Literature (additional / continuing) 2/2
Chapter 3: Fuzzy Kiendl, H.: Fuzzy Control methodenorientiert. Oldenbourg Verlag, 1997.
"Ausführliche Darstellung mit kurzer Einführung in die Regelungstechnik und sehr sehr
ausführlichem Beispiel.„
Chapter 4: Neuro Zakharian, S.; Ladewiw-Riebler, P.; Thoer, S.: Neuronale Netze für Ingenieure. Vieweg Verlag,
1998.
"Kompakte und gut verständliche Darstellung mir Anwendungen in der Regelungstechnik."
Chapter 5: Genetic Algorithms Goley, D.A.: An Introduction to Genetic Algorithms for Scientists and Engineers. World Scientific
Publishing, 1999.
"Sehr ausfürliche Darstellung."
Fleming, P.J.; Purshouse, R.C.: Genetic algorithms in control systems engineering. IFAC
PROFESSIONAL BRIEF.
"Sehr gute Übersicht.„
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Acknowledgements
Thank you for your interest during the semester