12. lecture stochastic optimization - uni-saarland.de · r2,x r3) 1 vektor (x) 1 test vector (u)...

12. Lecture

Stochastic Optimization

Differential Evolution

Soft Control

(AT 3, RMA)

http://www.uni-saarland.de/en/

SC

315WS 18/19 Georg Frey

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

xr1xr2

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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Minimum: 1,664


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Minimum: 0,348


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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


12. lecture stochastic optimization - uni-saarland.de · r2,x r3) 1 vektor (x) 1 test vector (u)...

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