Chap 7:Derivative-Free Optimization

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Chap 7Derivative-Free Optimization
Genetic Algorithms
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Genetic Algorithms

• Motivation
• Look at what evolution brings us?
• Vision
• Hearing
• Smelling
• Taste
• Touch
• Learning and reasoning
• Can we emulate the evolutionary process with
  today's fast computers?

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Genetic Algorithms

• Terminology
• Fitness function
• Polulation
• Encoding schemes
• Selection
• Crossover
• Mutation
• Elitism

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Genetic Algorithms

• Binary encoding

Chromosome
(11, 6, 9) 1011 0110 1001
Gene
Crossover
1 0 0 1 1 1 1 0
1 0 0 1 0 0 1 0
1 0 1 1 0 0 1 0
1 0 1 1 1 1 1 0
Crossover point
Mutation
1 0 0 1 1 1 1 0
1 0 0 1 1 0 1 0
Mutation bit
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Genetic Algorithms

• Flowchart

10010110 01100010 10100100 10011001 01111101 . .
. . . . . . . . . .
10010110 01100010 10100100 10011101 01111001 . .
. . . . . . . . . .
Elitism
Selection
Crossover
Mutation
Current generation
Next generation
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Genetic Algorithms

• Example Find the max. of the peaks function
• z f(x, y) 3(1-x)2exp(-(x2) - (y1)2) -
  10(x/5 - x3 - y5)exp(-x2-y2)
  -1/3exp(-(x1)2 - y2).

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Genetic Algorithms

• Derivatives of the peaks function
• dz/dx -6(1-x)exp(-x2-(y1)2) -
  6(1-x)2xexp(-x2-(y1)2) -
  10(1/5-3x2)exp(-x2-y2) 20(1/5x-x3-y5)
  xexp(-x2-y2) - 1/3(-2x-2)exp(-(x1)2-y2)
• dz/dy 3(1-x)2(-2y-2)exp(-x2-(y1)2)
  50y4exp(-x2-y2) 20(1/5x-x3-y5)yexp(-x
  2-y2) 2/3yexp(-(x1)2-y2)
• d(dz/dx)/dx 36xexp(-x2-(y1)2) -
  18x2exp(-x2-(y1)2) - 24x3exp(-x2-(y1)2
  ) 12x4exp(-x2-(y1)2) 72xexp(-x2-y2)
  - 148x3exp(-x2-y2) - 20y5exp(-x2-y2)
  40x5exp(-x2-y2) 40x2exp(-x2-y2)y5
  -2/3exp(-(x1)2-y2) - 4/3exp(-(x1)2-y2)x2
  -8/3exp(-(x1)2-y2)x
• d(dz/dy)/dy -6(1-x)2exp(-x2-(y1)2)
  3(1-x)2(-2y-2)2exp(-x2-(y1)2)
  200y3exp(-x2-y2)-200y5exp(-x2-y2)
  20(1/5x-x3-y5)exp(-x2-y2) -
  40(1/5x-x3-y5)y2exp(-x2-y2)
  2/3exp(-(x1)2-y2)-4/3y2exp(-(x1)2-y2)

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Genetic Algorithms

• GA process

Initial population
5th generation
10th generation
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Genetic Algorithms

• Performance profile

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Simulated Annealing

• Analogy

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Simulated Annealing

• Terminology
• Objective function E(x) function to be
  optiimized
• Move set set of next points to explore
• Generating function to select next point
• Acceptance function h(DE, T) to determine if the
  selected point should be accept or not. Usually
  h(DE, T) 1/(1exp(DE/(cT)).
• Annealing (cooling) schedule schedule for
  reducing the temperature T

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Simulated Annealing

• Flowchart

Select a new point xnew in the move sets via
generating function
Compute the obj. function E(xnew)
Set x to xnew with prob. determined by h(DE, T)
Reduce temperature T
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Simulated Annealing

• Example Travel Salesperson Problem (TSP)

How to transverse n cities once and only once
with a minimal total distance?
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Simulated Annealing

• Move sets for TSP

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12
10
10
3
3
Translation
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1
Inversion
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7
7
2
2
9
11
9
11
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8
4
5
4
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1-2-3-4-5-6-7-8-9-10-11-12
1-2-3-4-5-9-8-7-6-10-11-12
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12
10
10
3
3
Switching
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1
6
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7
2
2
9
11
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4
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4
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1-2-11-4-8-7-5-9-6-10-3-12
1-2-3-4-8-7-5-9-6-10-11-12
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Simulated Annealing

• A 100-city TSP using SA

Initial random path
During SA process
Final path
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Simulated Annealing

• 100-city TSP with penalities when crossing the
  circle

Penalty 0
Penalty 0.5
Penalty -0.3