Derivative-Free Optimization

1
Derivative-Free Optimization
Derivative-Free Optimization

• J.-S. Roger Jang (???)
• CS Dept., Tsing Hua Univ., Taiwan
• http//www.cs.nthu.edu.tw/jang
• jang_at_cs.nthu.edu.tw

2
Outline

• Genetic algorithms (GA)
• Simulated Annealing (SA)
• Downhill simplex search
• Random search

3
The Big Picture
Soft Computing
4
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?

5
Genetic Algorithms

• Terminology
• Fitness function
• Population
• 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
MATLAB file go_ga.m
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Genetic Algorithms

• Performance profile

MATLAB file go_ga.m
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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

12
12
10
10
3
3
Translation
1
1
Inversion
6
6
7
7
2
2
9
11
9
11
8
8
4
5
4
5
1-2-3-4-5-6-7-8-9-10-11-12
1-2-3-4-5-9-8-7-6-10-11-12
12
12
10
10
3
3
Switching
1
1
6
6
7
7
2
2
9
11
9
11
8
8
4
5
4
5
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
MATLAB file tsp.m
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Simulated Annealing

• 100-city TSP with penalities when crossing the
  circle

Penalty 0
Penalty 0.5
Penalty -0.3
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Random Search

• Properties
• Intuitive
• Simple
• Analogy
• Get down to a valley blindfolded
• Two heuristics
• Reverse step
• Bias direction

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Downhill Simplex Search

• Simplex a set of n1 points in n-dim. space
• A triangle in a 2D space
• A tetrahedron in a 3D space
• Concept of downhill simplex search
• Repeatedly replaces the highest points with a
  lower one
• Consecutive successful replacements lead to the
  enlargement of the simplex
• Consecutive unsucessful replacements lead to the
  shrinkage of the simplex

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Downhill Simplex Search

• Flowchart
• Figure 7.9 in page 188
• Behavior
• The simplex can adapt itself to the objective
  function landscape (just like an amoeba), and
  eventually converges to a nearby local minimum.
• Program
• The search procedure is implemented as a function
  fmins.m that comes with MATLAB.

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Downhill Simplex Search

• Example Find the min. 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).

MATLAB file go_simp.m
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Random Search

• Flowchart

Select a random dx
f(xbdx)ltf(x)?
yes
x x b dx b 0.2 b 0.4 dx
no
f(xb-dx)ltf(x)?
yes
x x b - dx b b - 0.4 dx
no
b 0.5 b
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Random Search

• Example Find the min. 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).

MATLAB file go_rand.m