The first part of this presentation is a tutorial level introduction to the principles of genetic search and models of simple genetic algorithms. The second half covers the combination of genetic algorithms with local search methods to produce hybrid genetic algorithms. Hybrid algorithms can be modeled within the existing theoretical framework developed for simple genetic algorithms. An application of a hybrid to geometric model matching is given. The hybrid algorithm yields results that improve on the current state-of-the-art for this problem

Whitley, Darrell

English

NASA Technical Reports Server

Genetic Algorithms and Local Search
Darrell Whitley
Computer Science Department
Colorado State University
Fort Collins, CO
225
https://ntrs.nasa.gov/search.jsp?R=19960047556 2020-06-16T03:22:37+00:00Z

GENETIC ALGORITHMS
LOCAL SEARCH
Darrell Whitley
Computer Science Department
Colorado State University
Fort Collins, CO 80523
whitley @ cs.colostate.edu
(970) 491-5373
AND
ABSTRACT
The first part of this presentation is a tutorial level introduction to the principles of genetic
search and models of simple genetic algorithms. The second half covers the combination of genetic
algorithms with local search methods to produce hybrid genetic algorithms. Hybrid algorithms can be
modeled within the existing theoretical framework developed for simple genetic algorithms.
An application of a hybrid to geometric model matching is given. The hybrid algorithm yields
results that improve on the current state-of-the-art for this problem.l
1Ross Beveridge collaborated in the geometric matching application. This work was supported by NSF grants IRI-
9312748 and IRI-9503366 and by the Advanced Research Projects Agency (ARPA) under grant DAAH04-93-G-422
monitored by the U.S. Army Research Office.
227
THE CANONICAL GENETIC ALGORITHM
In the canonical genetic algorithm, fitness is defined by: f,. / f where f_ is the evaluation
associated with string i, and f is the average evaluation of all the strings in the population. It is
helpful to view the execution of the genetic algorithm as a two-stage process. It starts with the current
population. Selection is applied to the current population to create an intermediate population. Then
recombination and mutation are applied to the intermediate population to create the next population.
F.
t _ 1 Generation _ t + 1
)producti, )n_ .,
Make I_i Relcombinat
Fi
m
Copies
each of
string i
Perform
cross
at rate
Pc
on
1. Generate a random population of strings.
2. Evaluate all strings in the population.
3. Reproduce (duplicate) strings according to
fitness.
4. Recombine strings according to the probability
of crossover.
5. Repeat steps 2 to 4.
228
AN EXAMPLE OF HYPERPLANE PARTITIONING
A function over a single variable is plotted as a one-dimensional space, with function
maximization as a goal. The hyperplane 0"***...** spans the first half of the space and 1"***...**
spans the second half of the space. Since the strings in the 0"***...** partition are on average better
than those in the 1"***...** partition, we would like the search to be proportionally biased toward this
partition. In the second graph the portion of the space corresponding to ** 1"*...** is shaded, which
also highlights the intersection of 0"***...** and ** 1"*...**, namely, 0* 1"...**. Finally, in the third
graph, 0* 10"*...** is highlighted.
F(X)
0 K/2 K
Variable X
F(X_
K/8 K/4 K/2 K
Variable X
F{X}
K/8 K/4 K/2 K
Variable X
Figure 3 - A function and various partitions of hyperspace. Fitness
is scaled to a 0 to 1 range in this diagram.
229
SELECTION IN A SAMPLE POPULATION
This example population contains only 21 (partially specified) strings. The table includes
information on the fitness of each string and the number of copies to be placed in the intermediate
population. This table enumerates the 27 hyperplanes (33) that can be defined over the first three bits of
the strings in the population and explicitly calculates the fitness associated with the corresponding
hyperplane partition. The true fitness of the hyperplane partition corresponds to the average fitness of
all strings that lie in that hyperplane partition.
Strings and Fitness Values
String-ID [I String [Fitness Random I copies
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
001#...#
111#...#
OLO#...#
2.0
1.9
1.8
1.7
111#...#
101:_...#
011#...#
001#...#
o00#...#
100#...#
010#...#
011#...#
0o0#...#
11o#...#
11o#_.#
lOO#...#
011#...#
ooo#...#
oo1#...#
100#...#
010#...#
1.6
1.5
1.4
1.3
1.2
I.I
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
O.l
O0
0.93
0.65
0.02
0.51
0.20
0.93
0.20
0.37
0.79
0.28
0.13
0.70
0.80
0.51
0.76
0.45
0.61
0.07
m
2
2
2
1
2
1
2
1
1
1
1
1
0
1
1
1
1
0
0
0
0
Table 1 - A population with fitness assigned
to strings according to rank.
230
PROCESSING OF HYPERPLANES
The expected number of strings sampling a hyperplane partition after selection can be calculated
by multiplying the number of hyperplane samples in the current population before selection by the
average fitness of the strings in the population that fall in that partition. The observed number of copies
actually allocated by selection is also given. In most cases, the match between expected and observed
sampling rate is fairly good: the error is a result of sampling error due to the small population size.
Schenla
101"...*
111"...*
1"1"...*
*01"...*
*11"...*
11"*...*
001"...*
1_... _
0"1"...*
10"*...*
...
Mean
1.70
1.70
1.70
1.38
1.30
1.22
1.175
1.136
1.089
1.033
1.020
1.010
1.000
Count
2
2
4
5
10
5
4
3
9
6
5
10
21
I
Schemata and Fitness Values
Expect Obs Schema
3.4 3 *0"*...*
3.4 4 00"*...*
6.8 7 0"**...*
6.9" 6 011"...*
13.0 14 010"...*
6.: 8 01"*...*
4.7 6 0"0"...*
3.5 3 *10'...*
9.8 11 000"...*
6.2 7 **0"...*
5.1 5 *00"...*
10.1 1_ 110"...*
21.0 2: l*0*...*
100"...*
Mean
0.991
0.967
0.933
0.900
0.900
0.900
0.833
0.800
0.767
0.727
0.667
0.650
0.600
0.566
Count Expect Obs
II 10.9 9
6 5.8 4
12 11.2 I0
3 2.7 4
3 2.7 2
6 5.4 6
6 5.0 3
5 4.0 4
3 2.3 I
II 8.O 7
6 4.0 3
2 1.3 2
5 3.0 4
3 1.70 2
Table 2 - The average fitnesses (Mean) associated with the samples from the 27 hyperplanes
defined over the first three bit positions are explicitly calculated. The Expected representation
(Expect) and Observed representation (Obs) are shown. Count refers to the number of
strings in hyperplane H before selection.
231
THE VOSE AND LIEPINS MODEL
In the Vose and Liepins model, the vector s' e 9_ represents the t th generation of the genetic
algorithm and the i th component of s' is the probability that the string i is selected for the gene pool.
The standard form of the executable equations corresponds to the following portion of the Liepins and
Vose model (T denotes transpose):
srMs
A permutation function, _, is defined as follows:
..., >T
_i < So" Sv- l = < Silo, ..., sj_(v- 1)>r
where the vectors are treated as columns and V = 2 L , the size of the search space.
The Vose and Liepins Model
The i th component of vector s' is the probability that
the string i is selected for the gene pool.
s:- P(ai,t) f (Si) / :
The function ri,j(k) is used to construct a mixing matrix
M where the i,j th entry mij = rij(0). This matrix gives the
probabilities that crossing strings i and j will produce So.
The proportional representation for string 0 at time t+l
is given by:
srMs
where T denotes transpose.
If recombination is a combination of crossover and
mutation then
rij(k_q)=ri_,,2_,(q)
232
THE VOSE AND LIEPINS MODEL (Cont.)
space.
A general operator M can be defined over s which remaps s TM s to cover all strings in the
_(s) = < ("0s) _g'os ..... (,,v_, s) _M,,v_ts>r
Fitness information is explicitly introduced to transform the population at the beginning of
iteration t + I to the next intermediate population. A fitness matrix F is defined such that fitness
information is stored along the diagonal; the i,i th element is given by f(i) where f is the evaluation
function.
The transformation from the vector p'+ _to the next intermediate population represented by s '÷ 1
is given as follows:
Fp' +l
st+ 1 :
I r Fp,+ i
The Vose and Liepins Model
A general operator M can now be defined over s
which remaps s TM s to cover all strings in the search
space.
=< s)Mo0s,..., is)T M Or-1 S >T
A fitness matrix F is defined such that fitness
information is stored along the diagonal; following Vose
and Wright, the (i,i)'helement is given by f(i) where f
is the fitness function.
st+ 1
F pt+ 1
1 r Fpt+l
233
BASINS OF ATTRACTION
This slide illustrates the basins of attraction for a four-bit space under steepest ascent local
search. Note that equivalence classes of functions are created when one considers the set of functions
that have the same basins of attraction. These equivalence classes carry over into hybrid genetic
algorithms that combine genetic search with local search, also referred to as Lamarckian search.
0000 1111
.,d>-..
1000 0100 0111
0011 0101 1001 0110 1010 1100
1011 1101 1110
234
UPDATES TO REPRESENTATION UNDER LOCAL SEARCH
Local search can be added to the existing models of genetic algorithms. This slide shows how
one step of steepest ascent changes the distribution of samples in the population.
How Population Distributions Change Under Lamarckian Search
P'(0000,t) = P(0000,t) + P(0001,t) + P(0010,t) + P(0100,t) + P(1000,t)
P'(0001,t) = P(00tl,t) + P(0t0t,t) + P(t00l,t)
P'(OO10,t) : P(Ol10,t) + P(lO10,t)
P'(OlOO,t) = P(llOO,t)
P'(llll,t) : P(llll,t) + P(lllO,t) + P(llOl,t) + P(lO11,t) + P(Olll,t)
P'(OO11,t) : 0
P'(OlOl,t) = 0
P'(Ol10,t) = 0
P'(O111,t) = 0
P'(lOOO,t) : 0
P'(lOOl,t) : 0
P'(1OlO,t) = 0
P'(lOll,t) : 0
P'(ll00,t) : 0
P'(llO1,t) : 0
P'(1110,t) = 0
Function 1:
e(0000) : 28 e(0100) = 20 e(1000) : 12 e(ll00) : 4
e(0001) = 26 e(0101) : 18 e(1001) = 10 e(1101) = 2
e(O010) = 24 e(0110) : 16 e(1010) : 8 e(1110) : 0
e(0011) = 22 e(0111) = 14 e(1011) = 6 e(1111) = 30
The following Function is Equivalent to Function 1 Under ltybrid Search.
e(O000) : 28
e(O0Ol) = 26
e(O010) : 24
e(O011) = 24
e(OlO0) = 20
e(0101) : 24
e(0110) = 22
e(0111) = 22
e(lO00) : 18
e(lO01) = 24
e(1010) = 22
e(1011) = 22
e(llO0) : 18
e(llO1) = 18
e(1110) = 18
e(1111) = 30
235
A MATRIX REPRESENTATION OF LOCAL UPDATES
The same redistribution of samples seen below can be captured in matrix form and used to
model multiple steps of local search.
Lamarckian Updates to the p Vector
10000 0
10000 0
10000 0
01000 0
10000 0
01000 0
00100 0
00000 ...0
10000 0
01000 0
00100 0
00000 0
00001 0
00000 0
00000 0
00000 0
pT
For one step of local search: pTL
For 2 steps of local search: (pTL)L = pTL2
For k steps of local search: prLk
236
THE GEOMETRIC MATCHING PROBLEM
The geometric matching problem matches a model to a set of data. For example, the model may
be a line drawing and the data might be line segments extracted from a photograph. Problems of this
type arise in a number of areas and are specifically relevant to problems associated with computer
vision. Variations of local search are currently being used to solve geometric matching problems in the
context of semi-autonomous photo-interpretation and robot navigation.
Rectangle Pole
I
I /
Deer Tree
Figure 1 - Four stick figure models used in geometric matching tests.
(AI (B) (C)
Figure 2 - Sample of synthetic test data for matching: a) rectangle model randomly placed,
scaled and corrputed, b) corrupted deer model with random clutter, c) multiple corrupted
instances of the tree model.
237
AN EXAMPLE MATCH
An example of a best match is shown for the tree model. The line segments making up the
model are labeled with letters and are shown on the left. The data line segments include three instances
of the tree and are shown to the right. The model is overlaid on top of the data in the best match
position. The correspondence matrix indicates which pairs of model and data segments are part of this
best match. Each square in the table may be thought of as representing one bit in the bit string encoding
of the match. The filled in squares correspond to 1s in the bit string. Details for this application can be
found in Whitley, Beveridge, Graves and Mathias (1995), "Testing Driving Three 1995 Genetic
Algorithms," Journal of Heuristics.
_ 17 1 13 14
--.-.az. W- 8/
g_
0 I
IAe
ID
IE
IF
I:
h
IJ
IK
IL
o
oe
O•
I
I
I
I
I
I
Figure 3 - Example of a best match for one of the
32 matching problems.
238
THE GENITOR ALGORITHM
The Genitor Algorithm. Offspring are produced one at a time and replace the worst member of
the population.
Genitor
• Ranking used to determine fitness
• One-at-a-time recombination and replacement
The Genitor Algorithm
_ 1 Trial _t+l
Crossover
Select 2 Choose I
strings offspring
Displace
worst
239
COMPARISON OF LOCAL SEARCH AND GENITOR
Neither the local search algorithms nor the genetic algorithms converge upon the globally
optimal match in all cases. Therefore, in comparing algorithms for model matching, two things must
be taken into consideration. First is the likelihood that the algorithm once initiated will converge on the
optimal solution. The second is how much work must be performed in order to reach convergence.
LS Inc uses a fast incremental evaluation while LS Full uses a full evaluation.
Model I Params
Pole 12C
Pole 24M
Rectangle 28C
Pole 42C
Pole 42M
Rectangle 52M
Rectangle 68C
Pole 72C
Deer 81C
Pole 81 M
Pole 96M
Deer 99M
Pole 102C
Rectangle 108C
Rectangle 108M
Rectangle 124 M
Rectangle 148C
Tree 156C
Rectangle 168M
Deer 171C
Deer 180M
'Free 216M
Deer 261C
Deer 261M
Tree 276C
Deer 342M
Deer 351C
Tree 396C
Tree 432M
Tree 516C
Tree 552M
Tree 780M
Trials Required: tgs
LS Inc. ] LS Full I Genitor+
6 5 1
8 7 2
6 6 3
26 20 5
12 10 7
5 5 4
11 9 4
35 39 20
13 11 2
37 29 11
45 24 29
19 15 3
96 72 13
47 17 8
12 13 18
14 15 49
42 26 8
42 32 9
20 24 24
29 34 4
31 33 6
51 52 8
43 67 6
59 46 6
51 52 5
Estimated Run Time
LS Inc. I LS Full [ Genitor+
106 67
64 66
52 70
135 99
99 96
149 106
299 175
7 159.0
6 96.0
3 114.4
7 310.5
5 297.0
75 461.9
- 1,554.8
0.2 * 0.2 * 0.3
0.8 o 0.7 * 1.4
0.6 * 0.6 * 3.6
2.6 o 2.0 * 4.0
1.2 o 1.0 * 9.1
1.0 * 1.5 o 7.2
1.1 * 2.7 o 8.4
3.5 * 7.8 o 20.0
3.9 * 8.8 o 12.2
3.7 * 8.7 o 20.9
9.0 * 9.6 o 52.2
7.6 * 18.0 o 25.5
9.6 * 21.6 20.8
9.4 * 10.2 o 41.6
3.6 * 10.4 o 138.6
5.6 * 13.5 o 240.1
12.6 * 23.4 o 67.2
33.6 * 118.4 o 361.5
10.0 * 72.0 o 206.4
17.4 * 85.0 o 137.6
21.7 * 99.0 o 152.4
66.3 * 291.2 o 331.5
43.0 * 341.7 o 414.0
59.0 * 239.2 o 372.0
76.5 * 473.2 o 617.5
• 576.2 o 984.9
• 607.2 517.2
• 1,113.0 829.2
• 1,871.1 o 2,405.2
, 2,438.4 2,156.5
• 3,307.2 o 35,317.5
• 10,202.5 o
O
Table 8 - Comparison of steepest ascent local search using partial incremental evaluation (LS-Inc),
steepest ascent local search using full evaluation (LS-Full), and Genitor+algorithm on 32 test problems.
240
COMPARISON OF A HYBRID ALGORITHM
A "Hybrid Genetic Algorithm" uses local search as an operator to improve the strings in the
population. The comparison in this slide suggests there is little difference in the hybrid genetic
algorithm compared to local search using incremental evaluation. Aside from concerns about the
execution speed of the implementations of the different algorithms, there is another factor which very
much influences the results shown here: What criterion should be used to terminate genetic search
when evaluation of the global optimum is unknown? This issue typically does not arise for artifically
constructed test functions and this issue has not received a great deal of attention in the genetic
algorithm literature.
Success Prob.: P, Trials: tgs Estimated Run Time
Model t Params
Pole 12C
Pole 24M
Rectangle 28C
Pole 42C
Pole 42M
Rectangle 52M
Rectangle 68C
Pole 72C
Deer 81C
Pole 81M
Pole 96M
Deer 99M
Pole 102C
Rectangle 108C
Rectangle 108M
Rectangle 124M
Rectangle 148C
Tree 156C
Rectangle 168M
Deer 171C
Deer 180M
Tree 216M
Deer 261C
Deer 261M
Tree 276C
Deer 342M
Deer 351 C
Tree 396C
Tree 432M
Tree 516C
Tree 552M
Tree 780M
LS Inc. I Hybrid LS Inc. [ Hybrid LS Inc. [ Hybrid ] % Diff.
0.2 0.2 83%
0.8 0.3 38%
0.6 0.5 83%
2.6 2.0 77%
1.2 1.2 100%
1.0 1.4 140%
1.1 1.1 100%
3.5 4.8 137%
3.9 7.2 185%
3.7 6.4 173%
9.0 9.6 107%
7.6 6.0 79%
9.6 8.4 88%
9.4 6.0 64%
3.6 8.0 222%
5.6 7.6 136%
12.6 14.0 111%
33.6 36.8 110%
10.0 18.3 183%
17.4 29.2 168%
21.7 16.8 77%
66.3 47.7 72%
43.0 80.5 187%
59.0 35.4 60%
76.5 78.3 102%
159.0 105.6 66%
96.0 145.2 151%
114.4 125.2 109%
310.5 235.5 76%
297.0 390.6 132%
461.9 482.4 104%
1,554.8 1,735.0 112%
0.42 0.94
0.32 0.98
0.44 1.00
0.11 0.52
0.23 0.80
0.47 0.98
0.24 1.00
0.08 0.42
0.21 0.66
0.08 0.62
0.07 0.40
0.15 0.84
0.03 0.40
0.06 0.64
0.23 0.94
0.19 0.92
0.07 0.60
0.07 0.58
0.14 0.76
0.10 0.60
0.10 0.90
0.O6 0.70
0.07 0.52
0.05 0.82
0.06 0.64
0.03 0.64
0.05 0.54
0.06 0.80
0.02 0.64
0.03 0.74
0.O2 O.74
0.01 0.46
6 2
8 1
6 1
26 5
12 2
5 1
11 1
35 6
13 3
37 4
45 6
19 2
96 6
47 3
12 2
14 2
42 4
42 4
20 3
29 4
31 2
51 3
43 5
59 2
51 3
106 3
64 4
52 2
135 3
99 3
149 3
299 5
Table 9 - Comparison of steepest ascent local search using partial evaluation (LS-Inc) and
the hybrid genetic algorithm. % Diff refers to the amount of time required by the
hybrid to find a solution with 95% reliability compared to local search. 241
EVALUATION TIME FOR THE HYBRID GA
The data presented in this slide indicates that the hybrid genetic algorithm is finding the solution
to the problems long before the stopping criterion is satisfied. The data in Table 10 is broken down
into the number of evaluations used by the local search algorithm (which can exploit incremental
evaluation) as well as the full-evaluation calls used by the genetic algorithm after mating. The number
of evaluations required to find the best solution is less than half the number of evaluations required to
satisfy the stopping criterion. Cutting the number of evaluations in half can be achieved easily by
picking a better stopping criteria. This would then represent a new state-of-the-art in general geometric
matching.
Model I Params
Pole 12C
Pole 24M
Rectangle 28C
Pole 42C
Pole 42M
Rectangle 52M
Rectangle 68C
Pole 72C
Deer 81C
Pole 81M
Pole 96M
Deer 99M
Pole 102C
Rectangle 108C
Rectangle 108M
Rectangle 124M
Rectangle 148C
Tree 156C
Rectangle 168M
Deer 171C
Deer 180M
Tree 216M
Deer 261C
Deer 261M
Tree 276C
Deer 342M
Deer 351C
Tree 396C
Tree 432M
Tree 516C
Tree 552M
Tree 780M
P, Best Match FoundInc-Evais I Full-Evals
94 161.4
96 452.0
100 438.8
54 1311.2
90 1284.0
100 1462.4
I00 1920.9
42 2324.5
84 6175.3
72 4288.2
56 3999.0
92 9740.3
48 4161.0
80 4888.9
96 6251.5
94 6286.8
62 7684.1
68 37024.8
90 8978.0
78 24980.9
88 21017.0
74 52856.6
62 50067.9
84 37051.1
72 89913.4
66 65857.8
64 85626.9
70 148512.5
76 175521.8
56 245209.8
66 263373.7
42 429647.0
Convergence
Inc-Evals
22.1 727.6
16.3 2540.8
12.2 3693.9
47.6 3264.2
30.2 4748.0
13.1 11597.6
22.1 10663.4
49.8 5949.4
40.6 23095.7
45.8 12057.3
38.6 12707.8
60.3 30829.1
58.3 10026.1
37.9 18074.7
28.7 28152.4
33.8 29511.1
74.2 19792.1
154.9 83272.4
48.0 41066.9
151.2 62196.5
116.8 73928.3
271.0 129904.2
285.2 113533.1
206.2 122304.8
370.7 202536.3
274.4 184221.0
371.5 179823.7
504.8 332537.4
693.4 387632.3
539.8 571884.2
613.2 643095.3
701.4 1032966.4
I Full-Evalsl Secs
129.2 0.2
138.0 0.5
164.7 0.7
141.8 0.9
148.8 0.9
178.9 1.9
184.7 1.8
150.8 1.2
252.9 3.6
162.4 2.4
163.3 2.5
280.7 4.7
163.7 2.1
197.9 3.4
205.8 5.0
233.6 5.7
227.0 4.3
505.6 13.4
291.4 8.4
520.5 10.6
602.3 12.0
897.4 21.1
796.8 20.7
925.7 22.7
1082.0 33.8
966.9 35.3
941.0 35.2
1392.8 62.9
1864.1 82.9
1590.3 113.7
1937.2 130.7
2191.5 245.9
Table 10 - Comparison of number of evaluations required to find solutions in contrast
to time required to reach the stopping criterion. This is further broken down into the
incremental evaluations (Inc-Evals) used by the local search component of the hybrid algorithm
and full evaluations (Full-Evals) of strings generated by recombination and mutlation.
242
OTHER APPLICATIONS
• Navy Air-Plane Simulator Scheduling
Developed by BBN using CSU algorithms
• Poor Prototype Warehouse Scheduler
A fast evaluation function is used that
approximates warehouse operations which is
orders of magnitude faster than a corresponding
detailed warehouse simulator.
Local methods yield poor results when the
evaluation function is noisy or approximate, while
genetic algorithms yield very good solutions.
• Artificial Genetically Grown Neural
Networks
Neural networks are grown using cell
development processes for control applications.
The grammar controlling the growth process is
optimized rather than the neural network itself.
The resulting networks have outperformed
temporal difference methods on several
applications.
• Seismic Data Interpretation
243



Genetic Algorithms and Local Search

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