Heuristic Optimization Methods
Scatter Search
Agenda
• Scatter Search (SS)
– For Local Search based Metaheuristics:
• SA based on ideas from nature
• TS based on problem-solving and learning
– For population based Metaheuristics:
• GA based on ideas from nature
• SS based on problem-solving and learning
– Nature works, but usually very slowly
– Being clever is better than emulating nature?
2
The following is a presentation previously
held at the conference ICS 2003...
Scatter Search:
Methodology and Applications
Manuel Laguna
University of Colorado
Rafael Martí
University of Valencia
Based on …
Laguna, M. and R. Martí (2003) Scatter Search:
Methodology and Implementations in C, Kluwer
Academic Publishers, Boston.
4
Scatter Search
Methodology
Metaheuristic
• A metaheuristic refers to a master strategy that
guides and modifies other heuristics to produce
solutions beyond those that are normally
generated in a quest for local optimality.
• A metaheuristic is a procedure that has the
ability to escape local optimality
6
Typical Search Trajectory
100
90
Objective Function
80
70
60
Value
50
Best Value
40
30
20
10
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
Iteration
7
Metaheuristic Classification
• x/y/z Classification
– x = A (adaptive memory) or M (memoryless)
– y = N (systematic neighborhood search) or S
(random sampling)
– Z = 1 (one current solution) or P (population of
solutions)
• Some Classifications
– Tabu search (A/N/1)
– Genetic Algorithms (M/S/P)
– Scatter Search (M/N/P)
8
Scatter Search
P
Diversification Generation
Method
Repeat until |P| = PSize
Improvement
Method
Improvement
Method
Reference Set
Update Method
Solution Combination
Method
Subset Generation
Method
RefSet
Stop if no more
new solutions
9
Scatter Search with Rebuilding
P
Diversification Generation
Method
Repeat until |P| = PSize
Improvement
Method
Improvement
Method
Solution Combination
Method
Reference Set
Update Method
Stop if MaxIter
reached
Improvement
Method
Subset Generation
Method
RefSet
No more new
solutions
Diversification Generation
Method
10
Tutorial
• Unconstrained Nonlinear Optimization Problem
Minimize

100 x 2 


2 2
x1
10.1 x 2  1   x 4  1
Subject to
2
10  x i  10

 1  x1   90 x 4 
2
2
  19.8x
2

2 2
x3
 1  x3  
2
 1x 4  1
for i  1,  ,4
11
Diversification Generation Method
Subrange 1
-10
Subrange 2
-5
Subrange 3
0
Subrange 4
+5
+10
Probability of selecting a subrange is proportional to a frequency count
12
Diverse Solutions
Solution
x1
x2
x3
x2
f(x)
1
2
3
4
5
6
7
8
9
10
1.11
-9.58
7.42
8.83
-6.23
1.64
0.2
-3.09
-6.08
-1.97
0.85
-6.57
-1.71
-8.45
7.48
3.34
-3.64
6.62
0.67
8.13
9.48
-8.81
9.28
4.52
6
-8.32
-5.3
-2.33
-6.48
-5.63
-6.35
-2.27
5.92
3.18
7.8
-8.66
-7.03
-3.12
1.48
8.02
835546.2
1542078.9
901878.0
775470.7
171450.5
546349.8
114023.8
7469.1
279099.9
54537.2
13
Improvement Method
Solution
1
2
3
4
5
6
7
8
9
10
x1
x2
x3
x2
f(x)
2.5
-0.52
-2.6
0.49
-3.04
-1.4
-0.36
-1.63
-0.8
-2.47
5.4
-0.5
5.9
0.53
9.45
2.46
-0.31
2.51
0.69
5.32
2.59
0.35
4.23
2.47
1.14
0.37
0.8
0.73
-1.16
-2.92
5.67
-0.14
10
5.89
0.41
-3.94
1.14
0.56
1.5
8.15
1002.7
138.5
7653.7
213.7
720.1
1646.7
57.1
21.5
11.2
1416.7
Nelder and Mead (1965)
14
Reference Set Update Method
(Initial RefSet)
b1 high-quality
solutions
Objective function
value to measure
quality
b2 diverse
solutions
Min-max criterion and
Euclidean distances to
measure diversity
RefSet of size b
15
Initial RefSet
High-Quality Solutions
Solution
number in P
35
46
34
49
38
x1
x2
x3
x4
-0.0444
1.133
-0.0075
1.1803
1.0323
0.0424
1.2739
0.0438
1.4606
0.9719
1.3577
-0.6999
1.4783
-0.344
-0.8251
1.8047
0.5087
2.2693
0.2669
0.695
f(x)
2.1
3.5
3.5
5.2
5.3
Diverse Solutions
Solution
37
30
45
83
16
x1
-3.4331
3.8599
-4.4942
-0.2414
6.1626
x2
10
10
10
-6.5307
10
x3
1.0756
-4.0468
3.0653
-0.9449
0.1003
f(x)
x4
0.3657
10
10
-9.4168
0.1103
1104.1
9332.4
13706.1
17134.8
78973.2
16
Subset Generation Method
• All pairs of reference solutions that include at
least one new solution
• The method generates (b2-b)/2 pairs from the
initial RefSet
17
Combination Method
y
x3 = x1 - r(x2 - x1)
x3 = (9,7)
r = 2/3
10
x4 = x1 + r(x2 - x1)
x5 = x2 + r(x2 - x1)
9
x1 = (5,7)
8
7
6
5
x4 = (6.5,5.5)
r = 1/2
x2 = (8,4)
4
3
x5 = (11,1)
r=1
2
1
1
2
3
4
5
6
7
8
9
10
11
12
13
x
18
Alternative Combination Method
y
x3 = x1 - r(x2 - x1)
10
x4 = x1 + r(x2 - x1)
x5 = x2 + r(x2 - x1)
9
x3
x1 = (5,7)
8
7
6
x4
5
x2 = (8,4)
4
3
x5
2
1
1
2
3
4
5
6
7
8
9
10
11
12
13
x
19
Reference Set Update Method
Quality
1
2
.
.
.
Best
1
2
.
.
.
Best
b
Worst
New trial
solution
b
Worst
RefSet of size b
Updated RefSet
20
Static Update
Pool of new trial solutions
Quality
1
2
.
.
.
Best
Updated RefSet = Best b from RefSet  Pool
b
Worst
RefSet of size b
21
RefSet after Update
x1
x2
x3
x4
f(x)
1.1383
1.2965
0.8306
0.715
0.14
0.7016
0.5297
1.2078
1.4633
0.36
0.5269
0.287
1.2645
1.6077
0.59
1.1963
1.3968
0.6801
0.446
0.62
0.3326
0.1031
1.3632
1.8311
0.99
0.3368
0.1099
1.3818
1.9389
1.02
0.3127
0.0949
1.3512
1.8589
1.03
0.7592
0.523
1.3139
1.7195
1.18
0.2004
0.0344
1.4037
1.9438
1.24
1.3892
1.9305
0.1252
-0.0152
1.45
22
Additional Strategies
• Reference Set
– Rebuilding
– Multi-tier
• Subset Generation
– Subsets of size > 2
• Combination Method
– Variable number of solutions
23
Rebuilding
RefSet
Rebuilt RefSet
b1
b2
Diversification
Generation Method
Reference Set
Update Method
24
2-Tier RefSet
Solution Combination
Method
Improvement
Method
RefSet
Try here first
b1
If it fails, then
try here
b2
25
3-Tier RefSet
Solution Combination
Method
Improvement
Method
RefSet
Try here first
b1
If it fails, then
try here
b2
Try departing
solution here
b3
26
Subset Generation
•
•
•
•
Subset Type 1: all 2-element subsets.
Subset Type 2: 3-element subsets derived from the 2element subsets by augmenting each 2-element
subset to include the best solution not in this subset.
Subset Type 3: 4-element subsets derived from the 3element subsets by augmenting each 3-element
subset to include the best solutions not in this subset.
Subset Type 4: the subsets consisting of the best i
elements, for i = 5 to b.
27
Subsets of Size > 2
90%
80%
70%
60%
50%
40%
30%
20%
10%
Random
0%
Type 1
Type 2
LOLIB
Type 3
Type 4
28
Variable Number of Solutions
Quality
1
2
.
.
.
Best
Generate 5 solutions
Generate 3 solutions
Generate 1 solution
b
Worst
RefSet of size b
29
Hybrid Approaches
• Use of Memory
– Tabu Search mechanisms for intensification and
diversification
• GRASP Constructions
• Combination Methods
– GA Operators
– Path Relinking
30
Multiobjective Scatter Search
• This is a fruitful research area
• Many multiobjective evolutionary approaches
exist (Coello, et al. 2002)
• SS can use similar techniques developed for
MOEA (multiobjective evolutionary approches)
31
Multiobjective EA Techniques
• Independent Sampling
– Search on f(x) = wi fi(x)
– Change weights and rerun
• Criterion Selection
– Divide reference set into k subsets
– Admission to ith subset is according to fi(x)
32
Advanced Designs
• Reference Set Update
– Dynamic / Static
– 2 Tier / 3 Tier
• Subset Generation
• Use of Memory
– Explicit Memory
– Attributive Memory
• Path Relinking
33
An Example
The Linear Ordering Problem
Given a matrix of weights E = {eij}mxm, the LOP consists
of finding a permutation p of the columns (and rows)
in order to maximize the sum of the weights in the
upper triangle
m1 m
CE ( p)    e pi p j
Maximize
i 1 j  i 1
Applications
 Triangulation for Input-Output Economic Tables.
 Aggregation of individual preferences
 Classifications in Sports
34
An Instance
1
1
2
3
4
2
 0 12
4 0

8 3

11 4
3
4
5 3
2 6

0 9

2 0
p=(1,2,3,4)
cE(p)=12+5+3+2+6+9=37
3
3
4
1
2
4
1
2
0 9 8 3 
2 0 11 4 


5 3 0 12


2
6
4
0


p*=(3,4,1,2)
cE(p*)=9+8+3+11+4+12=47
35
Diversification Generator
• Use of problem structure to create methods in order to
achieve a good balance between quality and diversity.
• Quality
– Deterministic constructive method
• Diversity
– Random Generator
– Systematic Generators (Glover, 1998)
• GRASP constructions.
– The method randomly selects from a short list of the most attractive
sectors.
• Use of Memory
– Modifying a measure of attractiveness proposed by Becker with a
frequency-based memory measure that discourages sectors from
occupying positions that they have frequently occupied.
36
Diversity vs. Quality
 Compare the different generators
 Create a set of 100 solutions with each one
1.6
d
1.4= Standardized Diversity
C
1.2 = Standardized Quality
 C+ d
1.0
0.8

0.6
d
0.4
C
0.2
0.0
DG05
DG04
DG02
DG01
DG09
DG08
DG06
DG03
DG07
Procedure
37
DG10
Improvement Method
• INSERT_MOVE (pj, i) consist of deleting pj from
its current position j to be inserted in
position i
• Apply a first strategy
– scans the list of sectors in search for the first
sector whose movement results in an
improvement
1
 0 12 5 3

 6 0 3 6 4
 8 5 0
5 7

E ( p)   2 7 2 0 3
 8 0 3 1 0

1 6 2 13
 9

 2 9 4 5 8
8
4
0
6
4
0
1
3

2
3

0
1

4

0
MoveValue = CE(p’) - CE(p)
CE (p’) = 78 + (1 - 4) + (6 - 0) + (2 - 6) +
(13 - 4) = 78 + 8 = 86
38
Solution Combination Method
• The method scans (from left to right) each reference
permutation.
– Each reference permutation votes for its first element that is still
not included in the combined permutation (“incipient element”).
– The voting determines the next element to enter the first still
unassigned position of the combined permutation.
– The vote of a given reference solution is weighted according to
the incipient element’s position.
Incipient element
(3,1,4,2,5) votes for 4
(1,4,3,5,2) votes for 4
(2,1,3,5,4) votes for 5
Solution under construction:
(3,1,2,4,_ )
39
Experiments with LOLIB
 49 Input-Output Economic Tables
GD
CK
CK10
TS
SS
Optima
deviation
0.15
%
0.15%
0.02%
0.04%
0.01%
Number
of optima
11
11
27
33
42
Run time
(seconds)
0.01
0.10
1.06
0.49
2.35
40
Another Example
A commercial SS implementation


OptQuest Callable Library (by OptTek)
As other context-independent methods separates the
method and the evaluation.
Output
Optimization
Procedure
Input
System
Evaluator
41
OptQuest based Applications
Solution Generator
OptQuest
Callable Library
User-written
Application
Solution Evaluator
System
Evaluator
42
Feasibility and Evaluation
User Implementation
The OptQuest engine
generates a new solution
Returns to
OptQuest
43
Comparison with Genocop
• Average on 28 hard nonlinear instances
1.0E+13
Average objective function value
(Logarithmic scale)
1.0E+12
Genocop
OCL
1.0E+11
1.0E+10
1.0E+09
1.0E+08
1.0E+07
1.0E+06
1.0E+05
1.0E+04
1.0E+03
0
1000
2000
3000
4000
5000
6000
Evaluations
7000
8000
9000
10000
44
Conclusions
• The development of metaheuristics usually
entails a fair amount of experimentation
(“skill comes from practice”).
• Code objectives:
– Quick Start
– Benchmark
– Advanced Designs
• Scatter Search provides a flexible
“framework” to develop solving methods.
45
Metaheuristic Classification
• x/y/z Classification
– x = A (adaptive memory) or M (memoryless)
– y = N (systematic neighborhood search) or S
(random sampling)
– Z = 1 (one current solution) or P (population of
solutions)
• Some Classifications
– Tabu search (A/N/1)
– Genetic Algorithms (M/S/P)
– Scatter Search (M/N/P)
46
Some Classifications
(local search)
Tabu Search
A/N/1
Simulated Annealing
M/S/1
(systematic)
(randomized)
Genetic Algorithm
M/S/P
Scatter Search
M/N/P
(population)
47
About the Classifications
• Our four main methods (SA, TS, GA, SS) all belong
far from the center (they are very randomized or very
systematic)
• Other methods have both some element of randomized
and some element of systematic behaviour
• Most implementations will mix the ingredients, and we
have an element of local search in population based
methods (e.g., Memetic Algorithms), or an element of
randomness in systematic approaches (such as random
tabu tenure in TS)
• The classifications highlight the differences between
methods, but there are also many similarities
48
GA vs. SS (1)
• GA has a ”long” history: proposed in the 1970s,
and immediately becoming popular
– Not initially used for optimization
– Gradually morphed into a metodology whose major
concern is the solution of optimization problems
• The concepts and principles of SS was also
proposed early (1970s), but was not popularized
until the 1990s
– The SS template most often used is from 1998
– Propsed to solve Integer Programming problems
49
GA vs. SS (2)
• GA is based on natural processes (genetics, the
”survival of the fittest”, and imitation of the
nature)
• SS is based on strategic ideas for how to use
adaptive memory
– Some TS concepts are critically linked with SS
50
GA vs. SS (3)
• Diversification
– GA: mutation
– SS: favoring diverse solutions in the reference set,
and generating diverse solutions in the initialization
• Intensification
– GA: probabilistic selection of parents, favoring the
fittest parents (but this is not really very
intensifying)
– SS: the improvement method
51
GA vs. SS (4)
• GA has a population that is usually 10x larger
than the reference set in SS
• GA applies operators (mutation, crossover) to
random solutions, SS applies operators
(combination, improvement) non-randomly
• Evolution in GA follows random ”survival of
the fittest”, in SS there are deterministic rules in
the reference set update method
52
GA vs. SS (5)
• Use of Local Search (improvement) is an
integral part of SS, but added to GA only to
create hybrid/improved approaches
• GA usually limited to combine only a pair of
solutions (parents), while SS allows
combination of any number of solutions
• GA uses full randomization to create initial
population, while SS balances diversity and
quality (diversification generation method)
53
Summary of Todays’s Lecture
• Scatter Search
– M/N/P (memoryless, systematic ”neighborhood”,
population of solutions)
• Components of Scatter Search:
–
–
–
–
–
Diversification Generation
Improvement
Reference Set Update
Subset Generation
Solution Combination
54