Scatter Search Methodology and Applications.pdf

Scatter Search:
Methodology and
Applications
Manuel Laguna
University of Colorado
Rafael Martí
University of Valencia

Based on …
Scatter Search: Methodology and
Implementations in C
Laguna, M. and R. Martí
Kluwer Academic Publishers, Boston, 2003.
2

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
4

Typical Search Trajectory
100
90
80
Objective Function
70
60
50
40
30
20
10
Value
Best Value
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Iteration
5

Metaheuristic Classification
x/y/z Classification
• x = A (adaptive memory) or M (memoryless(
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)
6

Scatter Search
Diversification Generation
Method
Repeat until |P| = PSize
P
Improvement
Method
Improvement
Method
Reference Set
Update Method
Solution Combination
Method
Subset Generation
Method
Stop if no more
new solutions
RefSet
7

Scatter Search with Rebuilding
Diversification Generation
Method
Repeat until |P| = PSize
P
Improvement
Method
Improvement
Method
Reference Set
Update Method
Solution Combination
Method
Stop if MaxIter
reached
Subset Generation
Method
Improvement
Method
No more new
solutions
RefSet
Diversification Generation
Method
8

Tutorial
Unconstrained Nonlinear Optimization
Problem
Minimize
100
10.1
(
2
)
2
2
( ) (
2
1 90 )
2
x − x + − x + x − x + ( 1 − x )
2
( ) ( ) )
2
2
x −1
+ x −1
+ 19.8( x −1)( x −1)
2
1
4
1
4
2
3
4
3
2
+
Subject
to
−10
≤
x
i
≤10
for
i
= 1, K,4
9

Diversification Generation
Method
Subrange 1 Subrange 2 Subrange 3 Subrange 4
-10 -5 0 +5 +10
Probability of selecting a subrange is proportional to a frequency count
10

11
11
Diverse Solutions
Diverse Solutions
x 1
1.11
-9.58
7.42
8.83
-6.23
1.64
0.2
-3.09
-6.08
-1.97
x 2
0.85
-6.57
-1.71
-8.45
7.48
3.34
-3.64
6.62
0.67
8.13
x 3
9.48
-8.81
9.28
4.52
6
-8.32
-5.3
-2.33
-6.48
-5.63
x 2
-6.35
-2.27
5.92
3.18
7.8
-8.66
-7.03
-3.12
1.48
8.02
f(x)
835546.2
1542078.9
901878.0
775470.7
171450.5
546349.8
114023.8
7469.1
279099.9
54537.2
Solution
1
2
3
4
5
6
7
8
9
10

Improvement Method
Solution x 1 x 2 x 3 x 2 f(x)
1 2.5 5.4 2.59 5.67 1002.7
2 -0.52 -0.5 0.35 -0.14 138.5
3 -2.6 5.9 4.23 10 7653.7
4 0.49 0.53 2.47 5.89 213.7
5 -3.04 9.45 1.14 0.41 720.1
6 -1.4 2.46 0.37 -3.94 1646.7
7 -0.36 -0.31 0.8 1.14 57.1
8 -1.63 2.51 0.73 0.56 21.5
9 -0.8 0.69 -1.16 1.5 11.2
10 -2.47 5.32 -2.92 8.15 1416.7
Nelder and Mead (1965)
12

Reference Set Update Method
(Initial RefSet)
b 1
high-quality
solutions
Objective function
value to measure
quality
b 2
diverse
solutions
Max-min criterion and
Euclidean distances to
measure diversity
RefSet of size b
13

Initial RefSet
High-Quality Solutions
Solution
number in P x 1 x 2 x 3 x 4 f(x)
35 -0.0444 0.0424 1.3577 1.8047 2.1
46 1.133 1.2739 -0.6999 0.5087 3.5
34 -0.0075 0.0438 1.4783 2.2693 3.5
49 1.1803 1.4606 -0.344 0.2669 5.2
38 1.0323 0.9719 -0.8251 0.695 5.3
Diverse Solutions
Solution x 1 x 2 x 3 x 4 f(x)
37 -3.4331 10 1.0756 0.3657 1104.1
30 3.8599 10 -4.0468 10 9332.4
45 -4.4942 10 3.0653 10 13706.1
83 -0.2414 -6.5307 -0.9449 -9.4168 17134.8
16 6.1626 10 0.1003 0.1103 78973.2
14

Subset Generation Method
All pairs of reference solutions that include
at least one new solution
The method generates (b2-b)/2 b)/2 pairs from
the initial RefSet
15

Combination Method
y
10
9
x 3 = (9,7)
r = 2/3
x 3 = x 1 - r(x 2 - x 1 )
x 4 = x 1 + r(x 2 - x 1 )
x 5 = x 2 + r(x 2 - x 1 )
8
x 1 = (5,7)
7
6
5
4
x 4 = (6.5,5.5)
r = 1/2
x 2 = (8,4)
3
2
x 5 = (11,1)
r = 1
1
1 2 3 4 5 6 7 8 9 10 11 12 13
x
16

Alternative Combination Method
y
10
9
8
x 3
x 1 = (5,7)
x 3 = x 1 - r(x 2 - x 1 )
x 4 = x 1 + r(x 2 - x 1 )
x 5 = x 2 + r(x 2 - x 1 )
7
6
x 4 x 5
5
4
x 2 = (8,4)
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13
x
17

Reference Set Update Method
Quality 1
2
.
.
.
Best
1
2
.
.
.
Best
New trial
solution
b
Worst
b
Worst
RefSet of size b
Updated RefSet
18

Static Update
Pool of new trial solutions
Quality 1
2
.
.
.
Best
b
Worst
RefSet of size b
Updated RefSet = Best b from RefSet ∪ Pool
19

RefSet after Update
x 1
x 2
x 3
x 4
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
20

Tutorial
0-11 Knapsack Problem
Maximize 10x 1 + 14x 2 + 9x 3 + 8x 4 + 7x 5 + 5x 6 + 9x 7 + 3x 8
S.t. 7x 1 + 12x 2 + 8x 3 + 9x 4 + 8x 5 + 6x 6 + 11x 7 + 5x 8 < 100
x i = { 0, 1} for i = 1, …, 8
21

Additional Strategies
Reference Set
• Rebuilding
• Multi-tier
tier
Subset Generation
• Subsets of size > 2
Combination Method
• Variable number of solutions
22

Rebuilding
RefSet
Rebuilt RefSet
b 1
b 2
Diversification
Generation Method
Reference Set
Update Method
23

2-Tier
RefSet
Solution Combination
Method
Improvement
Method
Try here first
RefSet
b 1
If it fails, then
try here
b 2
24

3-Tier
RefSet
Solution Combination
Method
Improvement
Method
RefSet
Try here first
b 1
If it fails, then
try here
b 2
Try departing
solution here
b 3
25

Subset Generation
Subset Type 1: all 2-element 2
subsets.
Subset Type 2: 3-element subsets derived
from the 2-element 2
subsets by augmenting
each 2-element 2
subset to include the best
solution not in this subset.
Subset Type 3: 4-element subsets derived
from the 3-element 3
subsets by augmenting
each 3-element 3
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.
26

Subsets of Size > 2
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Random
Type 1
Type 2
Type 3
Type 4
LOLIB
27

Variable Number of Solutions
Quality
1
2
.
.
.
Best
Generate 5 solutions
Generate 3 solutions
Generate 1 solution
b
Worst
RefSet of size b
28

Hybrid Approaches
Use of Memory
• Tabu Search mechanisms for intensification
and diversification
GRASP Constructions
Combination Methods
• GA Operators
• Path Relinking
29

Multiobjective Scatter Search
This is a fruitful research area
Many multiobjective evolutionary
approaches exist (Coello(
Coello, , et al. 2002)
SS can use similar techniques developed
for MOEA (multiobjective(
evolutionary
approches)
30

Multiobjective EA Techniques
Independent Sampling
• Search on f(x) = ∑w i f i (x)
• Change weights and rerun
Criterion Selection
• Divide reference set into k subsets
• Admission to i th subset is according to f i (x)
31

Scatter Search
Applications and Implementations

Scatter Search Elements
Diversification Generator Method
Improvement Method
Reference Set
• Initialization
• Update
• Rebuild
Subset Generation Method
Solution Combination Method
Problem Independent
33

C Code Objective
Illustrate the methods and strategies
Use it in different applications
Extend the previous version:
• http://www.uv.es/~rmarti/sscode.html
• Modularity
• Versatility
Non experts (use as it is with small changes)
Experts (can easily introduce important modifications)
• Ansi C (different platforms and systems)
34

Basic Design
Diversification Generation
Method
Repeat until |P| = PSize
P
Improvement
Method
Improvement
Method
Reference Set
Update Method
Solution Combination
Method
Subset Generation
Method
Stop if no more
new solutions
RefSet
35

RefSet Data Structure
typedef struct REFSET
{ int b; /* Size */
double **sol;
double *ObjVal;
/* Solutions */
/** Objective value of solutions */
int *order; /* Order of solutions */
int *iter; /* Entering iteration */
int NewSolutions; /* =1 if new element has been added */
} REFSET;
RefSet
b
sol
ObjVal
order
iter
NewSolutions
nvar
36

Main Function
int main(void)
{
SS *pb*
pb;
/* Pointer to problem data */
int nvar = 10; /* Number of variables */
int b = 10;
/* Size of reference set */
int PSize = 100; /* Size of P */
pb = SSProblem_Definition(nvar,b,PSize
(nvar,b,PSize);
/* Insert here the Scatter Search code */
SSFree_DataStructures(pb
(pb);
return 0;
}
37

Basic Design
pb = SSProblem_Definition(nvar,b,PSize);
SSCreate_P(pb
(pb);
SSCreate_RefSet(pb
(pb);
While(pb
pb->rs->NewSolutions)
SSUpdate_RefSet(pb
(pb);
SSBestSol(pb,sol,&value);
SSFree_DataStructures(pb);
38

Advanced Design
pb = SSProblem_Definition(nvar,b,PSize);
SSCreate_P(pb
(pb);
SSCreate_RefSet(pb
(pb);
for(Iter=1;
Iterrs
rs->NewSolutions)
SSUpdate_RefSet(pb
(pb);
else
SSRebuild_RefSet(pb
(pb);
}
SSBestSol(pb,sol,&value);
SSFree_DataStructures(pb);
39

SSCreate_P
while(currentPSize p
>p->PSize)
{
SSGenerate_Sol(pb,sol);
obj_val = sol_value(sol);
SSImprove_solution(pb,sol,&obj_val
(pb,sol,&obj_val);
/* Check whether sol is a new solution */
j=1;equal=0;
while(jp->sol[j++],pb->nvar);
/* Add improved solution to P */
if(!equal){
for(j=1;jnvar;j++)
pb->p
>p->sol[currentPSize][j]=sol[j];
]=sol[j];
pb->p
>p->ObjVal[currentPSize++] = obj_val;
}
}
40

SSCreate_RefSet
/* Order solutions in P */
p_order = SSOrder(pb
(pb->p->ObjVal, pb->p
>p->PSize, pb->Opt);
b1 = pb->rs
rs->b / 2;
/* Add the first (best)) b1 solutions in P to RefSet */
/* Compute minimum distances from P to RefSet */
for(i=1;ip->PSize;i++)
min_dist[i]= SSDist_RefSet(pb,b1,pb
(pb,b1,pb->p->sol[i]);
>sol[i]);
/* Add b-b1 b b1 diverse solutions to RefSet */
for(i=b1+1;irs->b;i++)
{
a=SSMax_dist_index(pb,min_dist
SSMax_dist_index(pb,min_dist);
/* Copy sol a from P to Refset */
SSUpdate_distances(pb,min_dist,i
(pb,min_dist,i);
}
/* Compute the order in RefSet: order */
pb->rs
rs->NewSolutions
= 1; pb->CurrentIter
= 1;
41

SSUpdate_RefSet
pb->rs
rs->NewSolutions=0;
SSCombine_RefSet(pb);
pb->CurrentIter
CurrentIter++;
for(a=1;apool_size;a++)
{
value=sol_value(pb
sol_value(pb->pool[a]);
SSImprove_solution(pb,pb->pool[a],&value);
SSTryAdd_RefSet(pb,pb->pool[a],value);
}
pb->pool_size
pool_size=0;
42

Advanced Designs
Reference Set Update
• Dynamic / Static
• 2 Tier / 3 Tier
Subset Generation
Use of Memory
• Explicit Memory
• Attributive Memory
Path Relinking
43

An Example
The Linear Ordering Problem
Given a matrix of weights E = {e ij } 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
• Applications
Maximize
• Triangulation for Input-Output Economic Tables.
• Aggregation of individual preferences
• Classifications in Sports
C ( p)=
e
E
m−1
i=
1
m
∑ ∑
j=+
i 1
p p
i
j

An Instance
1 2 3 4
3 4 1 2
1
2
3
4
⎡ 0 12 5 3⎤
⎢
4 0 2 6
⎥
⎢
⎥
⎢ 8 3 0 9⎥
⎢
⎥
⎣11 4 2 0⎦
3
4
1
2
⎡0 9 8 3⎤
⎢
2 0 11 4
⎥
⎢
⎥
⎢5 3 0 12⎥
⎢
⎥
⎣2 3 4 0⎦
p=(1,2,3,4)
c E (p)=12+5+3+2+6+9=37
p*=(3,4,1,2)
c E (p*)=9+8+3+11+4+12=47

SS for the LOP
Complete the “generic” code
Design specific methods for the “problem-
dependent” elements
• Diversification Generator Method
• Improvement Method
• Combination Method
46

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
47

Diversity vs. Quality
Compare several diversification generators
Create a set of 100 solutions with each one
1.6
1.4
1.2
1.0
0.8
∆d = Standardized Diversity
∆C = Standardized Quality
∆C+∆d
∆
0.6
0.4
∆d
0.2
∆C
0.0
DG05 DG04 DG02 DG01 DG09 DG08 DG06 DG03 DG07 DG10
Procedure
48

Improvement Method
INSERT_MOVE (pj(
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
⎛ 0 12 5 3 1 8 3⎞
⎜
⎟
⎜ 6 0 3 6 4 4 2⎟
⎜ 8 5 0 5 7 0 3⎟
⎜
⎟
Ep ( )= ⎜−2 7 2 0 −3 6 0⎟
⎜ 8 0 3 −1 0 4 1⎟
⎜
⎟
⎜ 9 1 6 2 13 0 4⎟
⎜
⎟
⎝ 2 9 4 −5 8 1 0⎠
MoveValue = C E (p’) - C E (p)
CE (p’) = 78 + (1 - 4) + (6 - 0) + (2 - 6) +
(13 - 4) = 78 + 8 = 86
49

Solution Combination Method
The method scans (from left to right) each reference
permutation.
• Each reference permutation votes for its first element that is still s
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 Solution under construction:
(1,4,3,5,2) votes for 4 (3,1,2,4,_ )
(2,1,3,5,4) votes for 5
50

Experiments with LOLIB
49 Input-Output Economic Tables
GD
CK
CK10
TS
SS
Optima
deviation
Number of
optima
Run time
(seconds
onds)
0.15% 0.15% 0.02% 0.04% 0.01%
11 11 27 33 42
0.01 0.10 1.06 0.49 2.35
51

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
52

OptQuest based Applications
Solution Generator
OptQuest
Callable Library
User-written
Application
Solution Evaluator
System
Evaluator
53

Feasibility and Evaluation
User Implementation
x x * F(x * )
Constraint
Mapping
Complex System
Evaluator
G(x * )
Penalty
Function
P(x * )
The OptQuest engine
generates a new solution
Returns to
OptQuest
54

Comparison with Genocop
Test on 28 hard nonlinear instances
1.0E+13
Average objective function value
(Logarithmic scale)
1.0E+12
1.0E+11
1.0E+10
1.0E+09
1.0E+08
1.0E+07
1.0E+06
1.0E+05
Genocop
OCL
1.0E+04
1.0E+03
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Evaluations
55

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.
56

Call for Papers
European Journal of Operational Research
Feature Issue on
SCATTER SEARCH METHODS
FOR OPTIMIZATION
Deadline for submissions: June 30, 2003
http://www.uv.es/~rmarti/ejor.html
57

Questions
58