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h.hajimirsadeghi@ece.ut.ac.ir http://khorshid.ut.ac.ir/~h.hajimirsadeghi
Ant Colony Optimization with a Genetic Restart Approach toward
Global OptimizationHossein Hajimirsadeghi, Mahdy Nabaee, Babak Nadjar-araabi
Control and Intelligent Processing Center of Excellence
School of Electrical and Computer engineering
University of Tehran, Tehran, IRAN
03/09/2008
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Outline
• Multiplicative Squares
• Ant Colony Optimization
• Local Search algorithms
• Genetic Algorithms
• Methodology
• Results
• Conclusion
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h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Multiplicative Squares
• Numbers 1 to
• :• MAX-MS: Max { }• MIN-MS: Min { }• Kurchan: Min (Max {} – Min {})
3
2n
j
ji
jji
jji
jji
DIAGONALsAnti
DIAGONALs
COLUMN
ROW
,
,
,
,
For each if
f
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Multiplicative Squares (3*3 example)
• Rows: 5*1*8 = 40, 3*9*4 = 108, 7*2*6 = 84• Columns: 5*3*7 = 105, 1*9*2 = 18, 8*4*6 = 192• Diagonals: 5*9*6 = 270, 1*4*7 = 28, 8*3*2 = 48 • Anti-diagonals: 8*9*7 = 504, 1*3*6 = 18, 5*4*2 = 40• MAX-MS/MIN-MS:
SF=40+108+84+105+18+192+270+28+48+504+18+40= 1455• Kurchan MS: SF= 504-18 = 486
4
518
394
726
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Why Multiplicative Squares?
• NP-hard Combinatorial Problem• Ill-conditioned
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• Complicated– precision of 20+ digits for dimensions greater than 10
12961354134332523412…???– Local Optima
5
1931616931
115136115136
215410215410
147128147128
(a) (b)
SF= 134355 SF=66045
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Introduction (ACO)
• Ant Colony Optimization (Marco Dorigo, 1992):
– bio-inspired– population-based– meta-heuristic– Evolutionary– Combinatorial Optimization problems.
• Used to solve
Traveling Salesman Problem
(TSP).
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http://iridia.ulb.ac.be/~mdorigo/ACO/ACO.html
Fig.1 TSP with 50 cities
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Ant System
• TSP
7
..0
.
.
,,
,,
,
wo
Njp
ki
Nllili
jiji
kji
ki
i
j
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Ant System
• : Heuristic Function
(attractiveness)
(visibility)
8
kji ,
i
jji
kji d ,
,
1
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Ant System
• : Pheromone Trails
9
kji ,
..0
),(
).1(
,
1,,,
wo
tourjiL
Q
k
kji
m
k
kjiji
kji
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Ant System Extensions
• ASrank• AS-elite• MMAS• Ant-Q• ACS• ACO-LBT• P-B ACO• Omicron ACO (OA)• …
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h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Local Search Algorithms
• Hill Climbing
• 2-opt and 3-opt
• K-opt
• Lin-Kernighan
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Fig. 3. With 2-opt algorithm dashed lines convert to solid lines: (a,b) (a,c) and (c,d) (b,d).
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Genetic Algorithms
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Encoding
GA OperatorsBinary Encoding
Permutation Encoding
Real Encoding
Tree Encoding
Selection
Cross Over
Mutation
Elitism
Selection Mutation
Cross OverElitism
Fig.4. Genetic Operators
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Proposed Method
1. Indices are selected
2. to 1 are put
according to the
indices
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18106
4911
321412
5137
Fig. 4. Graph representation for the MAX MS (4*4) problem, using ACO. Heavy lines show a feasible path for the problem.
1 3 2 15 16 …
1 3 2 15 16 …
1 3 2 15 16 …
1 3 2 15 16 …
1 3 2 15 16 …
start
2n
Index 13
Index 615
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h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
ACO Terms for MAX-MS
• Trails:
• Heuristic Function:
14
..0
),(,
wo
tourjiQ
SFkkji
Fig. 5. Heuristic function is illustrated for two sample conditions. The current position of the ant is displayed by .
( a)( b)
if
if
if
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
ACO Terms for MAX-MS
• Max and min trail like MAX-MIN Ant System (MMAS).
• iteration-best and global-best deposit pheromone
• Eating ants like Ant Colony System (ACS).
• Adaptive (decreasing with iterations)
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h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Local Search
• 2 opt for each iteration
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Fig.6. 2-opt
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Genetic Restart Approach
• Cross-over
• Mutation
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Fig. 7. An example of two cut cross over with 3 children.
Parent 113425
Parent 245123
Child 134512
Child 251234
Child 353412
Fig. 8. An example of a two cut mutation.
Parent 113425
Parent 245123
Child of parent 1
14325
Child of parent 2
25143
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Results
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TABLE 1Experiment results
(a )MS7
MethodBestAvg.Std. Dev.Std. Dev
%Best err.%
Avg. err%.
Adaptive heuristic
836927418654836545183884.3310273380.30.03700.046
Fixed heuristic836864383934836387896300.22827292770.0340.00750.064
No GA restart836590536598835890051299.2472719981.50.0570.04030.124
(b )MS8
MethodBestAvg.Std. Dev.Std. Dev
%Best err.%
Avg. err%.
Adaptive heuristic
402702517088866
402397450057731410397887424.80.10200.076
Fixed heuristic4026933164626
0239622889324340712487304223038.13.150.00231.608
No GA restart4026722455162
7837941167972993127191910644358.27.170.00755.784
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Results
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a bFig. 9. Evaluation of introduced algorithms.(a) Comparison between the proposed strategies on MS7. (b) Comparison between the proposed strategies on MS8 .
Zoom on iteration = 300 to 600
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Performance of the Genetic Restart Approach
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TABLE 2Genetic Semi-Random-Restart Performance
MethodAvg. number of successive genetic
restart (MS7)Avg. number of successive genetic
restart (MS8)
Fixed heuristic1.62.4
Flexible heuristic1.32.3
Fig. 10. Successful operation of the posed restart algorithm to evade local optimums .
SF Survivor semi-random-restart
h.hajimirsadeghi@ece.ut.ac.ir ECE Department, University of Tehran
Conclusion
• Novel algorithm to solve MAX-MS– Adaptive – Genetic Restart Algorithm
• Can be used for NP-hard combinatorial problems for global optimization
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