metaheuristic methods and their applications lin-yu tseng ( 曾怜玉 ) institute of networking and...
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Metaheuristic Methods and Their Applications
Lin-Yu Tseng ( 曾怜玉 )
Institute of Networking and MultimediaDepartment of Computer Science and EngineeringNational Chung Hsing University
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OUTLINE
I. Optimization ProblemsII. Strategies for Solving NP-hard Optimization
ProblemsIII. What is a Metaheuristic Method?IV. Trajectory MethodsV . Population-Based MethodsVI. Multiple Trajectory SearchVII. The Applications of Metaheuristic MethodsVIII. Conclusions
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I. Optimization Problems
Computer Science Traveling Salesman Problem Maximum Clique Problem
Operational Research Flow Shop Scheduling Problem P – Median Problem
Many optimization problems are NP-hard.
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Optimization problems
Calculus-based method
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Hill climbing
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How about this?
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An example : TSP (Traveling Salesman Problem)
18
1
2
3
5
4
23
10 5
3
7
15206
A solution A sequence
12345 Tour length = 31
13452 Tour length = 63
There are (n-1)! tours in total
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II. Strategies for Solving NP-hard Optimization Problems
• Branch-and-Bound Find exact solution
• Approximation Algorithms– e.g. There is an approximation algorithm for TSP which can f
ind a tour with tour length 1.5× (optimal tour length) in O≦(n3) time.
• Heuristic Methods Deterministic
• Metaheuristic Methods Heuristic + Randomization
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III. What is a Metaheruistic Method?
• Meta : in an upper level
• Heuristic : to findA metaheuristic is formally defined as an iterative generation process which guides a subordinate heuristic by combining intelligently different concepts for exploring and exploiting the search space, learning strategies are used to structure information in order to find efficiently near-optimal solutions. [Osman and Laporte 1996].
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Fundamental Properties of Metaheuristics [Blum and Roli 2003]
• Metaheuristics are strategies that “guide” the search process.
• The goal is to efficiently explore the search space in order to find (near-)optimal solutions.
• Techniques which constitute metaheuristic algorithms range from simple local search procedures to complex learning processes.
• Metaheuristic algorithms are approximate and usually non-deterministic.
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Fundamental Properties of Metaheuristics (cont.)
• They may incorporate mechanisms to avoid getting trapped in confined areas of the search space.
• The basic concepts of metaheuristics permit an abstract level description.
• Metaheuristics are not problem-specific.• Metaheuristics may make use of domain-specific k
nowledge in the form of heuristics that are controlled by the upper level strategy.
• Today’s more advanced metaheuristics use search experience (embodied in some form of memory) to guide the search.
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IV. Trajectory Methods
1. Basic Local Search : Iterative Improvement
Improve (N(S)) can be① First improvement② Best improvement③ Intermediate option
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Trajectory Methods
x1
x2
x3 X4
X5
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2. Simulated Annealing – Kirkpatrick (Science 1983)
• Probability
• Temperature T may be defined as
• Random walk + iterative improvement
T
sfsfssTp
)()'(exp),',(
10 ,1 αTT kk
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3. Tabu Search – Glover 1986
• Simple Tabu Search
• Tabu list• Tabu tenure: the length of the tabu list• Aspiration condition
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Tabu Search
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4. Variable Neighborhood Search – Hansen and Mladenović 1999
• Composed of three phase ① shaking ② local search ③ move
• A set of neighborhood structures, max
...21 kNNN
'max
' ,...,1, kkN k
'maxkk
'x ))('( ' xNxx k
''x ''xx
1k
1 kk1k
'x''x
Initialization: Select the set of neighborhood structures ; Find an initial solution x; Choose a stopping condition;
Repeat the following until no improvement is obtained: (1) Set ; (2) Until , repeat the following steps; (a) Shaking phase: Generate a random point of ;
(b) Local search: Apply some local search method with as initial solution; denote with the so obtained local optimum;
(c) Move or not: If the is better than x, set and , otherwise set
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Variable Neighborhood Search
x’’N1
x
x’
Bestx
x’’
N1
x
x’
Best
x’’
N2
x
Best
x’
x’’x x’’
N1
x
Best
x’
x’’
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V. Population-Based Methods
1. Genetic Algorithm – Holland 1975
Coding of a solution --- Chromosome
Fitness function --- Related to objective function
Initial population
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An example : TSP (Traveling Salesman Problem)
18
1
2
3
5
4
23
10 5
3
7
15206
A solution A sequence
12345 Tour length = 31
13452 Tour length = 63
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• Reproduction (Selection)
• Crossover
• Mutation
Genetic Algorithm
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Example 1
Maximize f(x) = x2 where x I and 0 x 31
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1. Coding of a solution : A five-bit integer, e.g. 01101
2. Fitness function : F(x) = f(x) = x2
3. Initial population : (Randomly generated) 0110111000 0100010011
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Reproduction
Roulette Wheel
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Reproduction
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Crossover
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Mutation
The probability of mutation
Pm = 0.001
20 bits * 0.001 = 0.02 bits
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Example 2 Word matching problem
• “to be or not to be” “tobeornottobe”
• (1/26)13 = 4.03038 10-19
• The lowercase letters in ASCII are represented by numbers in the range [97, 122]
[116,111,98,101,114,110, 116, 116, 111, 98,101]
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Initial population
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The corresponding strings
rzfqdhujardbeniedwrvrjahfjcyueisosqvcbfbfvgramtekuvskbuqrtjtjensbfwyqykktzyojhtbxblsoizggwmdtriusrgkmbgjvpbgemtpjalq
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2. Ant Colony Optimization – Dorigo 1992
Pheromone
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InitializationLoop
Loop Each ant applies a state transition rule to incrementally build a solution and applies a local updating rule to the pheromone
Until each of all ants has built a complete solution
A global pheromone updating rule is applied
Until End_Condition
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Example : TSPA simple heuristics – a greedy method: choose
the shortest edge to go out of a node
1
2
3
5
4
23
10 5
3
7
15206
18
Solution : 15432
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• Each ant builds a solution using a step by step constructive decision policy.
• How ant k choose the next city to visit?
otherwiseS
qqifs ruruJu r
,
})({maxarg 0
r
r
Jururu
rsrs
krs
Jsif
Jsifp
r
0
)(
)(
Where , be a distance measure associated with edge (r,s)
1 rs
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Local update of pheromone
10,)1( rsrsrs
,0 rs 0where is the initial pheromone level
Global update of pheromone
10,)1( rsrsrs
where globally best tour
otherwise
srifLgbrs,0
),(,1
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3. Particle Swarm Optimization – Kennedy and Eberhart 1995
• Population initialized by assigning random positpositionsions and velocitiesvelocities; potential solutions are then flown through hyperspace.
• Each particle keeps track of its “best” (highest fitness) position in hyperspace.– This is called “pBestpBest” for an individual particle.– It is called “gBestgBest” for the best in the population.
• At each time step, each particle stochastically accelerates toward its pBest pBest and gBestgBest.
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Particle Swarm Optimization Process
1. Initialize population in hyperspace. Includes position and velocity.
2. Evaluate fitness of individual particle.
3. Modify velocities based on personal best and global best.
4. Terminate on some condition.
5. Go to step 2.
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Initialization:- Positions and velocities
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Modify velocities - based on personal best and global best.
xt gBest
pBest
vxt+1
Here I am, now!
New Position !
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Particle Swarm Optimization
Modification of velocities and positions
vt+1 = vt + c1 * rand() * (pBest - xt ) + c2 * rand() * (gBest - xt ) xt+1 = xt + vt+1
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VI. Multiple Trajectory Search (MTS)
1.MTS begins with multiple initial solutions, so multiple trajectories will be produced in the search process.
2.Initial solutions will be classified as foreground solutions and background solutions, and the search will be focused mainly on foreground solutions.
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3.The search neighborhood begins with a large-scale neighborhood, then it contracts step by step until it reaches the pre-defined tiny size. At this time, it is reset to its original size.
4.Three candidate local search methods were designed. When we want to search the neighborhood of a solution, they will be tested first in order we can choose the one that best fits the landscape of the solution.
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• The MTS focuses its search mainly on foreground solutions. It does an iterated local search using one of three candidate local search methods. By choosing a local search method that best fits the landscape of a solution’s neighborhood, the MTS may find its way to a local optimum or the global optimum
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• Step 1 : Find M initial solutions that are uniformly distributed over the solution space.
• Step 2 : Iteratively apply local search to all these M initial solutions.
• Step 3 : Choose K foreground solutions based on the grades of the solutions.
• Step 4: Iteratively apply local search to K foreground solutions.
• Step 5 : If termination condition is met then stop else go to Step 3.
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• Three local search methods begin their search in a very large “neighborhood”. Then the neighborhood contracts step by step until it reaches a pre-defined tiny size, after then, it is reset to its original size.
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• Local Search 1This local search method searches along each of n dimensions in a randomly permuted order.
• Local Search 2This local search method searches along a direction by considering about n/4 dimensions.
Three Local Search Methods
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• Local Search 3This local search method also searches along each of n dimension in a randomly permuted order, but it evaluate multiple solutions along each dimension within a pre-specified range.
Three Local Search Methods
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Multiple Trajectory Search for Large Scale Global Optimization
Winner of 2008 IEEE World Congress on Computational Intelligence Competition on Large Scale Global Optimization
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Multiple Trajectory Search for Unconstrained/Constrained
Multi-Objective Optimization
The Second and the Third Places of 2009 IEEE Congress on Evolutionary ComputationCompetition on Performance Assessment of Constrained/Bound Constrained Multi-Objective Optimization Algorithm
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VII. The Applications of Metaheuristics
1. Solving NP-hard Optimization Problems• Traveling Salesman Problem• Maximum Clique Problem• Flow Shop Scheduling Problem• P-Median Problem
2. Search Problems in Many Applications• Feature Selection in Pattern Recognition• Automatic Clustering• Machine Learning (e.g. Neural Network
Training)
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VIII. Conclusions• For NP-hard optimization problems and compli
cated search problems, metaheuristic methods are very good choices for solving these problems.
• More efficient than branch-and-bound.• Obtain better quality solutions than heuristic me
thods.• Hybridization of metaheuristics• How to make the search more systematic ?• How to make the search more controllable ?• How to make the performance scalable?
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References1. Osman, I.H., and Laporte,G. “Metaheuristics:A bibliography”. Ann. Oper. Res. 63, 51
3–623, 1996.2. Blum, C., and Andrea R. “Metaheuristics in Combinatorial Optimization: Overview an
d Conceptual Comparison”. ACM Computing Surveys, 35(3), 268–308, 2003.3. Kirkpatrick, S., Gelatt. C. D., and Vecchi, M. P. “Optimization by simulated annealing
”, Science, 13 May 1983 220, 4598, 671–680, 1983.4. Glover, F. “Future paths for integer programming and links to artificial intelligence”,
Comput. Oper. Res. 13, 533–549, 1986.5. Hansen, P. and Mladenović, N. An introduction to variable neighborhood search. In M
etaheuristics: Advances and trends in local search paradigms for optimization, S. Voß, S. Martello, I. Osman, and C. Roucairol, Eds. Kluwer Academic Publishers, Chapter 30, 433–458, 1999.
6. Holland, J. H. Adaption in natural and artificial systems. The University of Michigan Press,Ann Harbor, MI. 1975.
7. Dorigo, M. Optimization, learning and natural algorithms (in italian). Ph.D. thesis, DEI, Politecnico di Milano, Italy. pp. 140, 1992.
8. Kennedy, J. and Eberhart, R. “Particle Swarm Optimization”, Proceedings of the 1995 IEEE International Conference on Neural Networks, pp. 1942-1948, IEEE Press, 1995.
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References
9. Lin-Yu Tseng* and Chun Chen, (2008) “Multiple Trajectory Search for Large Scale Global Optimization,” Proceedings of the 2008 IEEE Congress on Evolutionary Computation, June 2-6, 2008, Hong Kong.
10. Lin-Yu Tseng* and Chun Chen, (2009) “Multiple Trajectory Search for Unconstrained/Constrained Multi-Objective Optimization,” Proceedings of the 2009 IEEE Congress on Evolutionary Computation, May 18-21, Trondheim, Norway.
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謝謝 !
祝各位 身體健康 心情愉快 !