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IEEM 5119Genetic Algorithmsand Other

Nature-Inspired Metaheuristic Algorithms

Maw-Sheng Chern

http//chern.ie.nthu.edu.tw/gen/gen.htm

This course is tended to introduce the stochastic

and other extended local search methods (genetic

algorithms, simulated annealing, tabu search, ant

algorithms, particle swarm optimization, bee

algorithm, etc.) and their applications to

difficult-to-solve optimization problems in

industrial engineering and manufacturing systems

design. Genetic algorithms is based on the

concepts from population genetics and evolution

theory. The algorithm are constructed to optimize

fitness of a population of elements through

crossover (recombination) and mutation

(perturbation) operations on their genes.

Simulated annealing is based on an analog of

cooling the material in a heat bath a process

known as annealing. A solid material is heated in

a heat bath until it melts, then cooling it down

slowly until it crystallizes into a solid state

(low-energy state). The atoms in the material

have high energies at high temperatures and more

freedom to arrange themselves. As the temperature

is reduced, the atom energies decrease. The

structural properties of the solid depend on the

rate of cooling. From the point of view of search

methods for optimization problems, simulated

annealing is a stochastic local search method. It

always accepts a selected better-cost local

solution and it may also accept a worse-cost

local solution with a probability which is

gradually decreased in the course of algorithms

execution.

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The Ant algorithm is based on the observation of

real ants behavior. Ants can coordinate their

activities via stigmergy, a way of indirect

communication through the modification of the

environment. The main idea of ant algorithm is to

use self-organizing principles of artificial

agents which collaborate to solve the problems.

Tabu search is a deterministic iterative

improvement local search method with a

possibility to accept worse-cost local solution

in order to escape from local optimum. The set of

legal local solutions are restricted by a tabu

list which is designed to present from going back

to the recently visited solutions. The set of

solutions in tabe list are not accepted in the

next iteration.

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Particle Swarm Optimization (PSO) algorithm is a

population-based stochastic optimization method

proposed by James Kennedy and R. C. Eberhart in

1995. It is motivated by social behavior of

organisms such as bird flocking and fish

schooling. In the PSO algorithm, the potential

solutions called particles, are flown in the

problem hyperspace. Change of position of a

particle is called velocity. The particle changes

their position with time. During flight,

particles velocity is stochastically accelerated

toward its previous best position and toward a

neighborhood best solution. POS has been

successfully applied to solve various

optimization problems, artificial neural network

training, fuzzy system control, and others.

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The Bees Algorithm is a new population-based

search algorithm, first developed in 2005 by Pham

DT etc. 1 and Karaboga.D 2 independently. The

algorithm mimics the food foraging behaviour of

swarms of honey bees. In its basic version, the

algorithm performs a kind of neighbourhood search

combined with random search and can be used for

optimization problems.

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Textbook References

- Geng, Mitsuo and Cheng, Runwei, Genetic

Algorithms Engineering Design, John Wiley

Sons, New York, 1997. - Geng, Mitsuo and Cheng, Runwei, Genetic

Algorithms Engineering Optimization, John Wiley

Sons, New York, 2000. - Mitsuo Gen, Runwei Cheng, Lin Lin,,Network Models

and Optimization electronic resource

Multiobjective Genetic Algorithm Approach,

London Springer-Verlag, 2008.

3. Lecturer Slides

4. Word-Wide-Web

5. Course homepage

http//chern.ie.nthu.edu.tw/gen/gen.htm

References Genetic Algorithms Michalewicz ,

Zbigniew, Genetic Algorithms Data Structures

Evolution Programs, 1994, Third Revised and

Extended Edition, Springer, New York, 1999.

Goldberg, David E., Genetic Algorithms in

Search, Optimization Machine Learning,

Addison-Wesley Publishing Company, Inc. New York,

1989. Scott Robert Ladd., Genetic Algorithms

in C, MT Books, New York, 1996. Bagchi,

Tapan P., Multiobjective Scheduling by Genetic

Algorithms, Kluwer Academic Publishers, Boston,

Golderberg, David E., Genetic Algorithms in

Search, Optimization Machine Learning,

Addison-Wesley Publishing Company, Inc., New

York, 1989. Mitsuo Gen, Runwei Cheng, Lin

Lin,,Network Models and Optimization electronic

resource Multiobjective Genetic Algorithm

Approach, London Springer-Verlag, 2008. .

Bauer, J., Genetic Algorithms and Investment

Strategies, John Wiley Sons, New York, 1994.

Michell, M., An Introduction to Genetic

Algorithms, MIT Press, Cambridge, MA, 1996. Duc

Truong Pham and Dervis Karaboga, Intelligent

Algorithms, tabu search, simulated annealing and

neural networks, Springer, New York,

1998. Charles, L. Karr and L. Michael Freeman

(ed.), Industrial Applications of Genetic

Algorithms, CRC Press, New York, 1998. Man, K.

F., K.S. Tang and S. Kwong, Genetic Algorithms,

Springer, New York, 1999. (Refer a genetic game

in this book) Erick Cantu-Paz , Efficient and

Accurate Parallel Genetic Algorithms (Genetic

Algorithms and Evolutionary Computation Volume 1)

Randy L. Haupt, Sue Ellen Haupt, Practical

Genetic Algorithms George Lawton, A Practical

Guide to Genetic Algorithms in C/Book and

Disk Andrzej Osyczka, Evolution Algorithms for

single and Multicriteria Design Optimization,

Physica-Verlag, Heidelberg, 2002. Thomas Bäck,

Evolution Algorithms in Theory and Practice

Evolution Strategies, Evolution Programming,

Genetic Algorithms, Oxford University Press,

Oxford, 1996. Colin R. Reeves, Jonathan E. Rowe,

Evolution Algorithms Principles and

Perspectives, A Guide to GA Theory, Kluwer

Academic Publishers, Boston, 2003. David Corne,

Marco Dorigo, and Fred Glover (ed.), New Ideas in

Optimization, The McGraw-Hill Companies, NY,

1999.

Ant Algorithms Marco Dorigo, Thomas Stützle,

Ant colony optimization, Cambridge, Mass., MIT

Press, c2004 Eric Bonabeau, Marco Dorigo, and

Guy Theraulaz, Swarm Intelligence From Natural

to Artificial Systems, Oxford University Press,

NY, 1999. David Corne, Marco Dorigo, and Fred

Glover (ed.), New Ideas in Optimization, The

McGraw-Hill Companies, NY, 1999. James Kennedy,

Russell C. Eberhart, and Yuhui Shi, Swarm

Intelligence, Morgan Kaufmann Publishers, San

Francisco, 2001.

Simulated Annealing Emile Aarts and Jan Korst,

Simulate Annealing and Boltzmann Machines, John

Wiley Sons, Inc. NY, 1989.

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R.H.J.M. Otten and L.P.P.P. van Ginneken, The

Annealing Algorithm, Kluwer Academic Publishers,

Boston, MA, U.S.A, 1989.

Tabu Search F. Glover and M.Laguna, Tabu

Search, Kluwer Academic Publishers, Boston, MA,

U.S.A, 1997.

General Stochastic Search Methods Emile H. L.

Aarts and Jan Karel Lenstra, Local Search in

Combinatorial Optimization, John Wiley and Sons,

NY, 1997. Holger H. Hoos and Thomas Stutzle,

Stochastic Local Search Fundamentals and

Applications, Morgan Kaufman Publishers, NY, 2005.

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J. C. Spall, Introduction to Stochastic Search

and Optimization Solving Estimation, Simulation,

and Control, John Wiley and Sons Inc., NY,

2003. George S. Tarasenko, Stochastic

Optimization in the Soviet Union Random Search

Algorithms, Delphic Associates, Inc., VA, 1985.

Colin R. Reeves (ed.), Mordern Heuristic

Techniques for Combinatorial Problems, John

Wiley Sons, Inc. NY, 1993. Sadiq M. Sait and

Habib Youssef, Iterative Computer Algorithms with

Application to Engineering Solving Combinatorial

Optimization Problems, IEEE Computer Society, LA,

1999. Xin-She Yang, Nature-Inspired Metaheuristic

Algorithms, Luniver Press 2008. Raymond Chiong

(ed.), Nature-inspired algorithms for

optimisation electronic resource, Berlin,

Heidelberg Springer Berlin Heidelberg, 2009.

Randomized Algorithms Rajeev Motwani and

Prabhakar Raghavan, Randomized Alogorithms,

Cambridge University Press, NY, 1995.

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1. The Computational complexity We do not

expect NP-hard (and NP-complete) problems to be

solved in polynomial steps. Ambati et al.

(1991) An evolution algorithm that can achieve

heuristic solutions 25 worse than the expected

optimal solution on random Traveling Salesman

Problem in O(NlogN) time. Fogel (1993) An

evolution algorithm that can achieve heuristic

solutions 10 worse than the expected optimal

solution on random Traveling Salesman Problem in

O(N2) time.

The length of the minimal tour is 21134 km.

Aarts 1989

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The sizes of the Traveling Salesman

Problem 100,000 105 people in a

stadium. 5,500,000,000 5.5 ? 109 people on

earth. 1,000,000,000,000,000,000,000 1021

liters of water on the earth. 1010 years 3 ?

1017 seconds The age of the universe

of possible solutions

( 179 digits)

The shortest roundtrip through 120 German cities.

The length of this tour is 6942 km.

number of digits

2. Search Spaces (Solution Space)

Algorithm Problem Solving

How to do efficient search in the state space?

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e.g.

Karash-Kuhn-Tucker condition

1. To obtain a good initial state.

2. Goal identification (identify the optimal

condition).

For some problems, we are not able to identify

the goal state.

3. Control the search process.

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State Space Representation

( a partial solution or a complete solution )

( the desired solution )

How to search in the state space?

Simplex Method, . . . , Tabu Search, Simulated

Annealing, . . .

Evolution Process

. . . .

. . . .

?

?

?

n-th Generation

Initial Populations

1st Generation

2nd Generation

Genetic Algorithm, Ant Algorithm, Particle Swarm

Optimization, Bee Algorithm, Firefly Algorithm, .

. .

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Goal identification 1. The goal state can be

identified. (e.g. linear program, convex program

) 2. The goal state cannot be identified. (e.g.

traveling salesman problem, quadratic assignment

problem, knapsack problem, scheduling problems, .

. . )

Example 1 Simplex algorithm for linear program

Goal identification primal feasible dual

feasible condition

Control the search process moving to improved

adjacent basic solution

Example 2 Steepest descent algorithm for convex

program

Goal identification Karash-Kuhn-Tucker condition

Control the search process moving to the

steepest descent direction ( minimization

problem)

Example 3 Genetic algorithm for the Traveling

Salesman Problem

Goal identification The optimal condition can

not be identified efficiently. Using heuristic

rules to identify approximate solution.

Control the search process Crossover, mutation

and selection operations.

Example 4 Ant algorithm for the Traveling

Salesman Problem

Goal identification The optimal condition can

not be identified efficiently. Using heuristic

rules to identify approximate solution.

Control the search process pheromone state

transition probability

A state space of a Traveling Salesman Problem

with n 4.

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3. Search Methods

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Exact search methods Advantage To produce

an optimal solution and It is able to detect that

a given problem has no feasible

solution. Disadvantage It is time-consuming.

For example, it is infeasible for real-time

problems.

Traversal Search, Backtracking Algorithm, Branch

and Bound Algorithm, Dynamic Programming Method,

etc.

Local search methods Advantage It is

time-efficient and easy to write a program.

Disadvantage It may not produce an optimal

solution and is not able to detect that a given

problem has no feasible solution.

Traditional Local search does not provide a

mechanism for the search to escape from a local

optimum. The goal of local search is find a

solution which is as close as possible to the

optimum.

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neighborhood search, simulated annealing

population-based search It makes use the

information of a set of solutions. e.g. genetic

algorithms, ant algorithm, . . .

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RefinementThe best solution comes from a

process of repeatedly refining and inventing

alternative solutions.

Constructive Search Methods To generate a

complete solution by iteratively extending

partial solutions.

A ? J ? D ? E ? F ? K ? H ? I ? G ? C ? B ?

A (A J D E F K H I G C B) is a complete solution.

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Perturbation Search Methods For a complete

solution, we can easily change it into new

complete solution by modifying one or more

solution components.

For example, in TSP a complete solution (ABCD) is

changes into a new solution (ADCB) by interchange

the positions of B and D. ( neighborhood search

methods, mutation operations i.e. )

( The Liberty Times, July 27, 2005 )

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For a set of complete solutions, we can easily

change them into new complete solutions by

modifying one or more solution components among

the solutions. ( crossover and mutation

operations in genetic algorithms. etc. )

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Deterministic Algorithms In each search step, it

progresses toward the complete solution by making

deterministic decision. e.g. Simplex method,

Quasi-Newton algorithms, tabu search and many

other conventional algorithms.

Deterministic algorithm will produce the same

solution for a given problem instance. Even for

the same instance, the stochastic algorithm

usually product distinct solutions at each run.

Ackleys function

4. Stochastic Algorithms It make a random

decision at each search step. e.g. Monte Carlo

algorithms, simulated annealing, genetic

algorithms, ant algorithms, etc. There are two

cases. (1) The available information the

objective function to be optimized may be

considered possible erroneous or corrupted by

random noise. (2) For the case with perfect

information, we may introduce a random element to

guide us when searching for the optimum solution.

1

Hoos 2005

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Why stochastic algorithms?

1. They are efficient for the practical uses. 2.

They are simple to implement. For many

applications, stochastic algorithm is the

simplest algorithm available, or fastest, or

both. 3. They are every general and can be

implemented for a wide class of optimization. For

example, no differential function of real valued

parameters is required. It need not be

expressible in any particular constraint

language. 4. They can run in parallel. The

quality of solutions may be improved time by time.

trade-off

computing time ?? solution

quality

Configuration Graph

Transition probabilities for a deterministic

algorithm

Minimize f (x) subject to x ? S

f (xk) 2

Deterministic algorithm will produce the same

solution for a given problem instance.

Transition probabilities for a stochastic

algorithm

Remark Transition probabilities may be dependent

on the number of iterations.

Even for the same instance, the stochastic

algorithm usually product distinct solutions at

each run.

T the number of iterations.

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There are two ways to avoid getting trapped in a

local optimum.

- Accommodate nongreedy search move. It is allowed

to move to a neighborhood state with a worse

function value. ( Tabu search, simulated

annealing, . . . ) - To increase the number of edges in the

configuration graph. However, the denser the

configuration graph is, the more inefficient

search step will be.

To enlarge the neighborhood for each state.

5. Stochastic Quicksort Algorithm

Quicksort

Pick a number as pivot element.

21, 15, 36, 28, 32, 18, 84, 57, 72, 50

15

84

21

36

72

18

50

57

28

32

21, 15, 18, 28, 32, 36, 84, 57, 72, 50

50, 57, 72, 84

15 18 21 28 32

36 50 57 72 84

Deterministic Quick Sort Algorithm The pivot

element at each step is selected by using

deterministic rules.

Las Vegas Algorithm

Stochastic Quick Sort Algorithm The pivot

element at each step is selected randomly.

Theorem 5.1 Motwani 1995 The expected number

of comparisons in an execution of Stochastic

Quicksort algorithm is O(nlogn) where n elements

are to be sorted.

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6. A Stochastic Algorithm for Min-Cut

Problem For a multigraph G, a cut is a set of

edges whose removal results in G being broken

into two or more components. A min-cut is a cut

with minimum cardinality.

Multigraph G

a, e, g , f, g , a, b, d , c, d ,

a, b, e, g are cuts of G.

f, g , c, d are min-cuts of G.

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Contracting an edge

The effect of contracting edge e.

Important Observation An edge contraction does

not reduce the min-cut size of G.

Monte-Carlo algorithm

- A stochastic min-cut algorithm
- Pick an edge uniformly at random and merge its

two vertices. - With each contraction, the number of

vertices of G decreased by one. - 2. It continues the contraction process until

only two vertices remain. Then, the set of edges

between these two vertices is a cut of G.

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select e

c, d is a cut.

select a

select g

Theorem 6.1 Motwani 1995 The probability of

discovering a particular min-cut is larger than

2/n2 where n is the number of nodes.

Theorem 6.2 Motwani 1995 If we run the

algorithm n2/2 times, making independent random

choices each time, then the probability that a

min-cut is not found in any of the n2/2

attempts is at most

lt 1/e 1/3.1416.

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