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IEEM 5119 Genetic Algorithms and Other Nature-Inspired Metaheuristic Algorithms

IEEM 5119 Genetic Algorithms and Other Nature-Inspired Metaheuristic Algorithms. Maw-Sheng Chern. http://chern.ie.nthu.edu.tw/gen/gen.htm.

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IEEM 5119 Genetic Algorithms and Other Nature-Inspired Metaheuristic Algorithms

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  1. IEEM5119Genetic Algorithmsand Other Nature-Inspired Metaheuristic Algorithms Maw-Sheng Chern http://chern.ie.nthu.edu.tw/gen/gen.htm

  2. 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.

  3. 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 algorithm’s execution. 1

  4. The Ant algorithm is based on the observation of real ant’s 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. 1

  5. 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, particle’s 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. 1

  6. 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. 1

  7. 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

  8. 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++, M&T 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. .

  9. 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)

  10. 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.

  11. 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. 1

  12. # 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 Stűtzle, Stochastic Local Search: Fundamentals and Applications, Morgan Kaufman Publishers, NY, 2005. 1

  13. 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 algorithmsfor optimisation [electronic resource], Berlin, Heidelberg : Springer Berlin Heidelberg, 2009. Randomized Algorithms Rajeev Motwani and Prabhakar Raghavan, Randomized Alogorithms, Cambridge University Press, NY, 1995. 1

  14. Contents 1 Introduction 2 Genetic Algorithms 3 Non-linear Programming Problems 4 Foundation of Genetic Algorithms 5 Metaheuristic Algorithms

  15. 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.

  16. The length of the minimal tour is 21134 km. [Aarts 1989]

  17. 1

  18. 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

  19. # of possible solutions = ( 179 digits) The shortest roundtrip through 120 German cities. The length of this tour is 6942 km.

  20. number of digits

  21. 2. Search Spaces (Solution Space) Algorithm – Problem Solving How to do efficient search in the state space? 9

  22. 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. 1

  23. 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, . . .

  24. Evolution Process . . . . . . . .    n-th Generation Initial Populations 1st Generation 2nd Generation Genetic Algorithm, Ant Algorithm, Particle Swarm Optimization, Bee Algorithm, Firefly Algorithm, . . . 9

  25. 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 linearprogram 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)

  26. 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

  27. A state space of a Traveling Salesman Problem with n = 4. 1

  28. 3. Search Methods 1

  29. 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. 1

  30. neighborhood search, simulated annealing population-based search It makes use the information of a set of solutions. e.g. genetic algorithms, ant algorithm, . . . 1

  31. Refinement:The best solution comes from a process of repeatedly refining and inventing alternative solutions.

  32. 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. 1

  33. 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 ) 1

  34. 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. ) 1

  35. 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. Ackley’s function

  36. 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

  37. [ Hoos 2005] 1

  38. 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

  39. 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.

  40. 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.

  41. T = the number of iterations. 1

  42. 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.

  43. Get one friend to sort the first half. Get one friend to sort the first half. 84, 57, 72, 50 21, 15, 18, 28, 32 5. Stochastic Quicksort Algorithm: Quicksort Pick a number as pivot element. 21, 15, 36, 28, 32, 18, 84, 57, 72, 50 15 84 ≤36≤ 21 72 18 50 57 28 32 21, 15, 18, 28, 32, 36, 84, 57, 72, 50

  44. Glue pieces together. 36 15,18,21,28,32,36,50,57,72,84 15, 18, 21, 28, 32 50, 57, 72, 84

  45. 15 18 21 28 32 36 50 57 72 84

  46. 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. 1

  47. 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. 1

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