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![Multi-CriterionMulti-Criterion
FitnessFitness
n
k
kkk
Wts
ffwMin
:..
)]())([: ∑ −= 'Y'Y'µ
n
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![Metropolis Criterion
• Let
• X be the current solution and X’ be the new solution
• C(x) (C(x’))be the energy state (cost) of x (x’)
• Probability Paccept = exp [(C(x)-C(x’))/ T]
• Let N=Random(0,1)
• Unconditional accepted if
• C(x’) < C(x), the new solution is better
• Probably accepted if
• C(x’) >= C(x), the new solution is worse . Accepted only
when N < Paccept](https://image.slidesharecdn.com/0101-150520165556-lva1-app6892/75/0101-genetic-algorithm-59-2048.jpg)
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0101.genetic algorithm
- 1. Genetic Algorithms And other approachesfor similar applications Optimization Techniques
- 2. OptimizationOptimization TechniquesTechniques • Mathematical ProgrammingMathematicalProgramming • Network AnalysisNetwork Analysis • Branch & BoundBranch & Bound • Genetic AlgorithmGenetic Algorithm • Simulated AnnealingSimulated Annealing • Tabu SearchTabu Search
- 3. Genetic AlgorithmGenetic Algorithm •Based on Darwinian Paradigm • Intrinsically a robust search and optimization mechanism Reproduction Competition SelectionSurvive
- 4.
- 5. Genetic Algorithm Introduction 1 •Inspired by natural evolution • Population of individuals • Individual is feasible solution to problem • Each individual is characterized by a Fitness function • Higher fitness is better solution • Based on their fitness, parents are selected to reproduce offspring for a new generation • Fitter individuals have more chance to reproduce • New generation has same size as old generation; old generation dies • Offspring has combination of properties of two parents • If well designed, population will converge to optimal solution
- 6. Algorithm BEGIN Generate initial population; Computefitness of each individual; REPEAT /* New generation /* FOR population_size / 2 DO Select two parents from old generation; /* biased to the fitter ones */ Recombine parents for two offspring; Compute fitness of offspring; Insert offspring in new generation END FOR UNTIL population has converged END
- 7.
- 8. Introduction 2 • Reproductionmechanisms have no knowledge of the problem to be solved • Link between genetic algorithm and problem: • Coding • Fitness function
- 9. Basic principles 1 •Coding or Representation • String with all parameters • Fitness function • Parent selection • Reproduction • Crossover • Mutation • Convergence • When to stop
- 10. Basic principles 2 •An individual is characterized by a set of parameters: Genes • The genes are joined into a string: Chromosome • The chromosome forms the genotype • The genotype contains all information to construct an organism: the phenotype • Reproduction is a “dumb” process on the chromosome of the genotype • Fitness is measured in the real world (‘struggle for life’) of the phenotype
- 11. Coding • Parameters ofthe solution (genes) are concatenated to form a string (chromosome) • All kind of alphabets can be used for a chromosome (numbers, characters), but generally a binary alphabet is used • Order of genes on chromosome can be important • Generally many different codings for the parameters of a solution are possible • Good coding is probably the most important factor for the performance of a GA • In many cases many possible chromosomes do not code for feasible solutions
- 12. Genetic AlgorithmGenetic Algorithm •Encoding • Fitness Evaluation • Reproduction • Survivor Selection
- 13. EncodingEncoding • Design alternative individual (chromosome) • Single design choice gene • Design objectives fitness
- 14. ExampleExample • Problem • Schedulen jobs on m processors such that the maximum span is minimized. Design alternative: job i ( i=1,2,…n) is assigned to processor j (j=1,2,…,m) Individual: A n-vector x such that xi = 1, …,or m Design objective: minimize the maximal span Fitness: the maximal span for each processor
- 15. Reproduction • Reproduction operators •Crossover • Mutation
- 16. Reproduction • Crossover • Twoparents produce two offspring • There is a chance that the chromosomes of the two parents are copied unmodified as offspring • There is a chance that the chromosomes of the two parents are randomly recombined (crossover) to form offspring • Generally the chance of crossover is between 0.6 and 1.0 • Mutation • There is a chance that a gene of a child is changed randomly • Generally the chance of mutation is low (e.g. 0.001)
- 17. Reproduction Operators • Crossover •Generating offspring from two selected parents Single point crossover Two point crossover (Multi point crossover) Uniform crossover
- 18. One-point crossover 1 •Randomly one position in the chromosomes is chosen • Child 1 is head of chromosome of parent 1 with tail of chromosome of parent 2 • Child 2 is head of 2 with tail of 1 Parents: 1010001110 0011010010 Offspring: 0101010010 0011001110 Randomly chosen position
- 19. Reproduction Operators comparison •Single point crossover Cross point • Two point crossover (Multi point crossover)
- 20. One-point crossover -Nature 1 2 12 1 2 2 1
- 21. Two-point crossover Parents: 10100011100011010010 Offspring: 0101010010 0011001110 Randomly chosen positions • Randomly two positions in the chromosomes are chosen • Avoids that genes at the head and genes at the tail of a chromosome are always split when recombined
- 22. Uniform crossover • Arandom mask is generated • The mask determines which bits are copied from one parent and which from the other parent • Bit density in mask determines how much material is taken from the other parent (takeover parameter) Mask: 0110011000 (Randomly generated) Parents: 1010001110 0011010010 Offspring: 0011001010 1010010110
- 23. Reproduction Operators • Uniformcrossover • Is uniform crossover better than single crossover point? – Trade off between • Exploration: introduction of new combination of features • Exploitation: keep the good features in the existing solution
- 24. Problems with crossover •Depending on coding, simple crossovers can have high chance to produce illegal offspring • E.g. in TSP with simple binary or path coding, most offspring will be illegal because not all cities will be in the offspring and some cities will be there more than once • Uniform crossover can often be modified to avoid this problem • E.g. in TSP with simple path coding: Where mask is 1, copy cities from one parent Where mask is 0, choose the remaining cities in the order of the other parent
- 25. Reproduction Operators • Mutation •Generating new offspring from single parent • Maintaining the diversity of the individuals Crossover can only explore the combinations of the current gene pool Mutation can “generate” new genes
- 26. Reproduction Operators • Controlparameters: population size, crossover/mutation probability • Problem specific • Increase population size Increase diversity and computation time for each generation • Increase crossover probability Increase the opportunity for recombination but also disruption of good combination • Increase mutation probability Closer to randomly search Help to introduce new gene or reintroduce the lost gene • Varies the population • Usually using crossover operators to recombine the genes to generate the new population, then using mutation operators on the new population
- 27. Parent/Survivor Selection • Strategies • Survivorselection Always keep the best one Elitist: deletion of the K worst Probability selection : inverse to their fitness Etc.
- 28. Parent/Survivor Selection • Toostrong fitness selection bias can lead to sub- optimal solution • Too little fitness bias selection results in unfocused and meandering search
- 29. Parent selection Chance tobe selected as parent proportional to fitness • Roulette wheel To avoid problems with fitness function • Tournament Not a very important parameter
- 30. Parent/Survivor Selection • Strategies • Parentselection Uniform randomly selection Probability selection : proportional to their fitness Tournament selection (Multiple Objectives) Build a small comparison set Randomly select a pair with the higher rank one beats the lower one Non-dominated one beat the dominated one Niche count: the number of points in the population within certain distance, higher the niche count, lower the
- 31. Others • Global Optimal •Parameter Tuning • Parallelism • Random number generators
- 32. Example of codingfor TSP Travelling Salesman Problem • Binary • Cities are binary coded; chromosome is string of bits Most chromosomes code for illegal tour Several chromosomes code for the same tour • Path • Cities are numbered; chromosome is string of integers Most chromosomes code for illegal tour Several chromosomes code for the same tour • Ordinal • Cities are numbered, but code is complex • All possible chromosomes are legal and only one chromosome for each tour • Several others
- 33. Roulette wheel • Sumthe fitness of all chromosomes, call it T • Generate a random number N between 1 and T • Return chromosome whose fitness added to the running total is equal to or larger than N • Chance to be selected is exactly proportional to fitness Chromosome : 1 2 3 4 5 6 Fitness: 8 2 17 7 4 11 Running total: 8 10 27 34 38 49 N (1 ≤ N ≤ 49): 23 Selected: 3
- 34. Tournament • Binary tournament •Two individuals are randomly chosen; the fitter of the two is selected as a parent • Probabilistic binary tournament • Two individuals are randomly chosen; with a chance p, 0.5<p<1, the fitter of the two is selected as a parent • Larger tournaments • n individuals are randomly chosen; the fittest one is selected as a parent • By changing n and/or p, the GA can be adjusted dynamically
- 35. Problems with fitnessrange • Premature convergence ∀ ∆Fitness too large • Relatively superfit individuals dominate population • Population converges to a local maximum • Too much exploitation; too few exploration • Slow finishing ∀ ∆Fitness too small • No selection pressure • After many generations, average fitness has converged, but no global maximum is found; not sufficient difference between best and average fitness • Too few exploitation; too much exploration
- 36. Solutions for theseproblems • Use tournament selection • Implicit fitness remapping • Adjust fitness function for roulette wheel • Explicit fitness remapping Fitness scaling Fitness windowing Fitness ranking Will be explained below
- 37. Fitness Function Purpose • Parentselection • Measure for convergence • For Steady state: Selection of individuals to die • Should reflect the value of the chromosome in some “real” way • Next to coding the most critical part of a GA
- 38. Fitness scaling • Fitnessvalues are scaled by subtraction and division so that worst value is close to 0 and the best value is close to a certain value, typically 2 • Chance for the most fit individual is 2 times the average • Chance for the least fit individual is close to 0 • Problems when the original maximum is very extreme (super-fit) or when the original minimum is very extreme (super-unfit) • Can be solved by defining a minimum and/or a maximum value for the fitness
- 39. Example of FitnessScaling
- 40. Fitness windowing • Sameas window scaling, except the amount subtracted is the minimum observed in the n previous generations, with n e.g. 10 • Same problems as with scaling
- 41. Fitness ranking • Individualsare numbered in order of increasing fitness • The rank in this order is the adjusted fitness • Starting number and increment can be chosen in several ways and influence the results • No problems with super-fit or super-unfit • Often superior to scaling and windowing
- 42. Fitness EvaluationFitness Evaluation •A key component in GA • Time/quality trade off • Multi-criterion fitness
- 43. Multi-Criterion FitnessMulti-Criterion Fitness •Dominance and indifference • For an optimization problem with more than one objective function (fi, i=1,2,…n) • given any two solution X1 and X2, then X1 dominates X2 ( X1 X2), if fi(X1) >= fi(X2), for all i = 1,…,n X1 is indifferent with X2 ( X1 ~ X2), if X1 does not dominate X2, and X2 does not dominate X1
- 44. Multi-Criterion FitnessMulti-Criterion Fitness •Pareto Optimal Set • If there exists no solution in the search space which dominates any member in the set P, then the solutions belonging the the set P constitute a global Pareto-optimal set. • Pareto optimal front • Dominance Check
- 45. Multi-Criterion FitnessMulti-Criterion Fitness •Weighted sum • F(x) = w1f1(x1) + w2f2(x2) +…+wnfn(xn) • Problems? Convex and convex Pareto optimal front Sensitive to the shape of the Pareto- optimal front Selection of weights? Need some pre-knowledge Not reliable for problem involving uncertainties
- 46. Multi-Criterion FitnessMulti-Criterion Fitness •Optimizing single objective • Maximize: fk(X) Subject to: fj(X) <= Ki, i <> k X in F where F is the solution space.
- 47. Multi-Criterion FitnessMulti-Criterion Fitness •Weighted sum • F(x) = w1f1(x1) + w2f2(x2) +…+wnfn(xn) • Problems? Convex and convex Pareto optimal front Sensitive to the shape of the Pareto- optimal front Selection of weights? Need some pre-knowledge Not reliable for problem involving uncertainties
- 48. Multi-Criterion FitnessMulti-Criterion Fitness •Preference based weighted sum (ISMAUT Imprecisely Specific Multiple Attribute Utility Theory) • F(x) = w1f1(x1) + w2f2(x2) +…+wnfn(xn) • Preference Given two know individuals X and Y, if we prefer X than Y, then F(X) > F(Y), that is w1(f1(x1)-f1(y1)) +…+wn(fn(xn)-fn(yn)) > 0
- 49. All the preferencesconstitute a linear space Wn={w1,w2,…,wn} w1(f1(x1)-f1(y1)) +…+wn(fn(xn)-fn(yn)) > 0 w1(f1(z1)-f1(p1)) +…+wn(fn(zn)-fn(pn)) > 0, etc For any two new individuals Y’ and Y’’, how to determine which one is more preferable? Multi-Criterion FitnessMulti-Criterion Fitness
- 50. Multi-CriterionMulti-Criterion FitnessFitness n k kkk Wts ffwMin :.. )]())([: ∑ −='Y'Y'µ n k kkk Wts ffwMin :.. )]())([': ∑ −= Y''Y'µ
- 51. Multi-CriterionMulti-Criterion FitnessFitness 'Y'Y '0 ⇒>µ Y'Y''0' ⇒>µ Then, Otherwise, Y’ ~ Y’’ Construct the dominant relationship among some indifferent ones according to the preferences.
- 52. Other parameters ofGA 1 • Initialization: • Population size • Random • Dedicated greedy algorithm • Reproduction: • Generational: as described before (insects) • Generational with elitism: fixed number of most fit individuals are copied unmodified into new generation • Steady state: two parents are selected to reproduce and two parents are selected to die; two offspring are immediately inserted in the pool (mammals)
- 53. Other parameters ofGA 2 • Stop criterion: • Number of new chromosomes • Number of new and unique chromosomes • Number of generations • Measure: • Best of population • Average of population • Duplicates • Accept all duplicates • Avoid too many duplicates, because that degenerates the population (inteelt) • No duplicates at all
- 54. Example run Maxima andAverages of steady state and generational replacement 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 St_max St_av. Ge_max Ge_av.
- 55. Simulated Annealing • What • Exploitsan analogy between the annealing process and the search for the optimum in a more general system.
- 56. Annealing Process • AnnealingProcess • Raising the temperature up to a very high level (melting temperature, for example), the atoms have a higher energy state and a high possibility to re-arrange the crystalline structure. • Cooling down slowly, the atoms have a lower and lower energy state and a smaller and smaller possibility to re-arrange the crystalline structure.
- 57. Simulated Annealing • Analogy •Metal Problem • Energy State Cost Function • Temperature Control Parameter • A completely ordered crystalline structure the optimal solution for the problem Global optimal solution can be achieved as long as the cooling process is slow enough.
- 58. Metropolis Loop • Theessential characteristic of simulated annealing • Determining how to randomly explore new solution, reject or accept the new solution at a constant temperature T. • Finished until equilibrium is achieved.
- 59. Metropolis Criterion • Let •X be the current solution and X’ be the new solution • C(x) (C(x’))be the energy state (cost) of x (x’) • Probability Paccept = exp [(C(x)-C(x’))/ T] • Let N=Random(0,1) • Unconditional accepted if • C(x’) < C(x), the new solution is better • Probably accepted if • C(x’) >= C(x), the new solution is worse . Accepted only when N < Paccept
- 60. Algorithm Initialize initial solutionx , highest temperature Th, and coolest temperature Tl T= Th When the temperature is higher than Tl While not in equilibrium Search for the new solution X’ Accept or reject X’ according to Metropolis Criterion End Decrease the temperature T End
- 61. Simulated Annealing • Definitionof solution • Search mechanism, i.e. the definition of a neighborhood • Cost-function
- 62. Control Parameters • Definitionof equilibrium • Cannot yield any significant improvement after certain number of loops • A constant number of loops • Annealing schedule (i.e. How to reduce the temperature) • A constant value, T’ = T - Td • A constant scale factor, T’= T * Rd A scale factor usually can achieve better performance
- 63. Control Parameters • Temperaturedetermination • Artificial, without physical significant • Initial temperature 80-90% acceptance rate • Final temperature A constant value, i.e., based on the total number of solutions searched No improvement during the entire Metropolis loop Acceptance rate falling below a given (small) value • Problem specific and may need to be tuned
- 64. Example • Traveling SalesmanProblem (TSP) • Given 6 cities and the traveling cost between any two cities • A salesman need to start from city 1 and travel all other cities then back to city 1 • Minimize the total traveling cost
- 65. Example • Solution representation •An integer list, i.e., (1,4,2,3,6,5) • Search mechanism • Swap any two integers (except for the first one) (1,4,2,3,6,5) (1,4,3,2,6,5) • Cost function
- 66. Example • Temperature • Initialtemperature determination Around 80% acceptation rate for “bad move” Determine acceptable (Cnew – Cold) • Final temperature determination Stop criteria Solution space coverage rate • Annealing schedule Constant number (90% for example) Depending on solution space coverage rate
- 67. Others • Global optimalis possible, but near optimal is practical • Parameter Tuning –Aarts, E. and Korst, J. (1989). Simulated Annealing and Boltzmann Machines. John Wiley & Sons. • Not easy for parallel implementation • Randomly generator
- 68. Optimization Techniques • MathematicalProgramming • Network Analysis • Branch & Bound • Genetic Algorithm • Simulated Annealing • Tabu Search
- 69. Tabu Search • What • Neighborhoodsearch + memory Neighborhood search Memory Record the search history Forbid cycling search
- 70. Algorithm • Choose aninitial solution X • Find a subset of N(x) the neighbor of X which are not in the tabu list. • Find the best one (x’) in N(x). • If F(x’) > F(x) then set x=x’. • Modify the tabu list. • If a stopping condition is met then stop, else go to the second step.
- 71. Effective Tabu Search •Effective Modeling • Neighborhood structure • Objective function (fitness or cost) Example Graph coloring problem: Find the minimum number of colors needed such that no two connected nodes share the same color. • Aspiration criteria • The criteria for overruling the tabu constraints and differentiating the preference of among the neighbors
- 72. Effective Tabu Search •Effective Computing • “Move” may be easier to be stored and computed than a completed solution move: the process of constructing of x’ from x • Computing and storing the fitness difference may be easier than that of the fitness function.
- 73. Effective Tabu Search •Effective Memory UseEffective Memory Use • Variable tabu list size For a constant size tabu list Too long: deteriorate the search results Too short: cannot effectively prevent from cycling • Intensification of the search Decrease the tabu list size • Diversification of the search Increase the tabu list size Penalize the frequent move or unsatisfied constraints
- 74. Example • A hybridapproach for graph coloring problem • R. Dorne and J.K. Hao, A New Genetic Local Search Algorithm for Graph Coloring, 1998
- 75. Problem • Given anundirected graph G=(V,E) • V={v1,v2,…,vn} • E={eij} • Determine a partition of V in a minimum number of color classes C1,C2,…,Ck such that for each edge eij, vi and vj are not in the same color class. • NP-hard
- 76. General Approach • Transforman optimization problem into a decision problem • Genetic Algorithm + Tabu Search • Meaningful crossover • Using Tabu search for efficient local search
- 77. Encoding • Individual • (Ci1,Ci2, …, Cik) • Cost function • Number of total conflicting nodes Conflicting node having same color with at least one of its adjacent nodes • Neighborhood (move) definition • Changing the color of a conflicting node • Cost evaluation • Special data structures and techniques to improve the efficiency
- 78. Implementation • Parent Selection •Random • Reproduction/Survivor • Crossover Operator • Unify independent set (UIS) crossover Independent set Conflict-free nodes set with the same color Try to increase the size of the independent set to
- 79.
- 80. Implementation • Mutation • WithProbability Pw, randomly pick neighbor • With Probability 1 – Pw, Tabu search Tabu search Tabu list List of {Vi, cj} Tabu tenure (the length of the tabu list) L = a * Nc + Random(g) Nc: Number of conflicted nodes a,g: empirical parameters
- 81. Summary • Neighbor Search •TS prevent being trapped in the local minimum with tabu list • TS directs the selection of neighbor • TS cannot guarantee the optimal result • Sequential • Adaptive
- 82.
- 83. sources Jaap Hofstede, Beasly,Bull, Martin Version 2, October 2000 Department of Computer Science & Engineering University of South Carolina Spring, 2002