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This is a keynote talk at the 10th Symposium of Experimental Algorithms (SEA2011) in Greece, 2011.

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- 1. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Metaheristics Optimization: Algorithm Analysis and Open Problems Xin-She Yang National Physical Laboratory, UK @ SEA 2011Xin-She Yang 2011Metaheuristics and Optimization
- 2. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate.Xin-She Yang 2011Metaheuristics and Optimization
- 3. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate.Xin-She Yang 2011Metaheuristics and Optimization
- 4. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate. All models are wrong, but some are useful. - George Box, StatisticianXin-She Yang 2011Metaheuristics and Optimization
- 5. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate. All models are inaccurate, but some are useful. - George Box, StatisticianXin-She Yang 2011Metaheuristics and Optimization
- 6. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate. All models are inaccurate, but some are useful. - George Box, Statistician All algorithms perform equally well on average over all possible functions. - No-free-lunch theorems (Wolpert & Macready)Xin-She Yang 2011Metaheuristics and Optimization
- 7. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate. All models are inaccurate, but some are useful. - George Box, Statistician All algorithms perform equally well on average over all possible functions. How so? - No-free-lunch theorems (Wolpert & Macready)Xin-She Yang 2011Metaheuristics and Optimization
- 8. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate. All models are inaccurate, but some are useful. - George Box, Statistician All algorithms perform equally well on average over all possible functions. Not quite! (more later) - No-free-lunch theorems (Wolpert & Macready)Xin-She Yang 2011Metaheuristics and Optimization
- 9. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate. All models are inaccurate, but some are useful. - George Box, Statistician All algorithms perform equally well on average over all possible functions. Not quite! (more later) - No-free-lunch theorems (Wolpert & Macready)Xin-She Yang 2011Metaheuristics and Optimization
- 10. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOverviewOverview Introduction Metaheuristic Algorithms Applications Markov Chains and Convergence Analysis Exploration and Exploitation Free Lunch or No Free Lunch? Open ProblemsXin-She Yang 2011Metaheuristics and Optimization
- 11. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMetaheuristic AlgorithmsMetaheuristic Algorithms Essence of an Optimization Algorithm To move to a new, better point xi +1 from an existing known location xi . xi x2 x1Xin-She Yang 2011Metaheuristics and Optimization
- 12. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMetaheuristic AlgorithmsMetaheuristic Algorithms Essence of an Optimization Algorithm To move to a new, better point xi +1 from an existing known location xi . xi x2 x1Xin-She Yang 2011Metaheuristics and Optimization
- 13. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMetaheuristic AlgorithmsMetaheuristic Algorithms Essence of an Optimization Algorithm To move to a new, better point xi +1 from an existing known location xi . xi ? x2 x1 xi +1 Population-based algorithms use multiple, interacting paths. Diﬀerent algorithms Diﬀerent strategies/approaches in generating these moves!Xin-She Yang 2011Metaheuristics and Optimization
- 14. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOptimization AlgorithmsOptimization Algorithms Deterministic Newton’s method (1669, published in 1711), Newton-Raphson (1690), hill-climbing/steepest descent (Cauchy 1847), least-squares (Gauss 1795), linear programming (Dantzig 1947), conjugate gradient (Lanczos et al. 1952), interior-point method (Karmarkar 1984), etc.Xin-She Yang 2011Metaheuristics and Optimization
- 15. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksStochastic/MetaheuristicStochastic/Metaheuristic Genetic algorithms (1960s/1970s), evolutionary strategy (Rechenberg & Swefel 1960s), evolutionary programming (Fogel et al. 1960s). Simulated annealing (Kirkpatrick et al. 1983), Tabu search (Glover 1980s), ant colony optimization (Dorigo 1992), genetic programming (Koza 1992), particle swarm optimization (Kennedy & Eberhart 1995), diﬀerential evolution (Storn & Price 1996/1997), harmony search (Geem et al. 2001), honeybee algorithm (Nakrani & Tovey 2004), ..., ﬁreﬂy algorithm (Yang 2008), cuckoo search (Yang & Deb 2009), ...Xin-She Yang 2011Metaheuristics and Optimization
- 16. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang 2011Metaheuristics and Optimization
- 17. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang 2011Metaheuristics and Optimization
- 18. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang 2011Metaheuristics and Optimization
- 19. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang 2011Metaheuristics and Optimization
- 20. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang 2011Metaheuristics and Optimization
- 21. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang 2011Metaheuristics and Optimization
- 22. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Newton’s Method ∂2f ∂2f ∂x1 2 ··· ∂x1 ∂xn xn+1 = xn − H−1 ∇f , H= . . .. . . . . . . ∂2f ∂2f ∂xn ∂x1 ··· ∂xn 2Xin-She Yang 2011Metaheuristics and Optimization
- 23. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Newton’s Method ∂2f ∂2f ∂x1 2 ··· ∂x1 ∂xn xn+1 = xn − H−1 ∇f , H= . . .. . . . . . . ∂2f ∂2f ∂xn ∂x1 ··· ∂xn 2 Quasi-Newton If H is replaced by I, we have xn+1 = xn − αI∇f (xn ). Here α controls the step length.Xin-She Yang 2011Metaheuristics and Optimization
- 24. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Newton’s Method ∂2f ∂2f ∂x1 2 ··· ∂x1 ∂xn xn+1 = xn − H−1 ∇f , H= . . .. . . . . . . ∂2f ∂2f ∂xn ∂x1 ··· ∂xn 2 Quasi-Newton If H is replaced by I, we have xn+1 = xn − αI∇f (xn ). Here α controls the step length.Xin-She Yang 2011Metaheuristics and Optimization
- 25. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Newton’s Method ∂2f ∂2f ∂x1 2 ··· ∂x1 ∂xn xn+1 = xn − H−1 ∇f , H= . . .. . . . . . . ∂2f ∂2f ∂xn ∂x1 ··· ∂xn 2 Quasi-Newton If H is replaced by I, we have xn+1 = xn − αI∇f (xn ). Here α controls the step length. Generation of new moves by gradient.Xin-She Yang 2011Metaheuristics and Optimization
- 26. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSimulated AnneallingSimulated Annealling Metal annealing to increase strength =⇒ simulated annealing. Probabilistic Move: p ∝ exp[−E /kB T ]. kB =Boltzmann constant (e.g., kB = 1), T =temperature, E =energy. E ∝ f (x), T = T0 αt (cooling schedule) , (0 < α < 1). T → 0, =⇒p → 0, =⇒ hill climbing.Xin-She Yang 2011Metaheuristics and Optimization
- 27. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSimulated AnneallingSimulated Annealling Metal annealing to increase strength =⇒ simulated annealing. Probabilistic Move: p ∝ exp[−E /kB T ]. kB =Boltzmann constant (e.g., kB = 1), T =temperature, E =energy. E ∝ f (x), T = T0 αt (cooling schedule) , (0 < α < 1). T → 0, =⇒p → 0, =⇒ hill climbing. This is essentially a Markov chain. Generation of new moves by Markov chain.Xin-She Yang 2011Metaheuristics and Optimization
- 28. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAn ExampleAn ExampleXin-She Yang 2011Metaheuristics and Optimization
- 29. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksGenetic AlgorithmsGenetic Algorithms crossover mutationXin-She Yang 2011Metaheuristics and Optimization
- 30. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksGenetic AlgorithmsGenetic Algorithms crossover mutationXin-She Yang 2011Metaheuristics and Optimization
- 31. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksGenetic AlgorithmsGenetic Algorithms crossover mutationXin-She Yang 2011Metaheuristics and Optimization
- 32. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksXin-She Yang 2011Metaheuristics and Optimization
- 33. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksXin-She Yang 2011Metaheuristics and Optimization
- 34. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Generation of new solutions by crossover, mutation and elistism.Xin-She Yang 2011Metaheuristics and Optimization
- 35. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSwarm IntelligenceSwarm Intelligence Ants, bees, birds, ﬁsh ... Simple rules lead to complex behaviour. Swarming StarlingsXin-She Yang 2011Metaheuristics and Optimization
- 36. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPSOPSO xj g∗ xi Particle swarm optimization (Kennedy and Eberhart 1995) vt+1 = vt + αǫ1 (g∗ − xt ) + βǫ2 (x∗ − xit ), i i i i xt+1 = xt + vit+1 . i i α, β = learning parameters, ǫ1 , ǫ2 =random numbers.Xin-She Yang 2011Metaheuristics and Optimization
- 37. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPSOPSO xj g∗ xi Particle swarm optimization (Kennedy and Eberhart 1995) vt+1 = vt + αǫ1 (g∗ − xt ) + βǫ2 (x∗ − xit ), i i i i xt+1 = xt + vit+1 . i i α, β = learning parameters, ǫ1 , ǫ2 =random numbers.Xin-She Yang 2011Metaheuristics and Optimization
- 38. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPSOPSO xj g∗ xi Particle swarm optimization (Kennedy and Eberhart 1995) vt+1 = vt + αǫ1 (g∗ − xt ) + βǫ2 (x∗ − xit ), i i i i xt+1 = xt + vit+1 . i i α, β = learning parameters, ǫ1 , ǫ2 =random numbers. Without randomness, generation of new moves by weighted average or pattern search. Adding randomization to increase the diversity of new solutions.Xin-She Yang 2011Metaheuristics and Optimization
- 39. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPSO ConvergencePSO Convergence Consider a 1D system without randomness (Clerc & Kennedy 2002) vit+1 = vit + α(xit − xi∗ ) + β(xit − g ), xit+1 = xit + vit+1 .Xin-She Yang 2011Metaheuristics and Optimization
- 40. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPSO ConvergencePSO Convergence Consider a 1D system without randomness (Clerc & Kennedy 2002) vit+1 = vit + α(xit − xi∗ ) + β(xit − g ), xit+1 = xit + vit+1 . αxi∗ +βg Considering only one particle and deﬁning p = α+β , φ =α+β and setting y t = p − xit , we have v t+1 = v t + φy t , y t+1 = −v t + (1 − φ)y t .Xin-She Yang 2011Metaheuristics and Optimization
- 41. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPSO ConvergencePSO Convergence Consider a 1D system without randomness (Clerc & Kennedy 2002) vit+1 = vit + α(xit − xi∗ ) + β(xit − g ), xit+1 = xit + vit+1 . αxi∗ +βg Considering only one particle and deﬁning p = α+β , φ =α+β and setting y t = p − xit , we have v t+1 = v t + φy t , y t+1 = −v t + (1 − φ)y t . This can be written as vt 1 φ Ut = ,A = , =⇒Ut+1 = AUt , yt −1 (1 − φ) a simple dynamical system whose eigenvalues are φ φ2 − 4φ ± λ± = 1 − . 2 2 Periodic, quasi-periodic depending on φ. Convergence for φ ≈ 4.Xin-She Yang 2011Metaheuristics and Optimization
- 42. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAnt and Bee AlgorithmsAnt and Bee Algorithms Ant Colony Optimization (Dorigo 1992) Bee algorithms & many variants (Nakrani & Tovey 2004, Karabogo 2005, Yang 2005, Asfhar et al. 2007, ..., others.Xin-She Yang 2011Metaheuristics and Optimization
- 43. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAnt and Bee AlgorithmsAnt and Bee Algorithms Ant Colony Optimization (Dorigo 1992) Bee algorithms & many variants (Nakrani & Tovey 2004, Karabogo 2005, Yang 2005, Asfhar et al. 2007, ..., others. Advantages Very promising for combinatorial optimization, but for continuous problems, it may not be the best choice.Xin-She Yang 2011Metaheuristics and Optimization
- 44. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAnt & Bee AlgorithmsAnt & Bee Algorithms Pheromone based Each agent follows paths with higher pheromone concentration (quasi-randomly) Pheromone evaporates (exponentially) with timeXin-She Yang 2011Metaheuristics and Optimization
- 45. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFireﬂy AlgorithmFireﬂy Algorithm Fireﬂy Algorithm by Xin-She Yang (2008) (Xin-She Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, (2008).) Fireﬂy Behaviour and Idealization Fireﬂies are unisex and brightness varies with distance. Less bright ones will be attracted to bright ones. If no brighter ﬁreﬂy can be seen, a ﬁreﬂy will move randomly. 2 xt+1 = xt + β0 e −γrij (xj − xi ) + α ǫt . i i i Generation of new solutions by random walk and attraction.Xin-She Yang 2011Metaheuristics and Optimization
- 46. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFA ConvergenceFA Convergence For the ﬁreﬂy motion without the randomness term, we focus on a single agent and replace xt by g j 2 xt+1 = xt + β0 e −γri (g − xt ), i i i where the distance ri = ||g − xt ||2 . iXin-She Yang 2011Metaheuristics and Optimization
- 47. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFA ConvergenceFA Convergence For the ﬁreﬂy motion without the randomness term, we focus on a single agent and replace xt by g j 2 xt+1 = xt + β0 e −γri (g − xt ), i i i where the distance ri = ||g − xt ||2 . i √ In the 1-D case, we set yt = g − xt and ut = i γyt , we have 2 ut+1 = ut [1 − β0 e −ut ].Xin-She Yang 2011Metaheuristics and Optimization
- 48. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFA ConvergenceFA Convergence For the ﬁreﬂy motion without the randomness term, we focus on a single agent and replace xt by g j 2 xt+1 = xt + β0 e −γri (g − xt ), i i i where the distance ri = ||g − xt ||2 . i √ In the 1-D case, we set yt = g − xt and ut = i γyt , we have 2 ut+1 = ut [1 − β0 e −ut ]. Analyzing this using the same methodology for ut = λut (1 − ut ), we have a corresponding chaotic map, focusing on the transition from periodic multiple states to chaotic behaviour.Xin-She Yang 2011Metaheuristics and Optimization
- 49. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Convergence can be achieved for β0 < 2. There is a transition from periodic to chaos at β0 ≈ 4. Chaotic characteristics can often be used as an eﬃcient mixing technique for generating diverse solutions. Too much attraction may cause chaos :)Xin-She Yang 2011Metaheuristics and Optimization
- 50. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems Thanks Convergence can be achieved for β0 < 2. There is a transition from periodic to chaos at β0 ≈ 4. Chaotic characteristics can often be used as an eﬃcient mixing technique for generating diverse solutions. Too much attraction may cause chaos :)Xin-She Yang 2011Metaheuristics and Optimization
- 51. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksCuckoo Breeding BehaviourCuckoo Breeding Behaviour Evolutionary Advantages Dumps eggs in the nests of host birds and let these host birds raise their chicks. Cuckoo Video (BBC)Xin-She Yang 2011Metaheuristics and Optimization
- 52. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksCuckoo SearchCuckoo Search Cuckoo Search by Xin-She Yang and Suash Deb (2009) (Xin-She Yang and Suash Deb, Cuckoo search via L´vy ﬂights, in: Proceeings of e World Congress on Nature & Biologically Inspired Computing (NaBIC 2009, India), IEEE Publications, USA, pp. 210-214 (2009). Also, Xin-She Yang and Suash Deb, Engineering Optimization by Cuckoo Search, Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 1, No. 4, 330-343 (2010). ) Cuckoo Behaviour and Idealization Each cuckoo lays one egg (solution) at a time, and dumps its egg in a randomly chosen nest. The best nests with high-quality eggs (solutions) will carry out to the next generation. The egg laid by a cuckoo can be discovered by the host bird with a probability pa and a nest will then be built.Xin-She Yang 2011Metaheuristics and Optimization
- 53. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksCuckoo SearchCuckoo Search Local random walk: xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ). i i j k [xi , xj , xk are 3 diﬀerent solutions, H(u) is a Heaviside function, ǫ is a random number drawn from a uniform distribution, and s is the step size.Xin-She Yang 2011Metaheuristics and Optimization
- 54. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksCuckoo SearchCuckoo Search Local random walk: xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ). i i j k [xi , xj , xk are 3 diﬀerent solutions, H(u) is a Heaviside function, ǫ is a random number drawn from a uniform distribution, and s is the step size. Global random walk via L´vy ﬂights: e λΓ(λ) sin(πλ/2) 1 xit+1 = xt + αL(s, λ), i L(s, λ) = , (s ≫ s0 ). π s 1+λXin-She Yang 2011Metaheuristics and Optimization
- 55. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksCuckoo SearchCuckoo Search Local random walk: xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ). i i j k [xi , xj , xk are 3 diﬀerent solutions, H(u) is a Heaviside function, ǫ is a random number drawn from a uniform distribution, and s is the step size. Global random walk via L´vy ﬂights: e λΓ(λ) sin(πλ/2) 1 xit+1 = xt + αL(s, λ), i L(s, λ) = , (s ≫ s0 ). π s 1+λXin-She Yang 2011Metaheuristics and Optimization
- 56. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksCuckoo SearchCuckoo Search Local random walk: xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ). i i j k [xi , xj , xk are 3 diﬀerent solutions, H(u) is a Heaviside function, ǫ is a random number drawn from a uniform distribution, and s is the step size. Global random walk via L´vy ﬂights: e λΓ(λ) sin(πλ/2) 1 xit+1 = xt + αL(s, λ), i L(s, λ) = , (s ≫ s0 ). π s 1+λ Generation of new moves by L´vy ﬂights, random walk and elitism. eXin-She Yang 2011Metaheuristics and Optimization
- 57. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksApplicationsApplications Design optimization: structural engineering, product design ... Scheduling, routing and planning: often discrete, combinatorial problems ... Applications in almost all areas (e.g., ﬁnance, economics, engineering, industry, ...)Xin-She Yang 2011Metaheuristics and Optimization
- 58. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksPressure Vessel Design OptimizationPressure Vessel Design Optimization d1 L d2 r rXin-She Yang 2011Metaheuristics and Optimization
- 59. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOptimizationOptimization This is a well-known test problem for optimization (e.g., see Cagnina et al. 2008) and it can be written as minimize f (x) = 0.6224d1 rL+1.7781d2 r 2 +3.1661d1 L+19.84d1 r , 2 2 g1 (x) = −d1 + 0.0193r ≤ 0 g2 (x) = −d2 + 0.00954r ≤ 0 subject to g3 (x) = −πr 2 L − 4π r 3 + 1296000 ≤ 0 3 g4 (x) = L − 240 ≤ 0. Xin-She Yang 2011Metaheuristics and Optimization
- 60. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOptimizationOptimization This is a well-known test problem for optimization (e.g., see Cagnina et al. 2008) and it can be written as minimize f (x) = 0.6224d1 rL+1.7781d2 r 2 +3.1661d1 L+19.84d1 r , 2 2 g1 (x) = −d1 + 0.0193r ≤ 0 g2 (x) = −d2 + 0.00954r ≤ 0 subject to g3 (x) = −πr 2 L − 4π r 3 + 1296000 ≤ 0 3 g4 (x) = L − 240 ≤ 0. The simple bounds are 0.0625 ≤ d1 , d2 ≤ 99 × 0.0625, 10.0 ≤ r , L ≤ 200.0.Xin-She Yang 2011Metaheuristics and Optimization
- 61. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOptimizationOptimization This is a well-known test problem for optimization (e.g., see Cagnina et al. 2008) and it can be written as minimize f (x) = 0.6224d1 rL+1.7781d2 r 2 +3.1661d1 L+19.84d1 r , 2 2 g1 (x) = −d1 + 0.0193r ≤ 0 g2 (x) = −d2 + 0.00954r ≤ 0 subject to g3 (x) = −πr 2 L − 4π r 3 + 1296000 ≤ 0 3 g4 (x) = L − 240 ≤ 0. The simple bounds are 0.0625 ≤ d1 , d2 ≤ 99 × 0.0625, 10.0 ≤ r , L ≤ 200.0. The best solution found so far f∗ = 6059.714, x∗ = (0.8125, 0.4375, 42.0984, 176.6366).Xin-She Yang 2011Metaheuristics and Optimization
- 62. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksDome DesignDome DesignXin-She Yang 2011Metaheuristics and Optimization
- 63. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksDome DesignDome Design 120-bar dome: Divided into 7 groups, 120 design elements, about 200 constraints (Kaveh and Talatahari 2010; Gandomi and Yang 2011).Xin-She Yang 2011Metaheuristics and Optimization
- 64. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksTower DesignTower Design 26-storey tower: 942 design elements, 244 nodal links, 59 groups/types, > 4000 nonlinear constraints (Kaveh & Talatahari 2010; Gandomi & Yang 2011).Xin-She Yang 2011Metaheuristics and Optimization
- 65. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMonte Carlo MethodsMonte Carlo Methods Random walk – A drunkard’s walk: ut+1 = µ + ut + wt , where wt is a random variable, and µ is the drift. For example, wt ∼ N(0, σ 2 ) (Gaussian).Xin-She Yang 2011Metaheuristics and Optimization
- 66. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMonte Carlo MethodsMonte Carlo Methods Random walk – A drunkard’s walk: ut+1 = µ + ut + wt , where wt is a random variable, and µ is the drift. For example, wt ∼ N(0, σ 2 ) (Gaussian). 25 20 15 10 5 0 -5 -10 0 100 200 300 400 500Xin-She Yang 2011Metaheuristics and Optimization
- 67. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMonte Carlo MethodsMonte Carlo Methods Random walk – A drunkard’s walk: ut+1 = µ + ut + wt , where wt is a random variable, and µ is the drift. For example, wt ∼ N(0, σ 2 ) (Gaussian). 25 10 20 5 15 0 10 -5 5 -10 0 -15 -5 -10 -20 0 100 200 300 400 500 -15 -10 -5 0 5 10 15 20Xin-She Yang 2011Metaheuristics and Optimization
- 68. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMarkov ChainsMarkov Chains Markov chain: the next state only depends on the current state and the transition probability. P(i , j) ≡ P(Vt+1 = Sj V0 = Sp , ..., Vt = Si ) = P(Vt+1 = Sj Vt = Sj ), =⇒Pij πi∗ = Pji πj∗ , π ∗ = stionary probability distribution. Examples: Brownian motion ui +1 = µ + ui + ǫi , ǫi ∼ N(0, σ 2 ).Xin-She Yang 2011Metaheuristics and Optimization
- 69. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMarkov ChainsMarkov Chains Monopoly (board games) Monopoly AnimationXin-She Yang 2011Metaheuristics and Optimization
- 70. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMarkov Chain Monte CarloMarkov Chain Monte Carlo Landmarks: Monte Carlo method (1930s, 1945, from 1950s) e.g., Metropolis Algorithm (1953), Metropolis-Hastings (1970). Markov Chain Monte Carlo (MCMC) methods – A class of methods. Really took oﬀ in 1990s, now applied to a wide range of areas: physics, Bayesian statistics, climate changes, machine learning, ﬁnance, economy, medicine, biology, materials and engineering ...Xin-She Yang 2011Metaheuristics and Optimization
- 71. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConvergence BehaviourConvergence Behaviour As the MCMC runs, convergence may be reached When does a chain converge? When to stop the chain ... ? Are multiple chains better than a single chain? 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 800 900Xin-She Yang 2011Metaheuristics and Optimization
- 72. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConvergence BehaviourConvergence Behaviour −∞ ← t t=−2 converged U 1 2 t=2 t=−n 3 t=0 Multiple, interacting chains Multiple agents trace multiple, interacting Markov chains during the Monte Carlo process.Xin-She Yang 2011Metaheuristics and Optimization
- 73. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAnalysisAnalysis Classiﬁcations of Algorithms Trajectory-based: hill-climbing, simulated annealing, pattern search ... Population-based: genetic algorithms, ant & bee algorithms, artiﬁcial immune systems, diﬀerential evolutions, PSO, HS, FA, CS, ...Xin-She Yang 2011Metaheuristics and Optimization
- 74. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAnalysisAnalysis Classiﬁcations of Algorithms Trajectory-based: hill-climbing, simulated annealing, pattern search ... Population-based: genetic algorithms, ant & bee algorithms, artiﬁcial immune systems, diﬀerential evolutions, PSO, HS, FA, CS, ...Xin-She Yang 2011Metaheuristics and Optimization
- 75. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAnalysisAnalysis Classiﬁcations of Algorithms Trajectory-based: hill-climbing, simulated annealing, pattern search ... Population-based: genetic algorithms, ant & bee algorithms, artiﬁcial immune systems, diﬀerential evolutions, PSO, HS, FA, CS, ... Ways of Generating New Moves/Solutions Markov chains with diﬀerent transition probability. Trajectory-based =⇒ a single Markov chain; Population-based =⇒ multiple, interacting chains. Tabu search (with memory) =⇒ self-avoiding Markov chains.Xin-She Yang 2011Metaheuristics and Optimization
- 76. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksErgodicityErgodicity Markov Chains & Markov Processes Most theoretical studies uses Markov chains/process as a framework for convergence analysis. A Markov chain is said be to regular if some positive power k of the transition matrix P has only positive elements. A chain is call time-homogeneous if the change of its transition matrix P is the same after each step, thus the transition probability after k steps become Pk . A chain is ergodic or irreducible if it is aperiodic and positive recurrent – it is possible to reach every state from any state.Xin-She Yang 2011Metaheuristics and Optimization
- 77. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConvergence BehaviourConvergence Behaviour As k → ∞, we have the stationary probability distribution π π = πP, =⇒ thus the ﬁrst eigenvalue is always 1. Asymptotic convergence to optimality: lim θk → θ∗ , (with probability one). k→∞Xin-She Yang 2011Metaheuristics and Optimization
- 78. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConvergence BehaviourConvergence Behaviour As k → ∞, we have the stationary probability distribution π π = πP, =⇒ thus the ﬁrst eigenvalue is always 1. Asymptotic convergence to optimality: lim θk → θ∗ , (with probability one). k→∞ The rate of convergence is usually determined by the second eigenvalue 0 < λ2 < 1. An algorithm can converge, but may not be necessarily eﬃcient, as the rate of convergence is typically low.Xin-She Yang 2011Metaheuristics and Optimization
- 79. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConvergence of GAConvergence of GA Important studies by Aytug et al. (1996)1 , Aytug and Koehler (2000)2 , Greenhalgh and Marschall (2000)3 , Gutjahr (2010),4 etc.5 The number of iterations t(ζ) in GA with a convergence probability of ζ can be estimated by ln(1 − ζ) t(ζ) ≤ , ln 1 − min[(1 − µ)Ln , µLn ] where µ=mutation rate, L=string length, and n=population size. 1 H. Aytug, S. Bhattacharrya and G. J. Koehler, A Markov chain analysis of genetic algorithms with power of 2 cardinality alphabets, Euro. J. Operational Research, 96, 195-201 (1996). 2 H. Aytug and G. J. Koehler, New stopping criterion for genetic algorithms, Euro. J. Operational research, 126, 662-674 (2000). 3 D. Greenhalgh & S. Marshal, Convergence criteria for genetic algorithms, SIAM J. Computing, 30, 269-282 (2000).Xin-She Yang 2011 4Metaheuristics and Gutjahr, Convergence Analysis of Metaheuristics Annals of Information Systems, 10, 159-187 (2010). W. J. Optimization
- 80. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMultiobjective MetaheuristicsMultiobjective Metaheuristics Asymptotic convergence of metaheuristic for multiobjective optimization (Villalobos-Arias et al. 2005)6 The transition matrix P of a metaheuristic algorithm has a stationary distribution π such that |Pij − πj | ≤ (1 − ζ)k−1 , k ∀i , j, (k = 1, 2, ...), where ζ is a function of mutation probability µ, string length L and population size. For example, ζ = 2nL µnL , so µ < 0.5.Xin-She Yang 6 2011 M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics aMetaheuristics and Optimization
- 81. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMultiobjective MetaheuristicsMultiobjective Metaheuristics Asymptotic convergence of metaheuristic for multiobjective optimization (Villalobos-Arias et al. 2005)6 The transition matrix P of a metaheuristic algorithm has a stationary distribution π such that |Pij − πj | ≤ (1 − ζ)k−1 , k ∀i , j, (k = 1, 2, ...), where ζ is a function of mutation probability µ, string length L and population size. For example, ζ = 2nL µnL , so µ < 0.5.Xin-She Yang 6 2011 M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics aMetaheuristics and Optimization
- 82. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMultiobjective MetaheuristicsMultiobjective Metaheuristics Asymptotic convergence of metaheuristic for multiobjective optimization (Villalobos-Arias et al. 2005)6 The transition matrix P of a metaheuristic algorithm has a stationary distribution π such that |Pij − πj | ≤ (1 − ζ)k−1 , k ∀i , j, (k = 1, 2, ...), where ζ is a function of mutation probability µ, string length L and population size. For example, ζ = 2nL µnL , so µ < 0.5. Note: An algorithm satisfying this condition may not converge (for multiobjective optimization) However, an algorithm with elitism, obeying the above condition, does converge!.Xin-She Yang 6 2011 M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics aMetaheuristics and Optimization
- 83. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOther resultsOther results Limited results on convergence analysis exist, concerning (ﬁnite states/domains) ant colony optimization generalized hill-climbers and simulated annealing, best-so-far convergence of cross-entropy optimization, nested partition method, Tabu search, and of course, combinatorial optimization.Xin-She Yang 2011Metaheuristics and Optimization
- 84. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksOther resultsOther results Limited results on convergence analysis exist, concerning (ﬁnite states/domains) ant colony optimization generalized hill-climbers and simulated annealing, best-so-far convergence of cross-entropy optimization, nested partition method, Tabu search, and of course, combinatorial optimization. However, more challenging tasks for inﬁnite states/domains and continuous problems. Many, many open problems needs satisfactory answers.Xin-She Yang 2011Metaheuristics and Optimization
- 85. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConverged?Converged? Converged, often the ‘best-so-far’ convergence, not necessarily at the global optimality In theory, a Markov chain can converge, but the number of iterations tends to be large. In practice, a ﬁnite (hopefully, small) number of generations, if the algorithm converges, it may not reach the global optimum.Xin-She Yang 2011Metaheuristics and Optimization
- 86. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConverged?Converged? Converged, often the ‘best-so-far’ convergence, not necessarily at the global optimality In theory, a Markov chain can converge, but the number of iterations tends to be large. In practice, a ﬁnite (hopefully, small) number of generations, if the algorithm converges, it may not reach the global optimum.Xin-She Yang 2011Metaheuristics and Optimization
- 87. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksConverged?Converged? Converged, often the ‘best-so-far’ convergence, not necessarily at the global optimality In theory, a Markov chain can converge, but the number of iterations tends to be large. In practice, a ﬁnite (hopefully, small) number of generations, if the algorithm converges, it may not reach the global optimum. How to avoid premature convergence Equip an algorithm with the ability to escape a local optimum Increase diversity of the solutions Enough randomization at the right stage ....(unknown, new) ....Xin-She Yang 2011Metaheuristics and Optimization
- 88. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksAllAll So many algorithms – what are the common characteristics? What are the key components? How to use and balance diﬀerent components? What controls the overall behaviour of an algorithm?Xin-She Yang 2011Metaheuristics and Optimization
- 89. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksExploration and ExploitationExploration and Exploitation Characteristics of Metaheuristics Exploration and Exploitation, or Diversiﬁcation and Intensiﬁcation.Xin-She Yang 2011Metaheuristics and Optimization
- 90. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksExploration and ExploitationExploration and Exploitation Characteristics of Metaheuristics Exploration and Exploitation, or Diversiﬁcation and Intensiﬁcation. Exploitation/Intensiﬁcation Intensive local search, exploiting local information. E.g., hill-climbing.Xin-She Yang 2011Metaheuristics and Optimization
- 91. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksExploration and ExploitationExploration and Exploitation Characteristics of Metaheuristics Exploration and Exploitation, or Diversiﬁcation and Intensiﬁcation. Exploitation/Intensiﬁcation Intensive local search, exploiting local information. E.g., hill-climbing. Exploration/Diversiﬁcation Exploratory global search, using randomization/stochastic components. E.g., hill-climbing with random restart.Xin-She Yang 2011Metaheuristics and Optimization
- 92. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSummarySummary Exploration ExploitationXin-She Yang 2011Metaheuristics and Optimization
- 93. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSummarySummary uniform search Exploration ExploitationXin-She Yang 2011Metaheuristics and Optimization
- 94. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSummarySummary uniform search Exploration steepest Exploitation descentXin-She Yang 2011Metaheuristics and Optimization
- 95. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSummarySummary uniform search CS Ge net Exploration ic alg ori PS thms O/ SA EP FA Ant /E /Be S e Newton- Raphson Tabu Nelder-Mead steepest Exploitation descentXin-She Yang 2011Metaheuristics and Optimization
- 96. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksSummarySummary uniform search Best? CS Free lunch? Ge net Exploration ic alg ori PS thms O/ SA EP FA Ant /E /Be S e Newton- Raphson Tabu Nelder-Mead steepest Exploitation descentXin-She Yang 2011Metaheuristics and Optimization
- 97. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksNo-Free-Lunch (NFL) TheoremsNo-Free-Lunch (NFL) Theorems Algorithm Performance Any algorithm is as good/bad as random search, when averaged over all possible problems/functions.Xin-She Yang 2011Metaheuristics and Optimization
- 98. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksNo-Free-Lunch (NFL) TheoremsNo-Free-Lunch (NFL) Theorems Algorithm Performance Any algorithm is as good/bad as random search, when averaged over all possible problems/functions. Finite domains No universally eﬃcient algorithm!Xin-She Yang 2011Metaheuristics and Optimization
- 99. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksNo-Free-Lunch (NFL) TheoremsNo-Free-Lunch (NFL) Theorems Algorithm Performance Any algorithm is as good/bad as random search, when averaged over all possible problems/functions. Finite domains No universally eﬃcient algorithm! Any free taster or dessert? Yes and no. (more later)Xin-She Yang 2011Metaheuristics and Optimization
- 100. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksNFL Theorems (Wolpert and Macready 1997)NFL Theorems (Wolpert and Macready 1997) Search space is ﬁnite (though quite large), thus the space of possible “cost” values is also ﬁnite. Objective function f : X → Y, with F = Y X (space of all possible problems). Assumptions: ﬁnite domain, closed under permutation (c.u.p). For m iterations, m distinct visited points form a time-ordered x y x y set dm = dm (1), dm (1) , ..., dm (m), dm (m) . The performance of an algorithm a iterated m times on a cost y function f is denoted by P(dm |f , m, a). For any pair of algorithms a and b, the NFL theorem states y y P(dm |f , m, a) = P(dm |f , m, b). f fXin-She Yang 2011Metaheuristics and Optimization
- 101. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksNFL Theorems (Wolpert and Macready 1997)NFL Theorems (Wolpert and Macready 1997) Search space is ﬁnite (though quite large), thus the space of possible “cost” values is also ﬁnite. Objective function f : X → Y, with F = Y X (space of all possible problems). Assumptions: ﬁnite domain, closed under permutation (c.u.p). For m iterations, m distinct visited points form a time-ordered x y x y set dm = dm (1), dm (1) , ..., dm (m), dm (m) . The performance of an algorithm a iterated m times on a cost y function f is denoted by P(dm |f , m, a). For any pair of algorithms a and b, the NFL theorem states y y P(dm |f , m, a) = P(dm |f , m, b). f fXin-She Yang 2011Metaheuristics and Optimization
- 102. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksNFL Theorems (Wolpert and Macready 1997)NFL Theorems (Wolpert and Macready 1997) Search space is ﬁnite (though quite large), thus the space of possible “cost” values is also ﬁnite. Objective function f : X → Y, with F = Y X (space of all possible problems). Assumptions: ﬁnite domain, closed under permutation (c.u.p). For m iterations, m distinct visited points form a time-ordered x y x y set dm = dm (1), dm (1) , ..., dm (m), dm (m) . The performance of an algorithm a iterated m times on a cost y function f is denoted by P(dm |f , m, a). For any pair of algorithms a and b, the NFL theorem states y y P(dm |f , m, a) = P(dm |f , m, b). f f Any algorithm is as good (bad) as a random search!Xin-She Yang 2011Metaheuristics and Optimization
- 103. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksProof SketchProof Sketch Wolpert and Macready’s original proof by induction x y y x For m = 1, d1 = {d1 , d1 }, so the only possible value of d1 is f (d1 ), and thus y x )). This means δ(d1 , f (d1 y y P(d1 |f , m = 1, a) = δ(d1 , f (d1 )) = |Y||X |−1 , x f f which is independent of algorithm a. [|Y| is the size of Y.] y If it is true for m, or f P(dm |f , m, a) is independent of a, then for m + 1, we x y have dm+1 = dm ∪ {x, f (x)} with dm+1 (m + 1) = x and dm+1 (m + 1) = f (x). Thus, we get (Bayesian approach) y y y P(dm+1 |f , m + 1, a) = P(dm+1 (m + 1)|dm , f , m + 1, a)P(dm |f , m + 1, a). y So f P(dm+1 |f , m + 1, a) = m (m + 1), f (x))P(x|d y , f , m + 1, a)P(d y |f , m + 1, a). f ,x δ(dm+1 m m Using P(x|dm , a) = δ(x, a(dm )) and P(dm |f , m + 1, a) = P(dm |f , m, a), this leads to y 1 y P(dm+1 |f , m + 1, a) = P(dm |f , m, a), f |Y| f which is also independent of a.Xin-She Yang 2011Metaheuristics and Optimization
- 104. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksProof SketchProof Sketch Wolpert and Macready’s original proof by induction x y y x For m = 1, d1 = {d1 , d1 }, so the only possible value of d1 is f (d1 ), and thus y x )). This means δ(d1 , f (d1 y y P(d1 |f , m = 1, a) = δ(d1 , f (d1 )) = |Y||X |−1 , x f f which is independent of algorithm a. [|Y| is the size of Y.] y If it is true for m, or f P(dm |f , m, a) is independent of a, then for m + 1, we x y have dm+1 = dm ∪ {x, f (x)} with dm+1 (m + 1) = x and dm+1 (m + 1) = f (x). Thus, we get (Bayesian approach) y y y P(dm+1 |f , m + 1, a) = P(dm+1 (m + 1)|dm , f , m + 1, a)P(dm |f , m + 1, a). y So f P(dm+1 |f , m + 1, a) = m (m + 1), f (x))P(x|d y , f , m + 1, a)P(d y |f , m + 1, a). f ,x δ(dm+1 m m Using P(x|dm , a) = δ(x, a(dm )) and P(dm |f , m + 1, a) = P(dm |f , m, a), this leads to y 1 y P(dm+1 |f , m + 1, a) = P(dm |f , m, a), f |Y| f which is also independent of a.Xin-She Yang 2011Metaheuristics and Optimization
- 105. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksProof SketchProof Sketch Wolpert and Macready’s original proof by induction x y y x For m = 1, d1 = {d1 , d1 }, so the only possible value of d1 is f (d1 ), and thus y x )). This means δ(d1 , f (d1 y y P(d1 |f , m = 1, a) = δ(d1 , f (d1 )) = |Y||X |−1 , x f f which is independent of algorithm a. [|Y| is the size of Y.] y If it is true for m, or f P(dm |f , m, a) is independent of a, then for m + 1, we x y have dm+1 = dm ∪ {x, f (x)} with dm+1 (m + 1) = x and dm+1 (m + 1) = f (x). Thus, we get (Bayesian approach) y y y P(dm+1 |f , m + 1, a) = P(dm+1 (m + 1)|dm , f , m + 1, a)P(dm |f , m + 1, a). y So f P(dm+1 |f , m + 1, a) = m (m + 1), f (x))P(x|d y , f , m + 1, a)P(d y |f , m + 1, a). f ,x δ(dm+1 m m Using P(x|dm , a) = δ(x, a(dm )) and P(dm |f , m + 1, a) = P(dm |f , m, a), this leads to y 1 y P(dm+1 |f , m + 1, a) = P(dm |f , m, a), f |Y| f which is also independent of a.Xin-She Yang 2011Metaheuristics and Optimization
- 106. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksProof SketchProof Sketch Wolpert and Macready’s original proof by induction x y y x For m = 1, d1 = {d1 , d1 }, so the only possible value of d1 is f (d1 ), and thus y x )). This means δ(d1 , f (d1 y y P(d1 |f , m = 1, a) = δ(d1 , f (d1 )) = |Y||X |−1 , x f f which is independent of algorithm a. [|Y| is the size of Y.] y If it is true for m, or f P(dm |f , m, a) is independent of a, then for m + 1, we x y have dm+1 = dm ∪ {x, f (x)} with dm+1 (m + 1) = x and dm+1 (m + 1) = f (x). Thus, we get (Bayesian approach) y y y P(dm+1 |f , m + 1, a) = P(dm+1 (m + 1)|dm , f , m + 1, a)P(dm |f , m + 1, a). y So f P(dm+1 |f , m + 1, a) = m (m + 1), f (x))P(x|d y , f , m + 1, a)P(d y |f , m + 1, a). f ,x δ(dm+1 m m Using P(x|dm , a) = δ(x, a(dm )) and P(dm |f , m + 1, a) = P(dm |f , m, a), this leads to y 1 y P(dm+1 |f , m + 1, a) = P(dm |f , m, a), f |Y| f which is also independent of a.Xin-She Yang 2011Metaheuristics and Optimization
- 107. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFree LunchesFree Lunches NFL – not true for continuous domains (Auger and Teytaud 2009) Continuous free lunches =⇒ some algorithms are better than others! For example, for a 2D sphere function, an eﬃcient algorithm only needs 4 iterations/steps to reach the optimality (global minimum).7 7 A. Auger and O. Teytaud, Continuous lunches are free plus the design of optimal optimization algorithms, Algorithmica, 57, 121-146 (2010). 8 J. A. Marshall and T. G. Hinton, Beyond no free lunch: realistic algorithms for arbitrary problem classes, WCCI 2010 IEEE World Congress on Computational Intelligence, July 1823, Barcelona, Spain, pp. 1319-1324.Xin-She Yang 2011Metaheuristics and Optimization
- 108. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFree LunchesFree Lunches NFL – not true for continuous domains (Auger and Teytaud 2009) Continuous free lunches =⇒ some algorithms are better than others! For example, for a 2D sphere function, an eﬃcient algorithm only needs 4 iterations/steps to reach the optimality (global minimum).7 7 A. Auger and O. Teytaud, Continuous lunches are free plus the design of optimal optimization algorithms, Algorithmica, 57, 121-146 (2010). 8 J. A. Marshall and T. G. Hinton, Beyond no free lunch: realistic algorithms for arbitrary problem classes, WCCI 2010 IEEE World Congress on Computational Intelligence, July 1823, Barcelona, Spain, pp. 1319-1324.Xin-She Yang 2011Metaheuristics and Optimization
- 109. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksFree LunchesFree Lunches NFL – not true for continuous domains (Auger and Teytaud 2009) Continuous free lunches =⇒ some algorithms are better than others! For example, for a 2D sphere function, an eﬃcient algorithm only needs 4 iterations/steps to reach the optimality (global minimum).7 Revisiting algorithms NFL assumes that the time-ordered set has m distinct points (non-revisiting). For revisiting points, it breaks the closed under permutation, so NFL does not hold (Marshall and Hinton 2010)8 7 A. Auger and O. Teytaud, Continuous lunches are free plus the design of optimal optimization algorithms, Algorithmica, 57, 121-146 (2010). 8 J. A. Marshall and T. G. Hinton, Beyond no free lunch: realistic algorithms for arbitrary problem classes, WCCI 2010 IEEE World Congress on Computational Intelligence, July 1823, Barcelona, Spain, pp. 1319-1324.Xin-She Yang 2011Metaheuristics and Optimization
- 110. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMore Free LunchesMore Free Lunches Coevolutionary algorithms A set of players (agents?) in self-play problems work together to produce a champion – like training a chess champion – free lunches exist (Wolpert and Macready 2005).9 [A single player tries to pursue the best next move, or for two players, the ﬁtness function depends on the moves of both players.] 9 D. H. Wolpert and W. G. Macready, Coevolutonary free lunches, IEEE Trans. Evolutionary Computation, 9, 721-735 (2005). 10 D. Corne and J. Knowles, Some multiobjective optimizers are better than others, Evolutionary Computation,Xin-She Yang 4, 2506-2512 (2003). CEC’03, 2011Metaheuristics and Optimization
- 111. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMore Free LunchesMore Free Lunches Coevolutionary algorithms A set of players (agents?) in self-play problems work together to produce a champion – like training a chess champion – free lunches exist (Wolpert and Macready 2005).9 [A single player tries to pursue the best next move, or for two players, the ﬁtness function depends on the moves of both players.] 9 D. H. Wolpert and W. G. Macready, Coevolutonary free lunches, IEEE Trans. Evolutionary Computation, 9, 721-735 (2005). 10 D. Corne and J. Knowles, Some multiobjective optimizers are better than others, Evolutionary Computation,Xin-She Yang 4, 2506-2512 (2003). CEC’03, 2011Metaheuristics and Optimization
- 112. Intro Metaheuristic Algorithms Applications Markov Chains Analysis All NFL Open Problems ThanksMore Free LunchesMore Free Lunches Coevolutionary algorithms A set of players (agents?) in self-play problems work together to produce a champion – like training a chess champion – free lunches exist (Wolpert and Macready 2005).9 [A single player tries to pursue the best next move, or for two players, the ﬁtness function depends on the moves of both players.] Multiobjective “Some multiobjective optimizers are better than others” (Corne and Knowles 2003).10 [results for ﬁnite domains only] Free lunches due to archiver and generator. 9 D. H. Wolpert and W. G. Macready, Coevolutonary free lunches, IEEE Trans. Evolutionary Computation, 9, 721-735 (2005). 10 D. Corne and J. Knowles, Some multiobjective optimizers are better than others, Evolutionary Computation,Xin-She Yang 4, 2506-2512 (2003). CEC’03, 2011Metaheuristics and Optimization

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