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- 1. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks Nature-Inspired Metaheristics Algorithms for Optimization and Computational Intelligence Xin-She Yang National Physical Laboratory, UK @ FedCSIS2011Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 2. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 3. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksIntroIntro Computational science is now the third paradigm of science, complementing theory and experiment. - Ken Wilson (Cornell University), Nobel Laureate.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 4. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 5. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 6. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 7. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 8. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 9. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 10. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOverviewOverview Part I Introduction Metaheuristic Algorithms Monte Carlo and Markov Chains Algorithm AnalysisXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 11. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOverviewOverview Part I Introduction Metaheuristic Algorithms Monte Carlo and Markov Chains Algorithm Analysis Part II Exploration & Exploitation Dealing with Constraints Applications Discussions & BibliographyXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 12. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 13. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 14. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c ,Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 15. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 16. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 17. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 18. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2 Steepest DescentXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 19. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2 Steepest DescentXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 20. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2 Steepest Descent =⇒ d d 1 1 + y ′2 min t = ds = dx 0 v 0 2g [h − y (x)]Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 21. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2 Steepest Descent =⇒ d d 1 1 + y ′2 min t = ds = dx 0 v 0 2g [h − y (x)] =⇒Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 22. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2 Steepest Descent =⇒ d d 1 1 + y ′2 min t = ds = dx 0 v 0 2g [h − y (x)] =⇒ =⇒Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 23. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksA Perfect AlgorithmA Perfect Algorithm What is the best relationship among E , m and c? Initial state: m,E ,c , =⇒ =⇒ E =mc 2 Steepest Descent =⇒ d d 1 1 + y ′2 min t = ds = dx 0 v 0 2g [h − y (x)] A x= 2 (θ − sin θ) =⇒ =⇒ y = h − A (1 − cos θ) 2Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 24. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksComputing in RealityComputing in Reality A Problem & Problem Solvers ⇓ Mathematical/Numerical Models ⇓ Computer & Algorithms & Programming ⇓ Validation ⇓ ResultsXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 25. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksWhat is an Algorithm?What is an Algorithm? 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 26. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksWhat is an Algorithm?What is an Algorithm? 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 27. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksWhat is an Algorithm?What is an Algorithm? 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 28. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOptimization is Like Treasure HuntingOptimization is Like Treasure Hunting How to ﬁnd a treasure, a hidden 1 million dollars? What is your best strategy?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 29. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 30. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 31. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 32. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 33. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 34. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 35. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 36. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent/Hill ClimbingSteepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 37. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks Newton’s Method ∂2f ∂2f ∂x1 2 ··· ∂x1 ∂xn xn+1 = xn − H−1 ∇f , H= . . .. . . . . . . ∂2f ∂2f ∂xn ∂x1 ··· ∂xn 2 Generation of new moves by gradient.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 38. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 39. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSteepest Descent Method (Cauchy 1847, Riemann 1863)Steepest Descent Method (Cauchy 1847, Riemann 1863) From the Taylor expansion of f (x) about x(n) , we have f (x(n+1) ) = f (x(n) + ∆s) ≈ f (x(n) + (∇f (x(n) ))T ∆s, where ∆s = x(n+1) − x(n) is the increment vector. So f (x(n) + ∆s) − f (x(n) ) = (∇f )T ∆s < 0. Therefore, we have ∆s = −α∇f (x(n) ), where α > 0 is the step size. In the case of ﬁnding maxima, this method is often referred to as hill-climbing.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 40. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksConjugate Gradient (CG) MethodConjugate Gradient (CG) Method Belong to Krylov subspace iteration methods. The conjugate gradient method was pioneered by Magnus Hestenes, Eduard Stiefel and Cornelius Lanczos in the 1950s. It was named as one of the top 10 algorithms of the 20th century.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 41. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksConjugate Gradient (CG) MethodConjugate Gradient (CG) Method Belong to Krylov subspace iteration methods. The conjugate gradient method was pioneered by Magnus Hestenes, Eduard Stiefel and Cornelius Lanczos in the 1950s. It was named as one of the top 10 algorithms of the 20th century. A linear system with a symmetric positive deﬁnite matrix A Au = b, is equivalent to minimizing the following function f (u) 1 f (u) = uT Au − bT u + v, 2 where v is a vector constant and can be taken to be zero. We can easily see that ∇f (u) = 0 leads to Au = b.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 42. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksCGCG The theory behind these iterative methods is closely related to the Krylov subspace Kn spanned by A and b as deﬁned by Kn (A, b) = {Ib, Ab, A2 b, ..., An−1 b}, where A0 = I. If we use an iterative procedure to obtain the approximate solution un to Au = b at nth iteration, the residual is given by rn = b − Aun , which is essentially the negative gradient ∇f (un ).Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 43. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks The search direction vector in the conjugate gradient method is subsequently determined by dT Arn n dn+1 = rn − dn . dT Adn n The solution often starts with an initial guess u0 at n = 0, and proceeds iteratively. The above steps can compactly be written as un+1 = un + αn dn , rn+1 = rn − αn Adn , and dn+1 = rn+1 + βn dn , where rT rn n rT rn+1 n+1 αn = T , βn = . dn Adn rT r n n Iterations stop when a prescribed accuracy is reached.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 44. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksGradient-free MethodsGradient-free Methods Gradient-base methods Requires the information of derivatives. Not suitable for problems with discontinuities.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 45. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksGradient-free MethodsGradient-free Methods Gradient-base methods Requires the information of derivatives. Not suitable for problems with discontinuities.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 46. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksGradient-free MethodsGradient-free Methods Gradient-base methods Requires the information of derivatives. Not suitable for problems with discontinuities. Gradient-free or derivative-free methods BFGS, Downhill simplex, Trust-region, SQP ...Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 47. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksNelder-Mead Downhill Simplex MethodNelder-Mead Downhill Simplex Method The Nelder-Mead method is a downhill simplex algorithm, ﬁrst developed by J. A. Nelder and R. Mead in 1965. A Simplex In the n-dimensional space, a simplex, which is a generalization of a triangle on a plane, is a convex hull with n + 1 distinct points. For simplicity, a simplex in the n-dimension space is referred to as n-simplex.Xin-She Yang (a) (b) (c) FedCSIS2011Metaheuristics and Computational Intelligence
- 48. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksDownhill Simplex MethodDownhill Simplex Method xe xr xr ¯ x s s xc xn+1 xn+1 xn+1 The ﬁrst step is to rank and re-order the vertex values f (x1 ) ≤ f (x2 ) ≤ ... ≤ f (xn+1 ), at x1 , x2 , ..., xn+1 , respectively. Wikipedia AnimationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 49. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMetaheuristicMetaheuristic Most are nature-inspired, mimicking certain successful features in nature. Simulated annealing Genetic algorithms Ant and bee algorithms Particle Swarm Optimization Fireﬂy algorithm and cuckoo search Harmony search ...Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 50. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 51. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 52. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksAn ExampleAn ExampleXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 53. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksGenetic AlgorithmsGenetic Algorithms crossover mutationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 54. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksGenetic AlgorithmsGenetic Algorithms crossover mutationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 55. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksGenetic AlgorithmsGenetic Algorithms crossover mutationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 56. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 57. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 58. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks Generation of new solutions by crossover, mutation and elistism.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 59. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSwarm IntelligenceSwarm Intelligence Ants, bees, birds, ﬁsh ... Simple rules lead to complex behaviour. Go to Metaheuristic SlidesXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 60. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 61. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 xt+1 = xt + αL(s, λ), i i L(s, λ) = , (s ≫ s0 ). π s 1+λXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 62. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 xt+1 = xt + αL(s, λ), i i L(s, λ) = , (s ≫ s0 ). π s 1+λXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 63. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 xt+1 = xt + αL(s, λ), i 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 64. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMonte Carlo MethodsMonte Carlo Methods Almost everyone has used Monte Carlo methods in some way ... Measure temperatures, choose a product, ... Taste soup, wine ...Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 65. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMarkov ChainsMarkov Chains 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 66. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMarkov ChainsMarkov Chains 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 67. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMarkov ChainsMarkov Chains 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 68. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 69. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMarkov ChainsMarkov Chains Monopoly (board games) Monopoly AnimationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 70. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 71. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 72. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 73. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 74. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 75. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 76. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 77. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 78. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 79. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011 4Metaheuristics and Gutjahr, Convergence Analysis of Metaheuristics Annals of Information Systems, 10, 159-187 (2010). W. J. Computational Intelligence
- 80. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011 M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics aMetaheuristics and Computational Intelligence
- 81. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011 M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics aMetaheuristics and Computational Intelligence
- 82. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011 M. Villalobos-Arias, C. A. Coello Coello and O. Hern´ndez-Lerma, Asymptotic convergence of metaheuristics aMetaheuristics and Computational Intelligence
- 83. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 84. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 85. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 86. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 87. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 88. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications Thanks Coﬀee Break (15 Minutes)Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 89. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksAll and NFLAll and NFL 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 90. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksExploration and ExploitationExploration and Exploitation Characteristics of Metaheuristics Exploration and Exploitation, or Diversiﬁcation and Intensiﬁcation.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 91. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 92. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 93. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSummarySummary Exploration ExploitationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 94. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSummarySummary uniform search Exploration ExploitationXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 95. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSummarySummary uniform search Exploration steepest Exploitation descentXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 96. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSummarySummary uniform search CS Ge net Exploration ic alg ori PS th ms O/ SA EP FA A nt /E /Be S e Newton- Raphson Tabu Nelder-Mead steepest Exploitation descentXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 97. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksSummarySummary uniform search Best? CS Free lunch? Ge net Exploration ic alg ori PS th ms O/ SA EP FA A nt /E /Be S e Newton- Raphson Tabu Nelder-Mead steepest Exploitation descentXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 98. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 99. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 100. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 101. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 102. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 103. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications 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 FedCSIS2011Metaheuristics and Computational Intelligence
- 104. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOpen ProblemsOpen Problems Framework: Need to develop a uniﬁed framework for algorithmic analysis (e.g.,convergence). Exploration and exploitation: What is the optimal balance between these two components? (50-50 or what?) Performance measure: What are the best performance measures ? Statistically? Why ? Convergence: Convergence analysis of algorithms for inﬁnite, continuous domains require systematic approaches?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 105. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOpen ProblemsOpen Problems Framework: Need to develop a uniﬁed framework for algorithmic analysis (e.g.,convergence). Exploration and exploitation: What is the optimal balance between these two components? (50-50 or what?) Performance measure: What are the best performance measures ? Statistically? Why ? Convergence: Convergence analysis of algorithms for inﬁnite, continuous domains require systematic approaches?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 106. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOpen ProblemsOpen Problems Framework: Need to develop a uniﬁed framework for algorithmic analysis (e.g.,convergence). Exploration and exploitation: What is the optimal balance between these two components? (50-50 or what?) Performance measure: What are the best performance measures ? Statistically? Why ? Convergence: Convergence analysis of algorithms for inﬁnite, continuous domains require systematic approaches?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 107. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksOpen ProblemsOpen Problems Framework: Need to develop a uniﬁed framework for algorithmic analysis (e.g.,convergence). Exploration and exploitation: What is the optimal balance between these two components? (50-50 or what?) Performance measure: What are the best performance measures ? Statistically? Why ? Convergence: Convergence analysis of algorithms for inﬁnite, continuous domains require systematic approaches?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 108. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMore Open ProblemsMore Open Problems Free lunches: Unproved for inﬁnite or continuous domains for multiobjective optimization. (possible free lunches!) What are implications of NFL theorems in practice? If free lunches exist, how to ﬁnd the best algorithm(s)? Knowledge: Problem-speciﬁc knowledge always helps to ﬁnd appropriate solutions? How to quantify such knowledge? Intelligent algorithms: Any practical way to design truly intelligent, self-evolving algorithms?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 109. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMore Open ProblemsMore Open Problems Free lunches: Unproved for inﬁnite or continuous domains for multiobjective optimization. (possible free lunches!) What are implications of NFL theorems in practice? If free lunches exist, how to ﬁnd the best algorithm(s)? Knowledge: Problem-speciﬁc knowledge always helps to ﬁnd appropriate solutions? How to quantify such knowledge? Intelligent algorithms: Any practical way to design truly intelligent, self-evolving algorithms?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 110. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMore Open ProblemsMore Open Problems Free lunches: Unproved for inﬁnite or continuous domains for multiobjective optimization. (possible free lunches!) What are implications of NFL theorems in practice? If free lunches exist, how to ﬁnd the best algorithm(s)? Knowledge: Problem-speciﬁc knowledge always helps to ﬁnd appropriate solutions? How to quantify such knowledge? Intelligent algorithms: Any practical way to design truly intelligent, self-evolving algorithms?Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 111. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksConstraintsConstraints In describing optimization algorithms, we are not concern with constraints. Algorithms can solve both unconstrained and more often constrained problems. The handling of constraints is an implementation issue, though incorrect or ineﬃcient methods of dealing with constraints can slow down the algorithm eﬃciency, or even result in wrong solutions. Methods of handling constraints Direct methods Langrange multipliers Barrier functions Penalty methodsXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 112. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksAimsAims Either converting a constrained problem to an unconstrained one or changing the search space into a regular domainXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 113. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksAimsAims Either converting a constrained problem to an unconstrained one or changing the search space into a regular domain The ease of programming and implementation Improve (or at least not hinder) the eﬃciency of the chosen algorithm in implementation.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 114. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksAimsAims Either converting a constrained problem to an unconstrained one or changing the search space into a regular domain The ease of programming and implementation Improve (or at least not hinder) the eﬃciency of the chosen algorithm in implementation.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 115. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksAimsAims Either converting a constrained problem to an unconstrained one or changing the search space into a regular domain The ease of programming and implementation Improve (or at least not hinder) the eﬃciency of the chosen algorithm in implementation. Scalability The used approach should be able to deal with small, large and very large scale problems.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 116. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksCommon ApproachesCommon Approaches Direct method Simple, but not versatile, diﬃcult in programming.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 117. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksCommon ApproachesCommon Approaches Direct method Simple, but not versatile, diﬃcult in programming. Lagrange multipliers Main for equality constraints.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 118. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksCommon ApproachesCommon Approaches Direct method Simple, but not versatile, diﬃcult in programming. Lagrange multipliers Main for equality constraints. Barrier functions Very powerful and widely used in convex optimization.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 119. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksCommon ApproachesCommon Approaches Direct method Simple, but not versatile, diﬃcult in programming. Lagrange multipliers Main for equality constraints. Barrier functions Very powerful and widely used in convex optimization. Penalty methods Simple and versatile, widely used.Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 120. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksCommon ApproachesCommon Approaches Direct method Simple, but not versatile, diﬃcult in programming. Lagrange multipliers Main for equality constraints. Barrier functions Very powerful and widely used in convex optimization. Penalty methods Simple and versatile, widely used. OthersXin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 121. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksDirect MethodsDirect Methods Minimize f (x, y ) = (x − 2)2 + 4(y − 3)2 subject to −x + y ≤ 2, x + 2y ≤ 3. 2 ≤ y Optimal x+ − x+ 2y ≤3 Direct Methods: to generate solutions/points inside the region! (easy for rectangular regions)Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 122. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksMethod of Lagrange MultipliersMethod of Lagrange Multipliers Maximize f (x, y ) = 10 − x 2 − (y − 2)2 subject to x + 2y = 5. Deﬁning a combined function Φ using a multiplier λ, we have Φ = 10 − x 2 − (y − 2)2 + λ(x + 2y − 5). The optimality conditions are ∂Φ ∂Φ ∂Φ = 2x +λ = 0, = −2(y −2)+2λ = 0, = x +2y −5, ∂x ∂y ∂λ whose solutions become 49 x = 1/5, y = 12/5, λ = 2/5, =⇒ fmax = . 5Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 123. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksBarrier FunctionsBarrier Functions As an equality h(x) = 0 can be written as two inequalities h(x) ≤ 0 and −h(x) ≤ 0, we only use inequalities. For a general optimization problem: minimize f (x), subject to g (xi ) ≤ 0(i = 1, 2, ..., N), we can deﬁne a Indicator or barrier function 0 if u ≤ 0 I−1 [u] = ∞ if u > 0. Not so easy to deal with numerically. Also discontinuous!Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 124. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksLogarithmic Barrier FunctionsLogarithmic Barrier Functions A log barrier function ¯− (u) = − 1 log(−u), I u < 0, t where t > 0 is an accuracy parameters (can be very large).Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence
- 125. Intro Classic Algorithms Metaheuristic Markov Analysis All and NFL Constraints Applications ThanksLogarithmic Barrier FunctionsLogarithmic Barrier Functions A log barrier function ¯− (u) = − 1 log(−u), I u < 0, t where t > 0 is an accuracy parameters (can be very large). Then, the above minimization problem becomes N N ¯− (gi (x)) = f (x) + 1 minimize f (x) + I − log[−gi (x)]. t i =1 i =1 This is an unconstrained problem and easy to implement!Xin-She Yang FedCSIS2011Metaheuristics and Computational Intelligence

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