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A detailed and critical analysis of nature-inspired optimization algorithms from the mathematical and self-organization point of view.

A detailed and critical analysis of nature-inspired optimization algorithms from the mathematical and self-organization point of view.

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- 1. Analysis of Nature-Inspired Optimization Algorithms Xin-She Yang School of Science and Technology Middlesex University Seminar at Department of Mathematical Sciences University of Essex 20 Feb 2014 Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 1 / 48
- 2. Overview Overview Overview Introduction What is an Algorithm? Nature-Inspired Optimization Algorithms Applications in Engineering Design Self-Organization and Optimization Algorithms Conclusions Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 2 / 48
- 3. The Essence of an Algorithm The Essence of an Algorithm The Essence of an Algorithm Essence of an Optimization Algorithm To move to a new, better point x(t+1) from an existing location x(t) . x(t) ? x2 x(t+1) x1 Population-based algorithms use multiple, interacting paths. Diﬀerent algorithms Diﬀerent strategies/approaches in generating these moves! Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 3 / 48
- 4. The Essence of an Algorithm Optimization Algorithms Optimization 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 (Middlesex University) Algorithms 20 Feb 2014 4 / 48
- 5. The Essence of an Algorithm Steepest Descent/Hill Climbing Steepest Descent/Hill Climbing Gradient-Based Methods Use gradient/derivative information – very eﬃcient for local search. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 5 / 48
- 6. The Essence of an Algorithm Newton’s Method xt+1 = xt − H−1 ∇f , H= ∂2f ∂x1 2 . . . ∂2f ∂xd ∂x1 ··· .. . ··· ∂2f ∂x1 ∂xd . . . ∂2f ∂xd 2 . Quasi-Newton If H is replaced by I, we have xt+1 = xt − αI∇f (xt ). Here α controls the step length. Generation of new moves by gradient. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 6 / 48
- 7. The Essence of an Algorithm The Essence of an Algorithm The Essence of an Algorithm In essence, an algorithm can be written (mathematically) as xt+1 = A(xt , α), For any given xt , the algorithm will generate a new solution xt+1 . Functional or a Dynamical System? We can view the above equation as a functional, or a dynamical system, or a Markov chain, or a self-organizing system. The behavoir of the system (algorithm) can be controlled by A and its parameter α. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 7 / 48
- 8. Nature-Inspired Algorithms Stochastic/Metaheuristic Algorithms Stochastic/Metaheuristic Algorithms 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), artiﬁcial bee colony (Karaboga, 2005). Fireﬂy algorithm (Yang 2008), cuckoo search (Yang & Deb 2009), bat algorithm (Yang, 2010), ... Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 8 / 48
- 9. Simulated Annealling Simulated Annealling Simulated 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 equivalent to a random walk xt+1 = xt + p(xt , α). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 9 / 48
- 10. Simulated Annealling An Example An Example Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 10 / 48
- 11. Simulated Annealling Genetic Algorithms Genetic Algorithms Not easy to write a set of explicit equations! crossover Xin-She Yang (Middlesex University) mutation Algorithms 20 Feb 2014 11 / 48
- 12. Simulated Annealling Generation of new solutions by crossover, mutation and elistism. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 12 / 48
- 13. Swarm Intelligence Swarm Intelligence Ants, bees, birds, ﬁsh ... Simple rules lead to complex behaviour. Swarming Starlings Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 13 / 48
- 14. Swarm Intelligence PSO PSO Particle Swarm Optimization (Kennedy and Eberhart, 1995) xj g∗ xi vt+1 = vt + αǫ1 (g∗ − xt ) + βǫ2 (x∗ − xt ), i i i i i xt+1 = xt + vt+1 . i i i α, β = learning parameters, ǫ1 , ǫ2 =random numbers. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 14 / 48
- 15. Swarm Intelligence PSO Demo and Disadvantages PSO Demo and Disadvantages xi vi t+1 = 1 1 −(αǫ1 + βǫ2 ) 1 PSO Demo Xin-She Yang (Middlesex University) xi vi t + 0 αǫ1 g∗ + βǫ2 x∗ i . Premature convergence Algorithms 20 Feb 2014 15 / 48
- 16. Swarm Intelligence Fireﬂy Algorithm (Yang, 2008) Fireﬂy Algorithm (Yang, 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 (Middlesex University) Algorithms 20 Feb 2014 16 / 48
- 17. Swarm Intelligence What is FA so eﬃcient? What is FA so eﬃcient? Advantages of Fireﬂy Algorithm Automatically subdivide the whole population into subgroups, and each subgroup swarms around a local mode/optimum. Control modes/ranges by varying γ. Control randomization by tuning parameters such as α. Suitable for multimodal, nonlinear, global optimization problems. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 17 / 48
- 18. Swarm Intelligence Variants of Fireﬂy Algorithm Variants of Fireﬂy Algorithm About 1400 papers about ﬁreﬂy algorithm since 2008. Its literature is dramatically expanding. Variants for speciﬁc applications: Continuous optimization, mixed integer programming, ... Discrete ﬁreﬂy algorithm for scheduling, travelling-salesman problem, combinatorial optimization ... Image processing and compression ... Clustering, classiﬁcations and feature selection ... Chaotic ﬁreﬂy algorithm ... Hybrid ﬁreﬂy algorithm with other algorithms ... Multiobjective ﬁreﬂy algorithm. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 18 / 48
- 19. Swarm Intelligence Cuckoo Breeding Behaviour Cuckoo 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 (Middlesex University) Algorithms 20 Feb 2014 19 / 48
- 20. Swarm Intelligence Cuckoo Search (Yang and Deb, 2009) Cuckoo Search (Yang and Deb, 2009) 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 (Middlesex University) Algorithms 20 Feb 2014 20 / 48
- 21. Swarm Intelligence Cuckoo Search Cuckoo Search Local random walk: xt+1 = xt + s ⊗ H(pa − ǫ) ⊗ (xt − xt ). k i j i [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 xt+1 = xt + αL(s, λ), i i L(s, λ) = λΓ(λ) sin(πλ/2) 1 , (s ≫ s0 ). π s 1+λ Generation of new moves by L´vy ﬂights, random walk and elitism. e Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 21 / 48
- 22. Swarm Intelligence CS Demo: Highly Eﬃcient! CS Demo: Highly Eﬃcient! About 1000 papers about cuckoo search since 2009. Its literature is dramatically expanding. [See X. S. Yang, Cuckoo Search and Fireﬂy Algorithm: Theory and Applications, Springer, (2013).] CS Demo Xin-She Yang (Middlesex University) Eﬃcient search with a focus Algorithms 20 Feb 2014 22 / 48
- 23. Swarm Intelligence Bat Algorithm (Yang, 2010) Bat Algorithm (Yang, 2010) BBC Video Microbats use echolocation for hunting Ultrasonic short pulses as loud as 110dB with a short period of 5 to 20 ms. Frequencies of 25 kHz to 100 kHz. Speed up pulse-emission rate and increase loudness when homing at a prey. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 23 / 48
- 24. Swarm Intelligence Bat Algorithm Bat Algorithm Acoustics of bat echolocation v λ = ∼ 2 mm to 14 mm. f Rules used in the bat algorithm: fi = fmin + (fmax − fmin )β, vt+1 = vit + (xt − x∗ )fi , i i β ∈ [0, 1], xt+1 = xt + vt . i i i Variations of Loudness and Pulse Rate At+1 ← αAt , i i α ∈ (0, 1], rit+1 = ri0 [1 − exp(−γt)]. There are about 240 papers since 2010. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 24 / 48
- 25. Swarm Intelligence Other Algorithms Other Algorithms Genetic algorithms Diﬀerential evolution Artiﬁcial immune system Harmony search Memetic algorithm ... Reviews Yang, X. S., Engineering Optimization: An Introduction with Metaheuristic Applications, John Wiley & Sons, (2010). Yang, X. S., Nature-Inspired Metaheuristic Algorithms, Luniver Press, (2008). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 25 / 48
- 26. Applications Applications Applications 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 (Middlesex University) Algorithms 20 Feb 2014 26 / 48
- 27. Applications Pressure Vessel Design Optimization Pressure Vessel Design Optimization L d1 r Xin-She Yang (Middlesex University) d2 r Algorithms 20 Feb 2014 27 / 48
- 28. Applications Pressure Vessel Design Pressure Vessel Design This is a well-known test problem for optimization (e.g., see Cagnina et al. 2008) and it can be written as 2 2 minimize f (x) = 0.6224d1 rL + 1.7781d2 r 2 + 3.1661d1 L + 19.84d1 r , 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 (Yang, 2010; Gandomi and Yang, 2011) f∗ = 6059.714, Xin-She Yang (Middlesex University) x∗ = (0.8125, 0.4375, 42.0984, 176.6366). Algorithms 20 Feb 2014 28 / 48
- 29. Applications Speed Reducer/Gear Box Design Speed Reducer/Gear Box Design Mixed-Integer Programming: Continuous variables and integers. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 29 / 48
- 30. Applications 2 2 f (x1 , x2 , x3 , x4 , x5 , x6 , x7 ) = 0.7854x1 x2 (3.3333x3 + 14.9334x3 − 43.0934) 2 2 3 3 2 2 −1.508x1 (x6 + x7 ) + 7.4777(x6 + x7 ) + 0.7854(x4 x6 + x5 x7 ), subject to 27 − 1 ≤ 0, 2 x1 x2 x3 3 1.93x g3 = x x d4 − 1 ≤ 0, 4 2 3 1 1 g5 = 110x 3 ( 745x4 )2 + 16.9 × 106 − 1 ≤ 0, hx3 6 745x5 2 1 g6 = 85x 3 ( hx3 ) + 157.5 × 106 − 1 ≤ 0, 7 2x g7 = x403 − 1 ≤ 0, x1 g9 = 12x2 − 1 ≤ 0, 7 +1.9 g11 = 1.1xx5 − 1 ≤ 0. g1 = g2 = g4 = 397.5 2 2 x1 x2 x3 3 1.93x5 4 x2 x3 d2 − 1 ≤ 0, − 1 ≤ 0, g8 = 5x12 − 1 ≤ 0, x 6 +1.9 g10 = 1.5xx4 − 1 ≤ 0, Simple bounds are 2.6 ≤ x1 ≤ 3.6, 0.7 ≤ h ≤ 0.8, 17 ≤ x3 ≤ 28, 7.3 ≤ x4 ≤ 8.3, 7.8 ≤ x5 ≤ 8.3, 2.9 ≤ x6 ≤ 3.9, and 5.0 ≤ x7 ≤ 5.5. z must be integers. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 30 / 48
- 31. Applications Dome Design Dome Design 120-bar dome: Divided into 7 groups, 120 design elements, about 200 constraints (Gandomi and Yang 2011; Yang et al. 2012). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 31 / 48
- 32. Applications Tower Design Tower Design 26-storey tower: 942 design elements, 244 nodal links, 59 groups/types, > 4000 nonlinear constraints (Yang et al. 2011; Gandomi & Yang 2012). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 32 / 48
- 33. Applications Car Design Car Design Design better, safer and energy-eﬃcient cars Minimize weight, low crash deﬂection (< 32 mm), lower impact force (< 4 kN), .... Even a side barrier has 11 design variables! Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 33 / 48
- 34. The Essence of an Algorithm The Essence of an Algorithm The Essence of an Algorithm In essence, an algorithm can be written (mathematically) as xt+1 = A(xt , α), For any given xt , the algorithm will generate a new solution xt+1 . Functional or a Dynamical System? We can view the above equation as a functional, or a dynamical system, or a Markov chain, or a self-organizing system. The behavoir of the system (algorithm) can be controlled by A and its parameter α. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 34 / 48
- 35. The Essence of an Algorithm Self-Organization Self-Organization Self-organizing systems are everywhere, physical, chemical, biological, social, artiﬁcial ... Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 35 / 48
- 36. The Essence of an Algorithm Conditions for Self-Organization Conditions for Self-Organization Complex Systems: Large size, a suﬃcient number of degrees of freedom, a huge number of (possible) states S. Allow the systems to evolve for a long time. Enough diversity (perturbation, noise, edge of chaos, far-from-equilibrium). Selection (unchanging laws etc). That is α(t) S(θ) −→ S(π). S + α forms a larger system, so S is self-organizing, driven by α(t). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 36 / 48
- 37. The Essence of an Algorithm Optimization Process Optimization Process Optimization is a process to ﬁnd an optimal solution (state), from all possible states of a problem objective g , carried out by an algorithm A and driven by tuning some algorithm-dependent parameters p(t) = (p1 , ..., pk ). An Algorithm = Iterative Procedure xt+1 = A(xt , p(t)). Optimization A(t) g (xt=0 ) −→ g (x∗ ). An algorithm has to use information of problem states during iterations (e.g., gradient information, selection of the ﬁttest, best solutions etc). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 37 / 48
- 38. The Essence of an Algorithm Similarities Between Self-Organization and Optimization Similarities Between Self-Organization and Optimization Self-organization: Noise, perturbation =⇒ diversity. Selection mechanism =⇒ structure. Far-from-equilibrium and large perturbation =⇒ potentially faster to re-organize. Optimization Algorithms: Randomization, stochastic components, exploration =⇒ to escape local optima. Selection and exploitation =⇒ convergence & optimal solutions. High-degrees of randomization =⇒ more likely to reach global optimality (but may be slow). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 38 / 48
- 39. The Essence of an Algorithm But there are signiﬁcant diﬀerences! But there are signiﬁcant diﬀerences! Self-Organization: Avenues to self-organization may be unclear. Time may not be important. Optimization (especially metaheuristics): How to make an algorithm converge is very important. Speed of convergence is crucial (to reach truly global optimality with the minimum computing eﬀorts). However, we lack good theories to understand either self-organization or metaheuristic optimization. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 39 / 48
- 40. The Essence of an Algorithm Multi-Agent System (Swarm Intelligence?) Multi-Agent System (Swarm Intelligence?) For a multi-agent system or a swarm, an algorithm can be considered as a set of interacting Markov chain or a complex dynamical system t+1 t x1 x1 x2 x2 = A[x1 , ..., xn ; ǫ1 , ..., ǫm ; p1 (t), ..., pk (t)] . . . . . . . xn xn A population of solutions xt+1 (i = 1, ..., n) are generated from xt , i i controlled by k parameters and m random numbers. In principle, the behaviour of an algorithm is controlled by the eigenvalues of A, but in practice, it is almost impossible to ﬁgure out the eigenvalues (apart from very simple/rare cases). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 40 / 48
- 41. Some Interesting Results Genetic Algorithms Genetic Algorithms For a binary genetic algorithm with p0 = 0.5 with n chromosomes of length m, the probability of (premature) convergence at any time/iteration t is P(t, m) = 1 − 2 6p0 (1 − p0 ) 1− n n t m . For a population of n = 40, m = 100, t = 100 generations, we have P(t, m) = − 1 6 × 0.5(1 − 0.5) 2 (1 − )100 40 40 100 ≈ 0.978. For genetic algorithms with a given accuracy ζ, the number of iterations t(ζ) needed is (possibly converged prematurely) t(ζ) ≤ ln(1 − ζ) ln 1 − min[(1 − µ)nL , µnL ] , where µ =mutation rate, L =string length, and n =population size. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 41 / 48
- 42. Some Interesting Results Cuckoo Search with Guaranteed Global Convergence Cuckoo Search with Guaranteed Global Convergence Simplify the cuckoo search (CS) algorithm as t+1 if r < pa xi ← xt i t+1 xi ← xt + α ⊗ L(λ) if r > pa . i Markov Chain Theory The probability of convergence to the global optimal set G is lim P(xt ∈ G ) → 1. t→∞ See Wang et al. (2012) for details. Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 42 / 48
- 43. Exploration & Exploitation Key Components in All Metaheuristics Key Components in All Metaheuristics 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 (Middlesex University) Algorithms 20 Feb 2014 43 / 48
- 44. Exploration & Exploitation Exploration and Exploitation Exploration 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 (Middlesex University) Algorithms 20 Feb 2014 44 / 48
- 45. Exploration & Exploitation Summary Summary Exploration uniform search Ge net Best? Free lunch? CS ic alg ori th SA PS O/ FA EP /E A nt S /Be e ms NewtonRaphson Tabu Nelder-Mead Exploitation Xin-She Yang (Middlesex University) Algorithms steepest descent 20 Feb 2014 45 / 48
- 46. Open Problems Open Problems Open Problems Mathematical Analysis: A uniﬁed mathematical framework (e.g., Markov chain theory, dynamical systems) is needed. Comparison: What are the best performance measures? Parameter tuning: How to tune the algorithm-dependent parameters so that an algorithm can achieve the best performance? Scalability: Can the algorithms that work for small-scale problems be directly applied to large-scale problems? Intelligence: Smart algorithms may be a buzz word, but can truly intelligent algorithms be developed? Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 46 / 48
- 47. Open Problems Bibliography Bibliography Xin-She Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014). Xin-She Yang, Cuckoo Search and Fireﬂy Algorithm: Theory and Applications, Springer, (2013). Xin-She Yang and Suash Deb, Multiobjective cuckoo search for design optimization, Computers & Operations Research, 40(6), 1616–1624 (2013). A. H. Gandomi, X. S. Yang, S. Talatahari, S. Deb, Coupled eagle strategy and diﬀerential evolution for unconstrained and constrained global optimization, Computers & Mathematics with Applications, 63(1), 191–200 (2012). A. H. Gandomi, G. J. Yun, X. S. Yang, S. Talatahari, Chaos-enhanced accelerated particle swarm optimization, Communications in Nonlinear Science and Numerical Simulation, 18(2), 327–340 (2013). X. S. Yang and S. Deb, Two-stage eagle strategy with diﬀerential evolution, Int. Journal of Bio-Inspired Computation, 4(1), 1–5 (2012). X. S. Yang, Multiobjective ﬁreﬂy algorithm for continuous optimization, Engineering with Computers, 29(2), 175–184 (2013). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 47 / 48
- 48. Open Problems Thank you Thank you Xin-She Yang, Z. H. Cui, R. B. Xiao, A. H. Gandomi, M. Karamanoglu, Swarm Intelligence and Bio-Inspired Computation: Theory and Applications, Elsevier, (2013). Xin-She Yang, A. H. Gandomi, S. Talatahari, A. H. Alavi, Metaheuristics in Water, Geotechnical and Transport Engineering, Elsevier, (2012). A. H. Gandomi, Xin-She Yang, A. H. Alavi, Mixed variable structural optimization using ﬁreﬂy algorithm, Computers & Structures, vol. 89, no. 23, 2325–2336 (2011). Xin-She Yang and A. H. Gandomi, Bat algorithm: a novel approach for global engineering optimization, Engineering Computations, vol. 29, no. 5, 464–483 (2012). Xin-She Yang, A new metaheuristic bat-inspired algorithm, in: Nature Inspired Cooperative Strategies for Optimization (NISCO 2010) (Eds. J. R. Gonzalez et al.), Studies in Computational Intelligence (SCI 284), Springer, pp. 65–74 (2010). Xin-She Yang, Artiﬁcial Intelligence, Evolutionary Computing and Metaheuristics — In the Footsteps of Alan Turing, Studies in Computational Intelligence (SCI 427), Springer Heidelberg, (2013). Xin-She Yang (Middlesex University) Algorithms 20 Feb 2014 48 / 48

Full NameComment goes here.Minh David3 weeks ago