Particle Swarm optimisation: A mini tutorial

Transcription

1 : A mini tutorial

2 The inventors (1) Russell Eberhart

3 The inventors () Jim at work James Kennedy

4 Part 1: United we stand

5 Cooperation example

6 Memory and informers It is thanks to these eccentrics, whose behaviour is not conform to the one of the other bees, that all fruits sources around the colony are so quickly found. Karl von Frish 197 DPNP

7 Initialisation. Positions and velocities

8 Neighbourhoods geographical social

9 Psychosocial compromise Here I am! x i proximity p My best perf. g The best perf. of my neighbours v g proximity

10 The historical algorithm At each time step t for each particle for each component d update the velocity then move v ( ) d t+ 1 = α v () d t + βrand 0, ϕ1 pd xd t + βrand 0, ϕ gd xd t ( ) ( ) ( ) ( ) ( ) ( ) () t + v( 1) x( t + 1) = x t + Randomness inside the loop

11 Oscillations Initial v 1 Fitness 4 3 Search space

12 The circular neighbourhood Particle 1 s 3 neighbourhood Virtual circle 6 4 5

13 Random proximity Hyperparallelepiped => Biased i proximity p x g DPNP=Mayan pyramid g proximity v

14 Animated illustration Global optimum

15 Maths and parameters The right way This way Or this way

16 Neighbourhoods (topologies) Small world graph or? (informers)

17 Type 1 form vt ( + 1) = χ( vt ( ) + ϕ( q xt ( ))) xt ( + 1) = vt ( + 1) + xt ( ) with ϕ = rand( 0, ϕ rand + ϕ q ϕ ' p+ ϕ ' = ϕ' + ϕ' Global constriction coefficient ) + (0, ϕ) = ϕ' 1 ' κ for ϕ > 4 χ = ϕ + ϕ 4ϕ else κ g Usual values: κ=1 ϕ=4.1 Non divergence criterion => χ=0.73 swarm size=0 hood size=3

18 5D complex space A 3D section ϕ } } -4 Convergence Non divergence Re(v) 4 1 Re(y)

19 Move in a D section (attractor) Im(v) Re(v) -0.8

20 Beyond real numbers Bingo!

21 Minimun requirements Comparing positions in the search space H ( f ( x) < f ( x' )) ( f ( x) f ( ')) ( x, x') H H, x Algebraic operators ( coefficient, velocity) o ( velocity, velocity) velocity Θ ( position, position) velocity ( position, velocity) position velocity

22 Pseudo code form velocity = pos_minus_pos(position 1, position ) velocity = linear_combin(α,velocity 1,β,velocity ) position = pos_plus_vel(position, velocity) } algebraic (position,velocity) = confinement(position t+1,position t ) operators

23 Confinements Frontiers (ex. : interval) => Granularity =>

24 End of Part 1

25 Part : When the algo mutates

26 The PSO Family Empirical Weighted Constraints Combinatorial Multi objective Canonical Discrete Hybride Adaptive Multi swarm Spherical, Gaussian Pivot TRIBES OEP 0, 1,

27 Unbiased random proximity Hyperparallelepiped => Biased i proximity p x g proximity g Volume Hypersphere vs hypercube v Dimension

28 The three balls i proximity g proximity x p g

29 Think locally, act locally narchy

30 Adaptive coefficients Crisp or fuzzy rules αv The better I am the more I follow my own way rand(0 b)(p-x) The better is my best neighbour the more I tend to go towards him

31 Adaptive swarm size There has been enough improvement although I'm the worst Crisp or fuzzy rules I try to kill myself I'm the best but there has been not enough improvement I try to generate a new particle

32 TRIBES and strategies TRIBES Adaptive proximity distributions (DPNP) Adaptive information links

33 Energies: classical process Rosenbrock D. D. Swarm size=0, constant coefficients. 1,00E ,00E+03 8,00E ,00E+03 6,00E+03 5,00E+03 4,00E+03 3,00E+03,00E+03 1,00E+03 0,00E Potential energy Kinetic energy Swarm size Number of evaluations ε= after 40 evaluations

34 Energies: adaptive process Rosenbrock D. Adaptive swarm size, adaptive coefficients Rosenbrock D. Adaptive swarm size, adaptive coefficients. 1,00E ,00E+03 8,00E ,00E+03 6,00E+03 5,00E+03 4,00E+03 3,00E+03,00E+03 1,00E+03 0,00E Potential energy Kinetic energy Swarm size Number of evaluations ε= after 1414 evaluations

35 The simplest PSO Random informers K=3 1 8 Pivot method x g g proximity

36 End of Part

37 Part 3:Story of Optimisation

38 Classical results Optimum=0, dimension=30 Best result after evaluations 30D function PSO Type 1" Evolutionary algo.(angeline 98) Griewank [±300] Rastrigin [±5] Rosenbrock [±10]

39 Some small problems

40 Fifty fifty N=100, D=0. Search space: [1,N] D 105 evaluations: = (=450) x i { 1...N} i j x i x j D / D x i = x i granularity=1 1 D /+1

41 Knapsack N=100, D=10, S=100, 870 evaluations: granularity=1 x i { 1...N} i j x i x j x i = S i I, I = D, I { 1, N} run 1 => (9, 14, 18, 1, 16, 5, 6,, 1, 17) run => (9, 3, 16, 4, 1,, 6, 8, 6, 5)

42 Apple trees Best position Swarm size=3 n 1 n n 3 Evaluation n1 n n f = ( ) n n + ( n ) 1 n3

43 Graph Coloring Problem pos plus vel =

44 The Tireless Traveller Example of position: X=(5,3,4,1,,6) Example of velocity: v=((5,3),(,5),(3,1))

45 BR17, the movie Structured search space?

46 Ecological niche Non linear Multiobjective Fuzzy data Dynamical, real time Heterogeneous t1 t t3 t4 t5 t6 t7

47 End of Part 3

48 Part 4: Real applications Medical diagnosis Industrial mixer Electrical generator Electrical vehicle

49 Applications (1) Salerno, J. Using the particle swarm optimization technique to train a recurrent neural model. IEEE International Conference on Tools with Artificial Intelligence, 1997, p , He Z., Wei C., Yang L., Gao X., Yao S., Eberhart R. C., Shi Y., "Extracting Rules from Fuzzy Neural Network by PSO", IEEE IEC, Anchorage, Alaska, USA, Secrest B. R., Traveling Salesman Problem for Surveillance Mission using PSO, AFIT/GCE/ENG/01M-03, Air Force Institute of Technology, 001. Yoshida H., Kawata K., Fukuyama Y., "A PSO for Reactive Power and Voltage Control considering Voltage Security Assessment", IEEE TPS, vol. 15, 001, p Krohling, R. A., Knidel, H., and Shi, Y. Solving numerical equations of hydraulic problems using PSO. Proceedings of the IEEE CEC, Honolulu, Hawaii USA. 00.

50 Applications () Kadrovach, B.A., and Lamont G., A particle swarm model for swarm-based networked sensor systems, ACM symposium on Applied computing, Madrid, Spain, p , 00 Omran, M., Salman, A., and Engelbrecht, A. P. Image classification using PSO. Proceedings of the 4th Asia-Pacific Conference on Simulated Evolution and Learning 00 (SEAL 00), Singapore. p , 00. Coello Coello, C. A., Luna, E. H., and Aguirre, A. H. Use of PSO to design combinational logic circuits. LNCS No. 606, p , 003. Onwubolu, G. C. and Clerc, M., "Optimal path for automated drilling operations by a new heuristic approach using particle swarm optimization," International Journal of Production Research, vol. 4, p , 004. Onwubolu G.C., TRIBES application to the flowshop scheduling problem, New Optimization Techniques in Engineering. Heidelberg, Germany, Springer: p ,.

51 Neuronal network Test E i Wanted output S i Real output S' i (t) e i,1 e i,... e i,n τ Transfer functions ( entry, xkm) km,, s i,1... s i,p s' i,1 (t)... s' i,p (t) ( ) E = E 1...E nb _tests Xt ()=( x k, m () t ) S = S 1...S nb _ tests S ' ' ' ()= t S 1 t ( ) ( ()...S nb _tests () t ) 1 x + e km, 1 entry Function to minimise f( X)= S S '

52 To know more THE site: Particle Swarm Central, Kennedy, J., R. Eberhart, et al. (001). Swarm Intelligence, Morgan Kaufmann Academic Press. Self advert Clerc M., Kennedy J., "The Particle Swarm-Explosion, Stability, and Convergence in a Multidimensional Complex space", IEEE Transaction on Evolutionary Computation, 00,vol. 6, p IEEE TEC award

53 More self ad. My PSO site: If you read French Clerc M., "L'optimisation par essaim particulaire. Principes et pratique", Hermès, Techniques et Science de l'informatique, 00. Article de 5 p. Clerc M., L optimisation par essaims particulaires. Parution février 005

54 PSO in the world extended Particle Swarms (XPS) project

55 Some open questions Adaptive weighted relationships New mathematical ideas to model particle interactions Metamodel Better combinatorial approaches

56 Beat the swarm! Your current position Your best perf.best perf. of the swarm

57 APPENDIX

58 Canonical form ( ) v t+ 1 = v t + ϕ q x t x( t+ 1) = x( t) + v( t+ 1) ( ) ( ) ( ) M y( t) = q x( t) ( t + 1) ( t + 1) vt+1 ( )= αvα ( t)+ βϕy β ( t) yt+ ( 1)= γv γ ()+ t ( δδ ηϕ ηϕ)yt v y = 1 1 () ( t) ϕ v () 1 ϕ y t Eigen values e 1 and e

59 Constriction Constriction coefficients χ1 χ α + δ ηϕ + = α + δ ηϕ = ( ηϕ ) + ϕ ( αη δη βγ ) + ( α δ ) ϕ + ( ηϕ ) + ϕ ( αη δη βγ ) + ( α δ ) ϕ + ϕ ϕ 4ϕ 4ϕ

60 Convergence criterion χ 1 e 1 < 1 χ e < 1 χ 1 < 1 χ e < 1 κ = χ e ϕ

61 Robustness Performance map : NeededIterations(κ,ϕ) Iter f ( x, x ) ( x x ) + ( 1 x ) = 1 1

62 Clusters and queens Each particle is weighted by its perf. Dynamic clustering Centroids = queens = temporary new particles

63 Magic Square (1) D D D j i D m m m m m,,1, 1, 1,1 K M M K ( ) { } = + l k j i j i D i j j i m m N m x m,,, 1, 1L ( ) ( ) ,, ,, = + = = + = = + D j D i j i j i D i D j j i j i m m m m

64 Magic Square () D=3x3, N= runs evaluations solutions

65 Non linear system x1 + sin 10 ( x ) 1 x 1 x = = 0 0 Search space [0,1] 1 run 143 evaluations 10 runs 1430 evaluations 1 solution 3 solutions

66 Model fitting (ARMA +AIC) Autoregressive Moving Average + Akaike's Information Criterion σ n m φi yt i = i= 0 j= 0 N i= = 1 θ a ( y y ) f + t data N = t j ARMA t j nlog ( n + m) heterogenuous ( σ )

67 A binary PSO C code Information links are modified at random if there has been no improvement // Pivot method // Works pretty well on some problems.. and pretty bad on some others P[s]=P_m[g]; // Initialise the new position of particle s // at the position of the best known around dist=log(d); // We suppose here D>= r=alea(1,dist); // Radius for DPNP for (k=0;k<r;k++)// Switch at random some bits { d=alea_integer(0,d-1); P[s][d]=1- P[s][d][d]; // Around g }

68 End of ANNEXE