Ying TAN. Peking University, China. Fireworks Algorithm (FWA) for Optimization. Ying TAN. Introduction. Conventional FWA.

Transcription

1 () for () for Based Peking University, China 1 / 92

2 Contents () for Based Based / 92

3 Definition of Swarm Intelligence () for Based Swarm intelligence is an artificial intelligence technique based on the study of collective behavior in decentralized, self-organized systems.* Swarm intelligence is the property of a system whereby the collective behaviours of (unsophisticated) agents interacting locally with their environment cause coherent functional global patterns to emerge.** * ** 3 / 92

4 Ant Colony (ACO)* () for Based ACO is inspired by the phenomena of ants finding paths to food. *Colorni, A., Dorigo, M., & Maniezzo, V. (1991, December). Distributed optimization by ant colonies. In Proceedings of the first European conference on artificial life (Vol. 142, pp ). 4 / 92

5 Particle Swarm (PSO)* () for Based PSO is inspired by the birds flocking to find a food source. Both individual and social behavior are considered. *Kennedy, J., & Eberhart, R. (1995, November). Particle swarm optimization. In Neural Networks, Proceedings., IEEE International Conference on (Vol. 4, pp ). IEEE. 5 / 92

6 Fish School Search (FSS)* () for Based FSS is inspired by the nature fish to find food. Simple computation in all individuals with some diversity among individuals. *Bastos Filho, C. J., de Lima Neto, F. B., Lins, A. J., Nascimento, A. I., & Lima, M. P. (2009). Fish school search. In Nature-Inspired s for Optimisation (pp ). Springer Berlin Heidelberg. 6 / 92

7 ()* () for Based is inspired by the splendid fireworks in the sky. Good explosion is in a small range with plenty of sparks. *Tan, Y., & Zhu, Y. (2010). algorithm for optimization. In Advances in Swarm Intelligence (LNCS 6145, pp ). Springer Berlin Heidelberg. 7 / 92

8 Problem Description () for Origination Steps Pseudo Experiments Discussion Based Suppose the () is designed for the general optimization problem where x denotes a location in the potential space, f (x) is an objective function, and x min and x max denote the bounds of the potential space. 8 / 92

9 Mimicing the () for Origination Steps Pseudo Experiments Discussion The sparks of fireworks are used to search global best solution. However, the explosion of fireworks are distinguished from good to bad in fireworks algorithm. For a good explosion, the generated sparks are dense and numerous, and vice versa. Based Figure: Two types of fireworks explosion 9 / 92

10 Framework of () for Origination Steps Pseudo Experiments Discussion Based Figure: The flowchart of fireworks algorithm 10 / 92

11 Five Steps in () for Origination Steps Pseudo Experiments Discussion Based Calculation of Sparks Number Calculation of Explosion Amplitude Sparks Explosion Sparks Gaussian Explosion Selection 11 / 92

12 Five Steps in () for Origination Steps Pseudo Experiments Discussion Based Calculation of Sparks Number Calculation of Explosion Amplitude Sparks Explosion Sparks Gaussian Explosion Selection 12 / 92

13 Calculation of Sparks Number () for The number of sparks generated by each firework x i is defined as follow Origination Steps Pseudo Experiments Discussion Based where m is a parameter controlling the total number of sparks generated by the n fireworks, y max = max(f (x i ))(i = 1, 2,..., n) is the maximum (worst) value of the objective function among the n fireworks, and ξ, which denotes the smallest constant in the computer, is utilized to avoid zero-division-error. 13 / 92

14 Calculation of Sparks Number () for To avoid overwhelming effects of splendid fireworks, bounds are defined for ŝ i Origination Steps Pseudo Experiments Discussion Based where a and b are const parameters and function round(x) is to choose the nearest interger for variable x. 14 / 92

15 Five Steps in () for Origination Steps Pseudo Experiments Discussion Based Calculation of Sparks Number Calculation of Explosion Amplitude Sparks Explosion Sparks Gaussian Explosion Selection 15 / 92

16 Calculation of Explosion Amplitude () for Amplitude of explosion for each firework is defined as follows Origination Steps Pseudo Experiments Discussion Based where  denotes the maximum explosion amplitude and y min = min(f (xi)) (i = 1, 2,..., n) is the minimum (best) value of the objective function among the n fireworks. In contrast to the design of sparks number, the amplitude of a good firework explosion is smaller than a bad one. 16 / 92

17 Five Steps in () for Origination Steps Pseudo Experiments Discussion Based Calculation of Sparks Number Calculation of Explosion Amplitude Sparks Explosion Sparks Gaussian Explosion Selection 17 / 92

18 Sparks Explosion () for Explosion sparks are generated by calculating explosion displacement. Origination Steps Pseudo Experiments Discussion Based 18 / 92

19 Five Steps in () for Origination Steps Pseudo Experiments Discussion Based Calculation of Sparks Number Calculation of Explosion Amplitude Sparks Explosion Sparks Gaussian Explosion Selection 19 / 92

20 Sparks Gaussian Explosion () for Origination Steps Pseudo Experiments Discussion Gaussian Sparks are generated in a Gaussian explosion process. Gaussian Sparks conducts search in a local space around a firework. Based 20 / 92

21 Five Steps in () for Origination Steps Pseudo Experiments Discussion Based Calculation of Sparks Number Calculation of Explosion Amplitude Sparks Explosion Sparks Gaussian Explosion Selection 21 / 92

22 Selection () for The selection probability of a location x i is defined as follows Origination Steps Pseudo Experiments Discussion Based where 22 / 92

23 Pseudo Code of () for Origination Steps Pseudo Experiments Discussion Based 1 Randomly select n locations for fireworks; 2 while stop criteria=false do 3 Set off n fireworks respectively at the n locations: 4 for each firework xi do 5 Calculate the number of sparks ŝ i that the firework yields; 6 Obtain locations of m common sparks of the firework; 7 end for 8 for k = 1: m do 9 Randomly select a firework xi ; 10 Generate a specific spark for the firework using 2; 11 end for 12 Select the best location and keep it for next explosion generation; 13 Randomly select n-1 locations from the two types of sparks and the current fireworks according to the selection probability; 14 end while 23 / 92

24 Experiments () for Origination Steps Pseudo Experiments Discussion Benchmark Functions Experiment Setup Experiment Results Based 24 / 92

25 Experiments () for Origination Steps Pseudo Experiments Discussion Benchmark Functions Experiment Setup Experiment Results Based 25 / 92

26 Benchmark Functions () for The feasible bounds for all functions are set as [ 100, 100] D. Origination Steps Pseudo Experiments Discussion Table: Details of benchmark functions Based 26 / 92

27 Experiments () for Origination Steps Pseudo Experiments Discussion Benchmark Functions Experiment Setup Experiment Results Based 27 / 92

28 Experiment Setup () for Origination Steps Pseudo Experiments Discussion Based Parameters are set as follow. 1 Number of fireworks = 5 2 Parameter m = 50 3 Parameter a = Parameter b = Parameter  = 40 6 Parameter ˆm = 5 Each function runs for 20 times. 28 / 92

29 Experiments () for Origination Steps Pseudo Experiments Discussion Benchmark Functions Experiment Setup Experiment Results Based 29 / 92

30 Experiment Results () for Table: Mean and standard deviation of, CPSO and SPSO Origination Steps Pseudo Experiments Discussion Based 30 / 92

31 Experiment Results () for algorithm can reach the result within fewer function evaluations. Table: Accuracy of algorithms at function evaluations Origination Steps Pseudo Experiments Discussion Based 31 / 92

32 Discussion () for Origination Steps Pseudo Experiments Discussion Based Advantages Easy to understand and realize High population diversity Excellent results Disadvantages Large time consuming Weak on shifted functions Lack of mathematical foundation Complicated parameters To overcome the disadvantages, an enhanced fireworks algorithm is proposed. 32 / 92

33 Enhanced (E)* () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute explosion number and amplitude 4: Generate explosion sparks 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) *Zheng, S. Q., Janecek, A., & Tan, Y. (2013). Enhanced fireworks algorithm. IEEE International Conference on Evolutionary Computation (Vol. 1, pp ). 33 / 92

34 Operator 1 - Explosion Number and Amplitude () for at good locations:... will have a large number of sparks... will have a small explosion amplitude E I ES Comparison of and PSO Based 34 / 92

35 Operator 1 - Explosion Number and Amplitude () for E I ES Comparison of and PSO Based at good locations: Problems... will have a large number of sparks... will have a small explosion amplitude If explosion amplitude is [close to] zero, explosion sparks will be located at [almost] same location as the firework itself Location of the best firework cannot be improved until another firework find a better location 34 / 92

36 Operator 1 - New Minimal Explosion Amplitude Check () for E I ES Comparison of and PSO Based Figure: Non-linear decreasing minimal explosion amplitude* *Zheng, S. Q., Janecek, A., & Tan, Y. (2013). Enhanced fireworks algorithm. IEEE International Conference on Evolutionary Computation (Vol. 1, pp ). 35 / 92

37 Framework of E () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute explosion number and amplitude 4: Generate explosion sparks 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) 36 / 92

38 Operator 2 - Generating Explosion Sparks () for E I ES Comparison of and PSO Based Figure: Improvements of generate explosion sparks* *Zheng, S. Q., Janecek, A., & Tan, Y. (2013). Enhanced fireworks algorithm. IEEE International Conference on Evolutionary Computation (Vol. 1, pp ). 37 / 92

39 Framework of E () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute explosion number and amplitude 4: Generate explosion sparks 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) 38 / 92

40 Operation 3 - Generating Gaussian Sparks () for E I ES Comparison of and PSO Based Figure: Improvements of generate Gaussian sparks* *Zheng, S. Q., Janecek, A., & Tan, Y. (2013). Enhanced fireworks algorithm. IEEE International Conference on Evolutionary Computation (Vol. 1, pp ). 39 / 92

41 Operation 3 - Generating Gaussian Sparks () for E I ES Comparison of and PSO Based Figure: Sparks of no shift and shift function* *Zheng, S. Q., Janecek, A., & Tan, Y. (2013). Enhanced fireworks algorithm. IEEE International Conference on Evolutionary Computation (Vol. 1, pp ). 40 / 92

42 Framework of E () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute explosion number and amplitude 4: Generate explosion sparks 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) 41 / 92

43 Operator 4 - Mapping () for E I ES Comparison of and PSO Based Mapping in conventional : k X i = X k min + X i k % ( Xmax k Xmin k ) If search space is equally distributed X k min X k max and a new location exceeds allowed search space only by a small value, the new position is close to the origin point. Example: search space [-20, 20], the new spark is created at X k = 21 (dimension k) Mapping: X i k = %40 Xi k = 1 k Mapping in E: X i = X k min + rand ( Xmax k Xmin k ) Uniform random mapping operator Maps the sparks to any location in the search space with uniform distribution 42 / 92

44 Framework of E () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute explosion number and amplitude 4: Generate explosion sparks 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) 43 / 92

45 Operator 5 - Selection () for E I ES Comparison of and PSO Based : distance based selection strategy Favors to select fireworks/sparks in less crowded solution space Diversity increases with expensive computation The most time consuming part E: Elitism-Random Selection Optima of the set will be selected first, while other individuals are selected randomly. Linear complexity - reduces runtime of E significantly 44 / 92

46 Shifted Index () for Table: Shifted index (SI) and shifted value (SV) E I ES Comparison of and PSO Based 45 / 92

47 Experiment Results on Schwefel 1.2 Function () for E I ES Comparison of and PSO Based Figure: Comparison of, SPSO and two types of E 46 / 92

48 Another Improved (I)* () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute sparks number and amplitude 4: Generate sparks by explosion 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) *Liu, J., Zheng, S., & Tan, Y. (2013). The Improvement on Controlling Exploration and Exploitation of Firework. In Advances in Swarm Intelligence (Vol. 7928, pp ). Springer Berlin Heidelberg. 47 / 92

49 Sparks Number and Amplitude () for Sparks number is represented as S n E I ES Comparison of and PSO Based Amplitude is stated as A n The transfer function is defined as f (x) Parameter a varies from 20 to 1 evenly in the iteration. 48 / 92

50 Another Improved (I)* () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute sparks number and amplitude 4: Generate sparks by explosion 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) *Liu, J., Zheng, S., & Tan, Y. (2013). The Improvement on Controlling Exploration and Exploitation of Firework. In Advances in Swarm Intelligence (Vol. 7928, pp ). Springer Berlin Heidelberg. 49 / 92

51 Gaussian Sparks () for Random mutation is employed to replace Gaussian sparks. E I ES Comparison of and PSO Based 50 / 92

52 Another Improved (I)* () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute sparks number and amplitude 4: Generate sparks by explosion 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Select N locations for next iteration 9: until termination (time, max. # evals, convergence,...) *Liu, J., Zheng, S., & Tan, Y. (2013). The Improvement on Controlling Exploration and Exploitation of Firework. In Advances in Swarm Intelligence (Vol. 7928, pp ). Springer Berlin Heidelberg. 51 / 92

53 Selection () for Method 1 = Random Fitness Selection The best individual is selected for next generation. Other individuals are selected by possibility E I ES Comparison of and PSO Based where y max is the fitness of the worst individual and f (x i ) is the fitness of individual x i. Method 2 = Best Fitness Selection The best sparks are selected for next generation. 52 / 92

54 Experiment Results on Shifted Sphere Function () for E I ES Comparison of and PSO Based Figure: Comparison of PSO,, IFS and IBS 53 / 92

55 More Deatiled Results are in the Paper () for Table: Part of Statistic Results of Mean, Std and best of Benchmark Functions in 10 Dimension E I ES Comparison of and PSO Based *FES stands for the times of function evaluation. 54 / 92

56 Empirical Study on (ES)* () for E I ES Comparison of and PSO Based 1: Initialize N fireworks and constant parameters 2: repeat 3: Compute explosion number and amplitude 4: Generate explosion sparks 5: Generate Gaussian sparks 6: Bound particles back to search space 7: Evaluate fitness of newly created sparks 8: Resampling 9: Select N locations for next iteration 10: until termination (time, max. # evals, convergence,...) *Pei, Y., Zheng, S., Tan, Y., & Takagi, H. (2012, October). An empirical study on influence of approximation approaches on enhancing fireworks algorithm. In Systems, Man, and Cybernetics (SMC), 2012 IEEE International Conference on (pp ). IEEE. 55 / 92

57 Experiment Functions () for Table: The attributes of benchmark functions E I ES Comparison of and PSO Based 56 / 92

58 Shifted Value and Environment () for Table: The shifted index and value E I ES Comparison of and PSO Based The experimental platform is Visual Studio 2012 and the program is running on 64bit Window 8 operation system with a Intel Core i7-3820qm; 2.70GHz and 2GB RAM. Each experiment is all running 30 times of over 300,000 function evaluations. 57 / 92

59 Comparison of with PSO () for Table: Part of the experiment results E I ES Comparison of and PSO Based 58 / 92

60 Result Analysis () for E I ES Comparison of and PSO Based From the experiment results, it can be concluded as follows. E, IFS, IBS and LS2-BST10 are superior to conventional on most functions. SPSO achieves better results on large shifted indexes. E is fast on 11 functions and SPSO is quicker than other algorithms on 2 other functions. consumes more time than all the other algorithms. GPU- greatly reduced the computational time for each function. 59 / 92

61 The principle of GPU () for A graphics processing unit (GPU), is a specialized electronic circuit designed to rapidly manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display.* Based The Principle of GPU Experiments Figure: A graphics processing unit *Owens, J. D., Houston, M., Luebke, D., Green, S., Stone, J. E., & Phillips, J. C. (2008). GPU computing. Proceedings of the IEEE, 96(5), / 92

62 What is CUDA () for NVIDIA s Computing Unified Device Architecture (CUDA) is a high level general purpose parallel computing platform and programming model. Based The Principle of GPU Experiments Figure: Memory model on CUDA 61 / 92

63 Running GPU- on CUDA () for Based The Principle of GPU Experiments Figure: The Flowchart of the GPU- Implementation on CUDA 62 / 92

64 GPU- () for Based The Principle of GPU Experiments Two Novel Strategies Greedy fireworks search Attract repulse mutation Advantages The algorithm can find good solutions, compared to the state-of-the-art algorithms. As the problem gets complex, the algorithm can scale in a natural and decent way. Few control variables are used to steer the optimization. The variables are robust and easy to choose. 63 / 92

65 Experimental Environment () for Based The Principle of GPU Experiments Operation System: Windows 7 Professional x64 with 4G DDR3 Memory (1333 MHz) CPU: Intel core I (2.9 GHz, 3.1 GHz) GPU: NVIDIA GeForce GTX 560 Ti with 384 CUDA cores CUDA runtime version: / 92

66 Benchmark Functions for GPU- () for Based The Principle of GPU Experiments Table : Benchmark functions ID Function Expression Feasible bounds Dimension optima f1 Sphere f1 = D i=1 x2 i [ 5.12, 5.12] D 30 0 f2 Hyper-ellipsoid f2 = D i=1 i x2 i [ 5.12, 5.12] D 30 0 f3 Schwefel 1.2 f3 = D i=1 ( i j=1 xj ) 2 [ , ] D 30 0 f4 Rosenbrock f4 = [ D 1 i=1 100 (xi+1 x 2 ) 2 i + (1 xi) 2] [ 2.048, 2.048] D 30 0 f5 Rastrigin f5 = 10 D + D [ i=1 x 2 i 10 cos (2πxi) ] [ 5.12, 5.12] D 30 0 f6 Schwefel f6 = [ ( D xi )] i=1 xi sin [ 500, 500] D e+04 f7 Griewangk f7 = 1 D 4000 i=1 x2 i ( ) D i=1 cos i xi + 1 [ 600, 600] D 30 0 ( ) 1 ( f8 Ackley f8 = a exp b D D i=1 x2 i exp 1 ) D D i=1 cos (cxi) + a + exp(1) [ , ] D / 92

67 Experiment Results () for Based The Principle of GPU Experiments Table : Precision comparison Fun GPU- PSO Avg. Std. Avg. Std. Avg. Std. f1 1.31E E E E E E-07 f2 1.49E E E E E E-10 f3 3.46E E E E E E+04 f4 1.92E E E E E E+02 f5 7.02E E E E E E+02 f6-8.09e E E E E E+03 f7 1.33E E E E E E+00 f8 3.63E E E E E E+01 Table : p-values of t-test f1 f2 f3 f4 f5 f6 f7 f8 GPU- vs. 1.00E E E E E E E E+00 GPU- vs. PSO 3.46E E E E E E E E / 92

68 Experiment Results () for GPU- is the fastest algorithm among and PSO. Based The Principle of GPU Experiments Table : Running time and speedup of function Rosenbrock n (s) PSO(s) GPU-(s) SU() SU(PSO) / 92

69 Comparison of GPU- and () for The speedup is up to 190. Based The Principle of GPU Experiments Figure: Speedup of GPU- compared with conventional The horizontal axis represents the number of fireworks, while the vertical axis means the speedup. 68 / 92

70 Comparison of GPU- and PSO () for The speedup is up to 250. Based The Principle of GPU Experiments Figure: Speedup of GPU- compared with PSO Again, the horizontal axis represents the number of fireworks, while the vertical axis means the speedup. 69 / 92

71 The of () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Five applications are listed below. 1 for Non-negative Matrix Factorization (NMF) computing 2 on spam detection 3 on design of digital filters 4 solve non-linear equations 5 Multiobjective for variable-rate fertilization in oil crop production 70 / 92

72 for NMF computing [1 3] () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Definition of NMF The Non-negative Matrix Factorization (NMF) refers to as low-rank approximation and has been utilized in several various areas like content based retrieval and data mining applications, etc. NMF can reduce storage and runtime requirements, and also reduce redundancy and noise in the data representation while capturing the essential associations. [1] Janecek, A., & Tan, Y. (2011). Swarm Intelligence for Non-Negative Matrix Factorization. International Journal of Swarm Intelligence Research (IJSIR), 2(4), [2] Janecek, A., & Tan, Y. (2011). Using population based algorithms for initializing nonnegative matrix factorization. In Advances in Swarm Intelligence (pp ). Springer Berlin Heidelberg. [3] Janecek, A., & Tan, Y. (2011, July). Iterative improvement of the Multiplicative Update NMF algorithm using nature-inspired optimization. In Natural Computation (ICNC), 2011 Seventh International Conference on (Vol. 3, pp ). IEEE. 71 / 92

73 Problem and Solution () for Problem The nonlinear optimization problem underlying NMF can generally be stated as min W,H f (W, H) = min W,H 1 2 A WH 2 F. Solution Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization 72 / 92

74 Experiment Result () for The best algorithm is optimal fireworks search (opt-fs). Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Figure: Convergence curves for six different algorithms 73 / 92

75 Spam Detection* () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Definition Spam detection is an action to put those spams away from general s. Problem In previous research, parameters in the anti-spam process are set simply and manually. Solution A new framework of fireworks algorithm was proposed that automatically optimizes parameters in anti-spam model. *He, W., Mi, G., & Tan, Y. (2013). Parameter of Local-Concentration Model for Spam Detection by Using. In Advances in Swarm Intelligence (pp ). Springer Berlin Heidelberg. 74 / 92

76 Flowchart of on Spam Detection () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Figure: Flowchart of on spam detection 75 / 92

77 Experiment Result () for Table: Comparison of fireworks algorithm and local concentration method (LC) Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization 76 / 92

78 for Digital Filters Design () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Definition A digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal.* Problem Filters designed by other intellligent algorithms have disadvantages. Solution Design digital filters by fireworks algorithm is proposed.** *Rabiner, L. R., & Gold, B. (1975). Theory and application of digital signal processing. Englewood Cliffs, NJ, Prentice-Hall, Inc., p., 1. **Gao, H., & Diao, M. (2011). Cultural firework algorithm and its application for digital filters design. International Journal of Modelling, Identification and Control, 14(4), / 92

79 Solution () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Figure: The flow chart of design digital filters by culture fireworks algorithm 78 / 92

80 Experiment Result () for Table: Comparison of four algorithms on FIR filter Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization 79 / 92

81 for Solving Nonlinear Equations () for Four equations are listed below. Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Zhang J. (2012). Artificial bee colony algorithm for solving nonlinear equation and system. Computer Engineering and, 48(22), / 92

82 Solution () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Step 1: Randomly generates n individuals at initial. Step 2: Generate common sparks and Gaussian sparks the same as fireworks algorithm. Step 3: Choose the best individual for next generation and the next (N - 1) individuals are choose the same like fireworks algorithm. Step 4: If the terminal condition is met, end the procedure. If not, go back to step / 92

83 Experiment Result () for Table: Comparison of ABC and algorithms on four equations Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization 82 / 92

84 Multiobjective for variable-rate fertilization in oil crop production () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Oil crop fertilization is a multiobjective problem. Three objectives 1 Crop quality 2 Fertilizer cost 3 Energy consumption Solution Multiobjective fireworks algorithm Differential evolution strategies Zheng, Y. J., Song, Q., & Chen, S. Y. (2013). Multiobjective fireworks optimization for variable-rate fertilization in oil crop production. Applied Soft Computing. 83 / 92

85 Framework of Multiobjective () for Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization (1) Initialization. 1.1) Randomly generate a population P of p feasible solutions. 1.2) Create the empty non-dominated solution archive NP and select those non-dominated solutions from P to update NP. (2) Iterative improvement. 2.1) For each individual (2.1) For each individual x i in P do: 2.1.1) Calculate number of sparks s i for x i ) Calculate amplitude of sparks A i for x i ) Generate s i sparks of x i ) Generate a specific spark of x i ) Compute fitness for all sparks ) Update NP based on the new solutions (sparks). 2.2) Select p solutions from the fireworks and sparks, where the selection probability of each solution x i is f (x i )/sum(f (x i )); 2.3) For i = 1 to p 2.3.1) Apply the mutation, crossover and selection operators to x i and get a trial solution u i ) If the DE result indicates that x i is to be replaced by u i, then use u i to update NP. 2.4) Update P by including the best solution and other (p 1) ones randomly selected according to distance-based propability. 2.5) If the termination condition is satisfied, then the algorithm stops; else go to step 2.1). 84 / 92

86 Experiment Results () for Table: Solutions of Multiobjective random search (MORS) and Multiobjective fireworks algorithm (MOFOA) Based 1.NMF Computing 2.Spam Detection 3.Digital Filters Design 4.Nonlinear Equations 5.Crop Fertilization Figure: Distribution of the solutions in objective space 85 / 92

87 () for For fireworks algorithm, there are both advantages and disadvantages. Based Advantages 1 Effectively solve most optimization problems 2 Less running time when parallelized 3 Many variables algorithm and varied applications Disadvantages 1 Lack of necessary mathematical foundation 2 Difficult to choose proper parametersr 86 / 92

88 Future Works and Acknowledgement () for Based Future Works Mathematical foundation More improvements Research on parameters setting Deeply research on realization of parallelized algorithm The applications of fireworks algorithm Acknowledgement The following people offered great help. They are Chao Yu, Shaoqiu Zheng, Ke Ding, Zhongyang Zheng, Guyue Mi, Weiwei Hu, Xiang Yang and Lang Liu. 87 / 92

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