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Robust Immunological Algorithms for High-Dimensional Global Optimization
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Robust Immunological Algorithms for High-Dimensional Global Optimization Presentation Transcript

  • 1. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Robust Immunological Algorithms for High-Dimensional Global Optimization V. Cutello † G. Narzisi ‡ G. Nicosia † M. Pavone † † Department of Mathematics and Computer Science University of Catania Viale A. Doria 6, 95125 Catania, Italy (cutello, nicosia, mpavone)@dmi.unict.it ‡ Computer Science Department Courant Institute of Mathematical Sciences New York University New York, NY 10012, U.S.A. narzisi@nyu.edu Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 2. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Outline Introduction Global Optimization Numerical Minimization Problem Artificial Immune System Optimization Immunological Algorithm Cloning and Hypermutation Operators. Aging and Selection Operators. Metrics and Dynamic Behavior Influence of Different Potential Mutations. Tuning of the ρ parameter. Convergence and Learning Processes. Results and Comparison Conclusions Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 3. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Global Optimization ◮ Global Optimization (GO): finding the best set of parameters to optimize a given objective function ◮ GO problems are quite difficult to solve: there exist solutions that are locally optimal but not globally ◮ GO requires finding a setting x = (x1 , x2 , . . . , xn ) ∈ S, where S ⊆ Rn is a bounded set on Rn , such that a certain n-dimensional objective function f : S → R is optimized. ◮ GOAL: findind a point xmin ∈ S such that f (xmin ) is a global minimum on S, i.e. ∀x ∈ S : f (xmin ) ≤ f (x). ◮ It is difficult to decide when a global (or local) optimum has been reached ◮ There could be very many local optima where the algorithm can be trapped the difficulty increases proportionally with the problem dimension Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 4. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Numerical Minimization Problem Let be x = (x1 , x2 , . . . , xn ) the variables vector in Rn ; Bl = (Bl1 , Bl2 , . . . , Bln ) and Bu = (Bu1 , Bu2 , . . . , Bun ) the lower and the upper bounds of the variables, such that xi ∈ Bli , Bui (i = 1, . . . , n). GOAL: minimizing f (x) the objective function min(f (x)), Bl ≤ x ≤ Bu Benchmarks used to evaluate the performances and convergence ability: ◮ twenty-three functions taken by [Yao et al., IEEE TEC, 1999]; ◮ twelve functions taken by [Timmis et al., GECCO 2003] All these functions belong to three different categories: unimodal, multimodal with many local optima, and multimodal with few local optima Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 5. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Artificial Immune System Articial Immune Systems - AIS ◮ Immune System (IS) is the main responsible to protect the organism against the attack from external microorganisms, that might cause diseases; ◮ The biological IS has to assure recognition of each potentially dangerous molecule or substance ◮ Artificial Immune Systems are a new paradigm of the biologically-inspired computing ◮ Three immunological theory: immune networks, negative selection, and clonal selection ◮ AIS have been successfully employed in a wide variety of different applications [Timmis et al.: J.Ap.Soft.Comp., BioSystems, Curr. Proteomics, 2008] Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 6. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Artificial Immune System Clonal Selection Algorithms - CSA ◮ CSA represents an effective mechanism for search and optimization ◮ [Cutello et al.: IEEE TAC, J. Comb. Optimization, 2007] ◮ Cloning and Hypermutation operators: strength driven of CSA ◮ Cloning: triggers the growth of a new population of high-value B cells centered on a higher affinity value ◮ Hypermutation: can be seen as a local search procedure that leads to a faster maturation during the learning phase. Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 7. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Pseudo-code of Immunological Algorithm Immunological Algorithm(d, dup, ρ, τB , Tmax ) FFE ← 0; Nc ← d · dup; t ← 0; P (t) ← Initialize_Population(d); // xi = Bli + β · (Bui − Bli ) Compute_Fitness(P (t) ); FFE ← FFE + d; while (FFE < Tmax )do P (clo) ← Cloning (P (t) , dup); P (hyp) ← Hypermutation(P (clo) , ρ); Compute_Fitness(P (hyp) ); FFE ← FFE + Nc ; (t) (hyp) (Pa , Pa ) = Aging(P (t) , P (hyp) , τB ); P (t+1) ← (µ + λ)-Selection(P (t) , P (hyp) ); a a t ← t + 1; end_while Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 8. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Cloning and Hypermutation Operators. Cloning Operator ◮ Cloning operator clones each B cell dup times (P (clo) ) ◮ Each clone was assigned a random age chosen into the range [0, τB ] ◮ Using the cloning operator, an immunological algorithm produces individuals with higher affinities (higher fitness function values) ◮ Improvement: choosing the age of each clone into the range [0, 2 τB ] 3 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 9. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Cloning and Hypermutation Operators. Hypermutation Operator ◮ Tries to mutate each B cell receptor M times is not based on an explicit usage of a mutation probability. ◮ There exist several different kinds of hypermutation operator [Cutello et al.: LNCS 3239 and IEEE Press vol.1, 2004] ◮ We have used Inversely Proportional Hypermutation as the fitness function value of the current B cell increases, the number of mutations performed decreases ◮ two different potential mutations are used: ˆ e−f (x) ˆ α= , α = e−ρf (x) ρ α represents the mutation rate, and ˆ(x) the normalized f fitness function in [0, 1]. Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 10. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Cloning and Hypermutation Operators. How the Hypermutation Operator works ◮ Choose randomly a variable xi (i ∈ {1, . . . , ℓ = n}) ◮ replace xi with (t+1) (t) (t) xi = (1 − β)xi + βxrandom , (t) (t) xrandom = xi is a randomly chosen variable ◮ To normalize the fitness function value was used the best current fitness value decreased of a user-defined threshold θ, ◮ is not known a priori any additional information concerning the problem [Cutello et al., SAC 2006] Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 11. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Aging and Selection Operators. Aging Operator ◮ Eliminates all old B cells, in the populations P (t) , and P (hyp) ◮ Depends on the parameter τB : maximum number of generations allowed ◮ when a B cell is τB + 1 old it is erased ◮ GOAL: produce an high diversity into the current population to avoid premature convergence ◮ static aging operator: when a B cell is erased independently from its fitness value quality ◮ elitist static aging operator: the selection mechanism does not allow the elimination of the best B cell Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 12. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Aging and Selection Operators. (µ + λ)-Selection Operator ◮ The best survivors are selected to generate the new population P (t+1) , ◮ If d1 < d are survived then d − d1 are randomly selected (t) (hyp) among those “died”, i.e. (P (t) Pa ) ⊔ (P (hyp) Pa ) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 13. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Influence of Different Potential Mutations. ◮ Two different potential mutations were used to determine the number of mutations M; ◮ We present their comparisons, and hence their influence, in order to the performances ◮ The main goal is to determine best law to use to tackle optimization problems ◮ Next tables show the different “impact” in term of performance and quality of the solution ◮ We show the mean of the best B cells on all runs, and the standard deviation ◮ We used experimental protocol proposed in [Yao et al., 1999] ◮ Parameters used: ◮ d = 100, dup = 2, τB = 15 ◮ 1st mutation rate: ρ ∈ {50, 75, 100, 125, 150, 175, 200} ◮ 2nd mutation rate: ρ ∈ {4, 5, 6, 7, 8, 9, 10, 11} Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 14. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Influence of Different Potential Mutations. Unimodal Funtions ˆ e−f (x) ˆ α= ρ α = e−ρf (x) f1 4.663 × 10−19 0.0 7.365 × 10−19 0.0 f2 3.220 × 10−17 0.0 1.945 × 10−17 0.0 f3 3.855 0.0 5.755 0.0 f4 8.699 × 10−3 0.0 3.922 × 10−2 0.0 f5 22.32 16.29 11.58 13.96 f6 0.0 0.0 0.0 0.0 f7 1.143 × 10−4 1.995 × 10−5 1.411 × 10−4 2.348 × 10−5 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 15. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Influence of Different Potential Mutations. Multimodal Functions ˆ −f (x) ˆ ˆ −f (x) ˆ α= e ρ α = e−ρf (x) α= e ρ α = e−ρf (x) f8 −12559.69 −12535.15 f16 −1.017 −1.013 34.59 62.81 2.039 × 10−2 2.212 × 10−2 f9 0.0 0.596 f17 0.425 0.423 0.0 4.178 4.987 × 10−2 3.217 × 10−2 f10 1.017 × 10−10 0.0 f18 6.106 5.837 5.307 × 10−11 0.0 4.748 3.742 f11 2.066 × 10−2 0.0 f19 −3.72 −3.72 5.482 × 10−2 0.0 8.416 × 10−3 7.846 × 10−3 f12 7.094 × 10−21 1.770 × 10−21 f20 −3.293 −3.292 5.621 × 10−21 8.774 × 10−24 3.022 × 10−2 3.097 × 10−2 f13 1.122 × 10−19 1.687 × 10−21 f21 −10.153 −10.153 2.328 × 10−19 5.370 × 10−24 (7.710 × 10−8 ) 1.034 × 10−7 f14 0.999 0.998 f22 −10.402 −10.402 7.680 × 10−3 1.110 × 10−3 (1.842 × 10−6 ) 1.082 × 10−5 f15 3.27 × 10−4 3.2 × 10−4 f23 −10.536 −10.536 3.651 × 10−5 2.672 × 10−5 7.694 × 10−7 1.165 × 10−5 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 16. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Tuning of the ρ parameter. Potential Mutation behavior using different dimension values Mutation Rate for the Hypermutation Operator 200 dim=30, ρ=3.5 dim=50, ρ=4.0 180 dim=100, ρ=6.0 dim=200, ρ=7.0 160 140 10 120 9 Mutations 8 100 7 6 80 5 4 60 3 2 40 1 0.4 0.5 0.6 0.7 0.8 0.9 1 20 0 0.2 0.4 0.6 0.8 1 Fitness Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 17. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Tuning of the ρ parameter. Potential Mutation behavior on large dimension values Mutation Rate for the Hypermutation Operator 5000 dim=1000, ρ=9.0 dim=5000, ρ=11.5 4000 3 3000 Mutations 2.5 2 2000 1.5 1000 1 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0 0.2 0.4 0.6 0.8 1 Fitness Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 18. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Mean performance comparison curves for test function f1 Mean performance comparison curves for test function f6 70000 70000 Legend Legend Real Real 60000 Binary 60000 Binary 50000 50000 Function value Function value 40000 40000 30000 30000 20000 20000 10000 10000 0 0 100 200 300 400 500 600 700 5 10 15 20 25 30 35 40 45 50 Generation Generation Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 19. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Mean performance comparison curves for test function f8 Mean performance comparison curves for test function f10 -2000 25 Legend Legend Real Real -4000 Binary Binary 20 -6000 Function value Function value 15 -8000 10 -10000 5 -12000 -14000 0 100 200 300 400 500 600 700 800 900 1000 100 200 300 400 500 600 700 800 900 1000 Generation Generation Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 20. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Mean performance comparison curves for test function f18 Mean performance comparison curves for test function f21 70 0 Legend Legend Real -1 Real 60 Binary Binary -2 50 -3 Function value Function value -4 40 -5 30 -6 -7 20 -8 -9 10 -10 0 -11 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 Generation Generation Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 21. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Learning Process ◮ To analyze the learning process was used the Information Gain. ◮ measures the quantity of information the system discovers during the learning phase [Cutello et al.: journal of Combinatorial Optimization, 2007] ◮ B cells distribution function: (t) Bt Bmt fm = Ph m = t d m=0 Bm t with Bm the number of B cells at time step t with fitness function value m ◮ Information Gain: (t) (t ) X (t) K (t, t0 ) = fm log(fm /fm 0 ) m ◮ Entropy: (t) (t) X E(t) = fm log fm m Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 22. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Maximum Information-Gain Principle ◮ The gain is the amount of information the system has already learnt from the given problem instance with respect to the randomly generated initial population P (t=0) ◮ Once the learning process begins, the information gain increases monotonically until it reaches a final steady state ◮ Is consistent with the idea of a maximum information-gain principle [Cutello et al., GECCO 2003]: dK ≥0 dt Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 23. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Information Gain and Standard Deviation on function 5 * Clonal Selection Algorithm: opt-IMMALG 25 20 300 15 250 200 10 150 100 50 5 0 16 32 64 0 16 32 64 Generations Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 24. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Learning Processon functions f5 , f7 , and f10 Information Gain 25 f5 f7 f10 20 15 10 5 0 1 2 4 8 16 32 Generations Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 25. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Convergence and Learning Processes. Averge fitness versus Best fitness on function 5 * Clonal Selection Algorithm: opt-IMMALG 4e+09 avg fitness best fitness 3.5e+09 3e+09 25 2.5e+09 gain 20 entropy Fitness 2e+09 15 10 1.5e+09 5 1e+09 0 16 32 64 5e+08 0 0 2 4 6 8 10 Generations Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 26. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions ◮ IA was extensively compared against several nature inspired methodologies including differential evolution (DE) algorithms DE seems to perform better than many other EAs on the same test bed ◮ IA was compared on 33 optimization algorithms in the overall, including a deterministic global search algorithm DIRECT Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 27. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus DIRECT DIRECT: [Jones et al.,j. opt. theory appl., 1993] and [Finkel, Techn. Report, 2003]. ˆ −ˆ(x) f α = e−ρf (x) DIRECT α= e ρ f5 16.29 27.89 22.32 f7 1.995 × 10−5 8.9 × 10−3 1.143 × 10−4 f8 −12535.15 −4093.0 −12559.69 f12 1.770 × 10−21 0.03 7.094 × 10−21 f13 1.687 × 10−21 0.96 1.122 × 10−19 f14 0.998 1.0 0.999 f15 3.2 × 10−4 1.2 × 10−3 3.27 × 10−4 f16 −1.013 −1.031 −1.017 f17 0.423 0.398 0.425 f18 5.837 3.01 6.106 f19 −3.72 −3.86 −3.72 f20 −3.292 −3.30 −3.293 f21 −10.153 −6.84 −10.153 f22 −10.402 −7.09 −10.402 f23 −10.536 −7.22 −10.536 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 28. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus CLONALG1 and CLONALG2 - mut01 [De Castro et al., IEEE TEC, 2002] and [Cutello et al., ICARIS 2005] IA CLONALG1 CLONALG2 f1 4.663 × 10−19 3.7 × 10−3 5.5 × 10−4 (7.365 × 10−19 ) (2.6 × 10−3 ) (2.4 × 10−4 ) f2 3.220 × 10−17 2.9 × 10−3 2.7 × 10−3 (1.945 × 10−17 ) (6.6 × 10−4 ) (7.1 × 10−4 ) f3 3.855 1.5 × 10+4 5.9 × 10+3 (5.755) (1.8 × 10+3 ) (1.8 × 10+3 ) f4 8.699 × 10−3 4.91 8.7 × 10−3 (3.922 × 10−2 ) (1.11) (2.1 × 10−3 ) f5 22.32 27.6 2.35 × 10+2 (11.58) (1.034) (4.4 × 10+2 ) f6 0.0 2.0 × 10−2 0.0 (0.0) (1.4 × 10−1 ) (0.0) f7 1.143 × 10−4 7.8 × 10−2 5.3 × 10−3 (1.411 × 10−4 ) (1.9 × 10−2 ) (1.4 × 10−3 ) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 29. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus CLONALG1 and CLONALG2 - mut01 IA CLONALG1 CLONALG2 f8 −12559.69 −11044.69 −12533.86 (34.59) (186.73) (43.08) f9 0.0 37.56 22.41 (0.0) (4.88) (6.70) f10 1.017 × 10−10 1.57 1.2 × 10−1 (5.307 × 10−11 ) (3.9 × 10−1 ) (4.1 × 10−1 ) f11 2.066 × 10−2 1.7 × 10−2 4.6 × 10−2 (5.482 × 10−2 ) (1.9 × 10−2 ) (7.0 × 10−2 ) f12 7.094 × 10−21 0.336 0.573 (5.621 × 10−21 ) (9.4 × 10−2 ) (2.6 × 10−1 ) f13 1.122 × 10−19 1.39 1.69 (2.328 × 10−19 ) (1.8 × 10−1 ) (2.4 × 10−1 ) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 30. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus CLONALG1 and CLONALG2 - mut01 IA CLONALG1 CLONALG2 f14 0.999 1.0021 2.42 (7.680 × 10−3 ) (2.8 × 10−2 ) (2.60) f15 3.270 × 10−4 1.5 × 10−3 7.2 × 10−3 (3.651 × 10−5 ) (7.8 × 10−4 ) (8.1 × 10−3 ) f16 −1.017 −1.0314 −1.0210 (2.039 × 10−2 ) (5.7 × 10−4 ) (1.9 × 10−2 ) f17 0.425 0.399 0.422 (4.987 × 10−2 ) (2.0 × 10−3 ) (2.7 × 10−2 ) f18 6.106 3.0 3.46 (4.748) (1.3 × 10−5 ) (3.28) f19 −3.72 −3.71 −3.68 (8.416 × 10−3 ) (1.5 × 10−2 ) (6.9 × 10−2 ) f20 −3.293 −3.23 −3.18 (3.022 × 10−2 ) (5.9 × 10−2 ) (1.2 × 10−1 ) f21 −10.153 −5.92 −3.98 (7.710 × 10−8 ) (1.77) (2.73) f22 −10.402 −5.90 −4.66 (1.842 × 10−6 ) (2.09) (2.55) f23 −10.536 −5.98 −4.38 (7.694 × 10−7 ) (1.98) (2.66) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 31. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus CLONALG1 and CLONALG2 - mut02 IA CLONALG1 CLONALG2 f1 0.0 9.6 × 10−4 3.2 × 10−6 (0.0) (1.6 × 10−3 ) (1.5 × 10−6 ) f2 0.0 7.7 × 10−5 1.2 × 10−4 (0.0) (2.5 × 10−5 ) (2.1 × 10−5 ) f3 0.0 2.2 × 104 2.4 × 10+4 (0.0) (1.3 × 10−4 ) (5.7 × 10+3 ) f4 0.0 9.44 5.9 × 10−4 (0.0) (1.98) (3.5 × 10−4 ) f5 16.29 31.07 4.67 × 10+2 (13.96) (13.48) (6.3 × 10+2 ) f6 0.0 0.52 0.0 (0.0) (0.49) (0.0) f7 1.995 × 10−5 1.3 × 10−1 4.6 × 10−3 (2.348 × 10−5 ) (3.5 × 10−2 ) (1.6 × 10−3 ) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 32. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus CLONALG1 and CLONALG2 - mut02 IA CLONALG1 CLONALG2 f8 −12535.15 −11099.56 −1228.39 (62.81) (112.05) (41.08) f9 0.596 42.93 21.75 (4.178) (3.05) (5.03) f10 0.0 18.96 19.30 (0.0) (2.2 × 10−1 ) (1.9 × 10−1 ) f11 0.0 3.6 × 10−2 9.4 × 10−2 (0.0) (3.5 × 10−2 ) (1.4 × 10−1 ) f12 1.770 × 10−21 0.632 0.738 (8, 774 × 10−24 ) (2.2 × 10−1 ) (5.3 × 10−1 ) f13 1.687 × 10−21 1.83 1.84 (5.370 × 10−24 ) (2.7 × 10−1 ) (2.7 × 10−1 ) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 33. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus CLONALG1 and CLONALG2 - mut02 IA CLONALG1 CLONALG2 f14 0.998 1.0062 1.45 (1.110 × 10−3 ) (4.0 × 10−2 ) (0.95) f15 3.2 × 10−4 1.4 × 10−3 8.3 × 10−3 (2.672 × 10−5 ) (5.4 × 10−4 ) (8.5 × 10−3 ) f16 −1.013 −1.0315 −1.0202 (2.212 × 10−2 ) (1.8 × 10−4 ) (1.8 × 10−2 ) f17 0.423 0.401 0.462 (3.217 × 10−2 ) 8.8 × 10−3 ) (2.0 × 10−1 ) f18 5.837 3.0 3.54 (3.742) (1.3 × 10−7 ) (3.78) f19 −3.72 −3.71 −3.67 (7.846 × 10−3 ) (1.1 × 10−2 ) (6.6 × 10−2 ) f20 −3.292 −3.30 −3.21 (3.097 × 10−2 ) (1.0 × 10−2 ) (8.6 × 10−2 ) f21 −10.153 −7.59 −5.21 (1.034 × 10−7 ) (1.89) (1.78) f22 −10.402 −8.41 −7.31 (1.082 × 10−5 ) (1.4) (2.67) f23 −10.536 −8.48 −7.12 (1.165 × 10−5 ) (1.51) (2.48) Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 34. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus BCA and HGA [Timmis et al., GECCO 2003] ˆ −f (x) ˆ α= e ρ α = e−ρf (x) BCA HGA g1 −1.12 ± 1.17 × 10−3 −1.12 ± 1.62 × 10−3 −1.08 −1.12 g2 −1.03 ± 8.82 × 10−4 −1.03 ± 7.129 × 10−4 −1.03 −0.99 g3 −12.03 ± 8.196 × 10−4 −12.03 ± 9.28 × 10−4 −12.03 −12.03 g4 0.3984 ± 6.73 × 10−4 0.3985 ± 8.859 × 10−4 0.40 0.40 g5 −178.51 ± 11.49 −178.88 ± 9.83 −186.73 −186.73 g6 −179.27 ± 11.498 −179.12 ± 10.02 −186.73 −186.73 g7 −2.529 ± 0.2026 −2.571±0.253 0.92 0.92 g8 1.314 × 10−12 ± 4.668 × 10−12 1.314 × 10−12 ± 4.668 × 10−12 1.0 1.0 g9 −3.51 ± 1.464 × 10−3 −0.351 ± 1.62 × 10−3 −0.91 −0.99 g10 −186.67 ± 8.17 × 10−2 −186.65 ± 0.1158 −186.73 −186 g11 3.81 × 10−5 ± 5.58 × 10−15 3.81 × 10−5 ± 6.98 × 10−14 0.04 0.04 g12 0.0 ± 0.0 0.0 ± 0.0 1 1 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 35. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus PSO, arPSO and SEA - n = 30 [Thomsen et al., CEC 2004] IA PSO arPSO SEA ˆ e−f (x) ˆ α= ρ α= e−ρf (x) f1 0.0 0.0 0.0 6.8 × 10−13 1.79 × 10−3 0.0 0.0 0.0 5.3 × 10−13 2.77 × 10−4 f2 0.0 0.0 0.0 2.09 × 10−2 1.72 × 10−2 0.0 0.0 0.0 1.48 × 10−1 1.7 × 10−3 f3 0.0 0.0 0.0 0.0 1.59 × 10−2 0.0 0.0 0.0 2.13 × 10−25 4.25 × 10−3 f4 5.6 × 10−4 0.0 2.11 × 10−16 1.42 × 10−5 1.98 × 10−2 2.18 × 10−3 0.0 8.01 × 10−16 8.27 × 10−6 2.07 × 10−3 f5 21.16 12 4.026 3.55 × 10+2 31.32 11.395 13.22 4.99 2.15 × 10+3 17.4 f6 0.0 0.0 4 × 10−2 18.98 0.0 0.0 0.0 1.98 × 10−1 63 0.0 f7 3.7 × 10−5 1.52 × 10−5 1.91 × 10−3 3.89 × 10−4 7.11 × 10−4 5.62 × 10−5 2.05 × 10−5 1.14 × 10−3 4.78 × 10−4 3.27 × 10−4 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 36. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus PSO, arPSO and SEA - n = 30 [Thomsen et al., CEC 2004] IA PSO arPSO SEA ˆ −f (x) ˆ α= e ρ α = e−ρf (x) f8 −1.257 × 10+4 −1.256 × 10+4 −7.187 × 10+3 −8.598 × 10+3 −1.167 × 10+4 8.369 25.912 6.72 × 10+2 2.07 × 10+3 2.34 × 10+2 f9 0.0 0.0 49.17 2.15 7.18 × 10−1 0.0 0.0 16.2 4.91 9.22 × 10−1 f10 4.74 × 10−16 0.0 1.4 1.84 × 10−7 1.05 × 10−2 1.21 × 10−15 0.0 7.91 × 10−1 7.15 × 10−8 9.08 × 10−4 f11 0.0 0.0 2.35 × 10−2 9.23 × 10−2 4.64 × 10−3 0.0 0.0 3.54 × 10−2 3.41 × 10−1 3.96 × 10−3 f12 1.787 × 10−21 1.77 × 10−21 3.819 × 10−1 8.559 × 10−3 4.56 × 10−6 5.06 × 10−23 7.21 × 10−24 8.4 × 10−1 4.79 × 10−2 8.11 × 10−7 f13 1.702 × 10−21 1.686 × 10−21 −5.969 × 10−1 −9.626 × 10−1 −1.143 4.0628 × 10−23 1.149 × 10−24 5.17 × 10−1 5.14 × 10−1 1.34 × 10−5 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 37. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus PSO, arPSO and SEA - n = 30 [Thomsen et al., CEC 2004] IA PSO arPSO SEA ˆ −f (x) ˆ α= e ρ α = e−ρf (x) f14 9.98 × 10−1 9.98 × 10−1 1.157 9.98 × 10−1 9.98 × 10−1 5.328 × 10−4 2.719 × 10−4 3.68 × 10−1 2.13 × 10−8 4.33 × 10−8 f15 3.26 × 10−4 3.215 × 10−4 1.338 × 10−3 1.248 × 10−3 3.704 × 10−4 3.64 × 10−5 2.56 × 10−5 3.94 × 10−3 3.96 × 10−3 8.78 × 10−5 f16 −1.023 −1.017 −1.032 −1.032 −1.032 1.52 × 10−2 3.625 × 10−2 3.84 × 10−8 3.84 × 10−8 3.16 × 10−8 f17 4.19 × 10−1 4.2 × 10−1 3.98 × 10−1 3.98 × 10−1 3.98 × 10−1 2.9 × 10−2 3.5158 × 10−2 5.01 × 10−9 5.01 × 10−9 2.20 × 10−8 f18 4.973 5.371 3.0 3.516 3.0 2.9366 3.0449 0.0 3.65 0.0 f21 −10.15 −10.15 −5.4 −8.18 −8.41 1.81 × 10−6 1.018 × 10−7 3.40 2.60 3.16 f22 −10.4 −10.4 −6.946 −8.435 −8.9125 1.19 × 10−6 9.3 × 10−6 3.70 2.83 2.86 f23 −10.54 −10.54 −6.71 −8.616 −9.8 6.788 × 10−7 7.29 × 10−6 3.77 2.88 2.24 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 38. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus PSO, arPSO and SEA - n = 100 [Thomsen et al., CEC 2004] IA PSO arPSO SEA ˆ e−f (x) ˆ α= ρ α= e−ρf (x) f1 0.0 0.0 0.0 7.4869 × 10+2 5.229 × 10−4 0.0 0.0 0.0 2.31 × 10+3 5.18 × 10−5 f2 0.0 0.0 1.804 × 10+1 3.9637 × 10+1 1.737 × 10−2 0.0 0.0 6.52 × 10+1 2.45 × 10+1 9.43 × 10−4 f3 0.0 0.0 3.666 × 10+3 1.817 × 10+1 3.68 × 10−2 0.0 0.0 6.94 × 10+3 2.50 × 10+1 6.06 × 10−3 f4 7.32 × 10−4 6.447 × 10−7 5.312 2.4367 7.6708 × 10−3 2.109 × 10−3 3.338 × 10−6 8.63 × 10−1 3.80 × 10−1 5.71 × 10−4 f5 97.02 74.99 2.02 × 10+2 2.36 × 10+2 9.249 × 10+1 54.73 38.99 7.66 × 10+2 1.25 × 10+2 1.29 × 10+1 f6 0.0 0.0 2.1 4.118 × 10+2 0.0 0.0 0.0 3.52 4.21 × 10+2 0.0 f7 1.763 × 10−5 1.59 × 10−5 2.784 × 10−2 3.23 × 10−3 7.05 × 10−4 2.108 × 10−5 3.61 × 10−5 7.31 × 10−2 7.87 × 10−4 9.70 × 10−5 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 39. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus PSO, arPSO and SEA - n = 100 [Thomsen et al., CEC 2004] IA PSO arPSO SEA ˆ −f (x) ˆ α= e ρ α = e−ρf (x) f8 −4.176 × 10+4 −4.16 × 10+4 −2.1579 × 10+4 −2.1209 × 10+4 −3.943 × 10+4 2.08 × 10+2 2.06 × 10+2 1.73 × 10+3 2.98 × 10+3 5.36 × 10+2 f9 0.0 0.0 2.4359 × 10+2 4.809 × 10+1 9.9767 × 10−2 0.0 0.0 4.03 × 10+1 9.54 3.04 × 10−1 f10 1.18 × 10−16 0.0 4.49 5.628 × 10−2 2.93 × 10−3 6.377 × 10−16 0.0 1.73 3.08 × 10−1 1.47 × 10−4 f11 0.0 0.0 4.17 × 10−1 8.53 × 10−2 1.89 × 10−3 0.0 0.0 6.45 × 10−1 2.56 × 10−1 4.42 × 10−3 f12 5.34 × 10−22 5.3169 × 10−22 1.77 × 10−1 9.219 × 10−2 2.978 × 10−7 9.81 × 10−24 5.0655 × 10−24 1.75 × 10−1 4.61 × 10−1 2.76 × 10−8 f13 1.712 × 10−21 1.689 × 10−21 −3.86 × 10−1 3.301 × 10+2 −1.142810 9.379 × 10−23 9.877 × 10−24 9.47 × 10−1 1.72 × 10+3 2.41 × 10−8 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 40. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus IA∗ - unimodal class n = 30 n = 100 IA IA∗ IA IA∗ f1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f4 0.0 0.0 6.447 × 10−7 0.0 0.0 0.0 3.338 × 10−6 0.0 f5 12 0.0 74.99 22.116 13.22 0.0 38.99 39.799 f6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f7 1.521 × 10−5 7.4785 × 10−6 1.59 × 10−5 1.2 × 10−6 2.05 × 10−5 6.463 × 10−6 3.61 × 10−5 1.53 × 10−6 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 41. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA versus IA∗ - multimodal class n = 30 n = 100 IA IA∗ IA IA∗ f8 −1.256041 × 10+4 −9.05 × 10+3 −4.16 × 10+4 −2.727 × 10+4 25.912 1.91 × 10+4 2.06 × 10+2 7.627 × 10−4 f9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f10 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f12 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 42. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus several DE variants - n = 30 [Coello et al., GECCO 2006] Unimodal Functions f1 f2 f3 f4 f6 f7 ∗ IA 0.0 0.0 0.0 0.0 0.0 2.79 × 10−5 IA 0.0 0.0 0.0 0.0 0.0 4.89 × 10−5 DE rand/1/bin 0.0 0.0 0.02 1.9521 0.0 0.0 DE rand/1/exp 0.0 0.0 0.0 3.7584 0.84 0.0 DE best/1/bin 0.0 0.0 0.0 0.0017 0.0 0.0 DE best/1/exp 407.972 3.291 10.6078 1.701872 2737.8458 0.070545 DE current-to-best/1 0.54148 4.842 0.471730 4.2337 1.394 0.0 DE current-to-rand/1 0.69966 3.503 0.903563 3.298563 1.767 0.0 DE current-to-rand/1/bin 0.0 0.0 0.000232 0.149514 0.0 0.0 DE rand/2/dir 0.0 0.0 30.112881 0.044199 0.0 0.0 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 43. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus several DE variants - n = 30 [Coello et al., GECCO 2006] Multimodal Functions f5 f9 f10 f11 f12 f13 ∗ IA 16.2 0.0 0.0 0.0 0.0 0.0 IA 11.69 0.0 0.0 0.0 0.0 0.0 DE rand/1/bin 19.578 0.0 0.0 0.001117 0.0 0.0 DE rand/1/exp 6.696 97.753938 0.080037 0.000075 0.0 0.0 DE best/1/bin 30.39087 0.0 0.0 0.000722 0.0 0.000226 DE best/1/exp 132621.5 40.003971 9.3961 5.9278 1293.0262 2584.85 DE current-to-best/1 30.984666 98.205432 0.270788 0.219391 0.891301 0.038622 DE current-to-rand/1 31.702063 92.263070 0.164786 0.184920 0.464829 5.169196 DE current-to-rand/1/bin 24.260535 0.0 0.0 0.0 0.001007 0.000114 DE rand/2/dir 30.654916 0.0 0.0 0.0 0.0 0.0 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 44. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus DE rand/1/bin variants [Thomsen et al., CEC 2004] 30 dimension 100 dimension IA IA∗ DE rand/1/bin IA IA∗ DE rand/1/bin f1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f3 0.0 0.0 2.02 × 10−9 0.0 0.0 5.87 × 10−10 0.0 0.0 8.26 × 10−10 0.0 0.0 1.83 × 10−10 f4 0.0 0.0 3.85 × 10−8 6.447 × 10−7 0.0 1.128 × 10−9 0.0 0.0 9.17 × 10−9 3.338 × 10−6 0.0 1.42 × 10−10 f5 12 0.0 0.0 74.99 22.116 0.0 13.22 0.0 0.0 38.99 39.799 0.0 f6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f7 1.521 × 10−5 7.48 × 10−6 4.939 × 10−3 1.59 × 10−5 1.2 × 10−6 7.664 × 10−3 2.05 × 10−5 6.46 × 10−6 1.13 × 10−3 3.61 × 10−5 1.53 × 10−6 6.58 × 10−4 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 45. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus DE rand/1/bin variants [Thomsen et al., CEC 2004] 30 dimension 100 dimension IA IA∗ DE rand/1/bin IA IA∗ DE rand/1/bin f8 −1.256 × 10+4 −9.05 × 10+3 −1.2569 × 10+4 −4.16 × 10+4 −2.727 × 10+4 −4.189 × 10+4 25.912 1.91 × 104 2.3 × 10−4 2.06 × 10+2 7.63 × 10−4 1.06 × 10−3 f9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f10 0.0 0.0 −1.19 × 10−15 0.0 0.0 8.023 × 10−15 0.0 0.0 7.03 × 10−16 0.0 0.0 1.74 × 10−15 f11 0.0 0.0 0.0 0.0 0.0 5.42 × 10−20 0.0 0.0 0.0 0.0 0.0 5.42 × 10−20 f12 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 f13 0.0 0.0 −1.142824 0.0 0.0 −1.142824 0.0 0.0 4.45 × 10−8 0.0 0.0 2.74 × 10−8 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 46. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus rand/1/exp and best/1/exp variants [Iba et al., GECCO, 2005] IA∗ IA DE rand/1/exp DE best/1/exp n = 50 dimensional search space 0±0 309.74 ± 481.05 f1 0±0 0±0 0±0 0±0 0.0535 ± 0.0520 0.0027 ± 0.0013 79.8921 ± 102.611 3.69 × 10+5 ± 5.011 × 10+5 f5 1.64 ± 8.7 30 ± 21.7 52.4066 ± 19.9109 54.5985 ± 25.6652 90.0213 ± 33.8734 58.1931 ± 9.4289 0±0 0.61256 ± 1.1988 f9 0±0 0±0 0±0 0±0 0±0 0±0 0±0 0.2621 ± 0.5524 −6 f10 0±0 0±0 9.36 × 10 ± 3.67 × 10−6 6.85 × 10−6 ± 6.06 × 10−6 0.0104 ± 0.0015 0.0067 ± 0.0015 0±0 0.1651 ± 0.2133 f11 0±0 0±0 9.95 × 10−7 ± 4.3 × 10−7 0±0 0.0053 ± 0.010 0.0012 ± 0.0028 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 47. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus rand/1/exp and best/1/exp variants [Iba et al., GECCO, 2005] IA∗ IA DE rand/1/exp DE best/1/exp n = 100 dimensional search space 1.58 × 10−6 ± 3.75 × 10−6 0.0046 ± 0.0247 f1 0±0 0±0 59.926 ± 16.574 30.242 ± 5.93 2496.82 ± 246.55 1729.40 ± 172.28 120.917 ± 41.8753 178.465 ± 60.938 f5 26.7 ± 43 85.6 ± 31.758 12312.16 ± 3981.44 7463.633 ± 2631.92 3.165 × 10+6 ± 6.052 × 10+5 1.798 × 10+6 ± 3.304 × 10+5 0±0 0±0 f9 0±0 0±0 2.6384 ± 0.7977 0.7585 ± 0.2524 234.588 ± 13.662 198.079 ± 18.947 1.02 × 10−6 ± 1.6 × 10−7 9.5 × 10−7 ± 1.1 × 10−7 f10 0±0 0±0 1.6761 ± 0.0819 1.2202 ± 0.0965 7.7335 ± 0.1584 6.7251 ± 0.1373 0±0 0±0 f11 0±0 0±0 1.1316 ± 0.0124 1.0530 ± 0.0100 20.037 ± 0.9614 13.068 ± 0.8876 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 48. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus rand/1/exp and best/1/exp variants [Iba et al., GECCO, 2005] IA∗ IA DE rand/1/exp DE best/1/exp n = 200 dimensional search space 50.005 ± 16.376 26.581 ± 7.4714 f1 0±0 0±0 5.45 × 10+4 ± 2605.73 4.84 × 10+4 ± 1891.24 1.82 × 10+5 ± 6785.18 1.74 × 10+5 ± 6119.01 9370.17 ± 3671.11 6725.48 ± 1915.38 f5 88.65 ± 91.85 165.1 ± 71.2 4.22 × 10+8 ± 3.04 × 10+7 3.54 × 10+8 ± 3.54 × 10+7 3.29 × 10+9 ± 2.12 × 10+8 3.12 × 10+9 ± 1.65 × 10+8 0.4245 ± 0.2905 0.2255 ± 0.1051 f9 0±0 0±0 1878.61 ± 60.298 1761.55 ± 43.3824 5471.35 ± 239.67 5094.97 ± 182.77 0.5208 ± 0.0870 0.4322 ± 0.0427 f10 0±0 0±0 15.917 ± 0.1209 15.46 ± 0.1205 19.253 ± 0.0698 19.138 ± 0.0772 0.7687 ± 0.0768 0.5707 ± 0.0651 f11 0±0 0±0 490.29 ± 21.225 441.97 ± 15.877 1657.93 ± 47.142 1572.51 ± 53.611 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 49. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus memetic versions of rand/1/exp and best/1/exp DE variants [Iba et al., GECCO, 2005] IA∗ IA DEfirDE DEfirSPX n = 50 dimensional search space 0±0 0±0 f1 0±0 0±0 0±0 0±0 −4 0.0026 ± 0.0023 1 · 10 ± 4.75 · 10−5 72.0242 ± 47.1958 65.8951 ± 37.8933 f5 1.64 ± 8.7 30 ± 21.7 53.1894 ± 26.1913 45.8367 ± 10.2518 66.9674 ± 23.7196 52.0033 ± 13.6881 0±0 0±0 f9 0±0 0±0 0±0 0±0 0±0 0±0 0±0 0±0 f10 0±0 0±0 2.28 × 10−5 ± 1.45 × 10−5 3.0 × 10−6 ± 1.07 × 10−6 0.0060 ± 0.0015 0.0019 ± 4.32 × 10−4 0±0 0±0 f11 0±0 0±0 0±0 0±0 −4 4.96 · 10 ± 6.68 · 10−4 5.27 · 10−4 ± 0.0013 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 50. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus memetic versions of rand/1/exp and best/1/exp DE variants [Iba et al., GECCO, 2005] IA∗ IA DEfirDE DEfirSPX n = 100 dimensional search space 0±0 0±0 f1 0±0 0±0 11.731 ± 5.0574 1.2614 ± 0.4581 358.57 ± 108.12 104.986 ± 22.549 107.5604 ± 28.2529 99.1086 ± 18.5735 f5 26.7 ± 43 85.6 ± 31.758 2923.108 ± 1521.085 732.85 ± 142.22 2.822 × 10+5 ± 3.012 × 10+5 16621.32 ± 6400.43 0±0 0±0 f9 0±0 0±0 0.1534 ± 0.1240 0.0094 ± 0.0068 17.133 ± 7.958 27.0537 ± 20.889 1.2 × 10−6 ± 6.07 × 10−7 0±0 f10 0±0 0±0 0.5340 ± 0.1101 0.3695 ± 0.0734 3.7515 ± 0.2773 3.4528 ± 0.1797 0±0 0±0 f11 0±0 0±0 0.7725 ± 0.1008 0.5433 ± 0.1331 3.7439 ± 0.7651 2.2186 ± 0.3010 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 51. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA and IA∗ versus memetic versions of rand/1/exp and best/1/exp DE variants [Iba et al., GECCO, 2005] IA∗ IA DEfirDE DEfirSPX n = 200 dimensional search space 17.678 ± 9.483 0.8568 ± 0.2563 f1 0±0 0±0 9056.0 ± 1840.45 2782.32 ± 335.69 44090.5 ± 6122.35 9850.45 ± 1729.9 5302.79 ± 2363.74 996.69 ± 128.483 f5 88.65 ± 91.85 165.1 ± 71.2 2.39 × 10+7 ± 6.379 × 10+6 1.19 × 10+6 ± 4.10 × 10+5 3.48 × 10+8 ± 1.75 × 10+8 1.21 × 10+7 ± 4.73 × 10+6 0.1453 ± 0.2771 0.0024 ± 0.0011 f9 0±0 0±0 352.93 ± 46.11 369.88 ± 136.87 1193.83 ± 145.477 859.03 ± 99.76 0.3123 ± 0.0426 0.1589 ± 0.0207 f10 0±0 0±0 9.2373 ± 0.4785 6.6861 ± 0.3286 14.309 ± 0.3706 9.4114 ± 0.4581 0.5984 ± 0.1419 0.1631 ± 0.0314 f11 0±0 0±0 78.692 ± 11.766 28.245 ± 4.605 368.90 ± 41.116 85.176 ± 12.824 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 52. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions IA on Large Dimensional Search Space f1 f5 f9 f10 f11 Tmax = 10000 −1 +3 −2 −3 n = 1000 1.93 × 10 1.01 × 10 2.29 × 10 1.21 × 10 1.27 × 10−2 2.44 × 10−2 2.94 × 10+2 5.09 × 10−3 7.76 × 10−5 1.7 × 10−3 n = 5000 1.6 × 10+1 9.11 × 10+3 1.83 2.76 × 10−3 3.26 × 10−1 2.86 × 10+1 3.56 × 10+3 8.13 2.31 × 10−3 5.61 × 10−1 Tmax = 50000 n = 1000 1.97 × 10−2 8.93 × 10+2 9.48 × 10−12 8.53 × 10−16 3.28 × 10−8 9.81 × 10−2 2.76 × 10+2 4.70 × 20−11 1.52 × 10−15 1.58 × 10−7 n = 5000 4.44 × 10+1 5.95 × 10+3 1.71 1.13 × 10−3 2.79 × 10−1 9.6 × 10+1 2.76 × 10+3 2.73 3.3 × 10−3 8.18 × 10−1 Tmax = 100000 n = 1000 3.35 × 10−3 9.54 × 10+2 7.06 × 10−4 3.76 × 10−8 6.66 × 10−12 2.22 × 10−2 1.54 × 10+2 4.72 × 10−3 2.63 × 10−7 4.56 × 10−11 n = 5000 3.52 5.95 × 10+3 3.64 × 10−1 8.14 × 10−4 8.99 × 10−2 5.14 1.98 × 10+3 6, 34 × 10−1 1.59 × 10−3 3.33 × 10−1 Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 53. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Conclusion 1/3. ◮ We have presented an extensive comparative study illustrating the performance ◮ IA was compared with 33 state-of-the-art optimization algorithms (deterministic and nature inspired methodologies): FEP; IFEP; three versions of CEP; two versions of PSO and AR PSO; EO; SEA; HGA; immunological inspired algorithms, as BCA and two versions of CLONALG; CHC algorithm; Generalized Generation Gap (G3 − 1); hybrid steady-state RCMA (SW-100), Family Competition (FC); CMA with crossover Hill Climbing (RCMA-XHC); eleven variants of DE and two its memetic versions ◮ Two variants of IA were presented Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 54. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Conclusion 2/3. ◮ Main features of the designed immune algorithm: 1. cloning operator, which explores the neighborhood of a given solution 2. inversely proportional hypermutation operator, which perturbs each candidate solution as a function of its fitness function value 3. aging operator, that eliminates the oldest candidate solutions from the current population in order to introduce diversity and thus avoiding local minima during the search process ◮ A large set of experiments was used divided in two different categories of functions [Yao et al., IEEE TEC, 1999] and [Timmis et al., GECCO 2003] Robust Immunological Algorithms for High-Dimensional Global Optimization
  • 55. Introduction Optimization Immunological Algorithm Metrics and Dynamic Behavior Results and Comparison Conclusions Conclusion 3/3. ◮ The dimensionality of the problems was varied from small to high dimensions (5000 variabiles). ◮ The results suggest that the proposed immune algorithm is an effective numerical optimization algorithm (in terms of solution quality) ◮ All experimental comparisons show that IA and IA∗ are comparable, and often outperform, all nature inspired methodologies used, and one well-known deterministic optimization algorithm (DIRECT). Robust Immunological Algorithms for High-Dimensional Global Optimization