Hybrid Optimization Algorithms Resembling the Life ...

Hybrid Optimization Algorithms Resembling the Life Cycles of Organisms of Phylum Cnidaria Upeka Premaratne∗ , and Saman Halgamuge† ∗ Department

of Electronic and Telecommunication Engineering University of Moratuwa, Katubedda, Moratuwa 10400, Sri Lanka E-mail: [email protected] † Department of Mechanical Engineering University of Melbourne, Parkville 3010 VIC, Australia E-mail: [email protected]

Abstract—The biological world is an ideal place for seeking inspiration for developing mathematical optimization algorithms. In this paper we propose two hybrid stochastic optimization algorithms that bear resemblance to the sexual reproduction cycle of Jellyfish and asexual reproductive cycle of species of Hydra. The performance of these two algorithms are investigated against other common optimization algorithms on a set of benchmark optimization problems. The results show that the proposed algorithms perform well. Index Terms—Optimization Algorithms, Biological Inspiration, Hybrid Optimization Algorithms, Jellyfish Reproduction, Hydra Asexual Reproduction

I. I NTRODUCTION The nature is a realm that favors organisms that are the fittest. In other words, organisms that best suit their surroundings have the best chance of surviving, passing on their genes and continuing the cycle of life. Over the four billion years since life first evolved on planet earth, organisms have developed myriad strategies for adapting to best suit their environment. Finding the best solution for a problem is a common requirement in engineering and economics. Often it is necessary to produce a device such as an air conditioner that requires minimal capital to construct and provide the maximum cooling efficiency. In mathematics, this is known as optimization and it can be defined as seeking the maximum or minimum value of an objective function for a given domain for a given set of constraints. In most real world problems, finding the optimum solution is no trivial task due to the extent of the search space, nature of the objective function etc. Hence, most optimization problems have no analytical solution. Optimization has been solved in the natural world for four billion years by organisms adapting to best suit their environment. Therefore, seeking inspiration for solving optimization problems from biology is a worthwhile pursuit. A. Contribution In this paper we propose a hybrid optimization algorithm that resembles the life cycle of jellyfish (Superfamily Scyphozoa). When modified to exclude the fertilization stage, it resembles the asexual reproduction cycle of organisms of

genus Hydra (Superfamily Hydrozoa). Both of these organisms belong to Phylum Cnidaria. B. Outline of this Paper Initially we provide an overview of existing optimization algorithms in Section II. This is followed by a description of the Jellyfish Optimization Algorithm (JOA) (Section III) and the Hydra Optimization Algorithm (HOA) (Section IV). The two algorithms are compared to related algorithms in Section V. The performance of the two algorithms are tested on benchmark problems and compared against other common optimization algorithms in the subsequent section (Section VI). II. O PTIMIZATION A LGORITHMS Hill Climbing Algorithms (HCA) seek out the optimum point by starting from an initial point and incrementally adjusting the solution based upon a certain criteria such as the steepest gradient. A HCA will succeed only if the initial point falls within the basin of attraction of an isolated optimum point. However, the algorithm cannot distinguish between a local and global optimum. Similarly it is likely to fail on a plain, ridge or plateau. The Simplex or Nelder-Mead Algorithm [1] overcomes this handicap by using a N+1 simplex to find the optimum of an N dimensional problem instead of a single point. Alternatively, it is possible to randomly select a new initial point and restarting the HCA [2]. During a random restart it is possible to repeatedly locate the same local optimum. Such a situation is avoided in algorithms such as the Tabu search [3] [4] and BRST algorithm [5]. In Simulated Annealing (SA) [6], the optimization problem is solved based upon the physical phenomenon of the annealing process of a metal. In biologically inspired optimization algorithms, the process of solving the optimization is based upon a biological process or behaviour. Evolutionary Algorithms (EA) [7] [8] which seek out the global optimum using a strategy based upon evolutionary biology while Artificial Immune Systems (AIS) [9] which includes the Clonal Selection Algorithm (CSA) [10] do so by mimicking the vertabrate immune system. Algorithms such as Ant Colony Optimization (ACO) [11], Bee Colony Optimization (BCO) [12], Bacterial Foraging (BF) [13], Physarum polycephalum shortest path

solver [14] and Cuckoo Search (CS) [15] are based upon the foraging behaviours of organisms. Invasive Weed Optimization (IWO) [16] and the PFA [17] are derived from the life cycle of plants while Particle Swarm Optimization (PSO) [18] is inspired by animal flocking behaviour. A hybrid algorithm combines the searching strategies of multiple optimization algorithms for improved results. Often, a stochastic algorithm is coupled with a HCA. An example is the use of an EA to converge to the vicinity of the optimum solution and HCA to reach the point itself. The EA is strong on searching the search space but weak on converging to the optimum. On the other hand, the HCA is strong on converging to the optimum but weak on searching the search space. III. T HE J ELLYFISH O PTIMIZATION A LGORITHM A. The Scyphozoa (Jellyfish) Life Cycle Jellyfish belong to the superclass Scyphozoa of phylum Cnidaria and are closely related to corals and sea anenomes (Anthozoa). They are simple organisms with a fossil record dating back 505 million years [19]. They usually reproduce via sexual reproduction and have a high fecundity resulting in jellyfish blooms [20] [21]. They start their life cycle as fertilized ova [22]. These develop into free swimming pelagic planula larvae. The planulae migrate to the best spots on the local seabed and develop into benthic polyps. The next phase of the life cycle is when the polyps grow to form strobila which asexually bud. Each bud of the strobila develop into a free swimming ephyra. The ephyra then grow into the adult phase known as the medusa and restart the cycle. B. Algorithm Stages The JOA consists of five stages each analogous to phases of the life cycle (Fig. 2). 1) Initialization: In this stage, an initial population of n0 planulae are distributed at random in the search space and subsequently migrate to the best local positions using the HCA. 2) Polyp Selection: After migration, the fittest polyps are selected for strobilation according to a thresholding method. A polyp is selected for the next iteration if its fitness is greater than the threshold yt (i.e. y ≥ yt ). For optimization purposes a Population Bounding Threshold (PBT) is used. A PBT that selects the fittest nT individuals for the next iteration is used such that y > ynT , where ynT is the fitness of the nth T individual ranked in descending order of fitness. It is called a PBT because the population of the subsequent iteration (n) will be upper bounded such that n < nT qmax , where qmax is the maximum number of epharae of a strobilating polyp. 3) Strobilation: In this stage each polyp produces ephyrae proportional to its fitness. The fittest polyp produces the maximum number of ephyrae (qmax ). The formula of [17] is used to obtain the number of ephyrae (q) for an individual of fitness y in terms of the maximum fitness ymax of the population and the threshold fitness yt .   y − yt (1) q = qmax ymax − yt

4) Fertilization: Out of the ephyrae produced by the strobila, only a fraction will become viable adults due to fertilization. This will depend on the number of neighbors of the individual. This stage of the algorithm differs significantly from the natural Scyphozoa life cycle. The fertilizated ephyrae for a polyp with v neighbors is given by, qV

=

v qe[ vmax −1]

(2)

where vmax is the number of neighbors for the polyp with the most neighbors within the population. The number of neighbors is determined by calculating the Euclidean distance to every other member of the population and checking if it is within or outside a spherical perimeter [17] of radius RF . For two ephyrae xj and xk , the perimeter formula of, u(xj , xk )

= ||xj − xk || − RF

(3)

is used. If the two are within the sphere, then u < 0. From this for the number of neighbors (vj ) for an ephyra can be determined. 5) Ova Dispersion: The next generation of fertilized ova are then dispersed within the search space. When dispersing, the dimension values take a Gaussian distribution such that the new location in the parameter space is given by, xi+1 = Φ[xi + X i ]

(4)

where X i ∈ Rm is a Gaussian random vector of zero mean and variance σ. The operator Φ[] ensures that the new seed is within the parameter space and satisfies the required constraints by either limiting or resampling the random variable until all criteria are satisfied. The parameter σ (dispersion spread) will determine the spread of the dispersion. 6) Planula Migration: In this stage, the planulae resulting from the dispersed ova actively migrate using a HCA to the best local positions of the search space.

Polyp Selection Strobilation

Planula Migration

Fertilization Ova Dispersion

Fig. 2.

Initialization

Stages of the Jellyfish Optimization Algorithm

IV. T HE H YDRA O PTIMIZATION A LGORITHM When the fertilization stage of the JOA is excluded, it resembles the asexual reproduction cycle of organisms of

Fertilized Ovum Ephyra

Adult Medusa

Planula Larva Strobila Polyp

Fig. 1.

Scyphozoa (Jellyfish) Life Cycle

the genus Hydra (Superclass Hydrozoa). This cycle (Fig. 3) consists of the adult individual developing numerous buds which grow into miniature adults. These clones then disperse via ocean currents and then settle on the sea bed and migrate albeit slower than the planulae of Scyphozoa to the best position in the local environment. They then mature into adults and restart the cycle [23]. We call this algorithm the Hydra Optimzation Algorithm or HOA. To the best of the authors’ knowledge such an optimization algorithm has not been proposed before.

Bud Selection Adult Budding Bud Dispersion

Fig. 3.

Bud Migration Initialization

Stages of the Hydra Optimization Algorithm

V. C OMPARISON OF A LGORITHMS A LGORITHMS The closest stochastic optimization algorithms to the JOA and HOA are the PFA [17] and CSA [10]/IWO [16] respectively. Table I outlines the basic stages of each algorithm for comparison. The JOA differs from the hybrid PFA algorithm proposed in [24] where the HCA is used only on the fittest solution of an iteration instead of all solutions as in the JOA. VI. A LGORITHM P ERFORMANCE In this section the performance of both the JOA and HOA are compared with other stochastic and evolutionary optimization algorithms. It should be noted that according to the [25], no optimization algorithm can be deemed superior to another

for an arbitrary function due to the No Free Lunch Theorem. Instead, benchmark functions will only provide heuristics for the type of functions for which an algorithm is more suitable. A. Continuous Functions Eight benchmark functions with diverse characteristics are used for this experiment. Most of the functions used can be extended beyond two dimensions. In order to fit into a common range of xi ∈ [−5, 5], some of the functions are scaled by a factor s such that, f (z) = f (x)|x→sx . Table II contains a summary of the benchmark functions. For the rest of the paper the following default parameters are used for the JOA and HOA unless specified otherwise. n0

=

10

qmax

=

20

RF

=

0.05

σ

=

0.5

(5)

A total of 1000 trials are performed for a set of parameters of a particular algorithm for comparison purposes. The success rate is given in terms of the percentage of trials where the final solution is within 0.1 units (or 1% of the search space) of the Global Optimum (GO) unless specified otherwise. For the JOA, HOA, PFA and CSA/IWO where selection of individuals for the next iteration is needed, the threshold, nT = n0 . The JOA and HOA are compared with following algorithms; 1) the PFA and CSA/IWO with the same default parameters, 2) the Genetic Algorithm (GA) implementation from the MATLAB Genetic Algorithm and Direct Search Toolbox for the toolbox default parameters and different populations, 3) the Simulated Annealing (SA) implementation from the same toolbox for default parameters,

TABLE I C OMPARISON OF THE JOA, HOA, PFA, CSA AND IWO JOA Polyp Selection Strobilation Fertilization Ova Dispersion Planula Migration

HOA Bud Selection Adult Budding Not Present Bud Dispersion Bud Migration

PFA Plant Selection Seeding Pollination Seed Dispersion Not Present

4) the MATLAB PSO implementation [26] for varying populations and number of iterations (remaining parameters set to default values). The MATLAB implementation of the Nelder-Mead HCA fminsearch is used for both algorithms. The maximum number of iterations and function iterations are decreased from 200∗m to 20 ∗ m to improve the efficiency. The value of σ is chosen such that the it is close to 10% of the search space. This is based upon the heuristic that the dispersion would effectively cover a higher proportion of the search space provided that σ is sufficiently large.

CSA Antigen Selection Antigen Cloning Not Present Clonal Hypermutation Not Present

IWO Competitive Exclusion Reproduction Not Present Spatial Dispersion Not Present

It should also be noted that for the same number of iterations both the PFA and CSA/IWO exhibit negligible success. TABLE IV I MPROVED P ERFORMANCE FOR f6 BY T UNING σ AND RF (N = 5) Success Rate∗∗ (%) JOA HOA PFA CSA 0.010 0.200 97.8 98.2 0.9 0.7 0.050 0.200 95.5 96.7 0.6 1.2 0.100 0.200 86.7 92.4 1.7 1.7 0.010 0.005 98.3 98.1 0.5 0.8 0.050 0.005 95.6 96.8 1.2 1.6 0.100 0.005 88.4 91.0 2.4 0.9 ∗ where applicable ∗∗ within 0.1 units of GO ∗ RF

σ

TABLE II B ENCHMARK O PTIMIZATION F UNCTION D ETAILS Function Ackley Beale Booth m-Griewank Rastrigrin Rosenbrock Schwefel Zakharov

Range [-30,30] [-4.5,4.5] [-10,10] [-10,10] [-5.12,5.12] [-10,10] [-500,500] [-10,10]

s 6 1 2 2 1 2 100 2

x∗ (0,..,0) (3,0.5) (1,3) (0,..,0) (0,..,0) (1,..,1) (1,..,1) (0,..,0)

f (x∗ ) 0 0 0 0 0 0 0 0

B. Constrained Optimization Modality Multi Unimodal Multi Multi Multi Multi Multi Unimodal

The results of the algorithms for two variable functions are shown in Table III. According to the results, both the JOA and HOA perform very well beyond five iterations for all of the benchmark functions except f6 . For the successful functions at five iterations the JOA achieves above 90% while the HOA scores above 95%. Both algorithms outperform the closely related PFA and CSA/IWO. Since the population of the JOA and HOA varies with each iteration, it cannot be directly compared with the GAs and PSO. Therefore, the performance for the GAs and PSO is investigated for the given benchmarks for populations (p) under the upper bound nT qmax (20,50,100 and 200 individuals). GAs perform very poorly for f7 and have moderate success for f6 even when the GA population equals 200, the upper bound of the JOA and HOA. At five iterations, both the JOA and HOA outperform PSO even for a population of 200. However, in the long run, the PSO does better than the rest in all functions except for the modest success for f7 . The initial poor performance of the JOA and HOA on f6 is most likely due to the large value of σ, resulting in the dispersion of ova outside the valley region of f6 which contains the global optimum. By decreasing σ for localized dispersion, performance of both algorithms increase significantly (Table IV). Interestingly, the best results are obtained for low values of σ comparable to or lesser than the fertility radius RF . The remaining default parameters (5) are unchanged while N = 5.

The performance of the two algorithms for constrained optimization is demonstrated using the multimodal function g1 (x) = −x21 sin(4πx1 ) + x2 sin(4πx2 + π) − 1

(6)

for the range xi ∈ [−1, 2] with the constraint x1 +x2 ≤ 3. This function, modified from [10] has the global optimum value of 4.4037 at (1.88,1.88) when unconstrained. When constrained it becomes 3.4057 at (1.88,0.88). The results for 1000 trials for σ = 0.2 and the remaining parameter values of (5) are given in Table V. Both algorithms have a 100% success rate beyond five iterations. Figure 4 shows the constraint and distribution of the results of the 1000 trials of each algorithm for N = 1 and N = 2 total iterations. TABLE V R ESULTS FOR C ONSTRAINED O PTIMIZATION Algorithm JOA HOA

∗

Success Rate∗ (%) N=1 N=2 N=5 N=10 90.6 98.7 100.0 100.0 98.7 99.1 100.0 100.0 within 0.01 units of GO

VII. C ONCLUSIONS In this paper we propose two optimization algorithms that bare resemblance to the sexual reproduction cycle of species of Superclass Scyphozoa and asexual reproductive cycle of species of Genus Hydra. The results (Table III) show that the JOA and HOA outperform the algorithms from which they are most closely related to (the PFA and CSA/IWO respectively). The JOA and HOA implementations used the standard MATLAB Nelder-Mead HCA which can at times be highly inefficient due to the implementation of either algorithm. In terms of future work, developing computationally efficient

TABLE III C OMPARISON OF JOA AND HOA WITH OTHER O PTIMIZATION A LGORITHMS (m = 2) Algorithm JOA (N=1) JOA (N=2) JOA (N=5) JOA (N=10) JOA (N=20) HOA (N=1) HOA (N=2) HOA (N=5) HOA (N=10) HOA (N=20) PFA (N=5) PFA (N=10) PFA (N=20) PFA (N=50) CSA (N=5) CSA (N=10) CSA (N=20) CSA (N=50) GA (p=20) GA (p=50) GA (p=100) GA (p=200) PSO (N=5,p=20) PSO (N=5,p=50) PSO (N=5,p=100) PSO (N=5,p=200) PSO (N=10,p=20) PSO (N=10,p=50) PSO (N=10,p=100) PSO (N=10,p=200) PSO (N=20,p=20) PSO (N=20,p=50) PSO (N=20,p=100) PSO (N=20,p=200) SA

f1 38.7 65.3 96.4 100.0 100.0 40.9 68.4 97.8 100.0 100.0 26.0 51.6 70.6 92.8 40.0 57.0 78.3 97.5 83.0 98.2 99.9 100.0 21.2 46.4 69.8 87.7 59.9 87.2 96.8 99.8 99.1 100.0 100.0 100.0 100.0

Success Rate (% within 0.1 units of Global Optimum) f2 f3 f4 f5 f6 f7 f8 100.0 100.0 99.9 69.3 17.9 88.0 100.0 100.0 100.0 100.0 96.7 20.9 91.5 100.0 100.0 100.0 100.0 100.0 20.0 93.9 100.0 100.0 100.0 100.0 100.0 23.3 96.9 100.0 100.0 100.0 100.0 100.0 19.4 99.5 100.0 100.0 100.0 100.0 70.8 20.0 92.1 100.0 100.0 100.0 100.0 98.3 21.3 93.9 100.0 100.0 100.0 100.0 100.0 20.2 95.8 100.0 100.0 100.0 100.0 100.0 18.5 96.7 100.0 100.0 100.0 100.0 100.0 21.8 99.5 100.0 19.2 31.5 22.9 22.3 3.4 27.6 35.3 31.8 53.3 53.5 41.9 2.7 54.0 62.8 46.4 79.9 85.7 64.8 3.6 71.9 87.8 71.7 97.2 99.7 95.5 3.4 73.4 99.7 22.9 41.1 37.6 35.5 4.0 43.2 47.3 37.1 63.3 71.4 54.8 3.8 73.1 76.2 54.7 84.5 94.3 80.0 3.4 87.1 94.4 80.1 98.6 99.9 98.4 2.9 89.3 100.0 70.8 100.0 100.0 65.1 24.6 0.1 100.0 93.1 100.0 100.0 94.7 48.4 3.3 100.0 98.4 100.0 100.0 100.0 74.0 12.2 100.0 99.8 100.0 100.0 100.0 86.6 34.1 100.0 8.5 18.8 12.8 9.8 5.0 7.4 20.6 16.4 41.2 40.1 23.1 6.8 24.1 54.2 29.3 64.6 69.0 42.2 11.7 46.8 79.5 51.2 88.9 91.9 68.6 19.6 68.2 96.6 19.5 58.2 47.0 24.3 8.6 24.3 70.3 39.5 85.9 81.3 53.7 19.1 49.3 96.5 65.1 97.9 98.1 73.1 27.2 63.9 99.9 85.0 99.8 99.8 91.8 42.9 85.8 100.0 59.5 98.9 84.3 49.0 21.4 43.1 100.0 87.8 100.0 99.1 84.5 44.0 63.7 100.0 96.7 100.0 100.0 97.9 74.5 80.3 100.0 95.2 100.0 97.8 93.8 99.0 67.5 100.0 73.5 100.0 69.0 45.6 47.7 56.6 100.0

integrated code for HCAs is a necessary direction. The main reason for inefficient code is the resampling of the new value in case it does not satisfy the given constraints after the HCA has been executed. It is also necessary to look into more efficient implementations of the fertilization stages for the JOA for use when handling real world optimization problems. Obtaining heuristics for adjusting the JOA and HOA parameters based upon landscape characteristics of the objective function is another open problem that warrants further investigation. R EFERENCES [1] J. Nelder, R. Mead, A simplex method for function minimization, Computer Journal 7 (1965) 308–313. [2] S. J. Russell, P. Norvig, Artificial Intelligence: A Modern Approach, 2nd Edition, Pearson Education, 2003. [3] F. Glover, Tabu search - part 1, ORSA Journal on Computing 1 (2) (1989) 190206. [4] F. Glover, Tabu search - part 2, ORSA Journal on Computing 2 (1) (1990) 4–32. [5] C. G. E. Boender, A. H. G. R. Kan, G. T. Timmer, L. Stougie, A stochastic method for global optimization, Mathematical Programming 22 (1) (1982) 125–140. [6] S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, Optimization by simulated annealing, Science 220 (4598) (1983) 671–680. [7] L. J. Fogel, A. J. Owens, M. J. Walsh, Artificial Intelligence through Simulated Evolution, John Wiley, New York, 1966.

[8] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975. [9] J. Farmer, N. Packard, A. Perelson, The immune system, adaptation and machine learning, Physica D 2 (1-3) (1986) 187–204. [10] L. N. de Castro, F. J. V. Zuben, Learning and optimization using the clonal selection principle, IEEE Transaction on Evolutionary Computation 6 (3) (2002) 239–251. [11] M. Dorigo, Optimization, learning and natural algorithms, Ph.D. thesis, Politecnico di Milano, Milan, Italy (1992). [12] D. Karaboga, An idea based on honey bee swarm for numerical optimization, Tech. Rep. TR06, Erciyes University, Engineering Faculty, Computer Engineering Department (2005). [13] Y. Liu, K. Passino, Biomimicry of social foraging bacteria for distributed optimization: Models, principles, and emergent behaviors, Journal of Optimization Theory and Applications 115 (3) (2002) 603–628. [14] J. Siriwardana, S. Halgamuge, Fast shortest path optimization inspired by shuttle streaming of Physarum polycephalum, in: 2012 IEEE World Congress on Computational Intelligence, 2012. [15] X. S. Yang, S. Deb, Cuckoo search via lvy flights, in: World Congress on Nature and Biologically Inspired Computing (NaBIC 2009), 2009, p. 210214. [16] A. R. Mehrabian, C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics 1 (4) (2006) 355–366. [17] U. Premaratne, J. Samarabandu, T. S. Sidhu, A new biologically inspired optimization algorithm, in: International Conference on Industrial and Information Systems (ICIIS), 2009, pp. 279–284. [18] J. Kennedy, R. Eberhart, Particle swarm optimization, in: Proceedings of the IEEE International Conference on Neural Networks, Vol. 4, Piscataway, NJ, 1995, pp. 1942–1948. [19] P. Cartwright, S. L. Halgedahl, J. R. Hendricks, R. D. Jarrard, A. C.

Fig. 4.

[20] [21]

[22] [23] [24] [25] [26]

Constrained Optimization Results

Marques, A. G. C. B. S. Lieberman, Exceptionally preserved jellyfishes from the middle cambrian, PLoS ONE 2 (10) (2007) e1121. J. E. Purcell, S. I. Uye, W. T. Lo, Anthropogenic causes of jellyfish blooms and their direct consequences for humans: a review, Marine Ecology Progress Series 350 (2007) 153–174. A. J. Richardson, A. Bakun, G. C. Hays, M. J. Gibbons, The jellyfish joyride: causes, consequences and management responses to a more gelatinous future, Trends in Ecology and Evolution 24 (6) (2009) 312– 322. F. Boero, J. Bouillon, Zoogeography and life cycle patterns of mediterranean hydromedusae (Cnidaria), Biological Journal of the Linnean Society 48 (1993) 239–266. R. Schaible, F. Ringelhan, B. H. Ramer, T. Miethe, Environmental challenges improve resource utilization for asexual reproduction and maintenance in hydra, Experimental Gerontology 46 (2011) 794–802. X. Kong, Y. L. Chen, W. Xie, X. Wu, A novel paddy field algorithm based on pattern search method, in: International Conference on Information and Automation, 2012. D. Wolpert, W. Macready, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1 (1) (1997) 67–82. S. Chen, MATLAB Particle Swarm Toolbox, uRL: http://www.mathworks.com/matlabcentral/fileexchange/25986-anotherparticle-swarm-toolbox (Accessed 2012.07.30) (2010).