Population Resizing Using Nonlinear Dynamics in

Population Resizing Using Nonlinear Dynamics in an Ecology-Based Approach Rafael Stubs Parpinelli1,2, and Heitor Silv´erio Lopes2 1

Applied Cognitive Computing Group Santa Catarina State University Joinville, Brazil 2 Bioinformatics Laboratory Federal Technological University of Paran´ a Curitiba, Brazil [email protected], [email protected]

Abstract. It is well known that, in nature, populations are dynamic in space and time. This means that the size of populations oscillate across their habitats over time. This work uses the concepts of habitats, ecological relationships, ecological successions and population dynamics to build a cooperative search algorithm, named ECO. This work aims to explore the population sizing not as a parameter but as a dynamic process. The Artificial Bee Colony (ABC) was used in the experiments where benchmark mathematical functions were optimized. Results were compared with ABC running alone, with and without the use of population dynamics. The ECO algorithm with population dynamics performed better than the other approaches, possibly thanks to the ecological interactions (intra and inter-habitats) that enabled the co-evolution of populations and to a more natural survival selection mechanism by the use of population dynamics. Keywords: optimization, cooperative search, co-evolution, habitats, logistic chaos model, ecology, population dynamics.

1

Introduction

Nature has always been an endless source of inspiration for computational models and paradigms, in particular for the computer scientists of the area known as Natural Computing [1]. Two outstanding families of bio-inspired algorithms are evolutionary computation (EC) and swarm intelligence (SI) that currently offer a wide range of strategies for optimization [2][3]. The concept of optimization can be abstracted from several natural processes such as the evolution of the species, the behavior of social groups, the dynamics of the immune system, the strategies 

Authors would like to thank the Brazilian National Research Council (CNPq) for the research grant to H.S. Lopes; as well as to UDESC (Santa Catarina State University) and FUMDES program for the doctoral scholarship to R.S. Parpinelli.

H. Yin et al. (Eds.): IDEAL 2012, LNCS 7435, pp. 27–34, 2012. c Springer-Verlag Berlin Heidelberg 2012 

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of searching for food and in the ecological relationships of different populations. Most of these cases were the source of inspiration to the development of new algorithms for optimization. It is worth mentioning that most bio-inspired algorithms only focus on and take inspiration from specific aspects of the natural phenomena. However, in nature, biological systems are interlinked to each other, e.g. biological ecosystems [4][5]. Also, in nature, populations are always dynamic in such a way that the size of populations oscillate across their habitats over time. However, in most Evolutionary Computation applications, the population size is constant and does not change during the search [6]. Current practice of manual setting of population size in evolutionary computation is experience-based, but not robust. Hence, this work aims to explore the population sizing not as a parameter but as a dynamic process that changes the population size deterministically over time. In [7] the authors first introduce the potentiality of some ecological concepts (e.g., habitats, ecological relationships and ecological successions) presenting a simplified ecological-inspired algorithm. In this work we explore a more biologically plausible survival selection mechanism by the use of population dynamics where the logistic chaos model is applied to control the size of populations [8][5]. The aim is to compare the results obtained by the implementation of the algorithm with the use of ecological concepts, without the use of ecological concepts (application of stand alone algorithms), and with and without the use of population dynamics.

2

The Proposed Ecological-Inspired Approach

The ecological-inspired algorithm, named ECO, represents a new perspective to develop cooperative evolutionary algorithms. The ECO is composed of populations of individuals (candidate solutions for a problem being solved) and each population evolves according to an optimization strategy. Therefore, individuals of each population are modified according to the mechanisms of intensification and diversification, and the initial parameters, specific to each optimization strategy. The ECO system can be modeled in two ways: homogeneous or heterogeneous. A homogeneous model implies that all populations evolve in accordance to the same optimization strategy, configured with the same parameters. Any change in the strategies or parameters in at least one population characterises a heterogeneous model. The ecological inspiration stems from the use of some ecological concepts, such as: habitats, ecological relationships and ecological successions [4][5]. Once dispersed in the search space, populations of individuals established in the same region constitute an ecological habitat. For instance, in a multimodal hypersurface, each peak can become a promising habitat for some populations. A hyper-surface may have several habitats. As well as in nature, populations can move around through all the environment. However, each population may belong only to one habitat at a given moment of time t. Therefore, by definition, the intersection between all habitats at moment t is the empty set.

Population Resizing Using Nonlinear Dynamics

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With the definition of habitats, two categories of ecological relationships can be defined. Intra-habitats relationships that occur between populations inside each habitat, and inter-habitats relationships that occur between habitats [4][5]. In ECO, the intra-habitat relationship is the mating between individuals. Populations belonging to the same habitat can establish a reproductive link between their individuals, meshing the populations and favoring the co-evolution of the involved populations. Populations belonging to different habitats are called reproductively isolated. The inter-habitats relationship are the great migrations. Individuals belonging to a given habitat can migrate to other habitats aiming at identifying promising areas for survival and mating. In addition to the mechanisms of intensification and diversification specific to each optimization strategy, when considering the ecological context of the proposed algorithm, the intra-habitats relationships are responsible for intensifying the search and the inter-habitats relationships are responsible for diversifying the search. Inside the ecological metaphor, the ecological successions represent the transformational process of the system. In this process, populational groups are formed (habitats), relations between populations are established and the system stabilizes by means of the self-organization of its components. Algorithm 1 shows the pseudo-code of the proposed approach. For a detailed description refer to [3]. The metric chosen to define the region of reference is the centroid and represents the point in the space where there is a highest concentration of individuals of population i. 2.1

Population Dynamics

In this paper we applied the one-parameter logistic chaos map to drive the population dynamics between ecological successions. The logistic map is often

Algorithm 1. Pseudo-code for ECO 1: Consider i = 1, . . . , N Q, j = 1, . . . , N H and t = 0; 2: Initialize each population Qti with ni random candidate solutions; 3: while stop criteria not satisfied do {Ecological succession cycles} 4: Perform evolutive period for each population Qti ; 5: Apply metric Ci to identify the region of reference for each population Qti ; 6: Using the Ci values, define the N H habitats; 7: For each habitat Hjt define the communication topology CTjt between populations Qtij ; 8: For each topology CTjt , perform interactions between populations Qtij ; 9: Define communication topology T H t between Hjt habitats; 10: For T H t topology, perform interactions between Hjt habitats; 11: Check populational scenario and compute population size; 12: Increase t; 13: end while

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cited as an example of how chaotic behaviour can arise from very simple nonlinear dynamical equations. Also, it can be used as a discrete-time demographic model [8]. Equation 1 presents the logistic chaos map used, where 0 < a < 4. = aP opti (1 − P opti ) P opt+1 i

(1)

Three populational scenarios can occur between ecological successions (line 11 in Algorithm 1). The first is when there are no changes in the size of populations from moment t to t + 1. In this case the evolution proceeds as usual. The second is when there is increment in the size of populations from t to t + 1. In this case, new solutions are randomly generated using the current centroid as reference. The third scenario is when there is decrement in the size of populations from t to t + 1. In this case, the worst solutions are discarded. Hence, the second scenario favors exploration and the third scenario favors exploitation. In addition to the exploration and exploitation routines provided by the evolution of populations and by the ecological interactions (inter and intra-habitats), the use of population dynamics creates a new biologically plausible mechanism to diversify the search.

3

Experiments and Results

Experiments were conducted using four benchmark functions extensively used in the literature for testing optimization methods [9]. Each function to be minimized was tested with 10 and 200 dimensions. The first function (f1 (x)) is known as generalized F6 Schaffer function. The second function (f2 (x)) is the Rastrigin function. The third function (f3 (x)) is the Griewank function. Finally, the fourth function (f4 (x)) is the Rosenbrock function. Table 1 summarizes the informations about the functions used. The parameters of the ECO algorithm are: number of populations (N-POP ) that will be co-evolved, the initial population size (POP-SIZE ), number of cycles for ecological successions (ECO-STEP ), the size of the evolutive period (EVOSTEP ) that represents number of function evaluations in each ECO-STEP, the tournament size (T-SIZE ) used to choose solutions to perform intra and interhabitat communications and the proximity threshold ρ used to define the habitats. Studies about the adjustment of parameters have not been carried out yet. Hence, all the parameters of the algorithm were defined empirically [3]. Table 1. Benchmark Functions Function f1 (x)

n−1

f2 (x) f3 (x) f4 (x)

1 4000



Definition 

sin2

  2 x2 i+1 +xi −0.5

0.5 + i=1 (0.001(x2i+1 +x2i )+1)2 n 2 (xi − 10cos(2πxi )+ 10) i=1  n n x 2 √i +1 i=1 xi − i=1 cos i

n−1

2 2 2 i=1 (100(xi+1 − xi ) + (xi − 1) )

Domain

Global Optimun

−100 ≤ xi ≤ 100

f1 (0) = 0

−5.12 ≤ xi ≤ 5.12

f2 (0) = 0

−600 ≤ xi ≤ 600

f3 (0) = 0

−30 ≤ xi ≤ 30

f4 (1) = 0

Population Resizing Using Nonlinear Dynamics

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In all experiments, the Artificial Bee Colony Optimization (ABC) algorithm [10] was used in a homogeneous model, i.e. all populations use this algorithm with the same control parameters. For all experiments the initial population size was set to POP-SIZE = 10. The logistic chaos map (Equation 1) was applied to adjust the population size dynamically. The logistic chaos map parameter was set to a = 3.57 and it is called ‘route to chaos’ [8]. This choice was done based on the work of Ma [11] where experiments were performed with different values for the parameter a. Figure 1 shows the resizing projection for 100 ecological successions. For the number of dimensions (D) equal to 10, the parameters used were NPOP = 100, ECO-STEP = 100, EVO-STEP = 100, T-SIZE = 5 e ρ = 0, 5. With this configuration, the total number of function evaluations was 10,000 for each population. For D = 200, some parameters were redefined: N-POP = 200, ECO-STEP = 500, EVO-STEP = 200. With this adjustment of parameters, for 200 dimensions, the total number of function evaluations was 100,000 evaluations for each population.

Fig. 1. Population dynamics according to the logistic chaos model (a = 3.57)

The ecological-inspired framework (ECO) was tested using four configurations. The first configuration implements the Algorithm 1 as described in Section 2, with the definitions of habitats, topologies and ecological relations. The second configuration complements the first one by adding population dynamics (Section 2.1). The third configuration disables the ability to create habitats and, consequently, topologies and interactions are not defined. This third configuration simulates the evolution completely isolated populations, and they evolve without exchanging information. The fourth configuration complements the third one by adding population dynamics. For each configuration, the algorithm was run 30 times. Table 2 shows the averaged results obtained for the benchmark functions. For both dimensions, D = 10 and D = 200, the results obtained by each configuration of the algorithms are presented (columns 2 to 4). Column 2 shows the results obtained by the ABC algorithm running alone, without co-evolution.

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R.S. Parpinelli and H.S. Lopes Table 2. Obtained results for the benchmarck functions f1 (x) Model Global Best Model Global Best f2 (x) Model Global Best Model Global Best f3 (x) Model Global Best Model Global Best f4 (x) Model Global Best Model Global Best

D = 10 ABCLM ECOABC 1.4559 ± 0.2 1.1344 ± 0.2 D = 200 ABC ABCLM ECOABC 27.5936 ± 0.73 24.7426 ± 0.5 20.2792 ± 0.40 D = 10 ABC ABCLM ECOABC 10−11 ± 0.0 10−12 ± 0.0 0.0000 ± 0.0 D = 200 ABC ABCLM ECOABC 62.1453 ± 9.6 34.0388 ± 4.3 10−05 ± 0.0 D = 10 ABC ABCLM ECOABC 10−06 ± 0.0 10−17 ± 0.0 10−13 ± 0.0 D = 200 ABC ABCLM ECOABC 10−7 ± 0.0 10−7 ± 0.0 10−11 ± 0.0 D = 10 ABC ABCLM ECOABC 0.0098 ± 0.0 0.0098 ± 0.0 0.0086 ± 0.0 D = 200 ABC ABCLM ECOABC 13036.1 ± 4193.4 35.1444 ± 11.6 137.86 ± 42.0 ABC 4.6569 ± 0.8

ECOABC−LM 0.5193 ± 0.0 ECOABC−LM 11.2742 ± 0.4 ECOABC−LM 0.0000 ± 0.0 ECOABC−LM 10−10 ± 0.0 ECOABC−LM 10−18 ± 0.0 ECOABC−LM 10−15 ± 0.0 ECOABC−LM 0.0064 ± 0.0 ECOABC−LM 9.2568 ± 4.1

Column 3 shows the results obtained by the ABC algorithm running with the logistic chaos model for modelling the population dynamics (ABCLM ). Column 4 shows the results obtained by the ABC algorithm using the ecological-inspired approach (ECOABC ). Finally, column 5 shows the results obtained by the ABC algorithm using the ecological-inspired approach running with the logistic chaos model for the population dynamics (ECOABC−LM ). For each dimension, the third line (Global Best ) shows the average and standard deviation of the best result obtained by all populations in all runs. Analyzing the ABC and ABCLM configurations we can observe that the use of population dynamics improved the results in most cases and remained the same in two. Analyzing the ABC and ECOABC we can observe that the ecologicalinspired approach obtained much better results than the algorithm executed without the concepts of habitats for all functions. Analyzing the results for the ecological-inspired approach with population dynamics, ECOABC−LM , we can observe that the results were significantly better than the ecological-inspired approach without population dynamics (ECOABC ). This gain is mainly due to the ecological interactions (intra and inter-habitats) that enabled the coevolution of populations and to the use of a more natural survival selection mechanism afforded by population dynamics. Moreover, the ECOABC−LM was the best approach for all functions. In Figure 2 we can visually verify the results for D = 200, where the x-axis shows the different approaches and the y-axis represents the Global Best values of each approach and are shown at the top of each bar.

Population Resizing Using Nonlinear Dynamics

(a) Function f1 (x).

(b) Function f2 (x).

(c) Function f3 (x).

(d) Function f4 (x).

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Fig. 2. Bar graph of each benchmarck function with D = 200

4

Conclusions

This paper presents an ecological-inspired algorithm for optimization with population dynamics. The proposed algorithm uses cooperative search strategies where populations of individuals co-evolve and interact among themselves using some ecological concepts. Each population behaves according to the mechanisms of intensification and diversification, and the control parameters, specific to a given search strategy. The Artificial Bee Colony Optimization algorithm was used in all populations. In this work, a population dynamics model was used to set up population sizes in the computational ecosystem. The population dynamics model applied was the logistic chaos due to its simplicity and its rich dynamic behaviour as discrete-time demographic model. The main ecological concepts addressed are the definition of habitats, ecological relationships, ecological successions and population dynamics. These features bring a higher biological plausibility to the proposed algorithm, opposed to most bio-inspired algorithms that take inspiration only from one biological phenomenon. Thus, the proposed methodology opens the possibility for the

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insertion of several ecological concepts in the optimization process, bringing more biological plausibility to the system. In addition to the exploration and exploitation routines provided by the evolution of populations and by the ecological interactions (inter and intra-habitats), the use of population dynamics creates a new biologically plausible mechanism to diversify the search. The results showed that the use of habitats and ecological relationships influence significantly the co-evolution process of populations, leading to better solutions (when compared to the results not using the ecological concepts). Also, the use of a population dynamics model inside the ECO framework improved considerably the results. This work is still under development and as future work we intend to analyze the influence of the control parameters (number of ecological successions, evolutive period, number of populations, and ρ threshold for creation of habitats) on the quality of solutions, as well as to add other search strategies in the proposed model. Another future research is to use the ECO approach in an asynchronous model. Also, the application of other population dynamics models such as the Lotka-Volterra predator-prey can be an interesting direction. Currently, in order to bring more biological plausibility to the system, other ecological concepts are being modeled.

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