2. Aging mechanisms to alleviate the premature convergence

to their structures which are closely related to one or other features observed in nature [10]. Accuracy and repeatability are the prime objectives of every optimization algorithm. Therefore, modeling biological mechanisms may impact the outcome of the observed system,

Problem-solving algorithms inspired by one or more biological features have been developed after observing the behavior in humans, animals, and cells [10–13]. For instance, Genetic Algorithms (GA) [4] defined the basis for evolutionary computing using the early works of Darwin [14] and Mendel [15]; or the Ant Colony System (ACS) [6, 7] which considers the traveling behavior model of self-organized argentine ants published by Goss [5], solving in a fashion way travel salesman-type problems; or the Particle Swarm Optimization (PSO) [8] inspired on feeding behavior of bird flocks becoming a very popular optimizer nowadays [17]. Particle-based optimizers, like those described in [8]; or those presented in [18–22, 28] are very popular because instead of working with one candidate solution, they offer a subset of individual candidate solutions (particles), which are explored, exploited and improved. A relevant mechanism related to evolution that could play a central role in optimization algorithms is aging [23, 25]. Aging is a natural characteristic whose inclusion in a particle-based optimizer could give a mean of individual control over the particle without highly increasing the com-

To the authors' best knowledge, previous PSO algorithms did not have a measurement to control individual particle existence within the swarm by evaluating each particle performance. In PSO, because of the very nature of the algorithm, an effect called premature convergence appears when most (or all) of the particles within a swarm compromises their ability to explore and stay close to a local solution. The Particle Swarm Optimization with an Aging Leader and Challengers (ALC-PSO) [24] was the first approach to include aging processes to alleviate this unwanted effect. However, this was only leadership-oriented and not swarm-related; even more important: the aging dynamics were linear and bounded to static predefined values. In [29], it is used to design a high speed symmetric switching CMOS

Our PSO variant proposal, the Particle Swarm Optimization with Bio-inspired Aging Model (BAM-PSO) is based on a mathematical model that describes the telomeres shortening observed in the immune system cells, this model includes a form of aging effect over all the particles of the swarm; this mechanism provides a mean to control the existence of each particle within the swarm avoiding the premature convergence effect. Therefore, the PSO variant with aging model possesses the potential to outperform current optimization algorithms and have further implications in computational intelligence. Finally, optimization under uncertainty and complex functions is an area of special scientific interest, and many real problems and applications include some form of uncertainty; it is also known that collectiveintelligence algorithms perform excellent in this type of scenarios [11]; BAMPSO has been successfully implemented in several optimization applications: in time-series forecasting [30] it was implemented as a training algorithm for an artificial neural network and in [31] it was used over a Geometric Algebra (GA) framework in order to compute the rigid movement on images to improve the accuracy of Structure from Motion (SfM) algorithms, which comprises a

designing more accurate and efficient heuristic algorithms [13, 16].

plexity of algorithms [26].

10 Particle Swarm Optimization with Applications

inverter.

Aging is the process of becoming older, which consists on the accumulation of changes over time. This process affects all living systems: humans, cells, unicellular organisms, fruit flies and mammals like rodents [12, 32, 42]. Since the particles of the PSO optimizer algorithm can be treated as a living system, aging could represent a relevant mechanism to alleviate the premature convergence problem in heuristic algorithms. Nowadays, we have better understanding of the lifespan of human cells, which is determined by homoeostatic properties of the immune system. Homeostasis refers to the regulation of the lymphocytes pool in an organism. It is assumed that the number of cells is determined by the capacity of the peripheral immune system.

In the immune system, it is observed that cell death rate accelerates if the immune cells exceed the allocated free space [33]. For instance, in the course of a viral infection, immune system cells can undergo approximately 15–20 divisions. Total proliferative capacity of human T lymphocyte is about 40–45 divisions and depends on the telomere length [41]. Telomeres are the end parts of the chromosomes, which become shorter in every cell division; this can be appreciated in Figure 1. The cell can reach its unresponsiveness state when the telomere length completes about one half of its initial value.

Telomere dynamics can be interpreted in a mathematical model based on experimental observations. In this work, we consider the mathematical model proposed by [33] to represent the telomere dynamics. This model considers the following equation:

$$\frac{dT}{dt} = \alpha (p^\* - T) \text{N} \tag{1}$$

where T represents the remaining telomere divisions per cell. α defines the telomere consumption rate per iteration. p<sup>∗</sup> represents the length of telomere repeats in naive cells produced at the age <sup>t</sup> (with initial length <sup>p</sup>0= 8:<sup>3</sup> � 103 ) and N is the number of cells as defined in [33].

Eq. (1) is a differential non-ascending equation that defines the derivative in telomere division per cell depending on the consumption rate of the cell α, telomere capacity of the cell (p<sup>∗</sup>) and number of cells (N). Eq. (1) describes the dynamic of average telomere length T in the pool of naive cells. The rate of this process depends on <sup>p</sup>ð Þ <sup>∗</sup> � <sup>T</sup> , where <sup>p</sup><sup>∗</sup> is the telomere length in the cells and p<sup>0</sup> defines the telomere at initial age. This dynamic describes the self-sustaining process of regulation of total concentration of the T cells and how the telomere is affected by T cells concentration and iterations [33].

3. Particle swarm optimization with a bio-inspired aging model

The ALC-PSO variant [24] suggests an interesting approach using aging factors to define when to remove non-useful characteristics of the algorithm. However, this variant proposes a simple

Particle Swarm Optimization Algorithm with a Bio-Inspired Aging Model

http://dx.doi.org/10.5772/intechopen.71791

13

• The lifespan exists only for the leader and is controlled by linear means according to

• There is no lifespan controller for the rest of the particles of the swarm, meaning that the particles continue offering candidate solutions even if premature convergence has occurred.

• There is no communication between particles within the swarm about premature convergence and no evaluation is performed about particle concentration around local minima.

Consequently, there are several points that can be improved in ALC-PSO. For instance, including aging mechanisms to the rest of the particles within the swarm may help exploration without affecting the convergence. Moreover, in order to alleviate premature convergence in the PSO, there is an urgent need to include means of measuring the premature convergence in real time allowing the swarm to discard non-useful particles and to explore new candidate

Based on previous observations, we propose a variant of the PSO named Particle Swarm Optimization with a Bio-inspired Aging Model (BAM-PSO). Our proposed algorithm considers the aging leader and challengers in the same fashion as ALC-PSO, but it applies senescence to each particle within the swarm by using the mathematical model that describes

For BAM-PSO to implement senescence efficiently, it is necessary to implement a mechanism that allows the algorithm to interpret when the swarm has reached a local minimum; this can

In the aging model represented by Eq. (1), the number of cells can be interpreted as a measure of the particles numbers around the same one-dimensional location. In this sense, measuring the standard deviation among the swarm in each particle dimension can be computed as

> kj <sup>¼</sup> kmin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN i¼1 PD j¼1 ð Þ xij�xj 2

where D ∈ R is the dimension of the problem; kmin represents the deviation minimum for all dimensions; kj is the premature convergence around j-th element of the dimension D. Note that xij is the particle within the swarm and xj represents the mean value of all j-th elements of the

vuut

ð Þ D�1

(2)

solutions without losing the convergence inertia toward the global minimum.

be achieved by means of premature convergence measurement.

(BAM-PSO)

aging model with the following characteristics:

lifespan controller shown in [24].

aging dynamics in Eq. (1).

follows:

Figure 1. Telomere division process in human T cells.

The proposed scheme in Eq. (1) can provide PSO the same self-regulation capabilities during swarm's concentration around a local minimum (known as premature convergence) if we consider the swarm as lymphocytes, and particle concentration around a local minimum as T cells concentration. Finally, senescence of the particle will be translated as a new randomgenerated particle, replacing the unresponsive one. For this to be achieved, a given particle within the swarm will have a limited number of iterations to exist within the swarm, similar to telomere length at p0, and senescence of this particle will occur when particle's capacity for search space-exploitation approaches to 0 and swarm's concentration around a local minimum has exceeded a given limit (premature convergence indicator), similar to T cells in human immune systems [35].

Based on this aging mechanism, it is possible to include senescence to a given particle within the swarm, and the lifespan of each particle will be adjusted according to the error produced by its candidate solution and the premature convergence indicator of the swarm.
