A full-featured cooperative coevolutionary memory-based artificial immune system for dynamic optimization

doi:10.1016/j.asoc.2021.108389

Applied Soft Computing

Volume 117, March 2022, 108389

https://doi.org/10.1016/j.asoc.2021.108389 Get rights and content

Highlights

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Combining of negative selection and clonal selection mechanisms, for increasing the convergence speed.
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Producing the uniform random B-cell behavior simulation in bone marrow, for increasing diversity.
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Simulating Memory-cell, to avoid forgetting the optima.
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Scaping from local minima by multi epitope simulation and search space division.

Abstract

In this paper, a novel cooperative coevolutionary memory-based artificial immune system enhanced by a new clonal selection algorithm is proposed for dynamic optimization problems. In the proposed algorithm, the whole n-dimensional population is decomposed into n one-dimensional subpopulations. Then, each subpopulation is evaluated separately using a set of context vectors called short-term memory. Also, inspired by the production of new cells in bone marrow, each subpopulation is divided into multiple regions to track and locate multiple optima cooperatively. This division helps the algorithm exploit search space effectively. Additionally, inspired by the immune memory concept, a memory-based approach called long-term memory is proposed to store and retrieve essential information when a fitness change occurs. Furthermore, a new clonal selection method, a combination of negative selection and clonal selection mechanisms, is proposed. This proposed algorithm is faster than the basic clonal selection algorithm. Finally, compared to other immune-based algorithms, which usually are implemented based on one or two qualities of the biologic immune system, the proposed approach exploit almost all immune qualities. Several experiments are conducted on different configurations of the moving peaks benchmark to examine the efficiency of the proposed method. The experimental results confirm that the proposed method is competitive with other state-of-the-art algorithms to optimize dynamic problems.

Introduction

Dynamic optimization problems (DOPs) are a class of optimization problems in which optima change with time in different environments. Since most real-world optimization problems are naturally dynamic, DOPs have attracted many researchers recently [1], [2].

In comparison with static optimization problems, DOPs face more challenges. While in static optimization problems, algorithms should track fixed optima and have sufficient time to locate global optima, a DOP solution needs to locate multiple optima, track the global optimum movement, and explore the environment simultaneously in a dynamic landscape [3]. Controlling diversity before and after a new change and using memory to increase convergence speed are other DOPs’ challenges. As effective solutions to address these challenges in DOPs, evolutionary algorithms (EAs) have been increasingly employed [4]. To handle dynamic landscapes, these algorithms have used such strategies as increasing and maintaining diversity [5], [6], multi-population methods [7], [8], memory approaches [9], self-adaptive mechanism [10], [11], and predicting change. As a large class of EAs, bio-inspired algorithms have proved to be an ideal solution to DOPs. Inspired by biological immune system, artificial immune system (AIS) is one of these bio-inspired algorithms.

Biological immune system has outstanding qualities, which can help it adapt to various changes in the body. AISs have been utilized in a wide variety of application areas, such as anomaly detection [12], data classification [13], global optimization [14], multi-objective optimization [15], robotics [16], and recommender systems [17]. However, they have rarely exploited in the area of dynamic optimization problems. Indeed, most proposed AISs are based on clonal selection theory, which is only one of immune system’s qualities. While a classic clonal selection algorithm (CSA) is good at maintaining diversity, it suffers from low convergence speed and premature convergence. Considering these classic CSA’s weaknesses, AIS has attracted less attention in the area of DOPs compared to other bio-inspired algorithms, such as PSO and DE [5].

Motivated by inherent qualities of biological immune system and cooperative coevolutionary [18], [19], this study develops an innovative AIS for DOPs. The main objective of this research is to overcome AIS’s deficiencies in DOPs using all biologic immune system’s characteristics. Immune cells have similar-shaped receptors, antibodies, each of which can combine with a specific part of an antigen, called an epitope. As any antigen has several epitopes, several antibodies are needed to defend against a particular antigen. In response to multiple antigens invading the body, antibodies are increased and clustered around infected area. This natural decomposition-based method is effective in locating and tracking multiple optima in DOPs [5]. Inspired by this strategy, in this paper, the entire n-dimensional search space is divided into n one-dimensional smaller search space, which are explored cooperatively and simultaneously. In this case, each dimension and its location are considered an antibody and an epitope, respectively. Based on cooperative coevolution, an n-dimensional vector named context vector is necessary to evaluate each population. To enhance the convergence speed of the classic CSA, each dimension, or population, is decomposed into several areas in which immune cells are randomly generated. This is based on the idea of generating uniformly random immune cells in bone marrow. How these areas help in accelerating the convergence speed is essential. It is achieved by incorporating a negative selection algorithm (NSA) into CSA, which helps increase convergence speed along with hyper-mutation. According to this procedure, areas are labeled as ‘self’ and ‘non-self’, indicating non-optimal and optimal areas. It happens gradually during the following stages. At the first stage, all areas are labeled ‘non-self’. At the second stage, an affinity-based hyper-mutation proposed in [20] is applied to antibodies within each area. At the last stage, areas failing to improve themselves are eliminated. As the search proceeds, the sub-populations begin to converge to a small number of areas, resulting in accelerating convergence speed. Many researchers [2], [5], [21] have used two kinds of memories, short-term and long-term memories, to track optima before and after a change occurs. In this paper, the short-term and long-term memory contain the current and the best previous environment’s solutions, respectively. It is based on primary and secondary responses qualities of natural immune system. Individuals are stored in the long-term memory in two ways. First, when a change occurs, the best solutions in the previous environment are moved to the long-term memory. Second, by applying a network suppression process to the short-term memory, unimproved current solutions are considered local optima and transferred to the long-term memory. Inspired by immune network theory, this process helps avoid premature convergence and trapping in local optima. After a change is detected, how to use the long-term memory and rebuild the short-term memory to track global optima is important. Motivated by [21], in this paper, the difference between the average of current fitness and the previous fitness is used to make certain whether an environment reoccurs. If the difference is less than a threshold, it means an environment reoccurs. Thus, the best solutions belonging to that environment are inserted into the short-term memory. Otherwise, the short-term memory is re-initialized.

In short, this study proposes a new AIS for DOPs based on the following strategies. At first, inspired by cooperative coevolution, the n-dimensional search space is divided into n one-dimensional subpopulations to speed exploration. Second, each subpopulation is decomposed into several regions, which contain optimal and non-optimal solutions. Third, a negative selection algorithm is incorporated into CSA to speed exploitation. Fourth, a network suppression mechanism is applied to solutions to avoid trapping in a local optimum. Finally, previous solutions are stored in the memory and reused further to track the new global optimum better when a change occurs.

The rest of the paper is organized as follows. Section 2 presents an overview of the approaches that are a combination of memory-based and multi-population strategies as well as immune-based algorithms applied to DOPs. Section 3 describes the proposed CM-AIS in more detail. Experimental results of the performance of CM-AIS on the MPB compared to other algorithms are given in Section 4. Finally, Section 5 provides conclusions and suggests a few useful directions for future work.

Section snippets

Cooperative coevolutionary algorithms

Firstly proposed by Potter and De Jong [22], a cooperative coevolutionary (CC) has been embedded in GA to solve complex problems. The idea is to decompose an n-dimensional problem into n one-dimensional subproblems each of which is solved independently. An n-dimensional vector called a context vector (CV) was introduced, that each subproblem is responsible for optimizing one of the CV’s components. After that, Van den Bergh and Engelbrecht [18] used Potter’s idea in the basic PSO for

Proposed algorithm

Although IBAs effectively explore search space, they are not good at exploitation, which makes optimizing dynamic environments challenging [5]. Moreover, they tend to have low convergence speed and high computation. However, the biological immune system has features helping it adapt to changes in the human body. Thus, this paper has two purposes:

(1) Simulating all the immune system’s features to create a new artificial immune system the most similar to the natural one.

(2) Creating an AIS able

The moving peaks benchmark problem

The moving peaks benchmark (MPB) firstly introduced by Branke in [43] has been commonly used for evaluating dynamic optimization problems. In MPB, in a D-dimensional landscape, there are $N_{p}$ peaks, whose locations, heights, and widths are changed according to a certain frequency. The change of ith peak is determined as follows: $W_{i} (t) = W_{i} (t - 1) + w i d t h_s e v e r i t y \times σ$ $H_{i} (t) = H_{i} (t - 1) + h e i g h t_s e v e r i t y \times σ$ $X_{i} (t) X_{i} (t - 1) + υ$ $σ \in N (0, 1)$ where $W_{i} (t)$ , $H_{i} (t)$ , and $X_{i} (t)$ are the width, height, and location of peak i at time t,

Conclusion and future work

AIS-based algorithms have been scarcely used for optimizing dynamic environments because of their low convergence speed and large computation. A new cooperative coevolutionary memory-based artificial immune system improved by a new clonal selection algorithm is proposed to overcome these problems. In the proposed method, inspired by cooperative coevolution, an n-dimensional population is divided into n one- dimensional subpopulations. Then, motivated by cells’ production in bone marrow, each

CRediT authorship contribution statement

Bahareh Etaati: Conceptualization, Software, Data curation, Visualization, Writing – original draft, Investigation. Zahra Ghorrati: Methodology, Writing – review & editing. Mohammad Mehdi Ebadzadeh: Methodology, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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