applied sciences Article Extended Evolutionary Algorithms with Stagnation-Based Extinction Protocol Gan Zhen Ye and Dae-Ki Kang * Department of Computer Engineering, Dongseo University, Busan 47011, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-51-320-1724 Abstract: Extinction has been frequently studied by evolutionary biologists and is shown to play a significant role in evolution. The genetic algorithm (GA), one of popular evolutionary algorithms, has been based on key concepts in natural evolution such as selection, crossover, and mutation. Although GA has been widely studied and implemented in many fields, little work has been done to enhance the performance of GA through extinction. In this research, we propose stagnation-driven extinction protocol for genetic algorithm (SDEP-GA), a novel algorithm inspired by the extinction phenomenon in nature, to enhance the performance of classical GA. Experimental results on various benchmark test functions and their comparative analysis indicate the effectiveness of SDEP-GA in terms of avoiding stagnation in the evolution process. Keywords: stagnation; extinction; genetic algorithm 1. Introduction The biological metaphor of evolution was being applied to computation since as early as 1950 [1–3], and the genetic algorithm (GA) is one of the evolutionary algorithms inspired Citation: Gan, Z.Y.; Kang, D.-K. by this process of natural evolution. The theory of evolution was introduced by Darwin in Extended Evolutionary Algorithms 1858 [4] and together with Weismann’s theory of natural selection [5] and Mendel’s concept with Stagnation-Based Extinction of genetics [6] they formed the neo-Darwinian paradigm [7] which the genetic algorithm Protocol. Appl. Sci. 2021, 11, 3461. was based on. Although it is clear that in the process of evolution, there are fundamental https://doi.org/10.3390/app11083461 components such as natural selection, reproduction (crossover), and mutation, it is often neglected that extinction plays an important role in this process too. Several evolutionary Academic Editor: Mauro Castelli biologists have raised this question and argued that extinction plays a significant role in evolution [4,8], but little has been done in the field of evolutionary computation where Received: 10 February 2021 extinction is still being neglected or ignored. Accepted: 8 April 2021 Published: 12 April 2021 When we take a look at the efforts in improving the performance of GA, we can categorize the efforts into two groups: One tries to improve GA through modeling GA Publisher’s Note: MDPI stays neutral closer to natural evolution by introducing new biological operators or modifying existing with regard to jurisdictional claims in ones to closely mimic processes observable in nature [9–11]. The other group mainly published maps and institutional affil- designs operators that were tailored to suit specific problems, and these operators have iations. no correspondence to nature of any sort whatsoever [12–14]. Based on the increased performance and accuracy of GAs that were modeled to closely mimic the natural evolution process, we came to a deduction: the closer we model our GA towards natural evolution, the better it will perform. With this in mind, this research aims to introduce a novel genetic algorithm inspired by the extinction phenomenon in nature: stagnation-driven extinction Copyright: c 2021 by the authors. Licensee MDPI, Basel, Switzerland. protocol for genetic algorithm (SDEP-GA). This article is an open access article The idea of SDEP-GA is analogous to Dropout regularization [15]. In training the distributed under the terms and neural network, Dropout operation is randomly omitting a subset of hidden units at each conditions of the Creative Commons training iteration in the neural network. This random removal of hidden units at each train- Attribution (CC BY) license (https:// ing iteration turns out to be a combination of exponentially many different neural networks creativecommons.org/licenses/by/ which share the same weights. Thus, Dropout regularization basically attempts to average 4.0/). multiple models to improve the performance of the neural network. Srivastava et al. [16] Appl. Sci. 2021, 11, 3461. https://doi.org/10.3390/app11083461 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3461 2 of 15 have shown that dropout reduces overfitting and improves generalization. Removing chromosomes in SFEP-GA is similar to omitting hidden units in dropout, but in SDEP-GA, removing chromosomes with the extinction probability in a random way or a targeted way can allow greater possibility of exploration for new solutions. The remaining of this paper is organized as follows. Section2 presents a description of GA, specially focusing on simple GA. Section3 provides the design and architecture of our proposed algorithm, stagnation-driven extinction protocol for GA (SDEP-GA). Section4 will cover the experimental design of our proposed algorithm together with the experimental results and discussion of SDEP-GA compared against Simple GA in terms of performance. Finally, Section5 gives a summary, a review of our contribution, and possible directions for future work. 2. Genetic Algorithm Genetic algorithm (GA) was invented by Holland in the 1960s and formally introduced by Holland in 1975 [17]. GA is based on ideas from Darwinian evolution and thus it adopts some biological terminology. To assist in the understanding of SDEP-GA, we outline Simple GA algorithm. 2.1. Simple GA Simple GA is the simplest form of GA, which consists of three types of operators commonly used in other GAs. The three basic operators are as below, and they will be further explained later. • Selection • Crossover • Mutation Figure1 show the flowchart of Simple GA process, and it can be interpreted as follows: 1. Generate a random population of n chromosomes. 2. Evaluate the fitness of each chromosome. 3. If the termination criterion is not met, then continue the following steps. Otherwise, return the best chromosome found. 4. Repeat the following until n chromosomes are created: (a) Select a pair of parent chromosomes. (b) Perform crossover with the selected pair of parent chromosome. A pair of offspring is created in this process. (c) Apply mutation operator on the pair of offspring. 5. Replace current population with the newly created n offspring. 6. Go to step2. Each complete iteration (step1–5 or step2–5) performed is called a generation, and the entire set of generations is called a run. At the end of each run, the fittest chromosome or the best approximated candidate solution is returned to the user. Appl. Sci. 2021, 11, 3461 3 of 15 Figure 1. Flowchart of Simple genetic algorithm (Simple GA). 2.1.1. Fitness Evaluation The first rule required to use GA to solve a problem is that we need to be able to clearly evaluate the fitness of a chromosome, meaning we must have a clear method of measuring the accuracy of each candidate solution. The fitness of a chromosome is judged by its accuracy in solving the given problem. The chromosome that is most accurate in solving the problem will be the fittest chromosome and the least accurate one will be the least fit chromosome. 2.1.2. Selection The selection operator selects a pair of parent chromosomes from the current popula- tion, the probability of selection being an increasing function of fitness. Selection is done “with replacement”, meaning that the same chromosome can be selected more than once to become a parent. There are various types of selection methods, but in this research we will only use stochastic universal sampling method which will be explained in Section4. 2.1.3. Crossover With probability Pc (the “crossover probability”), we cross over the pair of parents at a randomly chosen point (chosen with uniform probability) to form two offspring. If no crossover takes place (probability of 1-Pc), we form two offspring that are exact copies of their respective parents. The crossover operator randomly chooses one or more loci in a pair of chromosomes and exchanges the subsequences after the locus to create two offspring. There are two types of crossover operator: single-point crossover and multi-point crossover. Crossover points are selected randomly, so it might happen that during a single-point crossover the locus is located before the first bit or after the last bit of a chromosome. In such cases, the offspring pair will be an exact replica of their parents. 2.1.4. Mutation Similar to the crossover operator, mutation is done with a probability Pm (the mutation probability). There are multiple ways to mutate a chromosome: single-bit mutation or n-bit mutation. For a binary vector case, the mutation operator randomly selects one or more bits in the chromosome and then flips it. Whether or not mutation takes place in the offspring created from crossover operation, the offspring will be placed in the population pool to replace the current population after n offspring are created. Appl. Sci. 2021, 11, 3461 4 of 15 3. SDEP-GA This section introduces SDEP-GA, a GA with an extinction protocol driven by stag- nation. The insight of this extinction protocol is to mimic the extinction phenomenon in nature to bring GA closer to how natural evolution takes place. Before going deep into the mechanism of the extinction protocol, it is useful to define what stagnation is and when it occurs. We define stagnation as a condition when there is no improvement in the best fitness after K generations. Stagnation usually takes place when the algorithm is stuck in a local optimum while trying to search for a global optimum. To overcome this problem, we suggest make the chromosomes in the population extinct with a probability of Pe (the extinction probability). This will by chance remove some chromosomes that lead towards the stagnation from the population before the selection process takes place. This step will also increase the chances of less fit chromosomes to be selected to reproduce if they have survived the extinction step due to the decrease in the population size.