Solving multimodal optimization problems using adaptive differential evolution with archive

Abstract

Evolutionary algorithms are widely used to solve multimodal optimization problems. The two main challenges faced while solving MMOPs are locating multiple optimal solutions and improving the accuracy of these solutions. In this paper, we have proposed an adaptive algorithm based on differential evolution using the distributed framework in mutation strategy and an elite archive mechanism termed Adaptive Differential Evolution with Archive to deal with these challenges. The following techniques have been proposed and integrated to locate multiple diverse optimal solutions with refined accuracy. Firstly, each individual in the population is treated as a possible exemplar and is expected to reach an optimal value by exploring the nearby search space. The search space is controlled by using an adjustable range mechanism. An adaptive mutation strategy is then used to ensure that all the good solutions or individuals of the population move to better positions. Next, an elite archive is constructed for stagnated individuals to avoid getting stuck in local optimas. The experimental results on the 20 multimodal functions from IEEE Congress on Evolutionary Computation 2013 illustrate that the performance of the proposed algorithm is better than the existing multimodal optimization algorithms in terms of finding more number of accurate solutions.

Introduction

Multiple global optimal solutions of a given Multimodal Optimization Problem (MMOP) are located using the Multimodal Optimization (MMO) algorithms. For example, electromagnetic design [1], protein structure prediction [2], job scheduling [3] and traveling salesman problem [4] are few MMOPs that require to find many solutions. Among several solutions provided by the algorithm, the user can check for feasibility according to the conditions that cannot be computed and then select the better solution among them.

The traditional Evolutionary Algorithms (EAs) such as genetic algorithm (GA) [5], [6], [7], [8], ant colony optimization (ACO) [9], [10], differential evolution (DE) [11], [12], [13], [14] and particle swarm optimization (PSO) [15], [16], [17] were developed to find global optimal solution. The traditional EAs maintain track of the population of individuals in every generation by evaluating their fitness and trying to improvise them by applying the evolutionary operators. However, these traditional EAs tend to find one single optimal solution. While in the case of locating multiple solutions for MMOPs, the algorithm needs to identify geometrically diverse solutions as well as refine the accuracy of the solutions found. In other words, the algorithm has to maintain the balance between exploration and exploitation simultaneously among the solutions found. Exploration is determined by the different multiple optimal solutions found by the algorithm, while exploitation is determined by the proximity of the solution obtained to the desired solution.

Several variations in EAs have been proposed by different researchers to solve MMOPs in few recent years, including PSO [18], [19], GA [20], [21], ACO [22] and DE [23], [24], [25], [26], [27]. Most of the MMO algorithms are based on the niching techniques, which allow to divide the whole population into subpopulations called species or niches. The main issue with the use of niching techniques is that it makes the algorithm dependent on the parameters involved in the niching technique [28], [29]. A slight change in the parameter value of niching technique can change the results abruptly. Therefore, there is a need for such algorithm which is not sensitive to the parameters and can find multiple optimal solutions with high accuracy.

In this paper, we have proposed a novel algorithm, Adaptive Differential Evolution with Archive (ADEA), which uses an adaptive mutation strategy in differential evolution. ADEA follows the distributed framework and considers that every individual in the population is a possible exemplar and can reach the optimal solution. The distributed framework does not divide the whole population into subpopulations. Therefore, we do not employ any niching technique which reduces the number of parameters involved in an MMOP algorithm. To achieve this, we have proposed the following techniques. Firstly, we have designed and implemented an Adjustable Range Method (ARM) which helps in reducing the search space and generating the vectors to be used while applying the mutation strategy. The ARM helps the individuals which are far from the peak to explore the search space, while it helps to refine the accuracy of the possible solutions which are near to the peak. Then we have proposed the Adaptive Mutation Strategy (AMS) that uses the ARM to find the best neighbours of the individual and generate the mutant vector. We have also proposed an elite archive technique (EAT) to keep track of the individuals in the no-more-improvement state or possible exemplars of optimal fitness value and to remove the worse neighbours of these potential exemplars.

To test the quantity and quality of solutions obtained using the proposed ADEA algorithm, we have performed exhaustive experimentation of our proposed ADEA algorithm on the IEEE CEC-2013 special session test suite [30]. The benchmark test suite consists of 20 multimodal functions. The results obtained by our proposed ADEA algorithm on all of the 20 multimodal functions show the superiority and feasibility of our proposed ADEA algorithm.

The rest of the paper has been organized as follows: First, we have described the DE algorithm along with its different designs for MMOPs in Section 2. Then in Section 3, we have explained the proposed ADEA algorithm in a detailed manner with all the components. The experimental results with the dataset description and the evaluation metrics are presented in Section 4. Then an analysis of different components, hyperparameters tuning, and comparison among results obtained with existing algorithms have been discussed in Section 5. Finally, Section 6 includes the conclusion of the proposed work and future work.