An improved teaching–learning-based optimization algorithm with a modified learner phase and a new mutation-restarting phase

Abstract

Teaching–learning-based optimization (TLBO) is a powerful metaheuristic algorithm for solving complex optimization problems pertaining to the global optimum. Many TLBO variants have been presented to improve the local optima avoidance capability and to increase the convergence speed. In this study, a modified learner phase and a new gradient-based mutation strategy are proposed in an improved TLBO algorithm (TLBO-LM). A restarting strategy is adopted to help TLBO-LM escape from local optima and is combined with the gradient-based mutation strategy to build a mutation-restarting phase in an improved TLBO algorithm (LMRTLBO). The modified learner phase integrates a mutation operator to ensure the exploration capability and a dynamic boundary constraint strategy to increase the convergence speed. A greedy gradient information estimation method is developed to accelerate the convergence rate and then is combined with a mutation operator to establish the gradient-based mutation strategy with a good balance between exploration and exploitation. The proposed algorithms are examined by the CEC 2014 and 2020 benchmark functions. The results suggest that both the modified learner phase and gradient-based mutation strategy significantly enhance the exploration and exploitation capabilities of TLBO and that the restarting strategy effectively avoids local optima but sacrifices the exploitation capability to some extent. TLBO-LM and LMRTLBO obtain better results compared with ten algorithms, including TLBO variants, and are competitive compared with five CEC winners.

Introduction

As the global optimization problems generated in engineering and science become increasingly complicated, the deterministic optimization methods based on the gradient seem inadequate [1]. Recently, metaheuristic algorithms (MAs) have received much attention since they show advantages in solving complex optimization problems due to their nonconvex nature, nonlinearity, and large scale. MAs can solve discontinuous problems as they do not depend on the gradient, and with a stochastic nature, they effectively avoid local optima. Therefore, many MAs, such as differential evolution (DE) [2], particle swarm optimization (PSO) [3], teaching–learning-based optimization (TLBO) [4], artificial bee colony (ABC) [5], and moth-flame optimization (MFO) [6], have emerged. The TLBO algorithm is simple, does not need any adjusted parameters, and is powerful. The algorithm has been applied to effectively solve optimization problems generated in engineering, such as the dispatch of electric vehicle loads [7], parameter identification of battery models [8], neural network training [9], multilevel production planning problem [10], and image segmentation for multilevel thresholding [11].

The basic TLBO algorithm will also suffer from difficulties such as a slow convergence rate and premature convergence when solving complex problems. There is still ample room for performance improvement in TLBO, and many studies have emerged in the last few years [12]. The most common improvement is that a new phase that does not depend on the framework of the TLBO algorithm is added to it. In addition, some improvements focus on the modification of the original learner phase and teacher phase. As Rao and Patel [13] point out, there are three drawbacks of the basic algorithm, i.e., a lack of a self-motivated learning phase, only one teacher will lead to a slower convergence rate, and the teaching factor in the teacher phase is either 2 or 1. Therefore, three modifications, a self-motivated learning phase, an adaptive teaching factor strategy, and a mechanism to generate multiple teachers, are proposed for the basic TLBO algorithm. As opposition-based learning (OBL) is a powerful learning strategy, it has been widely employed to enhance the performance of MAs, and some OBL variants with better performance have been proposed [14]. Similarly, the OBL strategy and its variants are employed as an additional phase in the TLBO algorithm. Roy et al. integrated the quasi-opposition based learning (QOBL) strategy [15] and the OBL strategy [16] into the basic TLBO algorithm to increase the convergence rate. Xu et al. [17] introduced a novel OBL-based learning strategy named dynamic opposite learning (DOL) to balance exploration and exploitation for the TLBO algorithm. Chen et al. [18] proposed utilizing generalized opposition-based learning (GOBL) to improve the performance of TLBO to solve the parameter identification problem of solar cell models. Apart from OBL learning variants, other effective learning strategies for TLBO have been presented. Certain modifications, such as the weight design and utilization of new mutation strategies, have been conducted in the two phases. Chen [19] et al. introduced a gradient information estimation method to establish a self-learning phase and modified the learner phase with a learner choice mechanism. Chen [20] et al. improved the two phases of the TLBO algorithm by a self-learning strategy and a hybrid self-adaptive mutation to formulate a new SHSLTLBO algorithm. This algorithm enhances the exploitation capability through the self-learning strategy and improves the exploration capability by using adaptive mutation to enrich the diversity of the population. Ahmad et al. [21] proposed a BTLBO algorithm with a modified teacher phase and two extra phases, including the tutoring phase and restarting phase. The combination of the two strategies in the teacher phase can maintain diversity and improve the exploitation capability. The tutoring phase improves the exploitation capability by a local search operator around the teacher, and the restarting phase avoids BTLBO trapping into the local optima. Satapathy et al. [22] proposed a TLBO variant with a new learner phase, which realizes a tradeoff between exploration and exploitation by introducing the teacher to the phase. Shukla et al. [23] proposed using an adaptive exponential distribution inertia weight to modify TLBO. Rao and Patel [24] employed the elitist strategy to increase the convergence rate of TLBO. Zhang et al. [25] proposed a logarithmic spiral strategy to modify the teacher phase with a better convergence rate and a triangular mutation rule for the learner phase to improve both exploration and exploitation. The advantage of other MAs is also utilized to improve TLBO, and TLBO-based hybrid algorithms are developed. An effective algorithm is the hybrid TLBO and ABC algorithm (TLABC) [26]. Other hybrid algorithms are hybrid teaching–learning-based PSO (HMTL-PSO) [27], hybrid TLBO and charged system search (CSS) algorithms (HTC) [28], TLBO improved chicken swarm optimization (ICSOTLBO) [29], the hybrid TLBO and DE algorithm (TLBO-DE) [30], and the hybrid MFO and TLBO algorithm (MFO-TLBO) [31].

In TLBO and its variants, many learning strategies and the two phases are based on the concept of mutation. Inspired by biological mutation, mutation is an operator to maintain the population diversity from one generation to the next [32]. As a key part of the DE algorithm, the mutation strategy has been widely investigated [33]. In the basic DE algorithm, the original mutation operator, named “DE/rand/1”, is the sum of a random population member and the weighted difference between two other random members [34]. It is one of the simplest forms and only ensures the exploration capability of the algorithm. The other four most frequently used mutation themes are “DE/rand/2”, “DE/best/2”, “DE/best/1” and “DE/rand-to-best/1” [35]. The first mutation theme focuses on global research, the second and third methods pay more attention to the convergence rate, and the last theme is aimed at simultaneously balancing exploration and exploitation. Some studies present mutation strategies equipped with exploitation capabilities to increase the convergence speed of DE. Zhang et al. [36] proposed the mutation “DE/Current-to-pbest”, where the difference between a random individual and the best individual is utilized. Mohamed et al. [37] introduced a greedy algorithm named “DE/current-to-ord_pbest/1” to enhance the exploitation capability of the DE algorithm. The algorithm randomly selects three individuals and then orders them according to the fitness value before their use. A new trigonometric mutation strategy was presented by Fan and Lampinen [38], which is a greedy local search operator that enhances the exploitation capability of the DE algorithm. Some studies attempt to develop mutation strategies with both exploration and exploitation. Most of these strategies combine two or more mutation operators. A neighborhood-based mutation [39] that combines the local mutation model and global mutation model was proposed by Das et al. to balance exploration and exploitation. Wu et al. [40] proposed integrating three mutation operators, including “current-to-rand/1”, “current-to-pbest/1” and “rand/1”, into a combined mutation strategy to improve DE. Different mutation strategies also mean that different individuals are selected as solution candidates. Therefore, selection methods have an important role in MAs, and recently, an effective selection method referred to as fitness-distance balance (FDB) has been widely investigated and applied in different MAs [41], [42], [43].

Since the mutation strategy with a tradeoff between exploration and exploitation has the ability to comprehensively improve the performance of MAs [44], the combination of multiple mutation operators with different characteristics is a trend in research on mutation strategies. The gradient information can increase the convergence speed of optimization algorithms, and a gradient estimation method has been proposed to enhance the TLBO algorithm [19]. In this paper, a greedier estimation method for gradient information is presented, and based on it, a new mutation strategy that considers both exploration and exploitation is proposed to enhance the TLBO algorithm. The gradient information estimation method calculates the difference between the best individuals in two different iterations. This approach will accelerate the convergence rate because the trajectory formed by the best individual at each iteration is much more likely to reflect the gradient information. The mutation strategy also calculates the difference between two randomly selected individuals, which can guarantee the exploration capability. In addition, the learner phase of TLBO is modified by a traditional mutation operator combined with a dynamic boundary constraint strategy. The mutation operator concentrates on the exploration capability, and the boundary update mechanism, which has a critical role in OBL learning strategies, is effective in accelerating convergence. Furthermore, a restarting strategy adopted from the literature [21] is utilized to help TLBO avoid trapping into local optima. This strategy uses boundaries to conduct a crossover with the population, which is able to escape from local optima. Therefore, an improved TLBO algorithm (TLBO-LM) with a modified learner phase and a new gradient-based mutation strategy is developed. An improved TLBO variant (LMRTLBO) with a new mutation-restarting phase and a modified learner phase is exhibited. In the new mutation-restarting phase, the gradient estimation-based mutation and the restarting strategy alternatively run at each iteration to balance their performance [45]. The contributions of this study are described as follows:

• The learner phase of TLBO is modified by a traditional mutation operator combined with a dynamic boundary constraint mechanism.

• A greedy gradient information estimation operator is designed, and based on it, a novel mutation strategy that considers both exploitation and exploration is proposed to enhance the TLBO algorithm.

• A restarting phase is adopted to avoid local optima, which is integrated into a mutation-restarting phase for TLBO.

• The performance of the TLBO-LM and LMRTLBO algorithms is verified by the CEC 2014 and 2020 benchmark functions and compared with some algorithms, including TLBO variants and CEC winners.

The remainder of the paper is organized as follows: In Section 2, the concepts of TLBO are introduced. The LMRTLBO algorithm is described in Section 3. The experiments and results are presented in Section 4. Section 5 presents the conclusion of this study.