Recently, teachinglearningbased optimization (TLBO), as one of the emerging natureinspired heuristic algorithms, has attracted increasing attention. In order to enhance its convergence rate and prevent it from getting stuck in local optima, a novel metaheuristic has been developed in this paper, where particular characteristics of the chaos mechanism and Lévy flight are introduced to the basic framework of TLBO. The new algorithm is tested on several largescale nonlinear benchmark functions with different characteristics and compared with other methods. Experimental results show that the proposed algorithm outperforms other algorithms and achieves a satisfactory improvement over TLBO.
Optimization problems are always associated with many kinds of difficult characteristics involving multimodality, dimensionality, and differentiability [
Several wellknown swarm algorithms have been proposed in the latest years. For example, ant colony optimization (ACO) is based on the metaphor of ants seeking food [
Teachinglearningbased optimization (TLBO) algorithm is a teachinglearning inspired algorithm proposed by Rao et al., which is based on the effect of influence of a teacher on the output of learners in a class [
Chaos is a universal phenomenon of nonlinear dynamic systems, which has been extensively studied since Lorent [
Lévy flight is another technique for speeding up the convergence rate of the algorithm and escaping from local optima [
An efficient optimization algorithm means it has both strong exploration ability and a fast exploitation rate; moreover, the method can be adapted to tackle a broad range of problems [
This paper is organized as follows. In Section
TLBO is a recently published populationbased method, which mimics the classic teachinglearning phenomenon within a classroom environment. In this novel optimization algorithm a group of learners is considered as population and different design variables are considered as different subjects offered to the learners and learners’ result is analogous to the fitness value of the optimization problem. In the entire population the best solution is considered as the teacher. The main procedure of TLBO consists of two phases: teacher phase and learner phase. These two phases will be explained in the following parts.
This is the first stage of the algorithm where learners learn from the teacher. During this phase a teacher tries to increase the mean of the whole class to his or her level (the new mean). The difference between the existing mean and the new mean is given as
Based on this Difference_Mean, the existing solution is updated according to the following expression:
It is the second part of the algorithm where learners increase their knowledge by interaction between themselves. A learner interacts randomly with another learner for enhancing his or her knowledge. A learner learns new things if the other one has more knowledge than him or her. Mathematically the learning phenomenon of this phase is expressed below.
At any iteration
The steps for implementing TLBO are as follows.
Initialize the population size (
Calculate the mean of the population columnwise, which will give the mean of each design variable as
Calculate the Difference_Mean according to (
Modify the solutions in the teacher phase based on (
Update the solution in the learner phase according to (
Repeat Steps
An effective optimization algorithm must have a strong global searching ability along with a fast convergence rate. TLBO is free from specific algorithm parameters and outperforms PSO, HS, and so on due to its simplicity and efficiency. However, several hard benchmarks with complicated landscapes pose challenges to TLBO in finding a satisfactory result and escaping from local optima.
In order to enhance the performance of TLBO as well as take advantage of the properties of the chaotic system and Lévy flight, we integrate the chaotic search mechanism and Lévy flight into TLBO to improve its search efficiency. Hence, a chaotic TLBO with Lévy flight (CTLBO) is proposed in this paper. In the algorithm, the population is divided into two parts: the part with better fitness is evolved by the teachinglearning process in TLBO, while another part is performed with a Lévy flight. Then the chaotic perturbation is implemented on a randomly selected part of the population in terms of the diversification of the population. The main steps of CTLBO are elaborated in the next sections.
Lévy flights, also called Lévy motion, represent a kind of nonGaussian stochastic process whose step sizes are distributed based on a Lévy stable distribution [
When generating new solutions
There are a few ways to implement Lévy flights; the method we chose in this paper is one of the most efficient and simple ways based on Mantegna algorithm; all the equations are detailed in [
Chaos is a deterministic, quasirandom process that is sensitive to the initial condition [
A chaotic map is a discretetime dynamical system running in a chaotic condition [
Chaotic sequences have been proved to be simple and fast to produce and reserve; it is unnecessary to store long sequences [
In this paper,
By introducing the Lévy flight and the chaotic search into the TLBO, a new algorithm is proposed in this paper. The pseudocode of the proposed CTLBO is shown in Pseudocode
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In order to verify the performance of the proposed CTLBO and to analyze its properties, two sets of optimization problems are selected for the test experiments. In each set of problems, several wellknown functions are used as benchmark problems to study the search behavior of the proposed CTLBO and to compare its performance with those of other algorithms.
Firstly, to demonstrate the performance of the proposed algorithm, eight benchmark optimization problems [
Benchmark functions considered in Experiment
Number  Function  Formulation  Dim.  Search range 

1  Sphere 

10  [–100, 100] 


2  Rosenbrock 

10  [–2.048, 2.048] 


3  Ackley 

10  [–32.768, 32.768] 


4  Griewank 

10  [–600, 600] 


5  Weierstrass 

10  [–0.5, 0.5] 


6  Rastrigin 

10  [–5.12, 5.12] 


7  NCRastrigin 

10  [–5.12, 5.12] 


8  Schwefel 

10  [–500, 500] 
In [
Comparative results of different algorithms over 30 independent runs.
Algorithm  Sphere  Rosenbrock  Ackley  Griewank  

Mean  SD  Mean  SD  Mean  SD  Mean  SD  
PSO 








PSOcf 








PSO 








PSOcflocal 








UPSO 








FDR 








FIPS 








CPSOH 








CLPSO 








ABC 








Modified ABC 








TLBO 








ITLBO (NT = 4) 








CTLBO 










Algorithm  Weierstrass  Rastrigin  NCRastrigin  Schwefel  
Mean  SD  Mean  SD  Mean  SD  Mean  SD  


PSO 








PSOcf 








PSO 








PSOcflocal 








UPSO 








FDR 








FIPS 








CPSOH 








CLPSO 








ABC 








Modified ABC 








TLBO 








ITLBO (NT = 4) 








CTLBO 








It can be seen from Table
It can also be seen from Table
In order to observe the performance of CTLBO visually, the convergence curves of six functions are drawn as shown in Figure
Convergence curve of six functions.
Rosenbrock
Ackley
Griewank
Weierstrass
Rastrigin
Schwefel
The Rosenbrock function is always used as a test function to test the performance of optimization algorithms. The global optimum lies inside a long, narrow, parabolic shaped flat valley, and it is very difficult to find the global optimum. It can be seen from Figure
The Ackley function is a continuous, rotating, and nonseparable multimodal function. The exterior region of the function is nearly flat while the centre is a high peak, and it has many widespread locally optimal points from the flat region to the centre peak. From Figure
It can be observed from Figures
From Figure
From the results and analysis, we can see that the proposed TLBO has good searching ability for most functions, and CTLBO has improved its performance. The convergence rate and accuracy of CTLBO are better than those of TLBO. In order to test the proposed algorithm comprehensively, more test functions will be introduced in the next section.
In this experiment, the performance of the proposed CTLBO algorithm is compared with those of the recently developed PSABC [
Benchmark functions considered in Experiment
Number  Function  Formulation  Search range 

1  Sphere 

[−100, 100] 


2  Schwefel 2.22 

[−10, 10] 


3  Schwefel 1.2 

[−100, 100] 


4  Schwefel 2.21 

[−100, 100] 


5  Rosenbrock 

[−30, 30] 


6  Step 

[−100, 100] 


7  Quartic 

[−1.28, 1.28] 


8  Schwefel 

[−500, 500] 


9  Rastrigin 

[−5.12, 5.12] 


10  Ackley 

[−32.768, 32.768] 


11  Griewank 

[−600, 600] 


12  Penalized 

[−50, 50] 


13  Penalized 2 

[−50, 50] 
This experiment is conducted from smallscale to largescale by considering 20, 30, and 50 dimensions for all the benchmark functions. The number of function evaluations is set as 120000 for all tested algorithms. Each benchmark function is tested 30 times and the results are obtained in the form of the mean solution and the standard deviation of the objective function after 30 independent runs of the algorithms.
Table
Comparative results of different algorithms over 30 independent runs.
Function  Dim.  PSABC [ 
TLBO [ 
ITLBO [ 
CTLBO  

Mean  SD  Mean  SD  Mean  SD  Mean  SD  
Sphere  20 








30 









50 











Schwefel 2.22  20 








30 









50 











Schwefel 1.2  20 








30 









50 











Schwefel 2.21  20  0.00  0.00 






30 









50  19.6683 










Rosenbrock  20  0.5190 

15.0536 

1.3785 



30 


25.4036 

15.032 

11.1767 


50 


45.8955 

38.7294 

36.9081 




Step  20 








30 









50 











Quartic  20 








30 









50 











Schwefel  20 




−8263.84 

−7817.05 

30 




−12519.92 

−10903.20 


50 




−20700.70 

−15744.41 




Rastrigin  20 








30 









50 











Ackley  20 

0.00 





0.00 
30 

0.00 





0.00  
50 

0.00 





0.00  


Griewank  20 








30 









50 











Penalized  20 








30 









50 











Penalized 2  20 








30 









50 








It can be observed from Table
In order to observe the performance of CTLBO visually, the convergence curves of algorithms for several functions in 20 dimensions are drawn as shown in Figure
Convergence curve of four functions.
Rosenbrock
Quartic
Penalized
Penalized 2
From Figure
The Quartic function is unimodal with random noise. Noisy functions are widespread in realworld problems, and every evaluation of the function is disturbed by noise, so the algorithms’ information is inherited and diffused noisily, which makes the problem hard to optimize. It can be observed from Figure
From Figures
From the above analysis, we can see that the proposed TLBO has good searching ability for most of the functions, and CTLBO has improved its performance based on TLBO. The convergence rate and accuracy of CTLBO show better performance compared to TLBO, which reveals that the proposed chaotic mechanism is effective and provides an improvement on TLBO.
This paper formulated a novel TLBO algorithm, based on its combination with chaotic search and Lévy flight. From a quick look, CTLBO resembles other swarmintelligence approaches such as GA and PSO in many aspects; for example,
It can be seen from the framework of the CTLBO that the population is first divided into two parts in each iteration, then these two subparts evolve with the Lévy flight and teachinglearning mechanisms, respectively, and then the population is perturbed by using chaotic searching. This can be viewed as a kind of coevolution to some extent; that is, two independent subpopulations evolve interactively and, due to this process, not only are the decision solutions diversely exploited but also the convergence rate of the algorithm is accelerated.
Taking a closer look at the CTLBO, we conclude that it essentially consists of three components: exploitation by mutation operator, a global exploration by Lévy flight, and diversification by chaos mapping. The mutation operators including the teacher phase and learner phase ensure the exploitation around the best solution obtained so far. Lévy flight makes the search move away from the worst place with a large step and, at the same time, samples the search space effectively so that the new solutions are thoroughly diversified. Chaos mapping can disturb the solution so as to maintain the population diversity as well as avoid falling into local optima. In general, a good integration of the above three components may thus lead to an efficient algorithm such as CTLBO.
Furthermore, from simulation studies in which the CTLBO algorithm’s controlling parameters were varied, we observed that the convergence rate is insensitive to algorithm parameters such as
Moreover, it can be seen that the proposed method may not find the global minima of a few specific functions. This has its root in the mechanism of the teacher phase in TLBO, where a mean value is used to update the solutions, which may lead solutions directly to the centre of a search region. For those functions whose global minima are not located in the centre of the feasible solution region, it is usually challenging to find the global optima of the tested functions. However, in the CTLBO, the divided population weakens the effect of the “mean mechanism.”
Finally, in this paper merely logistic map has been embedded to diversify the population of CTLBO algorithm; however, other different chaotic maps will be analyzed in the future work.
This paper proposes chaotic teachinglearningbased optimization with Lévy flight (CTLBO). The algorithm is improved via a Lévy walk and perturbed by chaotic searching, which can enhance the diversification of the algorithm. The experimental results demonstrate that the designed algorithm has better performance than other methods. In addition, the properties of the proposed algorithm are analyzed and the characteristics and features are discussed in the paper.
Future work is likely to apply this novel method to a wider spectrum of problems such as constrained optimization problems and many engineering applications in the real world. What is more, the parallel implementation mechanism of CTLBO and its application to multiobjective optimization as well as combinatorial optimization problems will also be studied.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the National Basic Research Program of China (973 Program) under Grant nos. 2011CB706804 and 2014CB046705 and by the National Natural Science Foundation of China under Grant no. 51121002.