Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 754562 | 19 pages | https://doi.org/10.1155/2015/754562

An Improved Teaching-Learning-Based Optimization with Differential Learning and Its Application

Academic Editor: Ralph B. Dinwiddie
Received22 Apr 2015
Revised24 Jun 2015
Accepted12 Jul 2015
Published31 Aug 2015

Abstract

The teaching-learning-based optimization (TLBO) algorithm is a population-based optimization algorithm which is based on the effect of the influence of a teacher on the output of learners in a class. A variant of teaching-learning-based optimization (TLBO) algorithm with differential learning (DLTLBO) is proposed in the paper. In this method, DLTLBO utilizes a learning strategy based on neighborhood search of teacher phase in the standard TLBO to generate a new mutation vector, while utilizing a differential learning to generate another new mutation vector. Then DLTLBO employs the crossover operation to generate new solutions so as to increase the diversity of the population. By the integration of the local search and the global search, DLTLBO achieves a tradeoff between exploration and exploitation. To demonstrate the effectiveness of our approaches, 24 benchmark functions are used for simulating and testing. Moreover, DLTLBO is used for parameter estimation of digital IIR filter and experimental results show that DLTLBO is superior or comparable to other given algorithms for the employed examples.

1. Introduction

TLBO is a recently proposed population-based algorithm which simulates the teaching-learning process in a classroom. Rao et al. proposed TLBO for the optimization of mechanical design problems [1] and then applied TLBO to find global solutions to large-scale nonlinear optimization problems [2]. Toĝan [3] presented a design procedure that employed TLBO for the discrete optimization of planar steel frames. In [4], elitist concept is introduced in the TLBO algorithm and its effect on the performance of the algorithm is investigated. The effects of common controlling parameters such as the population size and the number of generations on the performance of the algorithm are also investigated. Degertekin and Hayalioglu [5] applied TLBO in the optimization of four truss structures. Rao and Patel [6] proposed an improved TLBO by introducing the concepts of the number of teachers, adaptive teaching factor, tutorial training, and self-motivated learning. Niknam et al. [7] presented a θ-multiobjective TLBO algorithm to solve the dynamic economic emission dispatch problem, where the optimization procedure considers phase angles attributed to the real value of the design parameters, instead of the design parameters themselves. Zou et al. [8] proposed an improved teaching-learning-based optimization algorithm with dynamic group strategy for global optimization problems. In [9], a ring neighborhood topology is introduced into the original TLBO algorithm to maintain the exploration ability of the population. Details of the conceptual basis of TLBO were given by Waghmare [10]. TLBO has emerged as one of the simplest and most efficient techniques, as it has been empirically shown to perform well on many optimization problems. However, in evolutionary computation research there have been always attempts to improve any given findings further and further.

The problem of IIR system identification can also be viewed as a problem of parameter estimation of adaptive digital filtering. For adaptive digital IIR filter design, the objective of the adaptation is to adjust filter coefficients of a digital filter to estimate actual parameter values of an unknown system from its inputs and outputs. However, there are some potential disadvantages with the design of adaptive digital IIR filters [1113]. A major concern in these disadvantages is the mean square error between the desired response and estimated filter output of digital IIR filter. This error surface is generally nonquadratic and multimodal with respect to the filter coefficients, and hence optimization algorithms are required to find out better global solution. In order to avoid the local optima problem encountered by gradient descent methods in IIR system identification, considering IIR system identification as an optimization problem, recently, evolutionary algorithm and swarm intelligence, such as Genetic Algorithms (GA) [1416], Simulated Annealing (SA) [17], Tabu Search (TS) [18], Differential Evolution (DE) [1921], Ant Colony Optimization (ACO) [22], Artificial Bee Colony (ABC) algorithm [23], Particle Swarm Optimizer (PSO) [24, 25], Gravitation Search Algorithm (GSA) [26, 27], and Cuckoo search optimization [28], have been made for studying alternative structures and algorithms for adaptive digital IIR filters.

In order to find suitable filter coefficients for the digital IIR filter efficiently, this paper aims to develop a teaching-learning-based optimization (TLBO) approach. First, an improved TLBO with differential learning (DLTLBO) is proposed in paper. In the method, DLTLBO utilizes the integration of the learning strategy based on neighborhood search of teacher phase in the standard TLBO and the differential learning to generate two new mutation vectors, and DLTLBO employs the crossover operation to generate new solutions so as to increase the diversity of the population. Then, to evaluate the optimization performance of the proposed algorithm, the comparison study on 24 benchmark optimization problems is given. Moreover, DLTLBO is applied to the design of the digital IIR filter, and its performance is compared to that of other algorithms. The simulation results show that DLTLBO has better or comparable performance compared to the other algorithms for the employed examples.

The remainder of this paper is organized as follows. Section 2 describes the teaching-learning-based optimization. Section 3 presents the improved teaching-learning-based optimization with differential learning (DLTLBO). In Section 4, 24 benchmark functions are tested and the experiments are conducted along with statistical tests. DLTLBO is applied to the design of the digital IIR filter in Section 5. Conclusions are given in Section 6.

2. Teaching-Learning-Based Optimization

Inspired by the philosophy of the classical school teaching and learning process, Rao et al. [1, 2] first proposed a novel teaching-learning-based optimization (TLBO). In this method, population consists of learners in a class and design variables are courses offered. The brief description of the algorithm is given as follows.

2.1. Teacher Phase

During the teacher phase, learner will move their position toward the position of the best learner (), by taking into account the current mean value of the learners () that represents the average qualities of all learners in the population. The updating formula of the learning for a learner in teaching phase is given bywhere TF is a teaching factor that decides the value of to be changed and is a random vector where each element is a random number in the range . The value of TF can be either 1 or 2, which is again a heuristic step and is decided randomly with equal probability as

Learner modification is expressed as follows: for each learner  Accept if it gives a better function value endfor

2.2. Learner Phase

During the learner phase, learners increase their knowledge by interacting with each other. A learner learns something new if the other learner has more knowledge than him or her:where is a random vector where each element is a random number in the range , is the fitness value of the learner , and is the fitness value of the learner .

Learner modification is expressed as follows: for each learner of the class Randomly select one learner , such that ;  Accept if it gives a better function value end for

3. Proposed Algorithm: DLTLBO

The search in TLBO is mainly based on the global information, and DE is mainly based on the distance and direction information which is a kind of local information [29]. To balance the global and local search ability, a modified interactive learning strategy is proposed in teacher phase. In this method, DLTLBO utilizes the learning strategy based on neighborhood search of teacher phase in the standard TLBO to generate a new mutation vector, while utilizing the differential learning to generate another new mutation vector. Different from our previous method [30], DLTLBO employs the DE crossover operation to generate new solutions so as to increase the diversity of the population. The complete flowchart of the DLTLBO algorithm is shown in Figure 1.

In the proposed DLTLBO algorithm, for each learner , its associated mutant vector based on differential mutation can be generated as follows:where and are the teacher and the mean of the learner’s corresponding neighborhood, respectively. In this paper, ring neighborhood topology with -neighborhood radius is used to determine in the neighborhood area of each learner.

For each learner , its other associated mutant vector can be generated based on differential mutation as follows:

After the mutation phase, crossover operation is applied to each pair of the mutant vectors in order to enhance the potential diversity of the population. That is, the updating formula of the learning for a learner in teacher phase is given as follows: is a parameter called crossover probability; is the learners which is calculated according to (4) and is the learners which is calculated according to (5).

Finally, the one-to-one greedy selection is employed by means of comparing a parent and its corresponding offspring .

In learner phase, interactive learning of the original TLBO algorithm in the learner phase is still used in our DLTLBO algorithm. In this learning method, one of the learners randomly learns from the other learner in the population. This learning method can be treated as the global search strategy. As explained above, the pseudocode for the implementation of DLTLBO is summarized in Algorithm 1.

) Begin
() Initialize (number of learners), (number of dimensions) and CR
() Initialize learners and evaluate all learners
() Donate and of the learner’s corresponding neighborhood
() while (stopping condition not met)
() for each learner of the class    % Teacher phase
()      TF = round(1 + rand(0, 1))
()      for  
()         =   + rand ()
()     endfor
()     Select three learners randomly from the current class where
()     
()     for  
()       if ,   = ; else,   = ; endif
()     endfor
()     Accept if f() is better than
() endfor
() for each learner of the class    % Learner phase
()     Randomly select one learner , such that
()     if better
()      for  
()        = + rand ()
()      endfor
()     else
()      for
()        = + rand ()
()      endfor
()     endif
()     Accept if is better than
()  endfor
() endwhile
() end

As explained in Algorithm 1, it can be observed that the probability CR determines adopting the learning strategy of the learners in the teacher phase of the DLTLBO algorithm. That is, a random number between 0 and 1 is generated for each learner; if < CR, the learning strategy of the original TLBO will be adopted by the learner; otherwise, the differential learning strategy will be adopted by the learner. This will be helpful in balancing the advantages of fast convergence rate (the attraction of the learning strategy of TLBO learning) and exploration for local areas (neighborhood search strategy and the differential learning) in DLTLBO. By this means, DLTLBO achieves a tradeoff between exploration and exploitation, improves the global search capability, and accelerates the convergence.

4. Functions Optimization

In this section, to compare the performance of DLTLBO with some other methods, 7 algorithms are also simulated in the paper. The details of benchmark functions, the simulation settings, and experimental results are given as follows.

4.1. Benchmark Functions

The details of 24 benchmark functions are shown in Table 1. Among 24 benchmark functions, to are unimodal functions, to are multimodal functions, and to are the rotated version of to , respectively. to are taken from the CEC 2008 test suite. The searching range and theory optima for all functions are also shown in Table 1.


FunctionFormulaRangeOptima

Sphere0

Quadric0

Sum square0

Zakharov0

Rosenbrock0

Ackley0

Rastrigin0

Weierstrass0

Griewank0

Schwefel0

Rotated sum square0

Rotated Zakharov0

Rotated Rosenbrock0

Rotated Ackley0

Rotated Rastrigin0

Rotated Weierstrass0

Rotated Griewank0

Rotated Schwefel0

Shifted sphere
     :
o

Shifted Schwefel’s Problem 2.21
      : the  shifted  global  optimum
o

Shifted Rosenbrock
       : the  shifted  global  optimum
o

Shifted Rastrigin
     : the  shifted  global  optimum
o

Shifted Griewank
     : the  shifted  global  optimum
o

Shifted Ackley
     : the  shifted  global  optimum
o

4.2. Parameter Settings

In this paper, all the experiments are carried out on the same machine with a Celoron 2.26 GHz CPU, 2 GB memory, and Windows XP operating system with MATLAB 7.9. For the purpose of reducing statistical errors, each function is independently simulated 50 runs, and their mean results are used in the comparison. For all approaches, the population size was set to 50, and the maximal number of cost function evaluations (FEs) is used as termination condition of algorithm, namely, 100,000 for 10D problems. The parameter settings for all algorithms in comparison are extracted from their corresponding literatures and are described as follows:(i)jDE [31]: , ;(ii)SaDE [32]: , , , LP = 50;(iii)PSO-wFIPS [33]: w = 0.7298;(iv)PSO-FDR [34]: wmin = 0.4, wmax = 0.9, ψ1 = 1, ψ2 = 1, ψ3 = 2;(v)ETLBO [4]: elite size = 2;(vi)Proposed DLTLBO: , , neighborhood size = 3.

4.3. Comparisons on the Solution Accuracy

The results are shown in Table 2 in terms of the average best solution and the standard deviation of the solutions obtained in the 50 independent runs by each algorithm for 10D problems on 24 test functions, where “” summarizes the competition results among DLTLBO and other algorithms in the last row of the table, which means that DLTLBO wins in functions, ties in functions, and loses in functions. The best results among the algorithms are shown in bold. Figure 2 presents the convergence graphs of different benchmark functions in terms of the median fitness values achieved by 7 algorithms for 50 runs.


FunjDESaDEPSO-wFIPSFDR-PSOTLBOETLBODLTLBO