A self-adaptive differential evolution algorithm with an external archive for unconstrained optimization problems

Abstract

Differential evolution algorithm (DE) has yielded promising results for solving nonlinear, non-differentiable and multi-modal optimization issues. Due to its simple structure, fast convergence and strong robustness, DE has received increasing attention and wide application in a variety of fields. We propose a novel differential evolution approach (SE-DE) which uses an external archive for opposition-based learning, by this way, more high quality solutions can be selected for candidate solutions. In addition, the mutation factor (F) is adaptively controlled based on the success of offspring/trial solutions generated. An optimization factor α is proposed to select the crossover strategy, a combination of binomial and exponential crossover can effectively balance the exploration and exploitation ability of the algorithm. The performance of SE-DE is compared with the other five DE algorithms including DE, SADE, ODE, NDE and MDE-pBX. The comparison is carried out for a set of 30-, 50- and 100-dimensional test functions from CEC2005. The results show that our algorithm is better than, or at least comparable to, the algorithms from other literature.

Keywords

Differential evolution (DE)self adaptive external archive global optimization unconstrained single objective optimization

1 Introduction

Heuristic algorithms utilize various stochastic search strategies to obtain the global optimization, as it only requires information about the objective function itself and does not to know its continuity and differentiability, which has been widely applied into many practical optimization problems. Evolutionary algorithms (EAs) [1] are a broad class of stochastic optimization algorithms inspired by biological processes, which allows populations by genetic inheritance and survival of the fittest to adapt to their surrounding environments. Differential evolution (DE) is a particular instance of EA. The superiorities of DE are its simple structure, fast convergence, strong robustness and it can lends itself very well to parallel computation. Due to these advantages, DE has drawn much attention from researchers in recent years.

The global single-objective unconstrained optimization is an important field in optimization. As constrained problems can be converted to unconstrained problems by some constraints handling mechanism, a plethora of efficient optimization algorithms have been proposed to deal with single objective optimization problems in recent years. The unconstrained global optimization problems considered in this paper can be defined in the following form: $f (x *) = min_{x \in Ω} f (x)$ (1) where x is a M-dimensional parameter vector, x = {x ¹, . . . , x ^M}, which represents a solution for a real valued function f (x), the object is to find the global optimum vector x ^* within domain Ω ⊂ R ^M[2].

Since DE was first proposed by Storn and Price (1995, 1997) [3–4], this simple and straightforward method has been successfully applied to some simple benchmark functions [5 –14] and practical problems, such as forward kinematics of parallel manipulators [15], transient stability constrained optimal power flow [16], economic dispatch optimization [17], optimization of dynamic systems [18], and many others [19]. To improve its property, many modified DE algorithms have been proposed in various kinds of literature.

DE has some prominent advantages over other types of methods. Vesterstroem et al. [8] made a comparative study of DE, particle swarm optimization (PSO) and SEA by solving the numerical benchmark problems. Based on experimental results, it has been demonstrated that the DE algorithm outperforms PSO and SEA in terms of quality of the solutions.

In order to suit DE to more complex optimization problems, some hybrid algorithms combining DE with other algorithms was proposed by researchers in past years. A. Sedki et al. [22] combined DE and PSO for optimal design of water distribution systems, by linking to the hydraulic simulator, EPANET, the proposed algorithm performs well in minimizing the cost design of water distribution systems. Sun et al. [23] combined DE with estimation of distribution algorithm (EDA) and proposed a novel DE/EDA algorithm, which tried to sampling new solutions from a probability model in order to guide its search toward a promising area. The experimental results demonstrated that, the novel algorithm outperforms both the DE algorithm and the EDA. Omran et al. [24] combined DE and PSO proposed a barebones differential evolution (BBDE), in this literature, the original PSO parameters were not adopted and DE scale parameter was eliminated. In addition, a probability of recombination was introduced into BBDE, it made BBDE to update the particle position on the basis of a best personal position which was randomly selected, or a mutation of the particle attractor.

DE is easy to work with, which has only three control parameters: mutation factor F, crossover rate CR, and population size M. As in the earlier study, the parameters are kept fixed throughout the entire optimization process [3 , 21]. Nevertheless, constant control parameters, which might be optimal for one optimization problem, might be inefficient for other problem. The best settings for the control parameters can be different for different functions, and for different requirements of one same function may also be different. The settings of the control parameters make a great difference on the performance of the DE algorithm. A fixed value F = 0.5 is often used in earlier forms of DEs [7], there are other ways on setting mutation factor F, W. Qian et al. [25] proposed a selection mechanism based on crowding distance to solve multiobjective optimization problems. A Cauchy distribution with a mean of 0.5 and a standard deviation of 0.1 was used in adaptive differential evolution with optional external archive [11]. A.K. Qin et al. [26] use Gaussian distribution with mean of 0.5 and standard deviation of 0.3 in strategy adaptation for F. Because all these values are the result of trial and error, it is not surprising to come across that, finding a constant parameter that suit for all kinds of problems is time-consuming and not accurate.

Brest J et al. [27] presented a self-adaptive differential evolution algorithm with a small and varying population size on large scale global optimization. During the optimization process, both the two control parameters (F and CR) were decided by a self-adaptive control mechanism. Small population size were used at early generations while bigger population size were used in later generations. This jDEsps algorithm obtained competitive results than other algorithms for high-dimensional continuous optimization problems.

Elsayed S M et al. [28] improved SAMODE and proposed the ISAMODE-CMA to solve constrained problems. In ISAMODE-CMA, the population was divided into a number of subpopulations, where each subpopulation was allocated to its own mutation strategy and two crossover operators. The subpopulation sizes were changed adaptively, depending on the evolution success. The covariance adaptation matrix was used periodically to increase the convergence pattern. The algorithm showed a superior performance on solving complex real world problems.

Zheng Y J et al. [29] presented a new hybrid DE and BBO algorithm, named HSDB, which used a trial-and-error method to select among two DE mutation schemes and two BBO migration schemes. In order to provide a good balance of exploration and exploitation, HSDB performed more DE mutations in early search stage and more BBO migrations in later stage. Experiments shown that HSDB outperforms DE, SaDE and blended BBO on the benchmark set for the ICSI 2014 Competition.

Leandro dos Santos Coelho et al. [30] proposed a method (DEGC) based on chaotic sequences, in which generally based on ergodicity, stochastic properties and irregularity of chaotic signals. In this paper, a DEGC approach based on Gaussian probability distribution and chaotic sequence generated by logistic map to adapt the parameters CR and F was proposed. As demonstrated by the tests on several benchmark functions, the use of chaotic sequences in DE can be helpful.

Md. Asafuddoula et al. [31] introduced an adaptive hybrid DE algorithm (AH-DEa). In AH-DEa, crossover rate (CR) was adaptively controlled instead of being defined by user in conventional DE. In this algorithm, the population was logically divided into an active population and an archive population, which provided two and one parents, separately. It also employed a binomial crossover in early stages of evolution for exploration, while an exponential crossover in later stages for exploitation. Sequential quadratic programming (SQP) was utilized for gradient based search to explore possibilities of further improvement. By testing its performance on some scalable benchmark problems, the results verified that the proposed algorithm is able to get better results for constrained single objective optimization problems.

DE combines one parent individual and several other individuals of a population to create new candidate solutions. If the offspring candidate has better fitness than the parent, it will take the parent individual’s place in the population. To improve the diversity of population and accelerate the convergence of solution, a number of significant modifications of DE have been proposed in recent years. In 2008, S. Rahnamayan proposed an opposition-based differential evolution (ODE) [14] which adopted an external archive that was denoted with OP. Any variable P _i,j in the population can get its only opposite value OP _i,j. Even though OP _i,j may increase the possible for finding better solution, its searching space was restricted and not diverse enough. In 2013, Dexuan Zou introduced a modified differential evolution algorithm (MDE) [29]. MDE modified the opposition-based mechanism and obtained outstanding performance, more details about this method will be described in part 3. In recent study, there are some approaches referring to the idea of distributed evolutionary algorithms. In 2012, Adam P. Piotrowski presented a new algorithm called DE with Separated Groups (DE-SG) [2]. In DE-SG, population was distributed into small groups, then different rules were defined to exchange information inside and between the groups. Moreover, two different strategies were used to balance the exploration and exploitation capabilities of DE-SG. For vast majority of rotated multi-modal problems, which with local optimum scattered in the search space, DE-SG outperforms than other methods.

On the basis of the adaptive hybrid differential evolution algorithm [31] and MDE [32], we propose a self-adaptive differential evolution algorithm with external archive (SE-DE). In SE-DE, a parent may be selected from either the active population or the external archive. This external archive can explore wider searching space, which is efficient to escape from local optimum. In addition, SE-DE introduced a roulette wheel based F selection scheme, the mutation factor (F) is adaptively controlled based on the success of offspring solutions generated. The better values of control parameters lead to better individuals, in turn, are more likely to propagate and produce offspring. The algorithm also put forward an optimization factor α to select the crossover strategy in the next iteration. Binomial and exponential crossover strategies are used in different stages of evolution to balance the exploration and exploitation capabilities of SE-DE. The performance of SE-DE is investigated and compared with other five modified algorithms previously presented. The benchmark functions of CEC2005 used in our paper are similar to benchmark functions in [32]. The results show that SE-DE is a promising method for most scalable optimization problems.

The rest of the paper is organized as follows: Section 2 describes the fundamentals of traditional differential evolution algorithm (DE). In Section 3, a novel approach uniting self-adaptive and external archive (SE-DE) is introduced in detail. The performance of SE-DE on solving the mathematical benchmarks of CEC-2005 is discussed in Section 4 while conclusions are given in Section 5.

2 Fundamentals of DE algorithm

Differential evolution algorithm is an efficient evolutionary optimization technique relating to EAs. Since DE was first introduced by Storn and Price (1997), it has been successfully applied to variety of optimization problems, such as linear, non-linear, differentiable, non-convex, multi-modal functions and so on. DE is a stochastic direct search method based on the idea of evolution of population. This section will briefly illustrate the working procedure of DE.

2.1 Initialization

Reasonable parameter setup may be conductive to improving the performance of DE. The parameters of DE including population size M, mutation factor F, crossover rate CR and the maximal iteration number K _m. To find the optimal solution x ^* in the D-dimensional space Ω, an active population P with variables x _i,j (i = 1, . . . , M, j = 1, . . . , D) is initialized randomly from a uniform probability distribution range from to, where, $\underline{x}$ and $\bar{x}$ are the lower and upper bound of the search space, respectively. Population size M ranging from M > D to M < 10D in other literature, in general, M is depends on the problem and the dimension of the problem. Additionally, the value of crossover rate CR is another problem-dependent control parameter. For separable problem lower CR is suggested to use, while higher CR is often used to non-separable problem.

2.2 Mutation

A new vector is created by means of the mutation operator. At each iteration, for every target vector x _i, three other parameter vector x _a, x _b and x _c are chosen from the population P randomly to produce the mutant vector v _i according to the following equation: $v_{i, g + 1} = x_{a, g} + F (x_{b, g + 1} - x_{c, g + 1})$ (2)

In the above equation, mutation factor F is used to control the amplification of the difference vector (x _b,g+1 - x _c,g+1) so as to avoid search stagnation. If one element of a mutant vector goes off the bounds, this element is needed to be set to bound value.

Aimed at increasing the algorithm’s ability to adapt to specific problems, different variants of DE have been proposed in the last few years, such as DE/rand/1/exp, DE/best/1/bin, DE/current-to-best/1/bin, DE/best/2/ bin, and so on.

2.3 Crossover

Following the mutation operation, crossover is applied to the population to form the trial vector u _i,j. The crossover between target vector x _i,j and mutant vector v _i,j is defined component-wise as: $u_{ij} = {\begin{matrix} v_{ij} if rand (0, 1)_{j} \leq CR, or j = ran d_{j} \\ x_{ij} otherwise \end{matrix}$ (3) where rand (0, 1) _j is a uniform random real variable from range [0, 1] and rand _j is a random integer variable drawn from [1, 12]. The rand _j can ensure that at least one element of trial vector u _i,j is selected from the mutate vector v _i,j.

Note that the above crossover mechanism is called Binomial crossover, which is well suited for exploration. There has been plenty of crossover strategies were proposed since 1995. Due to its strong ability in exploitation, exponential crossover is widely used to deal with separable functions. The mode can be described for clarity by means of the pseudocode.

k = 1 ; j = randint (1, n) ;

do {u_ij = v_ij ;

k = k + 1 ; j = (j + 1) % n ;}

while (k≤ n && rand (0, 1) _k < CR) ;

while (k ≤ n )

{u_ij = x_ij ;

k = k + 1 ; j = (j + 1) % n ;}

The mathematical description of the exponential crossover is as follows: $u_{i j} = \{\begin{cases} v_{i j} if j = {〈n〉}_{D}, {〈n + 1〉}_{D}, \dots, {〈n + L - 1〉}_{D} \\ x_{i j} otherwise \end{cases}$ (4) where the acute brackets 〈〉 _D denote a modulo function with modulus D. The integer n is a random number and L is an integer drawn from [1, D]. Hence, picking a random starting point, the u _i,j is generated by copying the v _i,j from L=0 to D, if the random number rand (0, 1) _j less than or equal to CR. The remaining parameters are copied from the x _i,j.

2.4 Selection

In the selection operation, a greedy selection scheme is used to pick out the individuals who have better objective function value. The entire algorithm terminates when the maximal iteration number K _m is reached.

3 The proposed SE-DE algorithm

In this section, a new version of differential evolution algorithm is presented. To make the optimization technique less vulnerable to entrapment in a local optimum and to improve the algorithm’s ability to adapt to various types of problems, three new ideas are introduced to the SE-DE. The detailed features of SE-DE will be explained below:

3.1 SE-DE adopts an external archive to improve the diversity of the population.

As in traditional DE approaches, an active population P with M individuals are initialized as follows: $P = (\begin{matrix} x_{11} & x_{12} & \dots & x_{1 D} \\ x_{21} & x_{22} & \dots & x_{2 D} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{M 1} & x_{M 2} & \dots & x_{MD} \end{matrix})$ (5) where, D stands for the dimension of the question, x _i,j represents the jth dimension of the ith candidate solution in population P. Then calculate the minimal value x _j,min and the maximal value x _j,max of each dimension. Based on the above values, a new external archive P′ is produced, which can be stated as follows: $P^{'} = (\begin{matrix} y_{11} & y_{12} & \dots & y_{1 D} \\ y_{21} & y_{22} & \dots & y_{2 D} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{M 1} & y_{M 2} & \dots & y_{MD} \end{matrix})$ (6) where, y _i,j represents the jth dimension of the ith candidate solution in population P′. For any variable y _i,j in population P′, it is generated randomly in the uniformly distributed range [x _j,min - x _j,max]. DE has a prominent advantage over other types of numerical methods. $y_{ij} = x_{j, min} + rand () \times (x_{j, max} - x_{j, min})$ (7)

By contrasting the fitness values of corresponding individuals of P and P′, the first half best solutions can be selected out, then they are reassigned to population P. If that, individuals who have better fitness values will always be concentrated on the population P by iteration.

Comparing with other algorithms which adopt similar external archive, the method that put forward in this article has better performance. SE-DE produce variable y _i,j randomly in the range of [x _j,min - x _j,max]. In other words, SE-DE can obtain millions of possible candidate solutions, which expand the search space and diversify the population. Therefore, the introduction of this new external archive P′ effectively improves the quality of solutions. The external archive is similar to the archive proposed in [32].

Although other algorithms can also obtain millions of possible candidate solutions, but they need to expand the population size. However, big population size may increase computational cost in the after mutation, crossover and selection operations. So the external archive strategy in SE-DE can effectively improve the quality of candidate solutions only increases computational complexity in initialization part.

3.2 Self-adaptive parameters control strategy is used to mutation factor F

Choosing suitable value of control parameter is frequently a problem-dependent task. In traditional DE approaches, control parameter values are constant which are predefined by user via a time-consuming trial and error scheme. Recently, the design of optimization techniques using self-adaptive parameters control strategy has received a great deal of attention in literature.

In this proposed algorithm (SE-DE), a roulette wheel based selection scheme is introduced to select mutation factor F. In SE-DE, F are applied at the individual level, each individual in the population is extended with the value of F, which will be adjusted by means of evolution. Initially, the mutation factors are mapped to contiguous segments between 1/N _F to k/N _F where, N _F is the number of Fs in the set and k = 1, 2, . . . , N _F. From this set, a F valve is selected based on their selection values (PF). These selection values represent a set PF _k = k/N _PF, k = 1, 2, . . . , N _PF. The values of PF _k are updated based on success or failure of the offspring generated. Success refers to a condition when an offspring solution produces better fitness value than its parent. The success ratio which is used to update each F selection value PF _k is calculated as follows:

$P F_{k} = \frac{S F_{k}}{\sum_{i = 1}^{N_{F}} S F_{i}}$ (8) where k is the number of mutation factor and SF _k is the success ratio of kth F. The success ratio SF _k is calculated as follows: $S F_{k} = \frac{\sum_{i = 1}^{M} ns}{\sum_{i = 1}^{M} ns + \sum_{i = 1}^{M} nf}$ (9) where, M is the size of active population, ns is the number of successes and nf is the number of failures.

The success ratio can indicate promising mutation factor more preferred in subsequent generations. The utilization of self-adaptation strategy in selecting control parameter F is extremely flexible for various problems. This strategy makes individuals move away from the current point and escapes from local minima more easily than traditional DE.

3.3 Selection mechanism of crossover strategy based on the optimization factor α

As mentioned above, binomial crossover has generally performed well in exploration, which can improve the convergence speed of the algorithm. Exponential crossover has strong ability in exploitation. In SE-DE, a binomial crossover is employed for exploration in early stages of evolution. Starting from the third generation, after each iteration, the optimization factor α is needed to be calculate in the following way: $α_{k} = | \frac{best (k - 1) - best (k)}{best (k - 2) - best (k - 1)} |$ (10) where best (k), best (k - 1) and best (k - 2) are the best values of the objective function at current iteration k, at (k - 1) iteration and (k - 2) iteration earlier, respectively. α _k represents the optimization factor of the k iteration.

If the value of optimization factor α is greater than or equal to 0.3, in the next round of iteration, binomial crossover is continue used to avoid the search falls into the local optimum. Otherwise, it is think that the algorithm reaches nearby the global optimum, exponential crossover which has strong exploitation (local search) ability is adopted in the next iteration to improve the solution accuracy. It should be noted that when |best (k - 1) - best (k) | equals to 0 after the k iteration, there is no need to calculate the optimization factor α of the (k + 1) iteration, exponential crossover is used in the (k + 1) iteration.

Based on the above detail illustrations, the procedure of SE-DE algorithm can be described in Table 1.

4 Experimental results and analysis

The above three ideas can effectively improve the stability and the efficiency of traditional DE. In order to test the optimization performance of the proposed strategies, SE-DE is applied to a suite of scalable benchmark problems defined in CEC-2005. Among these selected benchmark problems, functions F1-F5 are unimodal problems. Functions F7-F14 are multimodal problems. Moreover, F7-F12 are basic functions while F13-F14 are expanded functions. A more detailed description of each function is given in [30]. Table 2 lists the features of these problems. In addition, the conventional DE [3] and other four varieties, including SADE [7], ODE [14], NDE [12] and MDE-pBX [31] are adopted to make comparison with the SE-DE on solving unconstrained optimization problems. The value set of these control parameters of the five DE approaches are all based on the recommend value in the literature. To show performance of SE-DE method in a broad context, its performance is evaluated on 30-, 50- and 100-dimensional problems, and the comparison is made based on the value of mean and standard deviation.

4.1 Thirty-dimensional problems

Independent 30 runs are performed for 14 problems. For 30-dimension problems, the population size M is set to 40, and the maximum number of iterations K _m is equal to 1000, a constant value 0.9 is used to the crossover rate. According to a uniform probability distribution, the active population P is randomly initialized within the boundaries. The means and standard deviations values of the optimization results are summarized in Table 3. Results for the reference algorithms are taken from [32]. The best entries are marked in boldface.

It is evident from Table 3 that the proposed SE-DE algorithm is able to give better results for most benchmark functions than other DEs. Using mean value as the measure, SE-DE obtained best value for 9 functions among 14 functions. Using standard deviation value as the measure, almost all considered algorithms can easily find global optima except MDE-pBX, nevertheless, SE-DE and ODE perform better than the other four DE approaches according to the criteria ąřstandard deviationąś. For F1, F2, F3, F4, F6, F9, and F12, SE-DE outperforms in both mean value and standard deviation value. In addition, SE-DE obtains the second optimal solution for function F5 being exceeded by ODE alone. However, even SE-DE can provide better standard deviations for functions F7 and F8, the corresponding means are obviously bad, so, SE-DE performs poor convergence for functions F7 and F8.

4.2 Fifty-dimensional problems

The same 14 problems are used to verify the property of the proposed algorithm. The dimensions of the problems are change to 50, the corresponding maximum number of iterations K _m is set to 1000, and the population size M is kept to 40. The means and standard deviations of the best solutions averaged over 30 independent experiments are reported in Table 4.

Results reported in Table 4 show that, SE-DE can yield both better means and better standard deviations than those from the other five DE approaches for functions F2, F3, F4, F12 and F13 with N = 50. Furthermore, SE-DE acquires the second best solution for function F8 being superior by NDE alone, and it also acquires the second best performance for function F14 being superior by MDE-pBX alone. In addition, for F9 and F11, SE-DE obtains the optimal solution in the criteria "mean", but performs bad in the criteria "standard deviation". With respect to function F6 and F7, SE-DE performs worse when compared to the other approaches, as it may stick into the local optimum, there are only several times of independent experiment can get the global optimum. In a word, SE-DE demonstrates obvious superiority than the other algorithms on solving unconstrained problems.

4.3 One hundred-dimensional problems

Lastly, in order to verify the scalability of SE-DE, eight functions, F1, F2, F4, F5, F6, F9, F12 and F13, which can extend their dimensional to 100 are utilized. Accordingly, the population sizes and the maximum iteration number are set to 60 and 4000, respectively. For each case 30 independent experiments are implemented, and the means and standard deviations are listed in Table 5.

Similarity to lower-dimensional versions described in previous sections, SE-DE outperforms the other algorithms for functions F9, F12 and F13 when adopt mean value as the measure. Based on standard deviation value measure, SE-DE can provide better values than the other algorithms for functions F2, F12 and F13. Furthermore, for functions F1, F2, F4 and F5, SE-DE acquires the second best solution among other algorithms. On the other hand, SADE can yield better solutions for functions F5 and F6, while MDE-pBX can produce better solutions for functions F2 and F4. In conclusion, the proposed SE-DE algorithm is better than, or at least comparable to, the algorithms from other literature, on solving unconstrained optimization with high dimensions.

Figure 1 shows the average best fitness curves for the six DE algorithms over 30 independent runs for six selected benchmark functions with N = 100. As it can be seen clearly in Fig. 1, SE-DE outstanding than the five reference algorithms in fast convergence, beyond that, SE-DE shows strong capacity in exploration and exploitation.

4.4 Execution time analysis

Table 6 and Table 7 record the average execution time in each case with N = 30 and N = 50.

According to the above two tables, it can be seen that, for most cases, SE-DE is more time consuming than the reference algorithms, and the computational overhead increases with the increase of problem dimensionality. It should be noticed that, even if the external archive strategy can ensure the quality of the population, it brings a large execution time for a given amount of evolutionary generations. Therefore, eliminate or relax computational burden brought by the external structure is the key emphasis in our future work.

5 Conclusions

As a straightforward strategy, DE is a very powerful algorithm and it is easy to use. However, spreading the individuals in the searching space as much as possible and speeding up the convergence to the global optima is pivotal. Choosing proper control parameters for DE can effectively improve performance of the algorithm.

In this context, a new method SE-DE is presented. The proposed algorithm improved the traditional DE in three aspects. Firstly, an external archive is utilized to increase diversity of the population, which can effectively expand the searching space. Then, a roulette wheel based F selection scheme is introduced to adaptively select the mutation factor (F). Finally, an optimization factor α is proposed to select the crossover strategy, the combination of binomial and exponential crossover can effectively balance the exploration and exploitation ability of the algorithm.

Compared with five algorithms in other literature, the proposed SE-DE method is less susceptible to premature convergence and less likely to be stuck in local optima. For most majority of the benchmark functions of CEC 2005, SE-DE reflects superior features both in high quality of the solution and strong robustness for various classes of problems. As a whole, the proposed algorithm is competitive with other forms of optimization algorithms.

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