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[EQUATION] where [MATH] denotes the resultant vector, [MATH] and [MATH] are nonnegative integers, [MATH] is a coefficient, [MATH] [MATH] [MATH] [MATH] [MATH] are real numbers, and [MATH] and [MATH] mean two vectors in the original coordinate system. Eq.( 14 ) can be considered as a general form of the evolutionary oper... |
[EQUATION] Clearly, Eq.( 15 ) is equivalent to Eq.( ) and both of them are the velocity updating equation in PSO. Indeed, apart from PSO and DE, Eq.( 14 ) is also an effective way to describe the evolutionary operators in other EA paradigms such as cultural algorithm |
, artificial bee colony algorithm , fireworks algorithm , and brain storm optimization algorithm Note that the right-hand side of Eq.( 14 ) involves two parts: [MATH] and [MATH] . Since the first part is a linear operation of different vectors, it is irrelevant to the coordinate system. In terms of the second part, [MA... |
Remark 1 : Each evolutionary operator in Section can be rewritten as Eq.( 14 ). It can be found that the right-hand side of the velocity updating equation in PSO (i.e., Eq.( )) and the crossover operator in DE (i.e., Eq.( 12 )) contains the second part (i.e., [MATH] ), which suggests that these two operators may fail t... |
3.1.2 The Eigen coordinate system In this paper, the Eigen coordinate system is established by the columns of an orthogonal matrix , which comes from the Eigen decomposition of the covariance matrix |
[EQUATION] where is an orthogonal matrix, [MATH] is the transposed matrix of , and is a diagonal matrix. Each column of is an eigenvector of , and each diagonal element of is the square root of an eigenvalue of . The fundamental issue in Eq.( 16 ) is how to construct the covariance matrix . In general, the coveriance m... |
Next, we will discuss how to construct an evolutionary operator in the Eigen coordinate system. It contains three steps. Firstly, [MATH] is applied to transform the vectors in the original coordinate system into the Eigen coordinate system. Subsequently, these vectors in the Eigen coordinate system are combined with th... |
[EQUATION] where [MATH] denotes the resultant vector. By comparing Eq.( 17 ) with Eq.( 14 ), it can be seen that: if we replace [MATH] with [MATH] on the right-hand side of Eq.( 14 ), then the evolutionary operator in the original coordinate system is transformed into the corresponding evolutionary operator in the Eige... |
Remark 2: PSO’s velocity updating equation (i.e., Eq.( )) and DE’s crossover operator (i.e., Eq.( 12 )) in the Eigen coordinate system can be expressed as Eq.( 18 ) and Eq.( 19 ), respectively. |
[EQUATION] [EQUATION] 3.1.3 The difference between the original coordinate system and the Eigen coordinate system Next, we will investigate EA’s search behaviors in the original and Eigen coordinate systems. To make a clear explanation, we take the basic PSO as an example. For simplicity, suppose that the velocity [MAT... |
[EQUATION] [EQUATION] where [MATH] , and [MATH] and [MATH] are two uniformly distributed random numbers between [MATH] and [MATH] . By replacing [MATH] with [MATH] in Eq.( 21 ), the new position [MATH] is generated in the Eigen coordinate system: |
[EQUATION] Fig. 1 shows the difference between PSO in the original coordinate system (Fig. 1(a)) and in the Eigen coordinate system (Fig. 1(b)) for an optimization problem with variable correlation. The original coordinate system is fixed and denoted as [MATH] . As pointed out, the Eigen coordinate system is dynamicall... |
[EQUATION] [EQUATION] Since [MATH] and [MATH] are two uniformly distributed random numbers between [MATH] and [MATH] [MATH] generated in the original and Eigen coordinate systems can be any point in the rectangular areas [MATH] and [MATH] , respectively. As shown in Fig. 1, [MATH] does not contain the global optimal so... |
3.2 The related work on the Eigen coordinate system In this paper, the related work on the Eigen coordinate system are classified into two categories, according to the way of conducting the evolutionary operators. |
In the first category, the evolutionary operators are implemented only in the Eigen coordinate system. In 2001, CMA-ES was proposed which samples the offspring population according to: |
[EQUATION] where [MATH] denotes the mean vector of the search distribution at generation [MATH] [MATH] denotes the step size, [MATH] refers to a covariance matrix, [MATH] is an orthogonal matrix, [MATH] is a diagonal matrix, [MATH] is a multivariate normal distribution with zero mean and covariance matrix [MATH] , and ... |
, are designed to adapt [MATH] . In the rank- [MATH] -update strategy, a weighted combination of the [MATH] best out of [MATH] offspring is used to compute [MATH] , which is an estimator of the distribution of the current population: |
[EQUATION] where [MATH] is the [MATH] th weight coefficient, [MATH] is the population size, and [MATH] means the [MATH] th best individual among the [MATH] offspring. Thereafter, the information from both the previous and current generations are used to compute the covariance matrix [MATH] |
[EQUATION] where [MATH] is the learning rate for the rank- [MATH] -update strategy. In terms of the rank-one-update strategy, it exploits correlation between consecutive generations and constructs an evolution path to update the covariance matrix. Thus, its implementation is much more complex than the rank- [MATH] -upd... |
is proposed. In AE [MATH] , a more general approach for covariance matrix adaptation is proposed, which can be applied to ES and estimation of distribution algorithm |
. Again, in AE [MATH] , the evolutionary operators are executed only in the Eigen coordinate system. In the second category, the evolutionary operators are considered in both the Eigen and original coordinate systems at each generation of EAs. For instance, DE/eig |
and CoBiDE implement the crossover operator of DE in both the Eigen and original coordinate systems in a random manner. As a result, similar to the classical DE, one trial vector is created for one target vector. In DE/eig, all individuals from the current generation are used to compute the covariance matrix: |
[EQUATION] where [MATH] is the population size, and [MATH] and [MATH] mean the [MATH] th and [MATH] th individuals, respectively. While in CoBiDE, the [MATH] best out of the individuals from the current population are employed to update the covariance matrix: |
[EQUATION] where [MATH] [MATH] is a user-defined parameter, and [MATH] and [MATH] denote the [MATH] th and [MATH] th best individuals, respectively. From Eq.( 28 ) and Eq.( 29 ), it can be seen that only the current population distribution information is utilized to compute the covariance matrix. Very recently, a novel... |
is proposed. In CPI-DE, DE’s crossover operator is executed in both the Eigen and original coordinate systems in a deterministic manner and, consequently, two trial vectors are generated for each target vector. Thereafter, the best one among the target vector and its two trial vectors will survive into the next generat... |
[EQUATION] where [MATH] is the [MATH] th weight coefficient and [MATH] represents the [MATH] th best individual in the offspring population. In the second step, the population distribution information from the current and historical generations are used to adapt the covariance matrix: |
[EQUATION] where [MATH] is the learning rate and [MATH] is the step size. It is claimed in CPI-DE that there is no necessary to adapt the step size for DE, since DE has a different search pattern with ES. In fact, [MATH] is set to 1 in CPI-DE, which means that the covariance matrix is of equal importance at each genera... |
Our work in this paper falls into the second category. Moreover, the Eigen and original coordinate systems are adaptively tuned as the evolution proceeds. |
Proposed Approach 4.1 Motivation and general framework We continue the work on the coordinate systems and propose a novel framework named ACoS. The motivation of ACoS comes from three aspects: |
A large population can provide more information to estimate the Eigen coordinate system, compared with a small population. However, given the maximum number of fitness evaluations, the increase of the population size will lead to the decrease of the generation number, which might cause incomplete convergence of EAs. Co... |
As introduced in Section 3.2 , some researchers have recognized the importance of combining the original coordinate system with the Eigen coordinate system in the evolutionary computation research community. However, the current methods adjust these two coordinate systems in either a random way or a deterministic way. ... |
The coordinate systems play a very important role in the performance of EAs. Note, however, that in current studies the coordinate systems have been applied to enhance the performance of few EA paradigms (e.g., ES and DE). It is an interesting topic to boost the research on the coordinate systems to other EA paradigms. |
ACoS aims at addressing the above three issues. In ACoS, an additional archiving mechanism is designed to maintain the offspring not only in the current generation but also in the past several generations. Therefore, sufficient information can be obtained to estimate an appropriate Eigen coordinate system without addin... |
Algorithm 1 The framework of ACoS 1: Initialize [MATH] [MATH] , archive [MATH] , and [MATH] 2: Initialize the probability vector [MATH] |
3: while the termination criterion is not met do 4: for [MATH] to [MATH] do 5: if [MATH] then 6: Implement the mutation and crossover operators of EAs in the Eigen coordinate system to generate the [MATH] th offspring; |
7: else 8: Implement the mutation and crossover operators of EAs in the original coordinate system to generate the [MATH] th offspring; |
9: end if 10: end for 11: Evaluate the offspring population; 12: Implement the selection operator of EAs to get [MATH] 13: Update [MATH] , and [MATH] based on Section 4.2 |
14: Update [MATH] according to Section 4.3 15: [MATH] 16: end while In Algorithm 1 [MATH] denotes a uniformly distributed random number on the interval [MATH] . In the initialization process, the population [MATH] is randomly sampled from the search space, the archive is initialized to be empty, the covariance matrix [... |
Obviously, ACoS is different from the canonical EAs due to the simultaneous use and adaptive tuning of the Eigen and original coordinate systems. Next, we will introduce two core components of ACoS: the updating of the Eigen coordinate system and the updating of [MATH] |
4.2 The updating of the Eigen coordinate system The Eigen coordinate system is updated by making use of an additional archiving mechanism and the rank- [MATH] -update strategy |
The additional archiving mechanism adopts an external archive to store the offspring in both the current generation and the past several generations. It is because the search area may not change dramatically in the continuous several generations of EAs, and thus the offspring in the past several generations, other than... |
Subsequently, the rank- [MATH] -update strategy extracts the population distribution information from . The previous research has demonstrated that the rank- [MATH] -update strategy is an efficient technique to adapt the covariance matrix |
. In this paper, the size of is larger than that of [MATH] . Therefore, the rank- [MATH] -update can benefit from this relatively larger size to get a reliable estimator of the covariance matrix. Before executing the rank- [MATH] -update strategy, we need to initialize the mean vector of the search distribution [MATH] ... |
[EQUATION] where [MATH] is the number of the selected solutions, [MATH] denotes the [MATH] th best solution out of (i.e., [MATH] ), and [MATH] refers to the [MATH] th weight coefficient computed as: |
[EQUATION] Afterward, an estimator of the current population distribution [MATH] is obtained by: [EQUATION] Finally, the covariance matrix [MATH] is updated by making use of the cumulative population distribution information: |
[EQUATION] where [MATH] denotes the learning rate, [MATH] is the variance effective selection mass, and [MATH] is the dimension of the search space. |
After [MATH] is obtained, an Eigen decomposition is performed on [MATH] according to Eq.( 16 ) to produce the orthogonal matrix [MATH] , the columns of which form the Eigen coordinate system. |
4.3 The updating of the probability vector The probability vector [MATH] determines the selection ratio of each coordinate system for each individual. Since there is no priori knowledge about the characteristics of the function landscapes, the Eigen and original coordinate systems are considered to be of equal importan... |
In this paper, we collect the information including which coordinate system is used to generate the offspring and how about the quality of the generated offspring. It is easy to identify which coordinate system is used to produce the offspring. However, how to measure the quality of the offspring is usually dependent o... |
Eigen coordinate system is better : the Eigen coordinate system is used to generate the offspring and the offspring performs better; |
Eigen coordinate system is worse : the Eigen coordinate system is used to generate the offspring but the offspring performs worse; |
Original coordinate system is better : the original coordinate system is used to generate the offspring and the offspring performs better; |
Original coordinate system is worse : the original coordinate system is used to generate the offspring but the offspring performs worse. |
Algorithm 2 The updating of the probability vector 1: switch (the case of the collected information from the offspring) 2: case Eigen coordinate system is better |
3: [MATH] 4: case Eigen coordinate system is worse 5: [MATH] 6: case original coordinate system is better 7: [MATH] 8: case original coordinate system is worse |
9: [MATH] 10: end switch These four cases have been considered fully in Algorithm 2 to adaptively update [MATH] . The main principle behind Algorithm 2 is the “use it or lose it” rule: if one coordinate system is used to generate the offspring and the offspring performs better, the selection ratio for this coordinate s... |
In the case of Eigen coordinate system is better , a reward [MATH] is added into [MATH] [MATH] denotes a reward function defined as [MATH] . In this reward function, [MATH] is a constriction factor to clamp the reward value into [MATH] , and [MATH] is a concave function whose value decreases from 1 to 0 when the variab... |
In the case of Eigen coordinate system is worse , a punishment [MATH] is added into [MATH] In [MATH] [MATH] is a punishment coefficient on the interval [MATH] . Therefore, [MATH] is smaller than [MATH] , which implies that the case Eigen coordinate system is worse has less influence on [MATH] than the case Eigen coordi... |
In the case of original coordinate system is better , the selection ratio of the original coordinate system will increase and, therefore, [MATH] will decrease. The reduced value is equal to [MATH] |
In the case of original coordinate system is worse , the selection ratio of the original coordinate system will decrease and [MATH] thus will increase. The increased value is equal to [MATH] |
4.4 The application of ACoS in PSO and DE ACoS has a simple structure and can be easily applied to various EAs. For a specific EA, if it is under the framework of ACoS, it will dynamically select one of the Eigen and original coordinate systems according to [MATH] to generate the offspring. Since the updating of the Ei... |
For PSO, the offspring are generated via the velocity updating equation and the position updating equation. These two equations in the original coordinate system have been given in Eq.( ) and Eq.( ), respectively. According to Section , Eq.( ) is irrelevant to the coordinate systems. With respect to Eq.( ), it depends ... |
For DE, the offspring are produced through the mutation and crossover operators. These two operators in the original coordinate system have been given in Eqs.( )-( 10 ) and Eq.( ), respectively. In fact, the mutation operator is independent of the coordinate systems, while the crossover operator relies on the coordinat... |
4.5 Characteristics of ACoS Next, we compare ACoS with other related work introduced in Section . Compared with CMA-ES which samples all the individuals in the Eigen coordinate system, ACoS has some advantages listed as follows: |
It makes use of both the Eigen and original coordinate systems. The Eigen coordinate system enables EAs to identify the modality of the fitness landscape and enhance the search efficiency, while the original coordinate system can maintain the superiority of the original EAs. |
The updating of the Eigen coordinate system in ACoS is simpler. ACoS eliminates the much more complex rank-one-update strategy and only adopts the rank- [MATH] -update strategy to estimate the Eigen coordinate system. In addition, an additional archiving mechanism with negligible computational cost is designed to impro... |
ACoS can be readily applied to other EAs. This can be attributed to the fact that the step-size control, which plays a very important role in CMA-ES, can be ignored in many other EAs due to their different search patterns with CMA-ES. |
Compared with DE/eig, CoBiDE and CPI-DE which focus on enhancing DE’s performance, ACoS has the following advantages: ACoS is designed to improve the performance of not only DE but also other EAs. |
To update the Eigen coordinate system, DE/eig and CoBiDE only utilize the current population distribution information, therefore the established Eigen coordinate system might be inappropriate due to insufficient information. In CPI-DE and ACoS, the cumulative population distribution information is used to update the Ei... |
Although both the Eigen and original coordinate systems are utilized in DE/eig, CoBiDE, CPI-DE, and ACoS, DE/eig and CoBiDE adjust these two coordinate systems in a random manner which ignores the feedback information from the evolutionary search and, therefore, is not well suited for different kinds of fitness landsca... |
Experimental Study In this section, our experiments were conducted on 30 test functions with 30 dimensions (30D) and 50 dimensions (50D) at IEEE CEC2014. These 30 test functions are denoted as [MATH] [MATH] , and their details can be available from |
. In general, these 30 test functions can be grouped into four classes: 1) Unimodal functions [MATH] [MATH] ; 2) Simple multimodal functions [MATH] [MATH] ; 3) Hybrid functions [MATH] [MATH] ; and 4) Composition functions [MATH] [MATH] |
In our experiments, [MATH] independent runs were carried out for each test function. A run will terminate if the maximum number of fitness evaluations (FEs) is reached, which was recommended to be [MATH] |
. At the end of a run, the function error value [MATH] was recorded, where [MATH] is the optimal solution and [MATH] denotes the searched best solution. If the function error value is less than [MATH] , it was taken as zero. The average and standard deviation of the function error values in all runs (denoted as “Mean E... |
For the sake of convenience, if a specific EA is under the framework of ACoS, the name of this EA will be modified by adding four letters “ACoS-”. For example, PSO-w under our framework is named as ACoS-PSO-w. |
5.1 ACoS for two popular PSO variants Firstly, we applied ACoS to two of the most popular PSO variants: PSO-w and PSO-cf, which have been introduced in Section 2.1 . The resultant methods are denoted as ACoS-PSO-w and ACoS-PSO-cf, respectively. |
The population size of these two PSO variants and their augmented algorithms was set to be 40 and 60 when the dimension of the search space was equal to [MATH] and [MATH] , respectively. The experimental results on [MATH] [MATH] with 30D and 50D are given in Tables I-II, where “ [MATH] ”, “ [MATH] ”, and “ [MATH] ” den... |
Important observations can be obtained from Tables I-II: In the case of [MATH] , ACoS-PSO-w and ACoS-PSO-cf have an edge over their original algorithms on 27 and 24 test functions, respectively. With respect to [MATH] , both ACoS-PSO-w and ACoS-PSO-cf achieve better performance than their original algorithms on 26 test... |
ACoS-PSO-w and ACoS-PSO-cf are never inferior to their original algorithms on any unimodal functions, hybrid functions, and composition functions, regardless of the number of the decision variables. |
ACoS is able to achieve great performance improvement toward PSO-w and PSO-cf on all the unimodal functions (i.e., [MATH] [MATH] ), five simple modal functions (i.e., [MATH] [MATH] , and [MATH] [MATH] ), four hybrid functions (i.e., [MATH] [MATH] [MATH] , and [MATH] ), and two composition functions (i.e., [MATH] and [M... |
It seems that the increase of the dimension (i.e., from [MATH] to [MATH] ) does not have a remarkable influence on the performance improvement of our framework. |
From the above observations, our framework significantly improves the performance of these two popular PSO variants, which indicates that: 1) there is a necessity to consider both the Eigen and original coordinate systems in the design of PSO variants, and 2) the adaptive scheme in ACoS is capable of effectively utiliz... |
5.2 ACoS for three state-of-the-art DE variants subsequently, we investigated the influence of ACoS on three famous DE variants: JADE, jDE, and SaDE. To ensure the comparison fair, the parameter settings of JADE, jDE, and SaDE were identical with their original papers, and remained unchanged when they were under the fr... |
As can be seen from Tables III-IV, ACoS significantly improves JADE, jDE, and SaDE on many test functions. Specifically, compared with their original algorithms, when [MATH] , ACoS-JADE, ACoS-jDE and ACoS-SaDE obtain significance on 17, 13, and 21 test functions, respectively; meanwhile in the case of [MATH] , they out... |
The above comparison confirms that ACoS is an effective framework to improve the performance of these three state-of-the-art DE variants, which verifies the necessity to consider both the Eigen and original coordinate systems in an adaptive fashion when designing DE. Two convergence graphs are given in Fig. 3 for the p... |
5.3 Comparison between ACoS with other Eigen coordinate system based methods The aim of this subsection is to compare ACoS with other Eigen coordinate system based methods: CoBiDE, DE/eig, and CPI-DE. Due to its outstanding performance, JADE was selected as the instance algorithm. Afterward, we applied ACoS, CoBiDE, DE... |
As shown in Table V, ACoS-JADE exhibits the best performance among the four compared methods. It outperforms CoJADE, JADE/eig, and CPI-JADE on 14, 16 and 10 test functions, respectively; while only loses on no more than two test functions. It is worth noting that ACoS-JADE is never inferior to the three competitors on ... |
5.4 The Benefit of ACoS’s Components We are interested in identifying the benefit of two distinguished components of ACoS: the additional archiving mechanism and the probability vector [MATH] . To this end, we still selected JADE as the instance algorithm and two groups of experiments were carried out. In the first gro... |
We conducted the experiments on [MATH] [MATH] with 30D. Experimental results are presented in Table VI, where “ [MATH] ”, “ [MATH] ”, and “ [MATH] ” denote that the performance of the corresponding algorithm is better than, worse than, and similar to that of ACoS-JADE, respectively. From Table VI, ACoS-JADE performs th... |
From the above discussion, one can conclude that both the additional archiving mechanism and the probability vector [MATH] play very important roles in ACoS. The former is beneficial to estimate a more reliable Eigen coordinate system, and the latter enables each individual to select a more appropriate coordinate syste... |
5.5 Evolution of the probability vector in ACoS Since the probability vector [MATH] determines the selection ratio of each coordinate system for each individual, one may be interested in investigating the dynamic changes of [MATH] during the evolutionary search. For this purpose, the mean value of [MATH] , referred as ... |
We still chose JADE as the instance algorithm and tested ACoS-JADE on three test functions with 30D from IEEE CEC2014: the unimodal function [MATH] , the simple multimodal function [MATH] , and the composite function [MATH] . These three different kinds of test functions aim to provide a comprehensive study on the chan... |
As shown in Fig. 4, there are three different types of curves. In the first type (see Fig. 4(a)), the Eigen coordinate system has a larger probability to be selected than the original coordinate system. Nevertheless, in the second type (see Fig. 4 (b)), the situation is opposite. For the third type (see Fig. 4(c)), the... |
Conclusion An adaptive framework for tuning the coordinate systems in EAs referred as ACoS has been proposed in this paper. ACoS provides a simple yet efficient approach to establish the Eigen coordinate system via an additional archiving mechanism and the rank- [MATH] -update strategy. Thereafter, it adopts a probabil... |
# Source: arxiv 1704.02191 # Title: The (1+$λ$) Evolutionary Algorithm with Self-Adjusting Mutation Rate # Sections: all # Downloaded: 2026-03-03T01:58:01.101609+00:00 |
The (1+ [MATH] ) Evolutionary Algorithm with Self-Adjusting Mutation Rate Abstract We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms in discrete search spaces. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current muta... |
We analyze how the [MATH] evolutionary algorithm with this self-adjusting mutation rate optimizes the OneMax test function. We prove that this dynamic version of the [MATH] EA finds the optimum in an expected optimization time (number of fitness evaluations) of [MATH] . This time is asymptotically smaller than the opti... |
This result shows that the new way of adjusting the mutation rate can find optimal dynamic parameter values on the fly. Since our adjustment mechanism is simpler than the ones previously used for adjusting the mutation rate and does not have parameters itself, we are optimistic that it will find other applications. |
Introduction Evolutionary algorithms (EAs) have shown a remarkable performance in a broad range of applications. However, it has often been observed that this performance depends crucially on the use of the right parameter settings. Parameter optimization and parameter control are therefore key topics in EA research. S... |
Theoretical research has contributed to our understanding of these algorithms with mathematically founded runtime analyses, many of which show how the runtime of an EA is determined by its parameters. The majority of these works investigate static parameter settings i. e., the parameters are fixed before the start of t... |
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