Parameter estimation of chaotic system with time-delay: A differential evolution approach

It is an important issue to estimate parameters of chaotic system in nonlinear science. In this paper, parameter estimation problem of chaotic system with time-delay is considered. Parameters and time-delay are estimated together by treating time-delay as an additional parameter. The parameter estimation problem is converted to an multi-dimensional optimization problem. A differential evolution (DE) algorithm, which possess a powerful searching capability for finding the solutions for a given optimization problem, is applied to solve this optimization problem. Two illustrative examples are given to verify the effectiveness of the proposed method.     

A self-adaptive differential evolution algorithm based on ant system with application to estimate kinetic parameters

A self-adaptive differential evolution (DE) algorithm, in which an ant system is used to implement the self-adaptation of mutation and crossover control parameters, called ant system self-adaptive differential evolution (ASSDE), is proposed. First, the spaces of the mutation and crossover control parameters are divided into several regions and all regions are given the same initial intensity of pheromone trails. The probability of selecting parameter regions for each individual is influenced by the intensity of the region pheromone trails and its visibility. An individual will reinforce the trail of the selected regions with its own pheromone, when the offspring is better than its parent. The experiment results show that the ASSDE clearly outperforms the original DE algorithm and other existing self-adaptive DE algorithms for 16 benchmark functions. Furthermore, ASSDE is applied to develop the global kinetic model for hydropurification of terephthalic acid (HPTA), and satisfactory results are obtained.     

Parameter analysis based on stochastic model for differential evolution algorithm

In this paper, a stochastic model is used to describe and analyze the evolution process of differential evolution (DE) for numerical optimization. With the model, it illustrates how the probability distribution of the whole population is changed by mutation, selection and crossover operations. Based on the theoretical analysis, some guidelines about the parameter setting for DE are provided. In addition, numerical simulations are carried out to verify the conclusions drawn from model analysis.     

求解全局优化问题的三角进化算法

This paper presents a hybrid heuristic-triangle evolution （TE） for global optimization. It is a real coded evolutionary algorithm. As in differential evolution （DE）, TE targets each individual in current population and attempts to replace it by a new better individual. However, the way of generating new individuals is different. TE generates new individuals in a Nelder- Mead way, while the simplices used in TE is 1 or 2 dimensional. The proposed algorithm is very easy to use and efficient for global optimization problems with continuous variables. Moreover, it requires only one （explicit） control parameter. Numerical results show that the new algorithm is comparable with DE for low dimensional problems but it outperforms DE for high dimensional problems.     

Bakhvalov-Shishkin网格上求解边界层问题的差分进化算法

该文在Bakhvalov-Shishkin网格上求解具有左边界层或右边界层的对流扩散方程,并采用差分进化算法对Bakhvalov-Shishkin网格中的参数进行优化,获得了该网格上具有最优精度的数值解.对三个算例进行了数值模拟,数值结果表明:采用差分进化算法求解具有较高的计算精度和收敛性,特别是边界层的数值解精度明显优于选择固定网格参数时的结果.     

Differential evolution with generalized differentials

In this paper, we study the mutation operation of the differential evolution (DE) algorithm. In particular, we propose the differential of scaled vectors, called the ‘generalized differential’, as opposed to the existing scaled differential vector in the mutation of DE. We derive the probability distribution of points generated by the mutation with ‘generalized differentials’. We incorporate a vector-projection-based exploratory method within the new mutation scheme. The vector projection is not mandatory and it is only invoked if trial points continue to be unsuccessful. An algorithm is then proposed which implements the mutation strategy based on the difference of the scaled vectors as well as the vector projection technique. A numerical study is carried out using a set of 50 test problems, many of which are inspired by practical applications. Numerical results suggest that the new algorithm is superior to DE.     

Adaptive variable space differential evolution algorithm based on population distribution

A novel memetic computing optimization algorithms, i.e. an adaptive variable space differential evolution algorithm (AVSDE), is proposed to improve the global optimization performance. AVSDE guides most individuals search in adaptive variable space (AVS) and employs adaptive mutation and adaptive control parameter. In AVSDE, AVS is determined by population global distribution information, and DE’s operators depend on the local information of the distance and direction. The performance of AVSDE is improved by integrating the global information with the local information. In addition, different mutation strategies are selected according to the evolution stage and random probability to balance AVSDE’s exploration and exploitation abilities, and adaptive control parameter is used to further enhance the performance of AVSDE. 19 scalable benchmark functions are employed to demonstrate the performance of AVSDE. Comparing with two well-tuned conventional DE and several state $-$ of-the $-$ art parameter adaptive DE variants, the whole performance of AVSDE is the best. Finally, two experiments are conducted to analyze the effect of the key parameters on AVSDE’s performance, and the optimal parameters are obtained.     

A penalty function-based differential evolution algorithm for constrained global optimization

We propose a differential evolution-based algorithm for constrained global optimization. Although differential evolution has been used as the underlying global solver, central to our approach is the penalty function that we introduce. The adaptive nature of the penalty function makes the results of the algorithm mostly insensitive to low values of the penalty parameter. We have also demonstrated both empirically and theoretically that the high value of the penalty parameter is detrimental to convergence, specially for functions with multiple local minimizers. Hence, the penalty function can dispense with the penalty parameter. We have extensively tested our penalty function-based DE algorithm on a set of 24 benchmark test problems. Results obtained are compared with those of some recent algorithms.     

Differential evolution using a superior–inferior crossover scheme

Differential evolution (DE) is a new population-based stochastic optimization, which has difficulties in solving large-scale and multimodal optimization problems. The reason is that the population diversity decreases rapidly, which leads to the failure of the clustered individuals to reproduce better individuals. In order to improve the population diversity of DE, this paper aims to present a superior–inferior (SI) crossover scheme based on DE. Specifically, when population diversity degree is small, the SI crossover is performed to improve the search space of population. Otherwise, the superior–superior crossover is used to enhance its exploitation ability. In order to test the effectiveness of our SI scheme, we combine the SI with adaptive differential evolution (JADE), which is a recently developed DE variant for numerical optimization. In addition, the theoretical analysis of SI scheme is provided to show how the population’s diversity can be improved. In order to make the selection of parameters in our scheme more intelligently, a self-adaptive SI crossover scheme is proposed. Finally, comparative comprehensive experiments are given to illustrate the advantages of our proposed method over various DEs on a suite of 24 numerical optimization problems.     

Empirical analysis of self-adaptive differential evolution

Differential evolution (DE) is generally considered as a reliable, accurate, robust and fast optimization technique. DE has been successfully applied to solve a wide range of numerical optimization problems. However, the user is required to set the values of the control parameters of DE for each problem. Such parameter tuning is a time consuming task. In this paper, a self-adaptive DE (SDE) algorithm which eliminates the need for manual tuning of control parameters is empirically analyzed. The performance of SDE is investigated and compared with other well-known approaches. The experiments conducted show that SDE generally outperform other DE algorithms in all the benchmark functions. Moreover, the performance of SDE using the ring neighborhood topology is investigated.     

A Trigonometric Mutation Operation to Differential Evolution

Previous studies have shown that differential evolution is an efficient, effective and robust evolutionary optimization method. However, the convergence rate of differential evolution in optimizing a computationally expensive objective function still does not meet all our requirements, and attempting to speed up DE is considered necessary. In this paper, a new local search operation, trigonometric mutation, is proposed and embedded into the differential evolution algorithm. This modification enables the algorithm to get a better trade-off between the convergence rate and the robustness. Thus it can be possible to increase the convergence velocity of the differential evolution algorithm and thereby obtain an acceptable solution with a lower number of objective function evaluations. Such an improvement can be advantageous in many real-world problems where the evaluation of a candidate solution is a computationally expensive operation and consequently finding the global optimum or a good sub-optimal solution with the original differential evolution algorithm is too time-consuming, or even impossible within the time available. In this article, the mechanism of the trigonometric mutation operation is presented and analyzed. The modified differential evolution algorithm is demonstrated in cases of two well-known test functions, and is further examined with two practical training problems of neural networks. The obtained numerical simulation results are providing empirical evidences on the efficiency and effectiveness of the proposed modified differential evolution algorithm.     

Differential evolution algorithms using hybrid mutation

Differential evolution (DE) has gained a lot of attention from the global optimization research community. It has proved to be a very robust algorithm for solving non-differentiable and non-convex global optimization problems. In this paper, we propose some modifications to the original algorithm. Specifically, we use the attraction-repulsion concept of electromagnetism-like (EM) algorithm to boost the mutation operation of the original differential evolution. We carried out a numerical study using a set of 50 test problems, many of which are inspired by practical applications. Results presented show the potential of this new approach.     

正则参数求解的微分进化算法

研究了正则化方法中正则参数的求解问题,提出了利用微分进化算法获取正则参数.微分进化算法属于全局最优化算法,具有鲁棒性强、收敛速度快、计算精度高的优点.把正则参数的求解问题转化为非线性优化问题,通过保持在解空间不同区域中各个点的搜索,以最大的概率找到问题的全局最优解,同时还利用数值模拟将此方法与广义交叉原理、L-曲线准则、逆最优准则等进行了对比,数值模拟结果表明该方法具有一定的可行性和有效性.     

Scale factor local search in differential evolution

This paper proposes the scale factor local search differential evolution (SFLSDE). The SFLSDE is a differential evolution (DE) based memetic algorithm which employs, within a self-adaptive scheme, two local search algorithms. These local search algorithms aim at detecting a value of the scale factor corresponding to an offspring with a high performance, while the generation is executed. The local search algorithms thus assist in the global search and generate offspring with high performance which are subsequently supposed to promote the generation of enhanced solutions within the evolutionary framework. Despite its simplicity, the proposed algorithm seems to have very good performance on various test problems. Numerical results are shown in order to justify the use of a double local search instead of a single search. In addition, the SFLSDE has been compared with a standard DE and three other modern DE based metaheuristic for a large and varied set of test problems. Numerical results are given for relatively low and high dimensional cases. A statistical analysis of the optimization results has been included in order to compare the results in terms of final solution detected and convergence speed. The efficiency of the proposed algorithm seems to be very high especially for large scale problems and complex fitness landscapes.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

Optimal Parameter Tuning for Multiclass Support Vector Machines in Machinery Health State Estimation

The increasing demand for high reliability, safety and availability of technical systems calls for innovative maintenance strategies. The use of prognostic health management (PHM) approach where maintenance action is taken based on current and future health state of a component or system is rapidly gaining popularity in the maintenance industry. Multiclass support vector machines (MC-SVM) has been identified as a promising algorithm in PHM applications due to its high classification accuracy. However, it requires parameter tuning for each application, with the objective of minimizing the classification error. This is a single objective optimization problem which requires the use of optimization algorithms that are capable of exhaustively searching for the global optimum parameters. This work proposes the use of hybrid differential evolution (DE) and particle swarm optimization (PSO) in optimally tuning the MC-SVM parameters. DE identifies the search limit of the parameters while PSO finds the global optimum within the search limit. The feasibility of the approach is verified using bearing run-to-failure data and the results show that the proposed method significantly increases health state classification accuracy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential evolution algorithm-based parameter estimation for chaotic systems

Parameter estimation for chaotic systems is an important issue in nonlinear science and has attracted increasing interests from various research fields, which could be essentially formulated as a multidimensional optimization problem. As a novel evolutionary computation technique, differential evolution algorithm (DE) has attracted much attention and wide applications, owing to its simple concept, easy implementation and quick convergence. However, to the best of our knowledge, there is no published work on DE for estimating parameters of chaotic systems. In this paper, a DE approach is applied to estimate the parameters of Lorenz system. Numerical simulation and the comparisons demonstrate the effectiveness and robustness of DE. Moreover, the effect of population size on the optimization performances is investigated as well.     

求解非线性互补问题的微分方程方法

本文构造了一种求解非线性互补问题的微分方程方法.在一定条件下,证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法.在适当的条件下,证明了此算法产生的序列解是收敛的.本文最后给出了数值结果,该结果表明了此微分方程方法的有效性.     

A dynamic clustering based differential evolution algorithm for global optimization

A dynamic clustering based differential evolution algorithm (CDE) for global optimization is proposed to improve the performance of the differential evolution (DE) algorithm. With population evolution, CDE algorithm gradually changes from exploring promising areas at the early stages to exploiting solution with high precision at the later stages. Experiments on 28 benchmark problems, including 13 high dimensional functions, show that the new method is able to find near optimal solutions efficiently. Compared with other existing algorithms, CDE improves solution accuracy with less computational effort.