An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization

Differential evolution (DE) is one of the most powerful stochastic real parameter optimizers of current interest. In this paper, we propose a new mutation strategy, a fitness-induced parent selection scheme for the binomial crossover of DE, and a simple but effective scheme of adapting two of its most important control parameters with an objective of achieving improved performance. The new mutation operator, which we call DE/current-to-gr_best/1, is a variant of the classical DE/current-to-best/1 scheme. It uses the best of a group (whose size is q% of the population size) of randomly selected solutions from current generation to perturb the parent (target) vector, unlike DE/current-to-best/1 that always picks the best vector of the entire population to perturb the target vector. In our modified framework of recombination, a biased parent selection scheme has been incorporated by letting each mutant undergo the usual binomial crossover with one of the p top-ranked individuals from the current population and not with the target vector with the same index as used in all variants of DE. A DE variant obtained by integrating the proposed mutation, crossover, and parameter adaptation strategies with the classical DE framework (developed in 1995) is compared with two classical and four state-of-the-art adaptive DE variants over 25 standard numerical benchmarks taken from the IEEE Congress on Evolutionary Computation 2005 competition and special session on real parameter optimization. Our comparative study indicates that the proposed schemes improve the performance of DE by a large magnitude such that it becomes capable of enjoying statistical superiority over the state-of-the-art DE variants for a wide variety of test problems. Finally, we experimentally demonstrate that, if one or more of our proposed strategies are integrated with existing powerful DE variants such as jDE and JADE, their performances can also be enhanced.     

Differential Evolution with Adaptive Mutation and Parameter Control Using Lévy Probability Distribution

Differential evolution (DE) has become a very popular and effective global optimization algorithm in the area of evolutionary computation. In spite of many advantages such as conceptual simplicity, high efficiency and ease of use, DE has two main components, i.e., mutation scheme and parameter control, which significantly influence its performance. In this paper we intend to improve the performance of DE by using carefully considered strategies for both of the two components. We first design an adaptive mutation scheme, which adaptively makes use of the bias of superior individuals when generating new solutions. Although introducing such a bias is not a new idea, existing methods often use heuristic rules to control the bias. They can hardly maintain the appropriate balance between exploration and exploitation during the search process, because the preferred bias is often problem and evolution-stage dependent. Instead of using any fixed rule, a novel strategy is adopted in the new adaptive mutation scheme to adjust the bias dynamically based on the identified local fitness landscape captured by the current population. As for the other component, i.e., parameter control, we propose a mechanism by using the Lvy probability distribution to adaptively control the scale factor F of DE. For every mutation in each generation, an F i is produced from one of four different Lvy distributions according to their historical performance. With the adaptive mutation scheme and parameter control using Lvy distribution as the main components, we present a new DE variant called Lvy DE (LDE). Experimental studies were carried out on a broad range of benchmark functions in global numerical optimization. The results show that LDE is very competitive, and both of the two main components have contributed to its overall performance. The scalability of LDE is also discussed by conducting experiments on some selected benchmark functions with dimensions from 30 to 200.     

Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization

Differential evolution (DE) is an efficient and powerful population-based stochastic search technique for solving optimization problems over continuous space, which has been widely applied in many scientific and engineering fields. However, the success of DE in solving a specific problem crucially depends on appropriately choosing trial vector generation strategies and their associated control parameter values. Employing a trial-and-error scheme to search for the most suitable strategy and its associated parameter settings requires high computational costs. Moreover, at different stages of evolution, different strategies coupled with different parameter settings may be required in order to achieve the best performance. In this paper, we propose a self-adaptive DE (SaDE) algorithm, in which both trial vector generation strategies and their associated control parameter values are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, a more suitable generation strategy along with its parameter settings can be determined adaptively to match different phases of the search process/evolution. The performance of the SaDE algorithm is extensively evaluated (using codes available from P. N. Suganthan) on a suite of 26 bound-constrained numerical optimization problems and compares favorably with the conventional DE and several state-of-the-art parameter adaptive DE variants.     

A novel adaptive control strategy for decomposition-based multiobjective algorithm

Recently, evolutionary algorithm based on decomposition (MOEA/D) has been found to be very effective and efficient for solving complicated multiobjective optimization problems (MOPs). However, the selected differential evolution (DE) strategies and their parameter settings impact a lot on the performance of MOEA/D when tackling various kinds of MOPs. Therefore, in this paper, a novel adaptive control strategy is designed for a recently proposed MOEA/D with stable matching model, in which multiple DE strategies coupled with the parameter settings are adaptively conducted at different evolutionary stages and thus their advantages can be combined to further enhance the performance. By exploiting the historically successful experience, an execution probability is learned for each DE strategy to perform adaptive adjustment on the candidate solutions. The proposed adaptive strategies on operator selection and parameter settings are aimed at improving both of the convergence speed and population diversity, which are validated by our numerous experiments. When compared with several variants of MOEA/D such as MOEA/D, MOEA/D-DE, MOEA/D-DE+PSO, ENS-MOEA/D, MOEA/D-FRRMAB and MOEA/D-STM, our algorithm performs better on most of test problems.     

Enhancing differential evolution with role assignment scheme

As one of the most popular evolutionary algorithms, differential evolution (DE) has been used for solving a wide range of real-world problems. The performance of DE highly depends on the chosen mutation strategy and control parameter settings. Although the conventional trial-and-error procedure can be used to elaborately select the proper strategy and to tune the parameter values, this procedure is often very time-consuming and is not suitable for practitioners without a priori experience. To tackle this problem, DE with a novel role assignment (RA) scheme is proposed in this paper. In the RA scheme, both the fitness information and positional information of individuals are utilized to dynamically divide the population into several groups. Each group is considered as a role, which has its own mutation strategy and parameter settings and is expected to play a different role in the evolution process. To verify the performance of our approach, experiments are conducted on 23 well-known benchmark functions. Results show that our approach is better than, or at least comparable to, several state-of-the-art DE variants.     

An improved differential evolution algorithm with fitness-based adaptation of the control parameters

Differential Evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms of current interest. DE operates through the similar computational steps as employed by a standard Evolutionary Algorithm (EA). However, unlike the traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used, which makes the scheme self-organizing in this respect. Scale Factor (F) and Crossover Rate (Cr) are two very important control parameters of DE since the former regulates the step-size taken while mutating a population member in DE and the latter controls the number of search variables inherited by an offspring from its parent during recombination. This article describes a very simple yet very much effective adaptation technique for tuning both F and Cr, on the run, without any user intervention. The adaptation strategy is based on the objective function value of individuals in the DE population. Comparison with the best-known and expensive variants of DE over fourteen well-known numerical benchmarks and one real-life engineering problem reflects the superiority of proposed parameter tuning scheme in terms of accuracy, convergence speed, and robustness.     

Adaptive differential evolution algorithm with novel mutation strategies in multiple sub-populations

Differential evolution (DE) algorithm has been shown to be a very effective and efficient approach for solving global numerical optimization problems, which attracts a great attention of scientific researchers. Generally, most of DE algorithms only evolve one population by using certain kind of DE operators. However, as observed in nature, the working efficiency can be improved by using the concept of work specialization, in which the entire group should be divided into several sub-groups that are responsible for different tasks according to their capabilities. Inspired by this phenomenon, a novel adaptive multiple sub-populations based DE algorithm is designed in this paper, named MPADE, in which the parent population is split into three sub-populations based on the fitness values and then three novel DE strategies are respectively performed to take on the responsibility for either exploitation or exploration. Furthermore, a simple yet effective adaptive approach is designed for parameter adjustment in the three DE strategies and a replacement strategy is put forward to fully exploit the useful information from the trial vectors and target vectors, which enhance the optimization performance. In order to validate the effectiveness of MPADE, it is tested on 55 benchmark functions and 15 real world problems. When compared with other DE variants, MPADE performs better in most of benchmark problems and real-world problems. Moreover, the impacts of the MPADE components and their parameter sensitivity are also analyzed experimentally.     

Adaptive control of the filling velocity of thermoplastics injection molding

Injection velocity, a key variable in injection molding, was controlled via an adaptive controller using a self-tuning regulator (STR) scheme. The pole-placement design was employed first, together with the performance enhancement techniques of anti-windup estimation, feed-forward control, and cycle-to-cycle adaptation. The pole-placement design with the enhancement techniques was found experimentally to work very well over different molding conditions. However, this design was also found to be sensitive to the model mismatch. To overcome this problem, a new adaptive controller based on a generalized predictive control (GPC) principle was designed to make the controller more robust. Experiments have shown that the adaptive GPC control of injection velocity has inherently good set-point tracking performance and excellent tolerance to model structure mismatch.     

An evolving surrogate model-based differential evolution algorithm

Differential evolution (DE) is a simple and effective approach for solving numerical optimization problems. However, the performance of DE is sensitive to the choice of mutation and crossover strategies and their associated control parameters. Therefore, to achieve optimal performance, a time-consuming parameter tuning process is required. In DE, the use of different mutation and crossover strategies with different parameter settings can be appropriate during different stages of the evolution. Therefore, to achieve optimal performance using DE, various adaptation, self-adaptation, and ensemble techniques have been proposed. Recently, a classification-assisted DE algorithm was proposed to overcome trial and error parameter tuning and efficiently solve computationally expensive problems. In this paper, we present an evolving surrogate model-based differential evolution (ESMDE) method, wherein a surrogate model constructed based on the population members of the current generation is used to assist the DE algorithm in order to generate competitive offspring using the appropriate parameter setting during different stages of the evolution. As the population evolves over generations, the surrogate model also evolves over the iterations and better represents the basin of search by the DE algorithm. The proposed method employs a simple Kriging model to construct the surrogate. The performance of ESMDE is evaluated on a set of 17 bound-constrained problems. The performance of the proposed algorithm is compared to state-of-the-art self-adaptive DE algorithms: the classification-assisted DE algorithm, regression-assisted DE algorithm, and ranking-assisted DE algorithm.     

Implementing support vector regression with differential evolution to forecast motherboard shipments

In this study, we investigate the forecasting accuracy of motherboard shipments from Taiwan manufacturers. A generalized Bass diffusion model with external variables can provide better forecasting performance. We present a hybrid particle swarm optimization (HPSO) algorithm to improve the parameter estimates of the generalized Bass diffusion model. A support vector regression (SVR) model was recently used successfully to solve forecasting problems. We propose an SVR model with a differential evolution (DE) algorithm to improve forecasting accuracy. We compare our proposed model with the Bass diffusion and generalized Bass diffusion models. The SVR model with a DE algorithm outperforms the other models on both model fit and forecasting accuracy.     

A preference multi-objective optimization based on adaptive rank clone and differential evolution

Evolutionary multi-objective optimization (EMO) algorithms have been used in various real-world applications. However, most of the Pareto domination based multi-objective optimization evolutionary algorithms are not suitable for many-objective optimization. Recently, EMO algorithm incorporated decision maker’s preferences became a new trend for solving many-objective problems and showed a good performance. In this paper, we first use a new selection scheme and an adaptive rank based clone scheme to exploit the dynamic information of the online antibody population. Moreover, a special differential evolution (DE) scheme is combined with directional information by selecting parents for the DE calculation according to the ranks of individuals within a population. So the dominated solutions can learn the information of the non-dominated ones by using directional information. The proposed method has been extensively compared with two-archive algorithm, light beam search non-dominated sorting genetic algorithm II and preference rank immune memory clone selection algorithm over several benchmark multi-objective optimization problems with from two to ten objectives. The experimental results indicate that the proposed algorithm achieves competitive results.     

基于动态自适应的双档案大规模稀疏优化算法

针对传统大规模优化算法维数过高、过度稀疏、难以平衡等问题,文中提出基于动态自适应的双档案大规模稀疏优化算法,平衡维数和稀疏性对算法的影响,提高算法在解决大规模优化问题上的多样性和收敛性.首先,改变种群分数生成策略,加入自适应参数和惯性权重,增加分数的动态性,改善种群的多样性,使搜索不易陷入局部最优.然后,改变算法的环境选择策略,引入角度截断的思想,有效生成子代.同时引入双档案,分开真实决策变量和二进制决策变量,减少算法的运行时间.在大规模优化问题、稀疏优化问题及实际应用上的测试表明,文中算法保持原有的稀疏性质,同时稳定提升多样性和收敛性,具有较强的竞争性.     

Accelerating Differential Evolution Using an Adaptive Local Search

We propose a crossover-based adaptive local search (LS) operation for enhancing the performance of standard differential evolution (DE) algorithm. Incorporating LS heuristics is often very useful in designing an effective evolutionary algorithm for global optimization. However, determining a single LS length that can serve for a wide range of problems is a critical issue. We present a LS technique to solve this problem by adaptively adjusting the length of the search, using a hill-climbing heuristic. The emphasis of this paper is to demonstrate how this LS scheme can improve the performance of DE. Experimenting with a wide range of benchmark functions, we show that the proposed new version of DE, with the adaptive LS, performs better, or at least comparably, to classic DE algorithm. Performance comparisons with other LS heuristics and with some other well-known evolutionary algorithms from literature are also presented.     

A Differential Covariance Matrix Adaptation Evolutionary Algorithm for real parameter optimization

Hybridization in context to Evolutionary Computation (EC) aims at combining the operators and methodologies from different EC paradigms to form a single algorithm that may enjoy a statistically superior performance on a wide variety of optimization problems. In this article we propose an efficient hybrid evolutionary algorithm that embeds the difference vector-based mutation scheme, the crossover and the selection strategy of Differential Evolution (DE) into another recently developed global optimization algorithm known as Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES). CMA-ES is a stochastic method for real parameter (continuous domain) optimization of non-linear, non-convex functions. The algorithm includes adaptation of covariance matrix which is basically an alternative method of traditional Quasi-Newton method for optimization based on gradient method. The hybrid algorithm, referred by us as Differential Covariance Matrix Adaptation Evolutionary Algorithm (DCMA-EA), turns out to possess a better blending of the explorative and exploitative behaviors as compared to the original DE and original CMA-ES, through empirical simulations. Though CMA-ES has emerged itself as a very efficient global optimizer, its performance deteriorates when it comes to dealing with complicated fitness landscapes, especially landscapes associated with noisy, hybrid composition functions and many real world optimization problems. In order to improve the overall performance of CMA-ES, the mutation, crossover and selection operators of DE have been incorporated into CMA-ES to synthesize the hybrid algorithm DCMA-EA. We compare DCMA-EA with original DE and CMA-EA, two best known DE-variants: SaDE and JADE, and two state-of-the-art real optimizers: IPOP-CMA-ES (Restart Covariance Matrix Adaptation Evolution Strategy with increasing population size) and DMS-PSO (Dynamic Multi Swarm Particle Swarm Optimization) over a test-suite of 20 shifted, rotated, and compositional benchmark functions and also two engineering optimization problems. Our comparative study indicates that although the hybridization scheme does not impose any serious burden on DCMA-EA in terms of number of Function Evaluations (FEs), DCMA-EA still enjoys a statistically superior performance over most of the tested benchmarks and especially over the multi-modal, rotated, and compositional ones in comparison to the other algorithms considered here.     

Assembly Line Balancing Using Differential Evolution Models

There is a growing research interest on the application of evolutionary computation-based techniques in manufacturing optimization due to the fact that this field is associated with a plethora of complex combinatorial optimization problems. Differential evolution (DE), one of the latest developed evolutionary algorithms, has rarely been applied on manufacturing optimization problems (MOPs). A possible reason for the absence of DE from this research field is that DE was introduced as global optimizer over continuous spaces, while most of MOPs are of combinatorial nature with discrete decision variables. DE maintains and evolves floating-point vectors and therefore its application to MOPs that have solutions represented by permutations is not straightforward. This paper investigates the use of DE for the solution of the simple assembly line balancing problem (SALBP), a well-known NP-hard MOP. Two basic formulation types for SALBP are examined, namely type-1 and type-2: the former attempts to minimize the number of workstations required to manufacture a product in an assembly line for a given fixed cycle time; while the latter attempts to minimize the cycle time of the line for a given number of stations. Extensive experiments carried out over public benchmarks test instances estimate the performance of DE approach.     

A constraint sampling approach for multi-stage robust optimization

We propose a tractable approximation scheme for convex (not necessarily linear) multi-stage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization. We obtain a-priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi-infinite problem arising from the decision rule approximation. We demonstrate that for a suitable choice of basis functions, the approximation converges as the size of the basis and the number of sampled constraints tend to infinity. The approach yields an algorithm parameterized in the basis size, the probability of constraint violation and the confidence that this probability will not be exceeded. These three parameters serve to tune the trade-off between optimality and feasibility of the decision rules and the computational cost of the algorithm. We assess the convergence and scalability properties of our approach in the context of two inventory management problems.     

A self-adaptive differential evolution algorithm for continuous optimization problems

This paper proposes a new self-adaptive differential evolution algorithm (DE) for continuous optimization problems. The proposed self-adaptive differential evolution algorithm extends the concept of the DE/current-to-best/1 mutation strategy to allow the adaptation of the mutation parameters. The control parameters in the mutation operation are gradually self-adapted according to the feedback from the evolutionary search. Moreover, the proposed differential evolution algorithm also consists of a new local search based on the krill herd algorithm. In this study, the proposed algorithm has been evaluated and compared with the traditional DE algorithm and two other adaptive DE algorithms. The experimental results on 21 benchmark problems show that the proposed algorithm is very effective in solving complex optimization problems.     

Improving the tracking performance of mechanical systems by adaptive extended friction compensation

The paper discusses a tracking control system and shows with simulation and experimental results that extended friction models can be successfully incorporated in a computed-torque-like adaptive control scheme. The friction model used includes Coulomb, viscous, and periodic friction with sense of direction dependent parameters. To get small tracking errors, adaptation of the friction model parameters is necessary. The tracking performance is an order of magnitude better than with PD control. The robustness of the scheme for parameter inaccuracies is sufficient, owing to the adaptation, but the controller gains are limited due to stability problems caused by unmodeled dynamics.     

基于可伸缩视频编码的率失真优化编码算法*

可伸缩视频编码（SVC）可实现视频流空间、时间和信噪比的完全伸缩,但在差错信道上传输容易引起误差扩散,为此提出一种基于SVC的率失真优化编码算法。该算法在分析了差错信道下传输可伸缩视频流误差扩散失真的基础上,在率失真优化模型中引入时间和信噪比分级的误差分级参数,并根据信道状态自适应地确定宏块编码模式。该算法能够有效地抑制误码在各分层的扩散,提高了可伸缩视频流的鲁棒性。仿真结果表明,算法与以往的算法相比具有更好的抗误码性能,适合视频数据在差错信道上的传输。     

基于差分演化的自适应参数控制蚁群算法

蚁群算法存在对参数的依赖、早熟和停滞等缺点但具有与其他算法容易结合的特点,据此,将差分演化算法应用到蚁群算法的参数选取中,提出一种改进的蚁群算法。将蚁群算法的参数作为差分演化算法解空间的向量元素,在自适应地寻找蚁群算法最优参数组合的同时求解问题的最优解。改进算法对蚁群算法中的参数进行自适应调整,可避免大量盲目的测试,扩大蚁群算法的搜索空间,提高全局搜索能力。在典型的旅行商问题上进行对比实验,结果验证了改进算法的优化性能高于传统的蚁群算法。