Self Balanced Differential Evolution

Differential Evolution (DE) is a well known and simple population based probabilistic approach for global optimization. It has reportedly outperformed a few Evolutionary Algorithms (EAs) and other search heuristics like the Particle Swarm Optimization (PSO) when tested over both benchmark and real world problems. But, DE, like other probabilistic optimization algorithms, sometimes behave prematurely in convergence. Therefore, in order to avoid stagnation while keeping a good convergence speed for DE, two modifications are proposed: one is the introduction of a new control parameter, Cognitive Learning Factor (CLF) and the other is dynamic setting of scale factor. Both modifications are proposed in mutation process of DE. Cognitive learning is a powerful mechanism that adjust the current position of individuals by a means of some specified knowledge. The proposed strategy, named as Self Balanced Differential Evolution (SBDE), balances the exploration and exploitation capability of the DE. To prove efficiency and efficacy of SBDE, it is tested over 30 benchmark optimization problems and compared the results with the basic DE and advanced variants of DE namely, SFLSDE, OBDE and jDE. Further, a real-world optimization problem, namely, Spread Spectrum Radar Polly phase Code Design, is solved to show the wide applicability of the SBDE.     

Distributed mixed variant differential evolution algorithms for unconstrained global optimization

This paper proposes a novel distributed differential evolution algorithm called Distributed Mixed Variant Differential Evolution (dmvDE). To alleviate the time consuming trial-and-error selection of appropriate Differential Evolution (DE) variant to solve a given optimization problem, dmvDE proposes to mix effective DE variants with diverse characteristics in a distributed framework. The novelty of dmvDEs lies in mixing different DE variants in an island based distributed framework. The 19 dmvDE algorithms, discussed in this paper, constitute various proportions and combinations of four DE variants (DE/rand/1/bin, DE/rand/2/bin, DE/best/2/bin and DE/rand-to-best/1/bin) as subpopulations with each variant evolving independently but also exchanging information amongst others to co-operatively enhance the efficacy of the distributed DE as a whole. The dmvDE algorithms have been run on a set of test problems and compared to the distributed versions of the constituent DE variants. Simulation results show that dmvDEs display a consistent overall improvement in performance than that of distributed DEs. The best of dmvDE algorithms has also been benchmarked against five distributed differential evolution algorithms. Simulation results reiterate the superior performance of the mixing of the DE variants in a distributed frame work. The best of dmvDE algorithms outperforms, on average, all five algorithms considered.     

Fitness based Differential Evolution

Differential Evolution (DE) is a well known and simple population based probabilistic approach for global optimization. It has reportedly outperformed a few Evolutionary Algorithms and other search heuristics like Particle Swarm Optimization when tested over both benchmark and real world problems. But, DE, like other probabilistic optimization algorithms, sometimes exhibits premature convergence and stagnates at suboptimal point. In order to avoid stagnation behavior while maintaining a good convergence speed, a new position update process is introduced, named fitness based position update process in DE. In the proposed strategy, position of the solutions are updated in two phases. In the first phase all the solutions update their positions using the basic DE and in the second phase, all the solutions update their positions based on their fitness. In this way, a better solution participates more times in the position update process. The position update equation is inspired from the Artificial Bee Colony algorithm. The proposed strategy is named as Fitness Based Differential Evolution ( $FBDE$ ). To prove efficiency and efficacy of $FBDE$ , it is tested over 22 benchmark optimization problems. A comparative analysis has also been carried out among proposed FBDE, basic DE, Simulated Annealing Differential Evolution and Scale Factor Local Search Differential Evolution. Further, $FBDE$ is also applied to solve a well known electrical engineering problem called Model Order Reduction problem for Single Input Single Output Systems.     

A novel differential evolution algorithm for binary optimization

Differential evolution (DE) is one of the most powerful stochastic search methods which was introduced originally for continuous optimization. In this sense, it is of low efficiency in dealing with discrete problems. In this paper we try to cover this deficiency through introducing a new version of DE algorithm, particularly designed for binary optimization. It is well-known that in its original form, DE maintains a differential mutation, a crossover and a selection operator for optimizing non-linear continuous functions. Therefore, developing the new binary version of DE algorithm, calls for introducing operators having the major characteristics of the original ones and being respondent to the structure of binary optimization problems. Using a measure of dissimilarity between binary vectors, we propose a differential mutation operator that works in continuous space while its consequence is used in the construction of the complete solution in binary space. This approach essentially enables us to utilize the structural knowledge of the problem through heuristic procedures, during the construction of the new solution. To verify effectiveness of our approach, we choose the uncapacitated facility location problem (UFLP)—one of the most frequently encountered binary optimization problems—and solve benchmark suites collected from OR-Library. Extensive computational experiments are carried out to find out the behavior of our algorithm under various setting of the control parameters and also to measure how well it competes with other state of the art binary optimization algorithms. Beside UFLP, we also investigate the suitably of our approach for optimizing numerical functions. We select a number of well-known functions on which we compare the performance of our approach with different binary optimization algorithms. Results testify that our approach is very efficient and can be regarded as a promising method for solving wide class of binary optimization problems.     

Boolean autoencoders and hypercube clustering complexity

We introduce and study the properties of Boolean autoencoder circuits. In particular, we show that the Boolean autoencoder circuit problem is equivalent to a clustering problem on the hypercube. We show that clustering m binary vectors on the n-dimensional hypercube into k clusters is NP-hard, as soon as the number of clusters scales like ${m^\epsilon (\epsilon >0 )}$ , and thus the general Boolean autoencoder problem is also NP-hard. We prove that the linear Boolean autoencoder circuit problem is also NP-hard, and so are several related problems such as: subspace identification over finite fields, linear regression over finite fields, even/odd set intersections, and parity circuits. The emerging picture is that autoencoder optimization is NP-hard in the general case, with a few notable exceptions including the linear cases over infinite fields or the Boolean case with fixed size hidden layer. However learning can be tackled by approximate algorithms, including alternate optimization, suggesting a new class of learning algorithms for deep networks, including deep networks of threshold gates or artificial neurons.     

基于种群自适应调整的多目标差分进化算法

为提高已有多目标进化算法在求解复杂多目标优化问题上的收敛性和解集分布性,提出一种基于种群自适应调整的多目标差分进化算法。该算法设计一个种群扩增策略,它在决策空间生成一些新个体帮助搜索更优的非支配解;设计了一个种群收缩策略,它依据对非支配解集的贡献程度淘汰较差的个体以减少计算负荷,并预留一些空间给新的带有种群多样性的扰动个体;引入精英学习策略,防止算法陷入局部收敛。通过典型的多目标优化函数对算法进行测试验证,结果表明所提算法相对于其他算法具有明显的优势,其性能优越,能够在保证良好收敛性的同时,使获得的Pareto最优解集具有更均匀的分布性和更广的覆盖范围,尤其适合于高维复杂多目标优化问题的求解。     

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.     

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 global best artificial bee colony algorithm for global optimization

The artificial bee colony (ABC) algorithm is a relatively new optimization technique which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in the ABC algorithm regarding its solution search equation, which is good at exploration but poor at exploitation. Inspired by differential evolution (DE), we propose a modified ABC algorithm (denoted as ABC/best), which is based on that each bee searches only around the best solution of the previous iteration in order to improve the exploitation. In addition, to enhance the global convergence, when producing the initial population and scout bees, both chaotic systems and opposition-based learning method are employed. Experiments are conducted on a set of 26 benchmark functions. The results demonstrate good performance of ABC/best in solving complex numerical optimization problems when compared with two ABC based algorithms.     

Self-adaptive differential evolution incorporating a heuristic mixing of operators

A considerable number of differential evolution variants have been proposed in the last few decades. However, no variant was able to consistently perform over a wide range of test problems. In this paper, propose two novel differential evolution based algorithms are proposed for solving constrained optimization problems. Both algorithms utilize the strengths of multiple mutation and crossover operators. The appropriate mix of the mutation and crossover operators, for any given problem, is determined through an adaptive learning process. In addition, to further accelerate the convergence of the algorithm, a local search technique is applied to a few selected individuals in each generation. The resulting algorithms are named as Self-Adaptive Differential Evolution Incorporating a Heuristic Mixing of Operators. The algorithms have been tested by solving 60 constrained optimization test instances. The results showed that the proposed algorithms have a competitive, if not better, performance in comparison to the-state-of-the-art algorithms.     

Step-size Adaptation in a Multi-objective Differential Evolution Algorithm

If Differential Evolution (DE) is applied to multi-objective optimization problems, some of its features like automatic problem-specific step-size adaptation gets lost. This can be resolved by modifying the DE equations. Three strategies are proposed and compared with existing algorithms. It is shown, that the proposed strategies deliver a superior convergence and preserve the positive characteristics of DE known from solving mono-objective optimization problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global optimization based on novel heuristics,low-discrepancy sequences and genetic algorithms

In this paper a new heuristic hybrid technique for bound-constrained global optimization is proposed. We developed iterative algorithm called GLP_τS that uses genetic algorithms, LP_τ low-discrepancy sequences of points and heuristic rules to find regions of attraction when searching a global minimum of an objective function. Subsequently Nelder–Mead Simplex local search technique is used to refine the solution. The combination of the three techniques (Genetic algorithms, LP_τO Low-discrepancy search and Simplex search) provides a powerful hybrid heuristic optimization method which is tested on a number of benchmark multimodal functions with 10–150 dimensions, and the method properties – applicability, convergence, consistency and stability are discussed in detail.     

Gbest-guided artificial bee colony algorithm for numerical function optimization

Artificial bee colony (ABC) algorithm invented recently by Karaboga is a biological-inspired optimization algorithm, which has been shown to be competitive with some conventional biological-inspired algorithms, such as genetic algorithm (GA), differential evolution (DE) and particle swarm optimization (PSO). However, there is still an insufficiency in ABC algorithm regarding its solution search equation, which is good at exploration but poor at exploitation. Inspired by PSO, we propose an improved ABC algorithm called gbest-guided ABC (GABC) algorithm by incorporating the information of global best (gbest) solution into the solution search equation to improve the exploitation. The experimental results tested on a set of numerical benchmark functions show that GABC algorithm can outperform ABC algorithm in most of the experiments.     

A new differential mutation base generator for differential evolution

A new differential mutation base strategy for differential evolution (DE), namely best of random, is proposed. The best individual among several randomly chosen individuals is chosen as the differential mutation base while the other worse individuals are donors for vector differences. Hence both good diversity and fast evolution speed can be obtained in DE using the new differential mutation base. A comprehensive comparative study is carried out over a set of benchmark functions. Numerical results show that a better balance of reliability and efficiency can be obtained in differential evolution implementing the new generator of differential mutation base, especially in functions with high dimension.     

Hybridizing local search algorithms for global optimization

In this paper, we combine two types of local search algorithms for global optimization of continuous functions. In the literature, most of the hybrid algorithms are produced by combination of a global optimization algorithm with a local search algorithm and the local search is used to improve the solution quality, not to explore the search space to find independently the global optimum. The focus of this research is on some simple and efficient hybrid algorithms by combining the Nelder–Mead simplex (NM) variants and the bidirectional random optimization (BRO) methods for optimization of continuous functions. The NM explores the whole search space to find some promising areas and then the BRO local search is entered to exploit optimal solution as accurately as possible. Also a new strategy for shrinkage stage borrowed from differential evolution (DE) is incorporated in the NM variants. To examine the efficiency of proposed algorithms, those are evaluated by 25 benchmark functions designed for the special session on real-parameter optimization of CEC2005. A comparison study between the hybrid algorithms and some DE algorithms and non-parametric analysis of obtained results demonstrate that the proposed algorithms outperform most of other algorithms and their difference in most cases is statistically considerable. In a later part of the comparative experiments, a comparison of the proposed algorithms with some other evolutionary algorithms reported in the CEC2005 confirms a better performance of our proposed algorithms.     

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.     

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.     

A combined local search and integer programming approach to the traveling tournament problem

The traveling tournament problem is a well-known combinatorial optimization problem with direct applications to sport leagues scheduling, that sparked intensive algorithmic research over the last decade. With the Challenge Traveling Tournament Instances as an established benchmark, the most successful approaches to the problem use meta-heuristics like tabu search or simulated annealing, partially heavily parallelized. Integer programming based methods on the other hand are hardly able to tackle larger benchmark instances. In this work we present a hybrid approach that draws on the power of commercial integer programming solvers as well as the speed of local search heuristics. Our proposed method feeds the solution of one algorithm phase to the other one, until no further improvements can be made. The applicability of this method is demonstrated experimentally on the galaxy instance set, resulting in currently best known solutions for most of the considered instances.     

A quantum-behaved evolutionary algorithm based on the Bloch spherical search

In order to enhance the optimization ability of the quantum evolutionary algorithms, a new quantum-behaved evolutionary algorithm is proposed. In this algorithm, the search mechanism is established based on the Bloch sphere. First, the individuals are expressed by qubits described on the Bloch sphere, then the rotation axis is established by Pauli matrixes, and the evolution search is realized by rotating qubits on the Bloch sphere about the rotating axis. In order to avoid premature convergence, the mutation of individuals is achieved by the Hadamard gates. Such rotation can make the current qubit approximate the target qubit along with the great circle on the Bloch sphere, which can accelerate optimization process. Taking the function extreme value optimization as an example, the experimental results show that the proposed algorithm is obviously superior to other similar algorithms.