A harmony search algorithm for high-dimensional multimodal optimization problems

Harmony search (HS) and its variants have been found successful applications, however with poor solution accuracy and convergence performance for high-dimensional (≥200) multimodal optimization problems. The reason is mainly huge search space and multiple local minima. To tackle the problem, we present a new HS algorithm called DIHS, which is based on Dynamic-Dimensionality-Reduction-Adjustment (DDRA) and dynamic fret width (fw) strategy. The former is for avoiding generating invalid solutions and the latter is to balance global exploration and local exploitation. Theoretical analysis on the DDRA strategy for success rate of update operation is given and influence of related parameters on solution accuracy is investigated. Our experiments include comparison on solution accuracy and CPU time with seven typical HS algorithms and four widely used evolutionary algorithms (SaDE, CoDE, CMAES and CLPSO) and statistical comparison by the Wilcoxon Signed-Rank Test with the seven HS algorithms and four evolutionary algorithms. The problems in experiments include twelve multimodal and four complex uni-modal functions with high-dimensionality.Experimental results indicate that the proposed approach can provide significant improvement on solution accuracy with less CPU time in solving high-dimensional multimodal optimization problems, and the more dimensionality that the optimization problem is, the more benefits it provides.     

A two‐phase harmony search algorithm for continuous optimization

Harmony search (HS) algorithm is inspired by the music improvisation process in which a musician searches for the best harmony and continues to polish the harmony to improve its aesthetics. The efficiency of evolutionary algorithms depends on the extent of balance between diversification and intensification during the course of the search. An ideal evolutionary algorithm must have efficient exploration in the beginning and enhanced exploitation toward the end. In this paper, a two‐phase harmony search (TPHS) algorithm is proposed that attempts to strike a balance between exploration and exploitation by concentrating on diversification in the first phase using catastrophic mutation and then switches to intensification using local search in the second phase. The performance of TPHS is analyzed and compared with 4 state‐of‐the‐art HS variants on all the 30 IEEE CEC 2014 benchmark functions. The numerical results demonstrate the superiority of the proposed TPHS algorithm in terms of accuracy, particularly on multimodal functions when compared with other state‐of‐the‐art HS variants; further comparison with state‐of‐the‐art evolutionary algorithms reveals excellent performance of TPHS on composition functions. Composition functions are combined, rotated, shifted, and biased version of other unimodal and multimodal test functions and mimic the difficulties of real search spaces by providing a massive number of local optima and different shapes for different regions of the search space. The performance of the TPHS algorithm is also evaluated on a real‐life problem from the field of computer vision called camera calibration problem, ie, a 12‐dimensional highly nonlinear optimization problem with several local optima.     

基于和声搜索算法求解组合优化问题

为了能够应用和声搜索算法（HSA）求解组合优化问题,基于HAS的三种操作的离散化实现提出了一种二进制和声搜索算法(BHSA),并将BHSA用于求解著名的k-可满足性(k-SAT)问题和0-1背包问题,通过与粒子群优化（BPSO）和遗传算法（GA）的实例计算对比验证了新算法的可行性与有效性。     

A coupled particle swarm and harmony search optimization algorithm for difficult geotechnical problems

Many solutions in geotechnical problems are the result of optimization analysis. There are many practical engineering problems where the objective function is non-convex, discontinuous with the presence of multiple strong local minima, and the classical optimization methods may sometime be trapped by the local minimum during the analysis. In this paper, a coupled optimization method is proposed for such difficult cases. The mixed optimization algorithm can takes the advantage of different algorithms, and the proposed algorithm is demonstrated to be effective and efficient in solving a very complicated hydropower problem with a high level of confidence. The proposed method can further be applied to different kinds of difficult engineering problems.     

An efficient modified harmony search algorithm with intersect mutation operator and cellular local search for continuous function optimization problems

This paper proposes a modified harmony search (MHS) algorithm with an intersect mutation operator and cellular local search for continuous function optimization problems. Instead of focusing on the intelligent tuning of the parameters during the searching process, the MHS algorithm divides all harmonies in harmony memory into a better part and a worse part according to their fitness. The novel intersect mutation operation has been developed to generate new -harmony vectors. Furthermore, a cellular local search also has been developed in MHS, that helps to improve the optimization performance by exploring a huge search space in the early run phase to avoid premature, and exploiting a small region in the later run phase to refine the final solutions. To obtain better parameter settings for the proposed MHS algorithm, the impacts of the parameters are analyzed by an orthogonal test and a range analysis method. Finally, two sets of famous benchmark functions have been used to test and evaluate the performance of the proposed MHS algorithm. Functions in these benchmark sets have different characteristics so they can give a comprehensive evaluation on the performance of MHS. The experimental results show that the proposed algorithm not only performs better than those state-of-the-art HS variants but is also competitive with other famous meta-heuristic algorithms in terms of the solution accuracy and efficiency.     

A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice

Most engineering optimization algorithms are based on numerical linear and nonlinear programming methods that require substantial gradient information and usually seek to improve the solution in the neighborhood of a starting point. These algorithms, however, reveal a limited approach to complicated real-world optimization problems. If there is more than one local optimum in the problem, the result may depend on the selection of an initial point, and the obtained optimal solution may not necessarily be the global optimum. This paper describes a new harmony search (HS) meta-heuristic algorithm-based approach for engineering optimization problems with continuous design variables. This recently developed HS algorithm is conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. Various engineering optimization problems, including mathematical function minimization and structural engineering optimization problems, are presented to demonstrate the effectiveness and robustness of the HS algorithm. The results indicate that the proposed approach is a powerful search and optimization technique that may yield better solutions to engineering problems than those obtained using current algorithms.     

A novel global harmony search algorithm for reliability problems

Inspired by the swarm intelligence of particle swarm, a novel global harmony search algorithm (NGHS) is proposed to solve reliability problems in this paper. The proposed algorithm includes two important operations: position updating and genetic mutation with a small probability. The former enables the worst harmony of harmony memory to move to the global best harmony rapidly in each iteration, and the latter can effectively prevent the NGHS from trapping into the local optimum. Based on a large number of experiments, the proposed algorithm has demonstrated stronger capacity of space exploration than most other approaches on solving reliability problems. The results show that the NGHS can be an efficient alternative for solving reliability problems.     

和声搜索-分布估计混合算法求解多目标优化问题

针对和声搜索算法不能很好地求解多目标优化问题的缺陷,提出一种多目标和声搜索—分布估计混合算法(MHS-EDA)。该算法一方面利用分布估计的采样操作对和声记忆库内进行搜索,拓宽了和声记忆库内空间;另一方面对和声记忆库外进行外部档案搜索,实现群体间信息交换,从而提高了多目标和声算法的全局搜索能力。数值实验选取六个常用测试函数,并与多目标遗传算法、多目标分布估计算法、多目标和声搜索算法进行比较,测试结果表明提出的混合算法能够有效地解决多目标优化问题。     

Modified harmony search optimization for constrained design problems

Constrained optimization is a major real-world problem. Constrained optimization problems consist of an objective function subjected to both linear and nonlinear constraints. Here a constraint handling procedure based on the fitness priority-based ranking method (FPBRM) is proposed. It is embedded into a harmony search (HS) algorithm that allows it to satisfy constraints. The HS algorithm is conceptualized using the musical process of searching for a perfect state of harmony. Here, the original heuristic HS was improved by combining both improved and global-best methods along with the FPBRM. The resulting modified harmony search (MHS) was then compared with the original HS technique and other optimization methods for several test problems.     

A hybrid OBL-based firefly algorithm with symbiotic organisms search algorithm for solving continuous optimization problems

The metaheuristic optimization algorithms are relatively new optimization algorithms introduced to solve optimization problems in recent years. For example, the firefly algorithm (FA) is one of the metaheuristic algorithms inspired by the fireflies' flashing behavior. However, its weakness in terms of exploration and early convergence has been pointed out. In this paper, two approaches were proposed to improve the FA. In the first proposed approach, a new improved opposition-based learning FA (IOFA) method was presented to accelerate the convergence and improve the FA's exploration capability. In the second proposed approach, a symbiotic organisms search (SOS) algorithm improved the exploration and exploitation of the first approach; two new parameters set these two goals, and the second approach was named IOFASOS. The purpose of the second method is that in the process of the SOS algorithm, the whole population is effective in the IOFA method to find solutions in the early stages of implementation, and with each iteration, fewer solutions are affected in the population. The experiments on 24 standard benchmark functions were conducted, and the first proposed approach showed a better performance in the small and medium dimensions and exhibited a relatively moderate performance in the higher dimensions. In contrast, the second proposed approach was better in increasing dimensions. In general, the empirical results showed that the two new approaches outperform other algorithms in most mathematical benchmarking functions. Thus, The IOFASOS model has more efficient solutions.

     

一种新的多目标改进和声搜索优化算法

针对标准和声搜索算法存在收敛不稳定及不能用于多目标优化问题的缺陷,通过引入交叉算子、自适应记忆内搜索概率和调节概率,改进了传统的和声搜索算法;根据Pareto支配关系,结合算法和声记忆库内信息完全共享的特性,提出了基于动态Pareto最优前沿的能够求解多目标优化问题的多目标改进和声搜索算法。通过几个典型函数的仿真测试表明,提出的算法能够高效稳定地收敛于Pareto最优前沿,获得分布均匀的Pareto解集。     

An improved global-best harmony search algorithm for faster optimization

In this paper, an improved global-best harmony search algorithm, named IGHS, is proposed. In the IGHS algorithm, initialization based on opposition-based learning for improving the solution quality of the initial harmony memory, a new improvisation scheme based on differential evolution for enhancing the local search ability, a modified random consideration based on artificial bee colony algorithm for reducing randomness of the global-best harmony search (GHS) algorithm, as well as two perturbation schemes for avoiding premature convergence, are integrated. In addition, two parameters of IGHS, harmony memory consideration rate and pitch adjusting rate, are dynamically updated based on a composite function composed of a linear time-varying function, a periodic function and a sign function in view of approximate periodicity of evolution in nature. Experimental results tested on twenty-eight benchmark functions indicate that IGHS is far better than basic harmony search (HS) algorithm and GHS. In further study, IGHS has also been compared with other eight well known metaheuristics. The results show that IGHS is better than or at least similar to those approaches on most of test functions.     

Hybrid parallel chaos optimization algorithm with harmony search algorithm

The application of chaotic sequences can be an interesting alternative to provide search diversity in an optimization procedure, named chaos optimization algorithm (COA). Since the chaotic motion is pseudo-randomness and chaotic sequences are sensitive to the initial conditions, the search ability of COA is usually effected by the starting values. Considering this weakness, parallel chaos optimization algorithm (PCOA) is studied in this paper. To obtain optimum solution accurately, harmony search algorithm (HSA) is integrated with PCOA to form a novel hybrid algorithm. Different chaotic maps are compared and the impacts of parallel parameter on the hybrid algorithm are discussed. Several simulation results are used to show the effective performance of the proposed hybrid algorithm.     

Block-matching algorithm based on harmony search optimization for motion estimation

Motion estimation is one of the major problems in developing video coding applications. Among all motion estimation approaches, Block-matching (BM) algorithms are the most popular methods due to their effectiveness and simplicity for both software and hardware implementations. A BM approach assumes that the movement of pixels within a defined region of the current frame can be modeled as a translation of pixels contained in the previous frame. In this procedure, the motion vector is obtained by minimizing a certain matching metric that is produced for the current frame over a determined search window from the previous frame. Unfortunately, the evaluation of such matching measurement is computationally expensive and represents the most consuming operation in the BM process. Therefore, BM motion estimation can be viewed as an optimization problem whose goal is to find the best-matching block within a search space. The simplest available BM method is the Full Search Algorithm (FSA) which finds the most accurate motion vector through an exhaustive computation of all the elements of the search space. Recently, several fast BM algorithms have been proposed to reduce the search positions by calculating only a fixed subset of motion vectors despite lowering its accuracy. On the other hand, the Harmony Search (HS) algorithm is a population-based optimization method that is inspired by the music improvisation process in which a musician searches for harmony and continues to polish the pitches to obtain a better harmony. In this paper, a new BM algorithm that combines HS with a fitness approximation model is proposed. The approach uses motion vectors belonging to the search window as potential solutions. A fitness function evaluates the matching quality of each motion vector candidate. In order to save computational time, the approach incorporates a fitness calculation strategy to decide which motion vectors can be only estimated or actually evaluated. Guided by the values of such fitness calculation strategy, the set of motion vectors is evolved through HS operators until the best possible motion vector is identified. The proposed method has been compared to other BM algorithms in terms of velocity and coding quality. Experimental results demonstrate that the proposed algorithm exhibits the best balance between coding efficiency and computational complexity.     

A stochastic local search algorithm for constrained continuous global optimization

This paper presents a new stochastic local search algorithm known as feasible–infeasible search procedure (FISP) for constrained continuous global optimization. The proposed procedure uses metaheuristic strategies for combinatorial optimization as well as combined strategies for exploring continuous spaces, which are applied to an efficient process in increasingly refined neighborhoods of current points. We show effectiveness and efficiency of the proposed procedure on a standard set of 13 well‐known test problems. Furthermore, we compare the performance of FISP with SNOPT (sparse nonlinear optimizer) and with few successful existing stochastic algorithms on the same set of test problems.     

改进和声搜索算法及其在连续函数优化中的应用

针对一般和声搜索(HS)算法在求解连续函数优化问题时存在的困难,提出一种改进的多样化和声搜索(IDHS)算法。该算法借鉴模拟退火算法的思想对参数的更新方式作出调整,并且限制保存在和声记忆矩阵中的一致和声的数量以增加解的多样性。数值仿真结果表明,与其他几种传统的和声搜索算法相比,该方法进一步提高了计算精度和收敛速度,以及全局寻优能力。     

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.     

A parallel boundary search particle swarm optimization algorithm for constrained optimization problems

During the past decade, considerable research has been conducted on constrained optimization problems (COPs) which are frequently encountered in practical engineering applications. By introducing resource limitations as constraints, the optimal solutions in COPs are generally located on boundaries of feasible design space, which leads to search difficulties when applying conventional optimization algorithms, especially for complex constraint problems. Even though penalty function method has been frequently used for handling the constraints, the adjustment of control parameters is often complicated and involves a trial-and-error approach. To overcome these difficulties, a modified particle swarm optimization (PSO) algorithm named parallel boundary search particle swarm optimization (PBSPSO) algorithm is proposed in this paper. Modified constrained PSO algorithm is adopted to conduct global search in one branch while Subset Constrained Boundary Narrower (SCBN) function and sequential quadratic programming (SQP) are applied to perform local boundary search in another branch. A cooperative mechanism of the two branches has been built in which locations of the particles near boundaries of constraints are selected as initial positions of local boundary search and the solutions of local boundary search will lead the global search direction to boundaries of active constraints. The cooperation behavior of the two branches effectively reinforces the optimization capability of the PSO algorithm. The optimization performance of PBSPSO algorithm is illustrated through 13 CEC06 test functions and 5 common engineering problems. The results are compared with other state-of-the-art algorithms and it is shown that the proposed algorithm possesses a competitive global search capability and is effective for constrained optimization problems in engineering applications.     

A quantum-inspired Tabu search algorithm for solving combinatorial optimization problems

In this study, we propose a novel quantum-inspired evolutionary algorithm (QEA), called quantum inspired Tabu search (QTS). QTS is based on the classical Tabu search and characteristics of quantum computation, such as superposition. The process of qubit measurement is a probability operation that increases diversification; a quantum rotation gate used to searching toward attractive regions will increase intensification. This paper will show how to implement QTS into NP-complete problems such as 0/1 knapsack problems, multiple knapsack problems and the traveling salesman problem. These problems are important to computer science, cryptography and network security. Furthermore, our experimental results on 0/1 knapsack problems are compared with those of other heuristic algorithms, such as a conventional genetic algorithm, a Tabu search algorithm and the original QEA. The final outcomes show that QTS performs much better than other heuristic algorithms without premature convergence and with more efficiency. Also on multiple knapsack problems and the traveling salesman problem QTS verify its effectiveness.