Complex Numerical Optimization Problems Research Articles

A triple population adaptive differential evolution

The Differential Evolution (DE) algorithm is one of the most efficient algorithms for complex numerical optimization. However, the nature of differential mutation and crossover hinders the individuals from a major change and always guides them toward their superior neighbors. There's a lack of useful directional information to help the population escape from early convergence. To solve the above problem, this paper proposes a novel Triple-population-based Adaptive Differential Evolution (TPADE) to enhance the evolutionary efficiency in solving various complex numerical optimization problems. First, a population division method with symmetrical linear reduction is designed to divide the parent population of each iteration into three sub-populations of different sizes, i.e., superior sub-population, medium sub-population, and inferior sub-population. Each sub-population adopts distinct differential mutation and crossover operators to maintain balanced search directions. Second, a superior-trial-preserved selection mechanism is proposed to screen useful directional information to guide the next iteration of evolution. Third, an effective parameter adaptation strategy is designed with the linear population size reduction strategy to avoid redundant search. Experiments are then conducted to show that the TPADE exhibits well performance compared with eleven state-of-the-art DE variants, CEC winners, and their variants on the CEC'2014, CEC'2017, and CEC'2022 benchmark suites. The C++ source code of TPADE can be downloaded from https://github.com/DoubleGong/TPADE.

Differential Evolution and Agglomerative-Clustering-Based Mutation Strategy for Complex Numerical Optimization Problems

Differential evolution is an evolutionary algorithm that is used to solve complex numerical optimization problems. Differential evolution balances exploration and exploitation to find the best genes for the objective function. However, finding this balance is a challenging task. To overcome this challenge, we propose a clustering-based mutation strategy called Agglomerative Best Cluster Differential Evolution (ABCDE). The proposed model converges in an efficient manner without being trapped in local optima. It works by clustering the population to identify similar genes and avoids local optima. The adaptive crossover rate ensures that poor-quality genes are not reintroduced into the population. The proposed ABCDE is capable of generating a population efficiently where the difference between the values of the trial vector and objective vector is even less than 1% for some benchmark functions, and hence it outperforms both classical mutation strategies and the random neighborhood mutation strategy. The optimal and fast convergence of differential evolution has potential applications in the weight optimization of artificial neural networks and in stochastic and time-constrained environments such as cloud computing.

Multi‑strategy Equilibrium Optimizer: An improved meta-heuristic tested on numerical optimization and engineering problems.

The Equilibrium Optimizer (EO) is a recently proposed intelligent optimization algorithm based on mass balance equation. It has a novel principle to deal with global optimization. However, when solving complex numerical optimization problems and engineering problems, the algorithm will get stuck into local optima and degrade accuracy. To address the issue, an improved Equilibrium Optimizer (IEO) based on multi-strategy optimization is proposed. First, Tent mapping is used to generate the initial location of the particle population, which evenly distributes the particle population and lays the foundation for diversified global search process. Moreover, nonlinear time parameter is used to update the position equation, which dynamically balances the exploration and exploitation phases of improved algorithm. Finally, Lens Opposition‑based Learning (LOBL) is introduced, which avoids local optimization by improving the population diversity of the algorithm. Simulation experiments are carried out on 23 classical functions, IEEE CEC2017 problems and IEEE CEC2019 problems, and the stability of the algorithm is further analyzed by Friedman statistical test and box plots. Experimental results show that the algorithm has good solution accuracy and robustness. Additionally, six engineering design problems are solved, and the results show that improved algorithm has high optimization efficiency achieves cost minimization.

An Enhanced Differential Evolution Algorithm with Bernstein Operator and Refracted Oppositional-Mutual Learning Strategy

Numerical optimization has been a popular research topic within various engineering applications, where differential evolution (DE) is one of the most extensively applied methods. However, it is difficult to choose appropriate control parameters and to avoid falling into local optimum and poor convergence when handling complex numerical optimization problems. To handle these problems, an improved DE (BROMLDE) with the Bernstein operator and refracted oppositional-mutual learning (ROML) is proposed, which can reduce parameter selection, converge faster, and avoid trapping in local optimum. Firstly, a new ROML strategy integrates mutual learning (ML) and refractive oppositional learning (ROL), achieving stochastic switching between ROL and ML during the population initialization and generation jumping period to balance exploration and exploitation. Meanwhile, a dynamic adjustment factor is constructed to improve the ability of the algorithm to jump out of the local optimum. Secondly, a Bernstein operator, which has no parameters setting and intrinsic parameters tuning phase, is introduced to improve convergence performance. Finally, the performance of BROMLDE is evaluated by 10 bound-constrained benchmark functions from CEC 2019 and CEC 2020, respectively. Two engineering optimization problems are utilized simultaneously. The comparative experimental results show that BROMLDE has higher global optimization capability and convergence speed on most functions and engineering problems.

An Improved Lion Swarm Optimization Algorithm With Chaotic Mutation Strategy and Boundary Mutation Strategy for Global Optimization

Lion swarm optimization (LSO) inspired by the natural division of labor among lion king, lionesses and lion cubs in a lion group, i.e., lion king guarding, lionesses hunting and lion cubs following, is a relatively novel swarm intelligent optimization technique. Due to its remarkable performance, the canonical LSO has been extensively researched. However, how to balance contradictions between the exploration and the exploitation and alleviate the premature convergence are two critical concerns that need to be dealt with in the LSO study. To address these two drawbacks, enhance the optimization performance, and broaden its application domain, an improved lion swarm optimization algorithm with chaotic mutation strategy and a boundary mutation strategy (CBLSO) is proposed in this paper. In the proposed algorithm, a chaotic mutation strategy based on chaotic cubic mapping is designed to enhance the exploration ability of the algorithm, while the boundary mutation strategy based on the concept of multilevel parallel is adopted to manage boundary constraint violations, which is beneficial for improving the exploitation ability of the algorithm. The proposed CBLSO is evaluated on 56 classic test functions and 30 CEC2014 benchmark functions, and is compared with quite a few state-of-the-art algorithms regarding often-used performance metrics. The experimental results demonstrate the superior performance of the embedded strategies on balancing the exploration and the exploitation. Furthermore, the proposed CBLSO is applied to the optimal dispatch problem of cascade hydropower stations based on a novel constraints handling method designed in this paper to validate its good practicability and performance. The experimental results of a case study on the optimal dispatch problem of China’s Wujiang cascade hydropower stations indicate that the proposed CBLSO can produce better and more reliable optimal results than the canonical LSO and other comparison algorithms with competitive speed. Thus, we can conclude that the proposed CBLSO is a competitive and effective alterative tool to solve complex numerical optimization problems and real-world optimization with complicated constraints.

An Improved Teaching–Learning-Based Optimization for Multilevel Thresholding Image Segmentation

Due to its successful application image processing or computer vision system, image segmentation plays a significant role and has become a hot research hotspot. In this paper, we propose an improved teaching–learning-based optimization (NFDR-TLBO) to segment grayscale images via multilevel thresholding. In the proposed teaching–learning-based optimization variant, the neighborhood topology is introduced into the original teaching–learning-based optimization algorithm to maintain the exploration ability of the population and the fitness-distance-ratio mechanism is introduced into the original teaching–learning-based optimization algorithm to improve its optimization performance on complex numerical optimization problems. Moreover, the experimental results on 18 typical benchmark functions with different characteristics verify the feasibility and effectiveness of the proposed algorithm. Furthermore, the proposed algorithm is used to optimize Kapur entropy function in order to find the optimal threshold for image segmentation. Finally, the experimental simulations on different benchmark images show that the proposed algorithm is effective and efficient in improving the image segmentation performance in terms of peak-signal-to-noise ratio, structure similarity index and feature similarity.

Adolescent Identity Search Algorithm (AISA): A novel metaheuristic approach for solving optimization problems

This paper proposes a novel population-based metaheuristic optimization algorithm, called Adolescent Identity Search Algorithm (AISA), which is inspired by the process of identity development/search of adolescents. AISA simulates the identity formation behavior of adolescents in the peer group. This behavior is modeled mathematically to solve optimization problems. The proposed algorithm is evaluated on thirty-nine well-known unimodal, multimodal, fixed-dimensional multimodal, composite and CEC 2019 benchmark functions to test exploration, exploitation, local optima avoidance, and convergence properties. The results are verified by an extensive comparative study with thirteen state-of-art metaheuristic algorithms. Furthermore, AISA has been used to solve IIR system identification and inverse kinematics problem of a seven Degrees Of Freedom (7-DOF) robot manipulator considered as the real-life engineering applications. The overall optimization results demonstrate that AISA possesses a strong and robust capability to produce superior performance over other competitor metaheuristic algorithms in solving various complex numerical optimization problems.

Quantum-behaved particle swarm optimization with generalized space transformation search

Nature-inspired algorithms have been proved to be very powerful methods for complex numerical optimization problems. Quantum-behaved particle swarm optimization (QPSO) is a typical member of nature-inspired algorithms, and it is a simple and effective population-based technique used in numerical optimization. Despite its efficiency and wide use, QPSO suffers from premature convergence and poor balance between exploration and exploitation in solving complex optimization problems. To address these issues, a new evolutionary technique called generalized space transformation search is proposed, and then, we introduce an improved quantum-behaved particle swarm optimization algorithm combined with this new technique in this study. The proposed generalized space transformation search is based on opposition-based learning and generalized opposition-based learning, which can not only improve the exploitation of the current search space but also strengthen the exploration of the neighborhood of the current search space. The improved quantum-behaved particle swarm optimization algorithm employs generalized space transformation search for population initialization and generation jumping. A comprehensive set of 16 well-known unconstrained benchmark functions is employed for experimental verification. The contribution of the generalized space transformation search is empirically verified, and the influence of dimensionality is also investigated. Besides, the improved quantum-behaved particle swarm optimization algorithm is also compared with some typical extensions of QPSO and several competitive meta-heuristic algorithms. Such comparisons suggest that the improved quantum-behaved particle swarm optimization algorithm may lead to finding promising solutions compared to the other algorithms.

A modified surrogate-assisted multi-swarm artificial bee colony for complex numerical optimization problems

Artificial bee colony (ABC) algorithm has been widely used in solving complex optimization due to its few control parameters and outstanding global search capability. However, ABC suffers from slow convergence rate, which limits its real-world applications. To overcome such disadvantage, this paper proposes a surrogate-assisted multi-swarm artificial bee colony (SAMSABC). The multiple swarm strategy is employed to further keep the diversity. To enhance the local exploitation capability, the orthogonal method is utilized to provide a guide vector. Moreover, to avoid wasting the computation resources, the fitness estimation strategy for artificial bee colony algorithm, as a surrogate-assistance technology, is designed. Finally, the experimental results of 20 benchmark functions verify its outstanding performance on solving complex numerical optimization problems.

Novel chaotic grouping particle swarm optimization with a dynamic regrouping strategy for solving numerical optimization tasks

Particle swarm optimization (PSO) has been widely applied to address various optimization problems, since it is easy to implement and only has a few control parameters. However, PSO often suffers from a lack of population diversity during the search process and is ineffective in balancing exploration and exploitation, especially in solving complex numerical optimization tasks. To overcome these disadvantages of PSO for complex numerical optimization problems, a new chaotic grouping PSO algorithm with a dynamic regrouping strategy (CGPSO-DRS) is proposed in this paper. The newly proposed CGPSO-DRS is based on a dynamic multiswarm PSO framework that cooperates with the chaotic grouping mechanism (CGM) and the dynamic regrouping strategy (DRS). First, the CGM divides the entire population into many subswarms via a chaotic sequence. The CGM not only improves the population grouping quality in the search process but also increases the diversity of the population. Second, the DRS is used to guide the regrouping of the population, and the population starts searching with a new configuration. Here, the DRS changes with the number of function evaluations. The DRS facilitates the effective utilization of information to balance the early exploration and the later exploitation performances. In addition, the DRS can increase the diversity of the population in the search process. Experiments have been conducted on 41 benchmark functions, and the numerical results demonstrate that the proposed CGPSO-DRS method outperforms similar population-based approaches and state-of-the-art PSO variants in accelerating the convergence speed and finding the global optimum.

A Data-Driven Approach for Simultaneous Mesh Untangling and Smoothing Using Pointer Networks

Poorly-shaped and/or inverted elements negatively affect numerical simulation accuracy and efficiency. Most current approaches consider mesh quality improvement and mesh untangling problems as numerical optimization problems and solve them using nonlinear optimization. However, these optimization-based approaches require users to set and solve complex numerical optimization problems with no guarantee that the output meshes are valid meshes with good element qualities. Therefore, this paper proposes a data-driven approach for simultaneous mesh untangling and smoothing using a Pointer network. The proposed approach generates various triangular meshes and employs a novel sequence generation algorithm to train the Pointer network and predicts competitive approximate solutions using the trained network. The strength of the proposed framework lies in its simplicity to predict the best free vertex candidate, providing high-quality output meshes, since it does not require solving complex numerical optimization for prediction. Experimental results show that the proposed framework successfully eliminates inverted elements on the meshes and improved average and worst element quality up to 85.9% and 97.8%, respectively, compared to current optimization-based methods.

Accelerated particle swarm optimization with explicit consideration of model constraints

Population based metaheuristic can benefit from explicit parallelization in order to address complex numerical optimization problems. Typical realistic problems usually involve non-linear functions and many constraints, making the identification of global optimal solutions mathematically challenging and computationally expensive. In this work, a GPU based parallelized version of the Particle Swarm Optimization technique is proposed. The main contribution is the explicit consideration of equality and inequality constraints of general type, rather than addressing only box constrained models as typically done in acceleration studies of optimization algorithms. The implementation is tested on a set of optimization problems that serve as benchmark. Speedups averaging 299x were obtained with a single GPU on a standard PC using the PyCUDA technology. Satisfactory feasibility and optimality rates are also achieved, although a standard parameterization was adopted for all the experiments. Additional results are reported on a small set of difficult problems involving bilinear non-linearities.

Structured Clanning-Based Ensemble Optimization Algorithm: A Novel Approach for Solving Complex Numerical Problems

In this paper, a novel swarm intelligence-based ensemble metaheuristic optimization algorithm, called Structured Clanning-based Ensemble Optimization, is proposed for solving complex numerical optimization problems. The proposed algorithm is inspired by the complex and diversified behaviour present within the fission-fusion-based social structure of the elephant society. The population of elephants can consist of various groups with relationship between individuals ranging from mother-child bond, bond groups, independent males, and strangers. The algorithm tries to model this individualistic behaviour to formulate an ensemble-based optimization algorithm. To test the efficiency and utility of the proposed algorithm, various benchmark functions of different geometric properties are used. The algorithm performance on these test benchmarks is compared to various state-of-the-art optimization algorithms. Experiments clearly showcase the success of the proposed algorithm in optimizing the benchmark functions to better values.

Multiobjective backtracking search algorithm: application to FSI

Fluid-structure interaction (FSI) problems play an important role in many technical applications, for instance, wind turbines, aircraft, injection systems, or pumps. Thus, the optimization of such kind of problems is of high practical importance. Optimization algorithms aim to find the best values for a system’s parameters under various conditions. In this paper, we present a new Backtracking Search Optimization Algorithm for multiobjective optimization, named BSAMO, a new evolutionary algorithm (EA) for solving real-valued numerical optimization problems. EAs are popular stochastic search algorithms that are widely used to solve nonlinear, nondifferentiable and complex numerical optimization problems. In order to test the performance of this algorithm, a well known benchmark multiobjective problem has been chosen from the literature, and for FSI optimization, using a partitioned coupling procedure. The method has been tested through a 2D plate and a 3D wing subjected to aerodynamic loads. The obtained Pareto solutions are then presented and compared to those of the Non-dominated Sorting Genetic Algorithm-II (NSGA-II). The numerical results demonstrate the efficiency of BSAMO and also its best performance in tackling real-world multiphysics problems.

Quantum multiverse optimization algorithm for optimization problems

In this paper, a new hybrid algorithm called quantum multiverse optimization (QMVO) is proposed. The proposed QMVO is based on quantum computing and multiverse optimization (MVO) algorithm. The main features of quantum theory and MVO were applied in a new algorithm to find the optimal trade-off between exploration and exploitation. QMVO algorithm depends on adopting a quantum representation of the search space and the integration of the quantum interference and operators in the multiverse optimization algorithm to obtain the optimal solution of the objective function. The performance of QMVO algorithm is evaluated by using 50 unimodal and multimodal benchmark functions. The experimental results show that the proposed algorithm has comprehensive superiority in solving complex numerical optimization problems. Also, the results show that the proposed QMVO is a promising optimization algorithm compared with other well-known and popular algorithms.

A new experiential learning electromagnetism-like mechanism for numerical optimization

The Electromagnetism-like Mechanism algorithm (EM) is a population-based search algorithm which has shown good achievements in solving various types of complex numerical optimization problems so far. To date, the study on experience-based local search mechanism is relatively limited, and there is no study in the literature to integrate experience-based features into the EM. This work introduces an experience-learning feature into the EM for the first time. A new Experiential Learning Electromagnetism-like Mechanism algorithm (ELEM) is proposed in this paper. The ELEM is integrated with two new components. The first component is the particle memory concept which allows the particles to remember the details of their past search experience. The second component is the experience analysing and decision making mechanisms which enables the particles to adjust the settings for the coming iterations. Combining the advantages of this strong exploitation strategy and the powerful exploration mechanism of the EM, the proposed ELEM strikes a good balance in providing well diversified solutions with high accuracy. The results from extensive numerical experiments carried out using 21 challenging test functions show that ELEM is able to provide very competitive solutions and significantly outperforms other optimization techniques. It can thus be concluded from the results that the proposed ELEM performs well in solving high dimensional numerical optimization problems.

A novel artificial bee colony algorithm based on modified search strategy and generalized opposition-based learning

Artificial bee colony ABC algorithm, which is inspired by the foraging behavior of honey bee swarm, is a biological-inspired optimization algorithm. It shows more effective than genetic algorithm GA, particle swarm optimization PSO and differential evolution DE. However, ABC algorithm can sometimes be slow to converge, and it is good at exploration but poor at exploitation regarding its solution search equation. To address these concerning issues, we propose a novel search strategy at the employed bees stage by introducing generalized opposition-based learning method as a search mechanism and an improved solution search equation by taking advantages of the local best solution at the onlookers stage. Both operations can balance the exploration and the exploitation for the proposed algorithm. Then, in order to enhance the global convergence, we modify dynamically frequency of perturbation at each iteration. In addition, we use a more robust calculation to determine and compare the quality of alternative solutions. Experiments are conducted on a set of 21 benchmark functions. The experimental results show that the proposed algorithm can outperform ABC-based algorithms and other significant evolutionary optimizers in solving complex numerical optimization problems.

Numerical Optimization Oriented Artificial Immune Network with Cloud-based Mutation Operator

This paper proposes an Artificial Immune Network with the Cloud-based Mutation Operator (AINetCMO) for complex numerical optimization problems. Introducing the cloud model into the mutation operator is expected to enhance the convergence of AINet-CMO. In the mutation process, the mutation step length can be dynamically adapted to the evolution of candidate antibodies by measure of the cloud model. A series of numerical simulations are arranged to compare such performance indices as solution accuracy and convergence speed between AINet-CMO and other existing algorithms. The results indicate that AINet-CMO outperforms the other three artificial immune systems, i.e., opt-aiNet, IA-AIS and AAIS-2S.

Quantum Artificial Bee Colony Algorithm for Numerical Function Optimization

Artificial Bee Colony (ABC) algorithm is a swarm intelligence based algorithm, which simulate the foraging behavior of honey bee colonies. It has been widely applied to solve the real-world problem. However, ABC has good exploration but poor exploitation abilities, and its convergence speed is also an issue in some cases. In order to overcome these issues, this paper presents a new metaheuristic algorithm called Quantum Artificial Bee Colony (QABC) algorithm for global optimization problems inspired by quantum physics concepts. Simulations are conducted on a suite of unimodal/multimodal continuous benchmark functions. The results demonstrate the good performance of the QABC algorithm in solving complex numerical optimization problems when compared with other popular algorithms.

Special issue on advances in nature inspired algorithms for complex numerical optimization problems

Optimization problems occur in almost all spheres of human activities. However, with the increasing competitiveness in the real world scenario, the mathematical models of such problems are getting more and more complex. In fact, cases may appear where it become difficult to place a problem under a specific category. In order to deal with such cases, sophisticated techniques/algorithms are needed. Nature has always been a source of inspiration for researchers to develop algorithms for the complex numerical optimization problems which are usually infeasible to solve using traditional techniques. The computational algorithms inspired from nature fall under the category of Nature Inspired Algorithms (NIA). In the past few years, researchers have given much attention to NIA which not only give a reasonably good solution but are also computationally fast. Some of the popular NIA for complex numerical optimization include genetic algorithms, simulated annealing, differential evolution, particle swarm optimization, bacterial foraging optimization, artifi-