Complex Numerical Optimization Research Articles

An improved multi-operator differential evolution with two-phase migration strategy for numerical optimization

Over the last decade, the multi-operator differential evolution (MODE) has become one of the most popular research areas in the differential evolution (DE) family. Among these MODEs, an improved multi-operator differential evolution (IMODE) has achieved success as the winner in CEC2020. Despite its good performance, the random information sharing strategy may not efficiently handle the balance between exploration and exploitation for the complex numerical optimization. To address this limitation, we propose a two-phase migration strategy (TMS) to improve the performance of IMODE, called IMODE-TMS. In the first phase, the top-ranked individuals in each sub-population are retained for exploitation, and all the bottom-ranked individuals are assigned to three sub-populations to maintain diversity. Furthermore, the second phase plays a crucial role in eliminating stagnation. When a sub-population is identified as stagnant by the stagnation indicator, the optimal individual will migrate to that sub-population following the uni-directional ring structure. This process is mainly used to increase the off-trap capability on some problems with complex fitness landscapes. IMODE-TMS is tested on the CEC2020, CEC2021 and CEC2022 benchmark functions, and seven well-known complex global trajectory optimization problems (GTOP). The experimental results show that IMODE-TMS significantly outperforms not only IMODE but also other state-of-the-art comparison algorithms.

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.

Real-Time Drift-Driving Control for an Autonomous Vehicle: Learning from Nonlinear Model Predictive Control via a Deep Neural Network

A drift-driving maneuver is a control technique used by an expert driver to control a vehicle along a sharply curved path or slippery road. This study develops a nonlinear model predictive control (NMPC) method for the autonomous vehicle to perform a drift maneuver and generate the datasets necessary for training the deep neural network(DNN)-based drift controller. In general, the NMPC method is based on numerical optimization which is difficult to run in real-time. By replacing the previously designed NMPC method with the proposed DNN-based controller, we avoid the need for complex numerical optimization of the vehicle control, thereby reducing the computational load. The performance of the developed data-driven drift controller is verified through realistic simulations that included drift scenarios. Based on the results of the simulations, the DNN-based controller showed similar tracking performance to the original nonlinear model predictive controller; moreover, the DNN-based controller can demonstrate stable computation time, which is very important for the safety critical control objective such as drift maneuver.

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.

RETRACTED: Ecological environment management system based on artificial intelligence and complex numerical optimization

Raw environmental management is an essential factor in the successful implementation of strategic power. Environmental management improves the level need to manage innovation. If get called substitution, the environmental management of change of, ecological to realization, must be sustainable economic development. With environmental and economic policies and market management, the portfolio management machine learns the ecological environment's essential benefits. Existing environmental protection measures and environmental management methods on (Field Programmable Gate Arrays) FPGA-based machine learning ecology ecological significance analysis of current development, must establish an ecological compensation mechanism by the current FPGA environmental management. Assumptions. Due to the lack of laws and regulations, an essential factor in the overall management of the ecological environment and environmental management practices. Therefore, it is necessary to establish a legal system of ecological compensation law. The establishment of ecological compensation is clear, eco-taxes, ecological compensation fund system, and ecological management and oversight point to explore current FPGA environmental management.

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.

Deep Coupled-Representation Learning for Sparse Linear Inverse Problems With Side Information

In linear inverse problems, the goal is to recover a target signal from undersampled, incomplete or noisy linear measurements. Typically, the recovery relies on complex numerical optimization methods; recent approaches perform an unfolding of a numerical algorithm into a neural network form, resulting in a substantial reduction of the computational complexity. In this letter, we consider the recovery of a target signal with the aid of a correlated signal, the so-called side information (SI), and propose a deep unfolding model that incorporates SI. The proposed model is used to learn coupled representations of correlated signals from different modalities, enabling the recovery of multi-modal data at a low computational cost. As such, our work introduces the first deep unfolding method with SI, which actually comes from a different modality. We apply our model to reconstruct near-infrared images from undersampled measurements given RGB images as SI. Experimental results demonstrate the superior performance of the proposed framework against single-modal deep learning methods that do not use SI, multi-modal deep learning designs, and optimization algorithms.

Hierarchical Learning Water Cycle Algorithm

In order to improve the global searching ability of Water Cycle Algorithm (WCA), the hierarchical learning concept is introduced and the Hierarchical Learning WCA (HLWCA) is proposed in this paper. The underlying idea of HLWCA is to divide the solutions into collections and give these collections with hierarchy differences. One of the collections has a higher hierarchy than others and utilizes an exploration-inclined updating mechanism. The solutions in this high hierarchy collection are the exemplars of other collections. The other collections are sorted according to the exemplars’ function value and the solutions in these collections actively choose whether to follow their own exemplar or not. Through different updating mechanisms of collections, the global searching ability is improved while the fast convergence and strong local search ability of WCA are retained. The proposed HLWCA is firstly experimented on IEEE CEC 2017 benchmark suite to testify its performance on complex numerical optimization tasks. Then, it is tested on four practical design benchmark problems to verify its ability of solving real-world problems. The experimental results illustrate the efficiency of the proposed algorithm.

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.

Analyzing the Scalability Performance of Crossover-First and Self-Adaptive Differential Evolution Algorithms for Complex Numerical Optimization

Two Crossover-first Differential Evolution (XDE) algorithms as well as four self-adaptive DE algorithms are compared in this study in terms of their optimization accuracy for solving a set of 15 complex, non-linear numerical optimization functions across 4 different dimensions of 10, 30, 50 and 100 optimization variables. XDE is a crossover-first variant of the original DE algorithm where XjDE is the crossover-first variant of the self-adaptive jDE algorithm. The original DE representing a fixed parameter strategy is tested against four self-adaptive algorithms, namely the DESACR, DESACRF, SDE and jDE algorithms. Although XDE is able to outperform XjDE in all 15 test problems for the lowest dimensional benchmark test setting of 10 variables, the crossover-first approach in XjDE is able to improve its performance and obtained better results over XDE in some of the test problems for the higher-dimensional benchmark test settings of 30, 50 and 100 variables. As such, this shows that there is some merit in adopting the crossover-first approach into the self-adaptive XjDE algorithm since the CR and F parameters are automatically adjusted and optimized by the algorithm itself as compared to the fixed CR and F in XDE which has to be manually tuned by hand. The results also show that different self-adaptive parameter tuning schemes have significantly different effects on the performance of DE as the number of optimization dimensions increases.