Multiple Local Optimal Solutions Research Articles

A coevolutionary algorithm using Self-organizing map approach for multimodal multi-objective optimization

Multimodal Multi-Objective Problems (MMOPs) are frequently encountered in the real world. Traditional Multimodal Multi-Objective Evolutionary Algorithms (MMEAs) often find multiple Pareto optimal solutions with the same objective values. However, in real-world problems, there often exist multiple global optimal solutions and local optimal solutions at the same time. Ensuring that these solution sets are obtained simultaneously is the concern of most current researchers. To address this issue, this paper proposes a novel multimodal multi-objective evolutionary algorithm named CoSOMEA. In the CoSOMEA, a Self-organizing map (SOM) neural network is used to extract the information of decision space to ensure better exploration of the global optima and exploitation of the local optima. Meanwhile, coevolutionary mechanism are used to ensure a balance between the exploration and exploitation in order to avoid the algorithm falling into local areas. The three test suites named IDMP, IDMP_ee and MMF are adopted to verify the effectiveness of proposed algorithm. Experimental results demonstrate that the CoSOMEA exhibits competitive performance in solving MMOPs compared to other state-of-the-art MMEAs.

A collective incentive strategy to manage ridership rebound and consumer surplus in mass transit systems

This paper models and analyzes a collective incentive strategy with passenger departure time equilibrium to maximize the total passenger surplus, which could further enhance the sustainability of transit systems in the post-pandemic era. In the proposed collective incentive strategy, passengers with transit passes have dynamic discounts where the discount value depends on the total number of passengers enrolled in the incentive program. A bi-level optimization model is established by considering passengers' demand elasticity and responses to the strategy, which eventually determines the starting time and duration of the collective incentive strategy during the time of the day. We further analyze the properties of the proposed collective incentive model to identify the analytical solutions and performance. Analytical results indicate that there exists a threshold demand elasticity below which the total surplus of commuters with and without transit passes can be improved simultaneously. Properties also show that there could exist multiple local optimal solutions of incentive strategy duration for a given starting time. A case study is conducted with the metro system in Toronto: the results show that the proposed collective incentive strategy could improve both the fare revenues and ridership by 33.31 % and 83.56 %, respectively.

Multi-Node Feature Learning Network Based on Maximum Spectral Harmonics-to-Noise Ratio Deconvolution for Machine Condition Monitoring

Since the cyclostationarity in vibration signals is the key to judge the rotating machine health state, spectral harmonics-to-interference ratio (SHIR) has been used to construct single node feature learning network (SHIR-based blind deconvolution, BD-SHIR) to realize condition monitoring of machine. However, BD-SHIR still has several obvious limitations, including the need for prior fault information and the tendency to fall into local optimal solutions which will affect its feature learning and state monitoring performance. Therefore, this paper proposes a multi-node feature learning network based on BDSHIR, which does not rely on prior information, multi-node adaptive BDSHIR (MABD-SHIR). It uses each filter in the 1/3 binary tree filter bank to design each initial node (filter) in the learning network correspondingly, and introduces an adaptive fault characteristic frequency (FCF) detection process. Since this initialization can give some filters close to the fault resonant band, iterations starting from these nodes are more likely to detect the precise FCF and converge to the optimal solution. Finally, MABD-SHIR can obtain multiple local optimal solutions, among which the one with the largest SHIR is generally closer to the global optimal solution than the one obtained by BD-SHIR. Simulation and experimental data of defective rotating machinery verify the effectiveness of MABD-SHIR in machinery condition monitoring and fault feature learning. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the problems that the rotating machine condition monitoring based on BDSHIR is easy to prematurely fall into local optimal solution and requires prior fault information. These problems can be effectively solved by extending BDSHIR from single node structure to multi-node structure and introducing adaptive period detection technique (A-DPT). The core idea is to start iterations from multiple initial filters and to detect the FCF adaptively during each iteration. Multiple features conforming to filtered signals’ SHIR maximization can be finally learned. Therefore, in this paper, MABD-SHIR is proposed, which is equivalent to a multi-node feature learning network. Since the objective function of BD is usually non-convex, MABD-SHIR can obtain multiple local optimal solutions without requiring prior FCFs. This is more likely to achieve a global optimal solution than BDSHIR, which starts iteration from only one initial filter and ends up with one local optimal solution. In addition, the objective function of MABD-SHIR has lower mathematical complexity, which reduces the difficulty of optimization.

Trust-Tech Source-Point Method for Systematically Computing Multiple Local Optimal Solutions: Theory and Method.

In this article, a Trust-Tech source-point method is proposed to systematically compute multiple local optimal solutions (LOSs) for continuous unconstrained nonlinear optimization problems. This proposed method consists of four stages. Stage I finds one LOS (in which existing effective optimizers can be applied), stage II is the stage of escaping an LOS while stage III is the stage for entering the stability region (SR) of another stable equilibrium point (SEP) (i.e., another LOS). Stage IV computes other SEPs (i.e., LOSs) in corresponding SRs. A theoretical foundation for both stages II and III is developed, and these theoretical results are quite general on their own. The proposed method is numerically evaluated to compute multiple LOSs. For instance, a total of 5085 LOSs have been computed by the proposed Trust-Tech source point method on a 50-D test function. In addition, the proposed method can find the global optimal solutions of several test functions with 50 dimensions and 100 dimensions.

Hybrid Multi-Population and Adaptive Search Range Strategy With Particle Swarm Optimization for Multimodal Optimization

This paper proposes an algorithm named hybrid multi-population and adaptive search range strategy with particle swarm optimization (ARPSO) for solving multimodal optimization problems. The main idea of the algorithm is to divide the global search space into multiple sub-populations searching in parallel and independently. For diversity increasing, each sub-population will continuously change the search area adaptively according to whether there are local optimal solutions in its search space and the position of the global optimal solution, and in each iteration, the optimal solution in this area will be reserved. For the purpose of accelerating convergence, at the global and local levels, when the global optimal solution or local optimal solution is found, the global search space and local search space will shrink toward the optimal solution. Experiments show that ARPSO has unique advantages for solving multi-dimensional problems, especially problems with only one global optimal solution but multiple local optimal solutions.

Adaptive Grouping Brain Storm Optimization for Multimodal Optimization Problems

Multimodal optimization problem exists in multiple global and many local optimal solutions. The difficulty of solving these problems is finding as many local optimal peaks as possible on the premise of ensuring global optimal precision. This article presents adaptive grouping brainstorm optimization (AGBSO) for solving these problems. In this article, adaptive grouping strategy is proposed for achieving adaptive grouping without providing any prior knowledge by users. For enhancing the diversity and accuracy of the optimal algorithm, elite reservation strategy is proposed to put central particles into an elite pool, and peak detection strategy is proposed to delete particles far from optimal peaks in the elite pool. Finally, this article uses testing functions with different dimensions to compare the convergence, accuracy, and diversity of AGBSO with BSO. Experiments verify that AGBSO has great localization ability for local optimal solutions while ensuring the accuracy of the global optimal solutions.

Particle Swarm Optimization Based on Fitness Considering Distance Information for Multiple Local Optimal Solutions Search

Particle Swarm Optimization Based on Fitness Considering Distance Information for Multiple Local Optimal Solutions Search

Optimized minimum generalized Lp/Lq deconvolution for recovering repetitive impacts from a vibration mixture

Enhancing repetitive transients is a key to bearing fault detection. Blind deconvolution aims to recover impulsive impacts from repetitive transients and their mixture. Nowadays, based on different optimization criteria, such as kurtosis, entropy, generalized Lp/Lq norm, etc., various deconvolution methods have been proposed for recovering repetitive impacts. Due to the non-convex of these optimization criteria, local optimal solutions to these optimization problems may be obtained frequently. In this paper, inspired by the fast realization of spectral kurtosis, e.g. the fast kurtogram, an optimized minimum generalized Lp/Lq deconvolution (OMGD) is proposed and investigated. The main idea of the proposed method utilizes the filters designed in the fast kurtogram to provide proper initializations for the minimum generalized Lp/Lq deconvolution (MGD). Hence, multiple local optimal solutions can be iteratively obtained. Then, the minimum of these local optimal solutions is used as a precise estimate to approximate the globally optimal solution of MGD. Results show that the proposed method has better deconvolution performance than MGD and fast kurtogram for enhancement of repetitive impacts caused by localized bearing faults.

Research on Multi-Alternatives Problem of Finite Element Model Updating Based on IAFSA and Kriging Model.

Due to insufficient test data, insufficient constraint equations and uncertain objective function, the local optimal solution and the global optimal solution of the objective function in finite element model updating may represent the actual parameters of the structure. Based on this, this paper proposes an improved artificial fish school algorithm. By combining the niche technology with the artificial fish school algorithm, the improved algorithm can systematically find multiple global optimal solutions and local optimal solutions of the objective function. Aiming at the difficulty of determining the niche radius, an adaptive niche radius mechanism is proposed. The improved algorithm is used to study the multi-alternatives problem of finite element model updating after verifying its feasibility through numerical simulation analysis. In the case of benchmark framework model updating, it is confirmed that multi-alternative problems exist and the global optimal solution of the objective function does not necessarily represent the true parameters of the structure. In case 2, the improved algorithm combined with the Kriging model is applied to the model updating of a cable-stayed footbridge, and 15 sets of solutions are obtained, in which the error objective function values of the measured and theoretical values of the bridge modes are close but the solutions are completely different. Combining with the actual bridge condition and reanalysis technology, the author takes the suboptimal solution 2 as the most representative solution of the bridge parameters, which reduces the possibility of misjudgment of structural parameters.

A Novel TRUST-TECH-Enabled Trajectory-Unified Methodology for Computing Multiple Optimal Solutions of Constrained Nonlinear Optimization: Theory and Computation

A novel trajectory-unified (TJU) methodology, enabled by transformation under stability-retraining equilibria characterization (TRUST-TECH, TT) method, is proposed to systematically and deterministically compute a set of multiple local optimal solutions (LOSs), if not all of them, for constrained nonlinear optimization problems. The proposed TT-TJU methodology is composed of two key phases. During Phase I, it robustly computes a LOS; while during Phase II, it systematically computes a set of neighboring LOSs. A theoretical foundation is developed for the proposed TT-TJU methodology. The existence of an optimal solution is shown to be equivalent to the existence of a stable equilibrium point of a class of nonlinear dynamical system derived from the constrained optimization problem under study. From a computational viewpoint, Phase I is numerically implemented in a 3-stage procedure for fast and robust computation of an optimal solution. The second phase makes use of the TRUST-TECH methodology to lead the search away from a computed optimal solution and continue to deterministically compute neighboring LOSs. The proposed methodology has been evaluated on an illustrative bivariate example and a benchmark problem with 110 variables and 55 constraints with promising results.

Identifying Multiple Local Optima for Optimal Power Flow Based on Nonlinear Dynamics

An optimal power flow (OPF) problem of power systems can have multiple local optimal solutions, which is worthwhile studying both in theory and practice. Based on the existing nonlinear dynamic systems, this paper proposes an efficient deterministic algorithm to solve multiple or all local optimal solutions of OPF, which takes some numerical improving measures to enhance the numerical convergence for integration process of dynamics and adapt to OPF problem. The steps of this algorithm are as follows: 1. The reflected gradient system (RGS) is used to calculate the decomposition points to locate different feasible components. 2. The quotient gradient system (QGS) is used to calculate feasible points in different feasible components, and we numerically integrate projected gradient system (PGS) with these feasible points as initial points forward until the trajectories approach the local optima. 3. Slack variable perturbation method (SVPM) is proposed to help escape from the saddle points to the adjacent local optima when the trajectories fall into saddle points. Compared with the interior point method (IPM) with random initialization, multiple IEEE test cases show that the proposed algorithm can identify much more local optimal solutions, and meanwhile, significantly reduce the calculation time.

Superior relation‐based firefly algorithm for superior solution set search problem

In this paper, we analyze the properties of the superior solution set search (SSSS) problem and point out the structural similarities between it and the multiobjective optimization problem. Based on the analyzed properties, we propose a new Firefly algorithm (FA) based on superior relations, which can search for the superior solution set by using the user's quantitative desire level as a search strategy. By introducing the distance of the problem space into the moving mechanism, FA makes it possible to search for multiple local optimal solutions by dividing the solution set into multiple groups. By analyzing this property of FA, we clarify the affinity between FA and the SSSS problem. We subsequently analyze the SSSS problem and FA, and discuss their properties together in a similar problem setting. We propose an FA based on superior relations as a new optimization technique for the superior solution set problem based on these analyses. Numerical experiments are then conducted using the SSSS problem and demonstrate the usefulness of the proposed method by comparing the performance of the proposed method with the conventional FA. © 2019 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.

Multi-surrogate-based Differential Evolution with multi-start exploration (MDEME) for computationally expensive optimization

In this paper, we present a new global optimization algorithm MDEME for black-box problems with computationally expensive objectives. Considering that Differential Evolution (DE) is an efficient global optimization algorithm but has difficulty in expensive optimization problems, we combine DE with three surrogate models Kriging, Radial Basis Function (RBF), and Quadratic Polynomial Response (QRS) to realize surrogate-based optimization. Although the three surrogates have different approximate effects that may generate diverse updating points, the surrogate-based DE may still get stuck in local optimal regions. In order to enhance its exploration capability, a multi-start optimization algorithm with a new selecting strategy is proposed. The multi-start optimization algorithm can capture and select several promising points from Kriging and RBF that always generate multiple local optimal solutions per optimization cycle. In the whole optimization process, DE and the proposed multi-start optimization are alternately carried out on the three surrogate models that are dynamically updated. Once no more satisfactory points can be obtained from Kriging and RBF, the multi-start optimization will explore the sparsely sampled area using the estimated mean square error of Kriging. After the comparison with 5 global optimization algorithms on 17 representative cases, MDEME shows its high efficiency, strong stability and good parallelism capability in dealing with expensive optimization problems. Finally, MDEME is used for the shape optimization of a blended-wing-body underwater glider, and the design performance gets significantly improved.

TRUST-TECH-Enhanced Differential Evolution Methodology for Box-Constrained Nonlinear Optimization

The differential evolution algorithm and its variants have been developed to solve box-constrained optimisation problems with encouraging results. However, differential evolution still suffers from its poor ability to zoom in on promising regions to find a high-quality or global optimal solution. The transformation under stability-retraining equilibrium characterisation (TRUST-TECH) methodology is a systematical and deterministic method to find a set of multiple local optimal solutions. This paper presents a TRUST-TECH-enhanced differential evolution methodology (TT-DEM) to improve the performance of differential evolution method. In the TT-DEM framework, a differential evolution method is carried out to identify promising regions containing a set of high-quality solution or even the global optimal solution, while TRUST-TECH exploits the identified promising regions to compute those high-quality optimal solutions. Following this framework, the original differential evolution (DE) and three adaptive DEs are enhanced by TRUST-TECH. Numerical studies are conducted on several benchmark functions with promising results.

TRUST-TECH-enhanced differential evolution methodology for box-constrained nonlinear optimisation

The differential evolution algorithm and its variants have been developed to solve box-constrained optimisation problems with encouraging results. However, differential evolution still suffers from its poor ability to zoom in on promising regions to find a high-quality or global optimal solution. The transformation under stability-retraining equilibrium characterisation (TRUST-TECH) methodology is a systematical and deterministic method to find a set of multiple local optimal solutions. This paper presents a TRUST-TECH-enhanced differential evolution methodology (TT-DEM) to improve the performance of differential evolution method. In the TT-DEM framework, a differential evolution method is carried out to identify promising regions containing a set of high-quality solution or even the global optimal solution, while TRUST-TECH exploits the identified promising regions to compute those high-quality optimal solutions. Following this framework, the original differential evolution (DE) and three adaptive DEs are enhanced by TRUST-TECH. Numerical studies are conducted on several benchmark functions with promising results.

Niche Brain Storm Optimization Algorithm for Multi-Peak Function Optimization

Brain Storm Optimization (BSO) is a novel proposed swarm intelligence optimization algorithm which has a fast convergent speed. However, it is easy to trap into local optimal. In this paper, a new model based on niche technology, which is named Niche Brain Storm Optimization (NBSO), is proposed to overcome the shortcoming of BSO. Niche technology effectively prevents premature and maintains population diversity during the evolution process. NBSO shows excellent performance in searching global value and finding multiple global and local optimal solutions for the multi-peak problems. Several benchmark functions are introduced to evaluate its performance. Experimental results show that NBSO performs better than BSO in global searching ability and faster than Niche Genetic Algorithm (NGA) in finding peaks for multi-peak function.

Applying More Initial Groups to the Semi-Supervised Clustering Based on one-Class Support Vector Machines

The traditional semi-supervised clustering based on one-class support vector machines used some labeled data called seeds for the clustering initialization. These seeds were partitioned into several initial groups according to their labels and the number of initial groups was equal to the number of clusters. However, the traditional semi-supervised clustering based on one-class support vector machines is sensitive to the initial groups and often obtained the local optimal solutions. In this paper, more initial groups produced by seeds are applied to the traditional semi-supervised clustering based on one-class support vector machines to get more local optimal solutions and the proposed algorithm can combine multiple local optimal solutions to obtain the better clustering performance at last. To investigate the effectiveness of our approach, experiments are done on two real datasets. Experimental results show that the presented method can improve the clustering accuracies compared to the traditional algorithm.

A memetic particle swarm optimization algorithm for multimodal optimization problems

Recently, multimodal optimization problems (MMOPs) have gained a lot of attention from the evolutionary algorithm (EA) community since many real-world applications are MMOPs and may require EAs to present multiple optimal solutions. In this paper, a memetic algorithm that hybridizes particle swarm optimization (PSO) with a local search (LS) technique, called memetic PSO (MPSO), is proposed for locating multiple global and local optimal solutions in the fitness landscape of MMOPs. Within the framework of the proposed MPSO algorithm, a local PSO model, where the particles adaptively form different species based on their indices in the population to search for different sub-regions in the fitness landscape in parallel, is used for globally rough exploration, and an adaptive LS method, which employs two different LS operators in a cooperative way, is proposed for locally refining exploitation. In addition, a triggered re-initialization scheme, where a species is re-initialized once converged, is introduced into the MPSO algorithm in order to enhance its performance of solving MMOPs. Based on a set of benchmark functions, experiments are carried out to investigate the performance of the MPSO algorithm in comparison with some EAs taken from the literature. The experimental results show the efficiency of the MPSO algorithm for solving MMOPs.

ELITE: Ensemble of Optimal Input-Pruned Neural Networks Using TRUST-TECH

The ensemble of optimal input-pruned neural networks using TRUST-TECH (ELITE) method for constructing high-quality ensemble through an optimal linear combination of accurate and diverse neural networks is developed. The optimization problems in the proposed methodology are solved by a global optimization a global optimization method called TRansformation Under Stability-reTraining Equilibrium Characterization (TRUST-TECH), whose main features include its capability in identifying multiple local optimal solutions in a deterministic, systematic, and tier-by-tier manner. ELITE creates a diverse population via a feature selection procedure of different local optimal neural networks obtained using tier-1 TRUST-TECH search. In addition, the capability of each input-pruned network is fully exploited through a TRUST-TECH-based optimal training. Finally, finding the optimal linear combination weights for an ensemble is modeled as a nonlinear programming problem and solved using TRUST-TECH and the interior point method, where the issue of non-convexity can be effectively handled. Extensive numerical experiments have been carried out for pattern classification on the synthetic and benchmark datasets. Numerical results show that ELITE consistently outperforms existing methods on the benchmark datasets. The results show that ELITE can be very promising for constructing high-quality neural network ensembles.

Multiple Solutions and Plateau Phenomenon in Measurement-Based Load Model Development: Issues and Suggestions

In the measurement-based approach, given a load model structure, its parameter values are derived using actual measurements. The issue of multiple local optimal solutions may arise in parameter estimation when the parameter estimation task is formulated as a nonlinear least squares problem. In this paper, the multiple local optimal solutions issue and a plateau phenomenon in developing load models are addressed in detail and linked to the local identifiability problem. Examples of multiple local optimal solutions in estimating parameter values of several static, dynamic and composite load models are demonstrated. In order to address the local identifiability at an obtained local optimal solution, output sensitivity and correlation between parameters are analyzed. Finally, several suggestions are proposed to resolve these issues.