A decomposition-based differential evolution with reinitialization for nonlinear equations systems

Abstract Solving nonlinear equations systems (NESs) is one of the most important challenges in numerical computation, especially to find multiple roots in one run. In this paper, a decomposition-based differential evolution with reinitialization is proposed to tackle this challenging task. The main advantages of our method are: (i) an improved parameter-free decomposition technique is exploited to partition the population into numerous sub-populations to locate multiple roots of NESs; (ii) to enhance the search ability of optimization algorithm, a sub-population control strategy is presented to control the number of solutions in the sub-populations; and (iii) the sub-population reinitialization mechanism is proposed to enrich the population diversity. To evaluate the performance of our approach, thirty NES problems with different characteristics are selected as the test suite. Moreover, to further indicate the superiority of our method, ten complex NESs with many roots are also tested. Experimental results show that the proposed approach can locate multiple roots in a single run. In addition, it is able to obtain better results compared with other state-of-the-art methods in terms of both root rate and success rate.

[1]  Jun Zhang,et al.  Automatic Niching Differential Evolution With Contour Prediction Approach for Multimodal Optimization Problems , 2020, IEEE Transactions on Evolutionary Computation.

[2]  Sanyang Liu,et al.  A Cluster-Based Differential Evolution With Self-Adaptive Strategy for Multimodal Optimization , 2014, IEEE Transactions on Cybernetics.

[3]  Ana Maria A. C. Rocha,et al.  Multiple Roots of Systems of Equations by Repulsion Merit Functions , 2014, ICCSA.

[4]  T.G.I. Fernando,et al.  Solving systems of nonlinear equations using a modified firefly algorithm (MODFA) , 2019, Swarm Evol. Comput..

[5]  Michael Kastner.  Phase transitions and configuration space topology , 2008 .

[6]  Xin Yao,et al.  Evolutionary Multiobjective Optimization-Based Multimodal Optimization: Fitness Landscape Approximation and Peak Detection , 2018, IEEE Transactions on Evolutionary Computation.

[7]  Bernard Mourrain,et al.  Computer Algebra Methods for Studying and Computing Molecular Conformations , 1999, Algorithmica.

[8]  Ajith Abraham,et al.  A New Approach for Solving Nonlinear Equations Systems , 2008, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[9]  Nélio Henderson,et al.  Novel approach for the calculation of critical points in binary mixtures using global optimization , 2004 .

[10]  Wagner F. Sacco,et al.  Finding more than one root of nonlinear equations via a polarization technique: An application to double retrograde vaporization , 2010 .

[11]  A. Stavrakoudis,et al.  On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods , 2010 .

[12]  Miroslav Ciric,et al.  Fuzzy relation equations and subsystems of fuzzy transition systems , 2013, Knowl. Based Syst..

[13]  Swagatam Das,et al.  A Cluster-Based Differential Evolution Algorithm With External Archive for Optimization in Dynamic Environments , 2013, IEEE Transactions on Cybernetics.

[14]  Xuesong Yan,et al.  Fuzzy neighborhood-based differential evolution with orientation for nonlinear equation systems , 2019, Knowl. Based Syst..

[15]  Jun Zhang,et al.  Multimodal Estimation of Distribution Algorithms , 2017, IEEE Transactions on Cybernetics.

[16]  Oguz Emrah Turgut,et al.  Chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations , 2014, Comput. Math. Appl..

[17]  Yong Wang,et al.  MOMMOP: Multiobjective Optimization for Locating Multiple Optimal Solutions of Multimodal Optimization Problems , 2015, IEEE Transactions on Cybernetics.

[18]  Qingfu Zhang,et al.  Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II , 2009, IEEE Transactions on Evolutionary Computation.

[19]  Wenyin Gong,et al.  Adaptive Ranking Mutation Operator Based Differential Evolution for Constrained Optimization , 2015, IEEE Transactions on Cybernetics.

[20]  Junjie Wu,et al.  Inclined Geosynchronous Spaceborne–Airborne Bistatic SAR: Performance Analysis and Mission Design , 2016, IEEE Transactions on Geoscience and Remote Sensing.

[21]  Anil K. Jain,et al.  Data clustering: a review , 1999, CSUR.

[22]  Weiping Zhang,et al.  Control of discrete chaotic systems based on echo state network modeling with an adaptive noise canceler , 2012, Knowl. Based Syst..

[23]  Francisco Facchinei,et al.  Generalized Nash equilibrium problems , 2007, 4OR.

[24]  Christodoulos A. Floudas,et al.  Recent advances in global optimization for process synthesis, design and control: Enclosure of all solutions , 1999 .

[25]  María José del Jesús,et al.  KEEL: a software tool to assess evolutionary algorithms for data mining problems , 2008, Soft Comput..

[26]  Xuesong Yan,et al.  Solving Nonlinear Equations System With Dynamic Repulsion-Based Evolutionary Algorithms , 2020, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[27]  Dongsheng Guo,et al.  The Application of Noise-Tolerant ZD Design Formula to Robots’ Kinematic Control via Time-Varying Nonlinear Equations Solving , 2018, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[28]  P. John Clarkson,et al.  Erratum: A Species Conserving Genetic Algorithm for Multimodal Function Optimization , 2003, Evolutionary Computation.

[29]  Jing J. Liang,et al.  Differential Evolution With Neighborhood Mutation for Multimodal Optimization , 2012, IEEE Transactions on Evolutionary Computation.

[30]  Ali Barati,et al.  A third-order Newton-type method to solve systems of nonlinear equations , 2007, Appl. Math. Comput..

[31]  Charles L. Karr,et al.  Solutions to systems of nonlinear equations via a genetic algorithm , 1998 .

[32]  Ling Wang,et al.  Finding Multiple Roots of Nonlinear Equation Systems via a Repulsion-Based Adaptive Differential Evolution , 2020, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[33]  Yudong Wang,et al.  Optimal targeting of nonlinear chaotic systems using a novel evolutionary computing strategy , 2016, Knowl. Based Syst..

[34]  Xiaodong Yin,et al.  A Fast Genetic Algorithm with Sharing Scheme Using Cluster Analysis Methods in Multimodal Function Optimization , 1993 .

[35]  Tao Wang,et al.  Novel Homotopy Theory for Nonlinear Networks and Systems and Its Applications to Electrical Grids , 2018, IEEE Transactions on Control of Network Systems.

[36]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[37]  Yong Wang,et al.  A Weighted Biobjective Transformation Technique for Locating Multiple Optimal Solutions of Nonlinear Equation Systems , 2017, IEEE Transactions on Evolutionary Computation.

[38]  Nélio Henderson,et al.  Finding all solutions of nonlinear systems using a hybrid metaheuristic with Fuzzy Clustering Means , 2011, Appl. Soft Comput..

[39]  Kun Wu,et al.  A new filled function method for an unconstrained nonlinear equation , 2011, J. Comput. Appl. Math..

[40]  Hamed Mojallali,et al.  Solving nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering , 2012, Swarm Evol. Comput..

[41]  János D. Pintér.  Solving nonlinear equation systems via global partition and search: Some experimental results , 2005, Computing.

[42]  Y. Ramu Naidu,et al.  Solving Multiobjective Optimization Problems Using Hybrid Cooperative Invasive Weed Optimization With Multiple Populations , 2018, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[43]  Panos M. Pardalos,et al.  Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm , 2014, J. Glob. Optim..

[44]  Yong Wang,et al.  Locating Multiple Optimal Solutions of Nonlinear Equation Systems Based on Multiobjective Optimization , 2015, IEEE Transactions on Evolutionary Computation.

[45]  René Thomsen,et al.  Multimodal optimization using crowding-based differential evolution , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[46]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .