Algebraic Stability Analysis of Particle Swarm Optimization Using Stochastic Lyapunov Functions and Quantifier Elimination

This paper adds to the discussion about theoretical aspects of particle swarm stability by proposing to employ stochastic Lyapunov functions and to determine the convergence set by quantifier elimination. We present a computational procedure and show that this approach leads to a reevaluation and extension of previously known stability regions for PSO using a Lyapunov approach under stagnation assumptions.

[1]  Xuerong Mao,et al.  Stability of stochastic delay neural networks , 2001, J. Frankl. Inst..

[2]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..

[3]  Maurice Clerc,et al.  The particle swarm - explosion, stability, and convergence in a multidimensional complex space , 2002, IEEE Trans. Evol. Comput..

[4]  Klaus Röbenack,et al.  Automatic Generation of Bounds for Polynomial Systems with Application to the Lorenz System , 2017, Chaos, Solitons & Fractals.

[5]  Riccardo Poli,et al.  Mean and Variance of the Sampling Distribution of Particle Swarm Optimizers During Stagnation , 2009, IEEE Transactions on Evolutionary Computation.

[6]  Shiyuan Yang,et al.  Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm , 2007, Inf. Process. Lett..

[7]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[8]  X. Chen,et al.  Discrete-time Indefinite LQ Control with State and Control Dependent Noises , 2002, J. Glob. Optim..

[9]  Changbo Chen,et al.  Quantifier elimination by cylindrical algebraic decomposition based on regular chains , 2016, J. Symb. Comput..

[10]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition--preliminary report , 1974, SIGS.

[11]  Petar V. Kokotovic,et al.  A brief look at the Tsypkin criterion: from analysis to design , 2001 .

[12]  Mikhail A. Semenov,et al.  Analysis of Convergence of an Evolutionary Algorithm with Self-Adaptation using a Stochastic Lyapunov function , 2003, Evolutionary Computation.

[13]  Elizabeth F. Wanner,et al.  Lyapunov Design of a Simple Step-Size Adaptation Strategy Based on Success , 2016, PPSN.

[14]  Wen Yong Dong,et al.  Order-3 stability analysis of particle swarm optimization , 2019, Inf. Sci..

[15]  Zbigniew Michalewicz,et al.  Stability Analysis of the Particle Swarm Optimization Without Stagnation Assumption , 2016, IEEE Transactions on Evolutionary Computation.

[16]  Christopher W. Brown.  QEPCAD B: a program for computing with semi-algebraic sets using CADs , 2003, SIGS.

[17]  Ioan Cristian Trelea,et al.  The particle swarm optimization algorithm: convergence analysis and parameter selection , 2003, Inf. Process. Lett..

[18]  Huanshui Zhang,et al.  Infinite horizon linear quadratic optimal control for discrete‐time stochastic systems , 2008 .

[19]  Andries Petrus Engelbrecht,et al.  A study of particle swarm optimization particle trajectories , 2006, Inf. Sci..

[20]  J. Michael Herrmann,et al.  Stochastic stability of particle swarm optimisation , 2017, Swarm Intelligence.

[21]  Rüdiger Loos,et al.  Applying Linear Quantifier Elimination , 1993, Comput. J..

[22]  Qunfeng Liu,et al.  Order-2 Stability Analysis of Particle Swarm Optimization , 2015, Evolutionary Computation.

[23]  Andries Petrus Engelbrecht,et al.  Particle swarm variants: standardized convergence analysis , 2015, Swarm Intelligence.

[24]  Visakan Kadirkamanathan,et al.  Stability analysis of the particle dynamics in particle swarm optimizer , 2006, IEEE Transactions on Evolutionary Computation.

[25]  Hirokazu Anai,et al.  The Maple package SyNRAC and its application to robust control design , 2007, Future Gener. Comput. Syst..

[26]  K. Swamy,et al.  On Sylvester's criterion for positive-semidefinite matrices , 1973 .

[27]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[28]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[29]  Hirokazu Anai,et al.  An effective implementation of symbolic-numeric cylindrical algebraic decomposition for quantifier elimination , 2013, Theor. Comput. Sci..

[30]  Vasile Dragan,et al.  Mean Square Exponential Stability for some Stochastic Linear Discrete Time Systems , 2006, Eur. J. Control.

[31]  Beatrice Paternoster,et al.  About stability of nonlinear stochastic difference equations , 2000, Appl. Math. Lett..

[32]  D. Grigor'ev.  Complexity of deciding Tarski algebra , 1988 .

[33]  Andries Petrus Engelbrecht,et al.  Particle swarm stability: a theoretical extension using the non-stagnate distribution assumption , 2018, Swarm Intelligence.