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International Conference on Computer and Information Application

Quantum-behaved Particle Swarm Optimization Algorithm for Solving Nonlinear Equations

Xiaofeng Zhang, Guifang Sui

Published Jan 1, 2012• 0 citations
Publisher

Abstract

A quantum-behaved particle swarm optimization algorithm is presented in this paper for solving nonlinear equations.The positions of particle are coded by probability amplitudes of qubits that are updated by quantum rotation gates in this method.The corresponding real number solution at specified interval can be extracted by this algorithm for solving nonlinear equations.Compared to real traditional method, the simulation results show that this algorithm is more accurate and effective.

DOI
10.2991/iccia.2012.417

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A quantum-behaved particle swarm optimization algorithm is presented in this paper for solving nonlinear equations.The positions of particle are coded by probability amplitudes of qubits that are updated by quantum rotation gates in this method.The corresponding real number solution at specified interval can be extracted by this algorithm for solving nonlinear equations.Compared to real traditional method, the simulation results show that this algorithm is more accurate and effective.
Aiming at the problems of the classical algorithms for solving the nonlinear equations such as high sensitivity to the initial value、poor convergence,a hybrid quantum genetic algorithm for solving nonlinear equations is proposed.The positions of particles are encoded by the probability amplitudes of quantum bits.The movements and mutations of particles are performed by quantum rotation gate and quantum non-gate.The algorithm has the abilities of population search for quantum swarm particle and global convergence.The quasi-newton method is combined into the algorithm and increases local searching ability and the high accuracy.Numerical simulation experiments show that this algorithm has reliable convergence,fast convergence rate and high precision.
To enhance the optimization ability of particle swarm algorithm, a novel quantum-inspired particle swarm optimization algorithm is proposed. In this method, the particles are encoded by the probability amplitudes of the basic states of the multi-qubits system. The rotation angles of multi-qubits are determined based on the local optimum particle and the global optimal particle, and the multi-qubits rotation gates are employed to update the particles. At each of iteration, updating any qubit can lead to updating all probability amplitudes of the corresponding particle. The experimental results of some benchmark functions optimization show that, although its single step iteration consumes long time, the optimization ability of the proposed method is significantly higher than other similar algorithms.
To enhance the performance of quantum-behaved PSO, some improvements are proposed. First, an encoding method based on the Bloch sphere is presented. In this method, each particle carries three groups of Bloch coordinates of qubits, and these coordinates are actually the approximate solutions. The particles are updated by rotating qubits about an axis on the Bloch sphere, which can simultaneously adjust two parameters of qubits, and can automatically achieve the best matching of two adjustments. The optimization process is employed in the n-dimensional space [-1, 1]n, so this approach fits to many optimization problems. The experimental results show that this algorithm is superior to the original quantum-behaved PSO.
To improve the efficiency of Particle Swarm Optimization(PSO),a quantum particle swarm optimization algorithm combined with quantum theory on the basis of Bloch sphere was proposed.In Bloch spherical coordinates,the particle automatically updated rotation angle and particle position,without setting the rotation angle in the form of look-up table(or setting fixed value of the interval),making up for the deficiency of quantum evolutionary algorithm and quantum genetic algorithm on the basis of Bloch sphere,and the algorithm is more generalizable.Using quantum Hadamard gate to realize the variation of particle enhanced the diversity of population,and prompted particle jump out of local extreme value.The simulation results of the typical function optimization problem show that the algorithm is stable with high precision and fast convergence rate,and it is practical.
In view of the shortcomings of particle swarm optimization such as premature convergence to local optimization, particle swarm optimization algorithm based on quantum gate and particle swarm optimization algorithm based on quantum behavior are studied in this paper. The first algorithm uses the random observation method of quantum bit coding particles to simulate quantum particle collapse for generating a population and uses the quantum rotating gate to generate a new population. The adaptive mutation operator is used for ensuring the diversity of the population, effectively reducing the impact of local optimization; therefore, the robustness of the algorithm is improved. The second algorithm uses the probability density function of quantum computation to make the particles jump out of the local extreme points and fulfils the global search, which is more suitable for continuous optimization. The results of computational experiments show that both the two particle swarm optimization algorithms based on quantum technology have a good global convergence.