Particle Swarm Optimization With Particles Having Quantum Behavior

The Swarm Intelligence (SI) optimization algorithm refers to a class of heuristic search methods that can solve the specified problems by simulating the collective behaviors of social organisms. It is characterized by its stochastic, parallel and distributed search in the problem space. Particle Swarm Optimization (PSO) algorithm, which inspired by social behavior of bird flocking or fish schooling, is one of the typical SI algorithms and it is developed by Eberhart and Kennedy in 1995. Since its origin, PSO algorithm has appealed to many researchers in various fields due to its low computation consumption, easy realization, and few parameters. However, PSO has some shortcomings, the first of which is that it is not a global convergent algorithm according to the criterion of global convergence. Another limitation is that the evolution equation of the velocity and position derived from behavior simulation of bird flocks make the swarm less intelligent and cooperative. Further, a complaint about PSO algorithm is that its global search ability relies on the up-limit of velocity, which leads the algorithm to lower robustness. Based on the deep investigation to the features of swarm intelligence and the thinking pattern of human being, we establish a quantum . potential model for a PSO system, and thus proposed a so-called Quantum-behaved Particle Swarm Optimization (QPSO) algorithm. This novel global convergent algorithm has fewer parameters, faster convergence speed as well as strong search ability for complex problems.This dissertation focuses on the fundamental principle, convergence analysis and parameter control method of QPSO algorithms. Some improvements and applications of QPSO will also be discussed. The main contents are outlined as follows.(1) The introduction of optimization method is given followed by the backgrounds of evolutionary computation and swarm intelligence optimization. After that, the context of PSO algorithm, including its theory and applications goes into particulars. Aiming at the shortcomings of PSO algorithm, we put forward the objectives, contents and methods of our research task.(2) After the principle and procedure of PSO algorithm is presented, two important versions, PSO with inertia weight and PSO with contraction coefficient, are discussed. Some improved PSO methods are also mentioned for reference. The motivation of our research is talked of, and with that, a quantum . potential well model is established for QPSO algorithm. Working out the wave function and probability distribution function for the particle's position, we deduce the fundamental evolution equation of QPSO by Monte Carlo method and the criterion of particle's convergence. To present a complete QPSO algorithm, two versions of search strategy are proposed . The conditions of convergence and boundedness for the particle are acquired by stochastic simulation. We also analyze the wait phenomena and learning pattern among particles in QPSO.(3) Global convergence of QPSO algorithm is analyzed with three theoretical methods. The first is the probability analysis approach proposed by Wets and Solis. Using their criterion for a global convergent algorithm, we present the proof that QPSO satisfies the criterion and is able to converge to the global optima. The second method is to using Discrete Absorbed Markov Process Model (ADMPM) for a random search algorithm. After the introduction of ADMPM and the criterion of global convergence in this model, we construct an ADMPM for QPSO algorithm and thus proved that QPSO can converge to the global optima in probability. Further, we establish a probability metrics space for QPSO algorithm which is considered as a self mapping in the space. Proving that QPSO is a contractive mapping, we conclude that QPSO can converge to the fixed point.(4) Generally speaking, a given algorithm's performance and efficiency is subject to the parameters of the algorithm. In QPSO algorithm, Contraction-Expansion (CE) coefficient is the most important parameter. We adopt two control methods for this parameter, one is setting it at a fixed value and the other is decreasing its value linearly on the course of the search. Testing the QPSO with the two parameter control methods on several widely used benchmark functions, we can conclude that the latter can lead the algorithm to the good performance in general.(5) Some improved versions of QPSO algorithm are proposed. One is the QPSO using a hybrid of double exponential distribution and normal distribution. The results of the improved QPSO on several benchmarks show that it may have better performance than QPSO. A Binary-encoded QPSO (BQPSO) is proposed for the problems discrete binary search space. The BQPSO is implemented by modifying the evolution equation in QPSO into the discrete operators. Aiming at the premature convergence encountered by QPSO, another improvement is to introduce the diversity measure to guide the search of the algorithm. Setting the CE coefficient to a large value to make the particle explode or exerting a mutation operation on global best (gbest) position can prevent the diversity from declining incessantly, and thus enhances the global search ability of the algorithm. The other modified QPSO is to exert a selection operation on the gbest position, since the particle's converging to the gbest position may result in particle's missing the better region of the problem. The results on benchmarks show that the method improves the search ability of QPSO.(6) Some applications of QPSO to real world problems are given. Economic Dispatch (ED) in power system is a constrained optimization problem that can be solved by QPSO. The simulation results show that QPSO can obtain better solution than PSO and Genetic Algorithm (GA). We apply QPSO to stochastic programming problem with multi-period financial planning problem as an instance. The simulation results also show the better performance of QPSO. To test QPSO on system identification problem, optimal design problem of 2-D digital filters is used as an instance, on which QPSO shows the advantage over PSO and GA. We also employ QPSO in design of H_∞optimal Control, and the simulation results also indicate that QPSO is a promising problem solver.The main contributions in this work are summarized at the end and future research directions are also put forward.