The success of the Particle Swarm Optimiza- tion (PSO) algorithm as a single-objective optimizer (mainly when dealing with continuous search spaces) has motivated re- searchers to extend the use of this bio-inspired technique to other areas. One of them is multi-objective optimization. De- spite the fact that the first proposal of a Multi-Objective Par- ticle Swarm Optimizer (MOPSO) is over six years old, a con- siderable number of other algorithms have been proposed since then. This paper presents a comprehensive review of the vari- ous MOPSOs reported in the specialized literature. As part of this review, we include a classification of the approaches, and we identify the main features of each proposal. In the last part of the paper, we list some of the topics within this field that we consider as promising areas of future research.

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Multi-Objective Particle Swarm Optimizers: A Survey of the State-of-the-Art

A study of particle swarm optimization particle trajectories

Particle swarm optimization (PSO) has undergone many changes since its introduction in 1995. As researchers have learned about the technique, they have derived new versions, developed new applications, and published theoretical studies of the effects of the various parameters and aspects of the algorithm. This paper comprises a snapshot of particle swarming from the authors' perspective, including variations in the algorithm, current and ongoing research, applications and open problems.

Particle Swarm Optimisation

Particle swarm optimization (PSO) is a stochastic population-based algorithm motivated by intelligent collective behavior of some animals. The most important advantages of the PSO are that PSO is easy to implement and there are few parameters to adjust. The inertia weight (w) is one of PSO's parameters originally proposed by Shi and Eberhart to bring about a balance between the exploration and exploitation characteristics of PSO. Since the introduction of this parameter, there have been a number of proposals of different strategies for determining the value of inertia weight during a course of run. This paper presents the first comprehensive review of the various inertia weight strategies reported in the related literature. These approaches are classified and discussed in three main groups: constant, time-varying and adaptive inertia weights. A new adaptive inertia weight approach is also proposed which uses the success rate of the swarm as its feedback parameter to ascertain the particles' situation in the search space. The empirical studies on fifteen static test problems, a dynamic function and a real world engineering problem show that the proposed particle swarm optimization model is quite effective in adapting the value of w in the dynamic and static environments.

A novel particle swarm optimization algorithm with adaptive inertia weight

Particle swarm approach for structural design optimization

A PSO with increasing inertia weight, distinct from a widely used PSO with decreasing inertia weight, is proposed in this paper. Far from drawing conclusions from sole empirical study or rule of thumb, this algorithm is derived from particle trajectory study and convergence analysis. Four standard test functions are used to confirm its validity finally. From the experiments, it is clear that a PSO with increasing inertia weight outperforms the one with decreasing inertia weight, both in convergent speed and solution precision, with no additional computing load.

On the convergence analysis and parameter selection in particle swarm optimization

A PSO with increasing inertia weight, distinct from a widely used PSO with decreasing inertia weight, is proposed in this paper. Four standard test functions with asymmetric initial range settings are used to confirm its validity. From the experiments, it is clear that a PSO with increasing inertia weight outperforms the one with decreasing inertia weight, both in convergent speed and solution precision, with no additional computing load compared with the PSO with a decreasing inertia weight.

Empirical study of particle swarm optimizer with an increasing inertia weight

A simple but feasible Robust PID (RPID) tuning methodology is proposed in this paper. Most robust controllers that have been developed in literature by now have complex structures. They are either hard to be implemented or easily broken down. The simplest controller, PID, is chosen to be the prototype of robust controller. Through parameter selection, robust performance is achieved. Minimax criterion guarantees a set of controller parameters to be most optimal to the worst operating condition that may occur within the uncertainty ranges of model parameters. To execute the minimax search, a co-evolutionary algorithm based on Particle Swarm Optimization is advanced. Simulations are presented to demonstrate the advantages of RPID compared with other PIDs.

Robust PID controller design using particle swarm optimizer

In process control fields it is hard to build precise model due to complexity and nonlinear of industrial process. To solve this problem, this paper proposes a framework for modeling and controlling systems integrating continuous variable linear model and rules. A predictive control scheme is proposed which is able to stabilize this framework on desired reference trajectories. This problem can be solved as Mixed-integer Linear Programming/ Constraint Programming (MILP/CP). We use MILP to optimize and find partial solutions and CP to check feasibility and produce complete solutions. An example demonstrates how problems can be modeled and solved using this framework.

Building hybrid industrial process model mixed with rules

This paper advanced mixed integer nonlinear programming (MINLP) algorithms mixed with constraint programming (CP) to solve industrial process hybrid model mixed with logical rule. CP methods have proved to be successful in solving highly constrained discrete optimization and feasibility problems. It can effectively solve hybrid model mixed with logical rule by integrating CP, nonlinear programming and mixed integer linear programming. Computational results demonstrated this algorithm.

Liyan Zhang

Papers

On the convergence analysis and parameter selection in particle swarm optimization

Empirical study of particle swarm optimizer with an increasing inertia weight

Robust PID controller design using particle swarm optimizer

Building hybrid industrial process model mixed with rules

MINLP algorithms mixed with constraint programming