Low-discrepancy sequence initialized particle swarm optimization algorithm with high-order nonlinear time-varying inertia weight

Highlights

• Initial population particles are generated by the Halton sequence to fill the search space efficiently.

• 1/π² order nonlinear function is selected to adjust the time-varying inertia weight.

• Two acceleration coefficients are set to be constants.

• Performance of the proposed method is tested by a set of well-known benchmark optimization problems.

Abstract

Particle swarm optimization (PSO) is a population-based stochastic optimization algorithm motivated by intelligent collective behavior of some animals such as flocks of birds or schools of fish. The most important features of the PSO are easy implementation and few adjustable parameters. A novel PSO method called LHNPSO, with low-discrepancy sequence initialized particles and high-order (1/π²) nonlinear time-varying inertia weight and constant acceleration coefficients, is proposed in this paper. The initial population particles are generated by using the Halton sequence to fill the search space efficiently. Nonlinear functions with orders varied within big ranges are employed to adjust the inertial weight, cognitive and social parameters. Based on the sensitivity analysis of PSO performance to the changes of the orders of these nonlinear functions, 1/π² order nonlinear function is selected to adjust the time-varying inertia weight and the two acceleration coefficients are set to be constants. A set of well-known benchmark optimization problems is then used to investigate the performance of the proposed LHNPSO algorithm and facilitate the comparison with other three types of PSO algorithms. The results show that the easily implemented LHNPSO can converge faster and give a much more accurate final solution for a variety of benchmark test functions.

Introduction

With the inspiration motived by the research results on the modeling and the simulations of the behavior reflected in flocks of birds, Kennedy and Eberhart proposed the particle swarm optimization (PSO) algorithm [1]. The algorithm is a stochastic population-based method and is regarded as a global search strategy. In the PSO, particles move through the problem space with a specified velocity in search of optimal solution. Each particle maintains a memory which helps it keep the track of its previous best position and the global best position. The most important advantages of PSO are that it is easy to implement and has few parameters to adjust [2]. Commonly, the adjustable parameters are inertial weight, cognitive and social parameters.

PSO algorithm has attracted considerable attention and has been used in many research areas over the past decades [3], [4], [5], [6], [7], [8], [9], [10], [11]. Generally, convergence speed and ability to find global optima are basic criteria for assessing the performance of an optimization method. A large number of improved PSO algorithms [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] have been proposed to achieve the two goals, that is, faster convergence speed and avoiding premature or local optima.

The convergence speed of a PSO algorithm is dependent on the inertia weight and two acceleration coefficients. Random [23], linear time-varying [24], nonlinear time-varying [25], [26] and adaptive strategies [27], [28], [29] have been used to adjust the inertia weight to improve the convergence speed. A review and summary of various inertia weight modification mechanisms reported in literature can be found in reference [2]. Nonlinear time-varying and adaptive inertia weights normally have better performance than others. Moreover, the principle and implementation of nonlinearly decreasing inertia weights are simpler than those of adaptive ones. Currently, third and less order nonlinear functions have been investigated extensively for adjusting inertia weight, however, the performance of higher order nonlinear time-varying weight needs to be further studied.

The concept for initializing the particles of most PSO algorithms is same as that for generating random numbers in the traditional Monte-Carlo simulation method which could be regarded as the original stochastic optimization method. In contrast to traditional Monte-Carlo methods using pseudo random numbers, the quasi-Monte Carlo method produces deterministic sequences of well-chosen points that provide the best-possible spread in the change ranges of variables [30]. These deterministic sequences are often referred to as low-discrepancy sequences filling the sample area efficiently and uniformly [31], which have been successfully used to solve globally optimal problems [30], [31], [32], [33].

The major objective of this paper is to propose an improved PSO algorithm to benefit by the advantages of high-order nonlinear time-varying inertia weight and low-discrepancy sequence. This paper is organized as follows. Section 2 reviews the classical PSO and some commonly accepted improved version of PSO. In Section 3, different-order nonlinear functions, varying from 1/18 to 9, are tested to adjust the inertia weight and two acceleration coefficients, and the corresponding parametric studies on the orders are also implemented. The improved PSO with adjusted coefficients are further test with different population sizes. Section 4 presents an improved PSO method with low-discrepancy sequence initialized particles and high-order (1/π²) nonlinear dynamic varying inertia weight (LHNPSO), and experimental results by using well-known benchmark test functions. Section 5 gives a brief conclusion about this study.