Adaptive Central Force Optimization Algorithm Based on the Stability Analysis
Adaptive Central Force Optimization Algorithm Based on the Stability Analysis
Weiyi Qian,1 Bo Wang,1 and Zhiguang Feng2
1College of Mathematics and Physics, Bohai University, Jinzhou 121000, China
2College of Information Science and Technology, Bohai University, Jinzhou 121000, China
Received 15 October 2014; Revised 20 February 2015; Accepted 26 February 2015
Academic Editor: Farhang Daneshmand
Copyright © 2015 Weiyi Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In order to enhance the convergence capability of the central force optimization (CFO) algorithm, an adaptive central force optimization (ACFO) algorithm is presented by introducing an adaptive weight and defining an adaptive gravitational constant. The adaptive weight and gravitational constant are selected based on the stability theory of discrete time-varying dynamic systems. The convergence capability of ACFO algorithm is compared with the other improved CFO algorithm and evolutionary-based algorithm using 23 unimodal and multimodal benchmark functions. Experiments results show that ACFO substantially enhances the performance of CFO in terms of global optimality and solution accuracy.
1. Introduction
Consider the following global optimization problem:where is a real-valued bounded function and , , and are -dimensional continuous variable vectors. Such problem arises in many applications, for example, in risk management, applied sciences, and engineering design. The function of interest may be nonlinear and nonsmooth which makes the classical optimization algorithms easily fail to solve these problems. Over the last decades, many nature-inspired heuristic optimization algorithms without requiring much information about the function became the most widely used optimization methods such as genetic algorithms (GA) [1], particle swarm optimization (PSO) [2], ant colony optimization (ACO) [3], cuckoo search (CS) algorithm [4], group search optimizer (GSO) [5], and glowworm swarm optimization (GSO1) [6]. These search methods all simulate biological phenomena. Different from these algorithms, some heuristic optimization algorithms based on physical principles have been developed, for example, simulating annealing (SA) algorithm [7], electromagnetism-like mechanism (EM) algorithm [8], central force optimization (CFO) algorithm [9], gravitational search algorithm (GSA) [10], and charged system search (CSS) [11]. SA simulates solid material in the annealing process. EM is based on Coulomb’s force law associated with electrical charge process. GSA and CFO utilize Newtonian mechanics law. CSS is based on Coulomb’s force and Newtonian mechanics laws. Unlike other algorithms, CFO is a deterministic method. In other words, there is not any random nature in CFO, which attracts our attention on the CFO algorithm in this paper.
CFO, which was introduced by Formato in 2007 [9], is becoming a novel deterministic heuristic optimization algorithm based on gravitational kinematics. In order to improve the CFO algorithm, Formato and other researchers developed many versions of the CFO algorithm [12–23]. In [12, 13], Formato proposed PR-CFO (Pseudo-Random CFO) algorithm. The improved implementations are made in three areas: initial probe distribution, repositioning factor, and decision space adaptation. Formato presented an algorithm known as PF-CFO (Parameter Free CFO) in [14, 15]. PF-CFO algorithm improves and perfects the PR-CFO algorithm in the aspect of the selection of parameter. Mahmoud proposed an efficient global hybrid optimization algorithm combining the CFO algorithm and the Nelder-Mead (NM) method in [16]. This hybrid method is called CFO-NM. An extended CFO (ECFO) algorithm was presented by Ding et al. by adding the historical information and defining an adaptive mass in [17] where the convergence of ECFO algorithm was proved based on the second order difference equation.
In aforementioned CFO algorithms, two updated equations were used: one for a probe’s acceleration and the other for its position. In the probe’s position updated equation which is established based on the laws of motion, the velocity is defined as zero. But the velocity has influence on the exploring ability of the CFO algorithm. Therefore, in this paper, we introduce the velocity in the probe’s position updated equation, which leads us to build a velocity updated equation like the CSS algorithm. Since the weight which can balance the global and local search ability is an important parameter in many heuristic algorithms, we introduce weight in probe’s position updated equation. If the value of weight is too large, then the probes may move erratically, going beyond a good solution. On the other hand, if weight is too small, then the probe’s movement is limited and the optimal solution may no (...truncated)