Mathematical Problems in Engineering

Volume 2016, Article ID 8627083, 19 pages

http://dx.doi.org/10.1155/2016/8627083

## A Framework for Constrained Optimization Problems Based on a Modified Particle Swarm Optimization

^{1}National Key Laboratory of Aerospace Flight Dynamics, Xi’an, Shaanxi 710072, China^{2}School of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Received 1 March 2016; Revised 19 May 2016; Accepted 15 June 2016

Academic Editor: László T. Kóczy

Copyright © 2016 Biwei Tang 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

This paper develops a particle swarm optimization (PSO) based framework for constrained optimization problems (COPs). Aiming at enhancing the performance of PSO, a modified PSO algorithm, named SASPSO 2011, is proposed by adding a newly developed self-adaptive strategy to the standard particle swarm optimization 2011 (SPSO 2011) algorithm. Since the convergence of PSO is of great importance and significantly influences the performance of PSO, this paper first theoretically investigates the convergence of SASPSO 2011. Then, a parameter selection principle guaranteeing the convergence of SASPSO 2011 is provided. Subsequently, a SASPSO 2011-based framework is established to solve COPs. Attempting to increase the diversity of solutions and decrease optimization difficulties, the adaptive relaxation method, which is combined with the feasibility-based rule, is applied to handle constraints of COPs and evaluate candidate solutions in the developed framework. Finally, the proposed method is verified through 4 benchmark test functions and 2 real-world engineering problems against six PSO variants and some well-known methods proposed in the literature. Simulation results confirm that the proposed method is highly competitive in terms of the solution quality and can be considered as a vital alternative to solve COPs.

#### 1. Introduction

Over the last few decades, constrained optimization problems (COPs) have rapidly gained increasing research interests, since they are frequently encountered in different areas such as path planning [1], resource allocation [2], and economic environmental scheduling [3] to name but a few. Generally, solving a constrained optimization problem is to optimize a predefined objective function under some equality and/or inequality constraints [4, 5]. Nevertheless, owing to the nonlinearity in either the objective or constraints, or both, efficiently solving COPs remains a big challenge [4, 5]. Therefore, far more effective optimization algorithms are always needed.

Due to their population-based nature and promising search ability to produce high-quality solutions, even for complex optimization problems [6], evolutionary algorithms (EAs), such as genetic algorithm (GA) [2], simulated annealing (SA) [3], and differential evolution (DE) [7], have been proposed for solving different COPs. As one of the most powerful EAs, thanks to its simplicity and high convergence speed, particle swarm optimization (PSO) has been widely and successfully applied to solve different COPs in recent years [8–12].

Yet, since the basic PSO algorithm suffers from some drawbacks such as stagnation and poor ability in balancing exploration and exploitation, its optimization efficiency may be restricted [13, 14]. In order to improve the performance of the PSO, these weaknesses must be overcome. Moreover, when designing a PSO algorithm, the convergence of PSO is paramount because this property significantly influences the performance of PSO [15, 16]. To date, despite some studies investigating the convergence to an equilibrium point of PSO [16–19], the optimality of this point is not clearly established. Actually, it is still difficult to theoretically analyze the global or local convergence (i.e., the global or local optimality of this equilibrium point) of PSO due to its stochastic nature [15, 16].

So far, many researchers have committed themselves to developing different PSO algorithms in order to enhance the performance of PSO. Liu et al., proposed a hybrid PSO algorithm which hybridizes PSO algorithm with differential evolution (DE) algorithm in [20]. To tackle the stagnation issue, they proposed a new DE algorithm to evolve the personal best experience of particles in their hybrid PSO [20]. For sufficiently balancing the exploration and exploitation capabilities of PSO, Taherkhani and Safabakhsh [21] proposed a novel stability-based PSO algorithm, in which an adaptive approach is developed to determine the inertia weight of each particle in different dimensions. Furthermore, by considering the stability condition and the adaptive inertial weight, the cognitive and social acceleration parameters of their proposed PSO algorithm are adaptively determined [21]. Through extensive simulations on different benchmark test functions and a real-world application, the effectiveness and superiority of their proposed PSO have been validated in [21].

Additionally, among the currently existing PSO variants, based on the best knowledge of the authors, the standard particle swarm optimization (SPSO 2011) algorithm [22, 23] may be one of the most recent standard versions for PSO. By randomly drawing a point in a hypersphere which is centered on three points, the current position of the particle, a point a little “beyond” the personal best position of the particle, and a point a little “beyond” the global best position of the swarm, the nonstagnation property can be achieved in SPSO 2011 [22, 23]. However, since its three control parameters (i.e., the inertial weight and the cognitive and social acceleration parameters) are constant and there is no distinction between the cognitive acceleration parameter and the social acceleration parameter, SPSO 2011 cannot dynamically adjust the exploration and exploitation abilities. Besides, the convergence and the stability of SPSO 2011 have not been investigated in [22, 23].

Considering the advantage and the disadvantage of SPSO 2011, we propose a modified PSO algorithm, called SASPSO 2011, which is developed based on SPSO 2011. The main consideration of the development of SASPSO 2011 is to exploit the advantage (i.e., nonstagnation property) and overcome the shortcoming (i.e., poor ability in balancing exploration and exploitation) of SPSO 2011, so that the performance of SASPSO 2011 can be enhanced. To this end, particles in SASPSO 2011 first follow the same moving rules defined in SPSO 2011 to update their velocities and positions to prevent stagnation in SASPSO 2011. Then, a new self-adaptive strategy is developed for fine-tuning the three control parameters of particles in SASPSO 2011 to well balance the exploration and exploitation abilities of SASPSO 2011.

Although SASPSO 2011 is developed based on SPSO 2011, there are significant differences between these two algorithms, since(1)a novel self-adaptive strategy is proposed for fine-tuning the three control parameters of particles in SASPSO 2011,(2)the stability and the local convergence of SASPSO 2011 are investigated,(3)the convergence behavior of particles in SASPSO 2011 is investigated,(4)a parameter selection principle that can guarantee the local convergence of SASPSO 2011 is provided.

After analytical investigation on SASPSO 2011, this paper designs a SASPSO 2011-based framework for solving COPs. In order to easily handle constraints of COPs and release the burden of implementing the optimization algorithm, the adaptive relaxation method [4, 5] is combined with the feasibility-based rule [24, 25] to handle constraints of COPs and evaluate candidate solutions in the framework established. To verify the proposed method, it is compared to six state-of-the-art PSO variants and some methods proposed in the literature by solving 4 benchmark test functions and 2 real-world engineering problems. The simulation results show that the proposed method is highly competitive in finding high-quality solutions. Furthermore, the search stability of the proposed method is comparable with that of SAIWPSO [21] and outperforms those of the other compared methods. Thus, the proposed method can be considered as an effective optimization tool to solve COPs.

The remainder of this paper is organized as follows. After briefly reviewing SPSO 2011, SASPSO 2011 is presented in Section 2. Section 3 theoretically investigates some properties such as the stability, the local convergence, the convergence behavior of particles, and parameter selection principle pertaining to SASPSO 2011. The SASPSO 2011-based framework for COPs is described in Section 4. Simulations and comparisons are performed in Section 5. Section 6 summarizes this paper by way of a conclusion and options of future work.

#### 2. Particle Swarm Optimization (PSO)

##### 2.1. Review of SPSO 2011

Inspired by birds flocking and fish schooling, Eberhart and Kennedy [26] first proposed PSO in 1995. The aim of original PSO is to reproduce the social interactions among agents to solve different optimization problems [27]. Each agent in PSO is called a particle and is associated with a velocity that is dynamically adjusted accordingly to its own flight experience, as well as those of its companions. Since the first introduction of PSO in 1995, many different PSO algorithms have been proposed, among which SPSO 2011 [22, 23] may be one of the most recently proposed PSO algorithms. Because our proposed PSO algorithm is developed on the basis of SPSO 2011, this subsection will briefly describe SPSO 2011.

Let and denote the position and velocity of particle in a swarm with particles in a search space for at iteration . stands for the personal best position of the th particle at iteration . represents the global best position of the swarm at iteration . The position of particle at iteration is obtained from three components: the current velocity , the personal best position , and the global best position . Let denote the isobarycenter of the th particles , , and , where and are two positive real coefficients denoting the cognitive and social acceleration parameters. The coordinates of the barycenter can be obtained as follows [22, 23]:

Then, a point is randomly drawn in a hypersphere that is centered on with a radius . After randomly obtaining a point in the hypersphere, each particle updates its velocity and position as follows [22, 23]: where is a real coefficient representing the inertia weight. In [22], it is suggested that if , .

In SPSO 2011, the particle can always explore the surroundings of the explored region with a nonnull velocity, since a random point is added to the particle’s velocity as shown in (2). Hence, the nonstagnation property can be achieved in SPSO 2011 [22, 23]. The authors in [22] also propose to set the three control parameters of SPSO 2011 as follows:

##### 2.2. Description of SASPSO 2011

When applying PSO to solve an optimization problem, it is necessary to properly control the exploration and exploitation abilities of PSO in order to find optimal solutions efficiently [13, 14, 28]. Ideally, on one hand, in the early stage of the evolution, the exploration ability of PSO must be promoted so that particles can wander through the entire search space rather than clustering around the current population-best solution [13, 14, 28]. On the other hand, in the later stage of the evolution, the exploitation ability of PSO needs to be strengthened so that particles can search carefully in a local region to find optimal solutions efficiently [13, 14, 28].

Proverbially, the exploration and exploitation capabilities of PSO heavily depend on its three control parameters. The basic philosophies concerning how the three control parameters influence such abilities of PSO can be summarized as follows: a large inertia weight enhances exploration, while a small inertia weight facilitates exploitation [13, 14, 28]; a large cognitive component, compared to the social component, results in the wandering of particles through the entire search space, which strengthens exploration [14, 27]; a large social component, compared with the cognitive component, leads particles to a local search, which strengthens exploitation [14, 27].

According to the basic philosophies noted above, although SPSO 2011 is a nonstagnation algorithm, it cannot strike a good balance between exploration and exploitation, since its three control parameters remain unchanged and there is no difference between and . Considering the weakness of SPSO 2011, we propose a modified PSO algorithm, which is developed based on SPSO 2011 and is named SASPSO 2011. The main purpose of the development of this PSO is to adaptively adjust the exploration and exploitation abilities of SASPSO 2011. To achieve this goal, a novel self-adaptive strategy that is used to update the three control parameters of particles in SASPSO 2011 is proposed as follows: wherewhere and denote the initial and final values of the inertia weight, respectively; and represent the initial and final values of the cognitive acceleration parameter, respectively; and stand for the initial and final values of the social acceleration parameter, respectively; is the maximum iteration number; denotes the Euclidean distance between the personal best position of particle and the global best position of the swarm at iteration . Note that , , and in SASPSO 2011. Also, note that particles in SASPSO 2011 update their velocities and positions based on (1)–(3) in order to avoid them falling into stagnation.

###### 2.2.1. Parametric Analysis for SASPSO 2011

From (5) to (7), with increasing iteration number , it is clear that and decrease, while increases in SASPSO 2011. Therefore, according to the aforementioned basic philosophies, SASPSO 2011 may start with high exploration, which will be reduced over time, so that exploitation may be favored in the later phase of the evolution. Note that, following the update rule of a fixed , the balance between exploration and exploitation varies only with respect to the iteration number .

We also adapt the balance of the search in SASPSO 2011 using an additional parameter . From (5) and (6), it is trivial that and decrease as increases. On the other hand, the variation in becomes larger as increases according to (7). This implies that, for large , the exploration capability of SASPSO 2011 tends to be retained. According to (11), a large indicates that the personal best position of the particle is far away from the global best position of the swarm. Therefore, in the case where is large, it is natural to facilitate the exploration capability of the algorithm, so that the particle is promoted to a global search and can quickly move closer to the global best position.

In contrast, in the case where is small, the exploitation ability of the algorithm takes over the exploration ability more rapidly as decreases. It is also natural to strengthen the exploitation ability of the algorithm in the case where is small, because, according to (11), small implies that the personal best position of the particle is close to the global best position of the swarm. In this case, through strengthening the exploitation ability of the algorithm, particles tend to a local search around the global best position, so that the possibility of improving the quality of the global best solution can be increased.

Briefly, by utilizing the proposed self-adaptive strategy, the three control parameters of the algorithm can be adaptively adjusted, complying with the basic philosophies of PSO development. Hence, SASPSO 2011 is expected to improve the ability in finding high-quality solutions. Figure 1 demonstrates the tendency of these changes in the three control parameters with respect to different values of . Note that , , , , and in Figure 1.