Research Article | Open Access
Biwei Tang, Zhanxia Zhu, Jianjun Luo, "A Framework for Constrained Optimization Problems Based on a Modified Particle Swarm Optimization", Mathematical Problems in Engineering, vol. 2016, Article ID 8627083, 19 pages, 2016. https://doi.org/10.1155/2016/8627083
A Framework for Constrained Optimization Problems Based on a Modified Particle Swarm Optimization
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.
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 , resource allocation , and economic environmental scheduling  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 , evolutionary algorithms (EAs), such as genetic algorithm (GA) , simulated annealing (SA) , and differential evolution (DE) , 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 . To tackle the stagnation issue, they proposed a new DE algorithm to evolve the personal best experience of particles in their hybrid PSO . For sufficiently balancing the exploration and exploitation capabilities of PSO, Taherkhani and Safabakhsh  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 . 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 .
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  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  first proposed PSO in 1995. The aim of original PSO is to reproduce the social interactions among agents to solve different optimization problems . 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 , 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  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.
(a) Changes of under different
(b) Changes of under different
(c) Changes of under different
3. Analytical Investigations on SASPSO 2011
3.1. Mathematical Theory of Convergence and Some Basic Concepts
When designing a PSO algorithm, the convergence of the algorithm remains a key issue . However, owing to the stochastic nature of PSO, it results in difficulties in theoretically investigating the global or local convergence of the algorithm [15, 17]. Fortunately, Solis and Wets  have studied under which conditions the stochastic algorithms such as PSO can be considered as either globally or locally convergent search algorithms. In , it has been proven that only if a stochastic algorithm satisfies the algorithm condition and the convergence condition can the algorithm be considered as a local convergence algorithm. Since we will use these two conditions to analyze the local convergence of SASPSO 2011, these two conditions and some relevant notations presented in [15, 29] are first reproduced below for convenience.
Optimality Region. Let be the mapping that integrates the position of a particle with the current global best solution . In a minimization optimization problem, the optimality region can be mathematically described as follows : where denotes the optimal solution of on ; denotes the objective function; denotes the search space; is a positive coefficient.
Algorithm Condition. In a minimization optimization problem, mapping must satisfy and if , then , where denotes the iteration number (monotonic condition).
The algorithm condition on mapping stipulates that the solution obtained at current iteration is not worse than the previous ones. In other terms, at each iteration, if the newly obtained fitness dominates the previously best one, then the new one replaces the previously best fitness. Otherwise, the best fitness remains unchanged.
Convergence Condition. Based on the definition of the optimal region given by (12), the convergence to a local optimum is obtained by the sufficient condition: for all , there exists and , such that where represents a probability measure; denotes the solution obtained by the optimization problem at iteration ; is the distance between a point and a set . For more details about , the reader is referred to .
The convergence condition implies that a stochastic algorithm can be considered as an optimization problem if the particle has a nonzero probability to move closer to the optimality region by a minimum distance at each iteration or if the particle is already located at the optimality region with a probability no less than .
Theorem 1. Suppose is a measurable function, represents a measure subset of , and a stochastic algorithm satisfies both the algorithm and convergence conditions. Then, considering the sequence searched by the algorithm, the following condition holds : where denotes the probability that the algorithm converges to the optimal region .
Theorem 1 indicates that only if a stochastic algorithm satisfies the algorithm condition and convergence condition stated above, can it be considered as a local convergence algorithm and the local optimality of the algorithm can be at least guaranteed. The proof of this theorem can be found in .
3.2. Local Convergence Proof of SASPSO 2011
3.2.1. Verification of the Algorithm Condition
The algorithm condition can be validated through applying mapping used in SASPSO 2011. Let represent the global best position set searched by particles in SASPSO 2011. In a minimization optimization problem, after each iteration, mapping in SASPSO 2011 is updated as follows: where denotes the obtained position of particle at iteration . According to (15), if the newly obtained solution is better than , replaces in SASPSO 2011. Otherwise, is kept in . Therefore, the obtained sequence searched by particles in SASPSO 2011 is monotonic. Thus, SASPSO 2011 satisfies the algorithm condition.
3.2.2. Stability Analysis for SASPSO 2011
Prior to verifying that SASPSO 2011 satisfies the convergence condition, we first investigate the stability of SASPSO 2011. The aim of the stability analysis is to find boundaries of the three control parameters to guarantee the trajectory convergence of particles in SASPSO 2011. Note that this study focuses on the deterministic model stability analysis  for SASPSO 2011 in the case where . Without loss of generality, by omitting the subscript of each variable in (1)–(3) for simplicity, the update rules of particles in SASPSO 2011 can be rewritten into a matrix form as follows: where
Solving , where is the identity matrix with the same size of , the characteristic equation of the dynamic system (16) is derived as follows: where two roots, denoted by , are obtained as follows:
In the context of the dynamic system theory, the necessary and sufficient condition for the convergence of system (16) is that magnitudes of and are less than 1 [15, 30]. Therefore, system (16) converges, if and only if
(1) The first case is where are two complex numbers, denoted as .
Lemma 2. For system (16), , if and only if
Now, let us find conditions on and guaranteeing the convergence of system (16) in the case where . It is trivial that the system converges if and only if .
Lemma 3. System (16) converges in the case where , if and only if
Proof. Note that the magnitude of a complex number can be calculated as , where and denote the real and imaginary parts of . Hence, for , it is clear that Therefore For , according to Lemma 2, (21) must be held. Hence, when considering conditions that and , for , system (16), that is, SASPSO 2011, converges, if and only if
In the case where , the convergent region of SASPSO 2011 is shown in Figure 2.
(2) The second case is where and are two real numbers, denoted as .
Lemma 4. For system (16), , if and only if
As , it is clear that
Solving the right-hand inequalities in (31) yields
Then, considering both cases where and together, system (16), that is, SASPSO 2011, converges, if and only if
Figure 3 illustrates the convergent region of SASPSO 2011 in both cases where and . Only if any parameter selection of and locates in this triangle area as shown in Figure 3 does SASPSO 2011 guarantee the trajectory convergence of particles. Figure 4 shows a 3-dimensional representation of the value of .
(a) The case where
(b) The case where
Now, let us find the equilibrium point of SASPSO 2011. Calculating limits on both sides of (16) yields
When SASPSO 2011 converges, and . Therefore, substituting these two equations into (35) yieldswhere and denote the personal best position of the particle and the global best position of the swarm, respectively.
Through the stability analysis, it can be concluded that if and only if the convergence condition given by (34) is satisfied, the trajectories of particles can converge to the equilibrium point given by (36). Here, note that only the trajectory convergence of the particle can be assessed through the stability analysis. The global or local optimality of the equilibrium point cannot be assessed through the stability analysis.
3.2.3. Verification of the Convergence Condition
As stated in Section 3.1, only if a stochastic algorithm satisfies the algorithm and convergence conditions can it be considered as local convergence algorithm; namely, the local optimality of the algorithm can be at least guaranteed. In Section 3.2.1, it is proven that SASPSO 2011 satisfies the algorithm condition. Therefore, we still need to assess that SASPSO 2011 satisfies the convergence condition in order to investigate the local convergence of the algorithm.
Before proving that SASPSO 2011 satisfies the convergence condition described in Section 3.1, we first reproduce some notations presented in [15, 29] for convenience. Let denote the worst particle in a swarm with particles. In a minimization optimization problem, the worst particle is the one with the largest cost function in the swarm. Hence, can be defined as , where . From the worst particle, the convex compact set can be defined as the convex compact set in which all points have a fitness value smaller than or equal to [15, 29]:
By the definition of , it can be assessed that and in (36) are in . Since is convex and and are in , any point along the line connecting and is thus in . Since the equilibrium point given by (36) is on the line connecting and , we can conclude that the equilibrium point given by (36) is in . Moreover, in Section 3.2.2, it is proven that the trajectories of particles in SASPSO 2011 converge to the equilibrium point given by (36), if and only if the convergent condition given by (34) is satisfied. Although the optimality of this equilibrium point cannot be stated from this proof, it allows us at least to conclude that, regardless of the initial position of the particle, the particle can converge to the equilibrium point given by (36), which is in . In other terms, SASPSO 2011 can always generate a new point that can be sampled arbitrarily close to the point given by (36), which is in . Thus, we can conclude that there always exists a nondegenerated sampling hypersphere which guarantees that a new point can be sampled arbitrarily close to the equilibrium point given by (36), which is in .
As stated in Section 2.1, the stagnation is prevented in SPSO 2011 by adding an arbitrary point to the particle’s velocity as given in (2) [22, 23]. Since the same moving rules defined in SPSO 2011 are used in SASPSO 2011, SASPSO 2011 is a nonstagnant algorithm. Considering the nonstagnation property of SASPSO 2011, the algorithm can always improve the fitness of the global best position with a nonzero probability. Thus, we can conclude that, given any starting point in , SASPSO 2011 guarantees a nondegenerate sampling volume with a nonzero probability of sampling a point closer to the optimality region , as described by the convergence condition in Section 3.1. Hence, it is sufficient to conclude that SASPSO 2011 satisfies the convergence condition. Note that the authors in  used the same method to prove that their PSO algorithm satisfies the convergence condition. Since SASPSO 2011 satisfies both the algorithm and convergence conditions, it is a locally convergent algorithm.
3.3. Convergence Behavior of Particles in SASPSO 2011
Before particles converge to the equilibrium point given in (36), they may oscillate in different ways around the equilibrium point as a result of different values of and . Since different convergence oscillations may influence the quality of the final solution searched by particles [17, 18], it is necessary to investigate the oscillation behavior of particles. Four typical oscillations of particles in SASPSO 2011 are shown in Figure 5.
(a) Nonoscillatory convergence
(b) Harmonic convergence
(c) Zigzagging convergence
(d) Harmonic convergence combined with zigzagging convergence
Nonoscillatory convergence behavior, as shown in Figure 5(a), leads particles to only search on one side of the equilibrium point, which may be useful when the search space is bounded. Particles exhibit nonoscillatory convergence when and are two real roots and at least one of them is positive, which is equivalent to and . Harmonic oscillation, as demonstrated in Figure 5(b), may be beneficial in the exploitation stage, since particles smoothly oscillate around the equilibrium point. Harmonic oscillation behavior occurs when and are complex; that is, . Zigzagging convergence, as illustrated in Figure 5(c), may also facilitate exploitation as particles zigzag around the equilibrium point, which may be suitable for solving optimization problems with rugged search spaces. Particles display zigzagging convergence behavior when at least one of and has a negative real part; that is, or . The combined harmonic behavior with zigzagging behavior, as visualized in Figure 5(d), may be beneficial for the transition from exploration to exploitation due to its mixed nature. The combined harmonic behavior with zigzagging behavior emerges when at least one of the two complex roots and has a negative real part; that is, or .
If the boundaries of the coefficients associated with these oscillations are known beforehand, one may easily design an adaptive method to change values of these coefficients, so that the convergence of PSO can be guaranteed and the quality of the final solution found by particles could be improved.
3.4. Parameter Selection Principle for SASPSO 2011
In Section 3.2.2, it is proven that if and only if the convergent condition given by (34) is satisfied, SASPSO 2011 can converge to the equilibrium point given by (36). Now, we need to answer how to set the initial and final values of , , and to guarantee the convergence of SASPSO 2011. Please note that, without loss of generality, the subscript is omitted from , , and for simplicity.
Lemma 5. SASPSO 2011 converges, only if the initial and final values of , , and satisfy the following:
Proof. Since , the necessary and sufficient condition given by (34) can be rewritten asWhen , it appears from (6), (7), (9), and (10) that for any particle at any iteration in SASPSO 2011. From (6) to (8), it is clear that , , and for any particle at any iteration in SASPSO 2011. Therefore Since the right-hand side of (41) is the necessary and sufficient condition for the convergence of SASPSO 2011, Lemma 5 can be easily proven based on the relationship given by (41).
Since , , , , , and are predefined parameters, the convergent condition given by (39) can be easily satisfied by setting proper values of these parameters. Figure 6 shows the convergent position and velocity trajectories of the particle in SASPSO 2011 under the suggested parameter selection: , , , and .
(a) Convergent position trajectory
(b) Convergent velocity trajectory
4. Applying SASPSO 2011 for Solving COPs
4.1. Modeling of COPs
Generally, a constrained optimization problem is to optimize an objective function under some equality/inequality constraints, which can be mathematically presented as where is the objective function; is the -dimensional vector of decision variables; and denote the th equality and th inequality constraints, where and ; and are the number of equality and inequality constraints, respectively; and represent the lower and upper bounds of .
4.2. Constraint Handling Technique
As shown in (43)–(45), COPs are usually constrained by some conditions. To easily and efficiently solve COPs, the task of how to handle constraints of COPs must be addressed. Among the existing handling constraint techniques, the penalty function method may be the most common approach . By adding penalty terms to the objective function, the penalty function method can transform a constrained optimization problem into an unconstrained one. The drawback of the method is that determining proper values for the penalty factors is time-consuming and problem-dependent, which requires the previous experience of the users .
To overcome the shortcoming of the penalty function method and increase the diversity of solutions, the adaptive relaxation method [4, 5], which is integrated with the feasibility-based rule [24, 25], is applied to handle constraints of COPs and evaluate candidate solutions in this paper.
In the adaptive relaxation method, in order to handle equality and inequality constraints of COPs, the total constraint violation value of particle is first calculated as follows [4, 5]: where , , , , , and have the same definitions as those in (43) and (44).
After calculating the total constraint violation value of each particle, the median of total constraint violation values of all particles is assigned to the initial constraint relaxation value . If the total constraint violation value of a particle is less than , the particle is temporarily considered as a feasible solution; otherwise, it is temporarily considered as a nonfeasible solution. During the evolution, is gradually reduced according to the fraction of feasible particles () with respect to the relaxed constraints. Given a swarm with particles, from iteration to iteration , is updated as follows :
From (47), it is clear that more nonfeasible particles are allowed into the next iteration in the early phase of the evolution, which can consequently add some diversification to the swarm . With the evolution continuing, the relaxation value adaptively decreases when more and more feasible solutions are found. Hence, the feasible region gradually reduces until it converges to the real feasible region, which may increase the possibility of finding optimal solutions . Since is adaptively updated, no additional penalty factor is needed in the adaptive relaxation method, which can thus reduce the optimization difficulties .
After calculating the fitness and constraint violation values of each particle, the feasibility-based rule [24, 25] is applied to evaluate and select the elite solution between any two candidate solutions. The feasibility-based rule can be described as follows: for any two solutions with same constraint violation value, the solution with better fitness value is preferred; for any two solutions with different constraint violation values, the solution with smaller constraint violation value is preferred.
As the fitness and constraint violation information is considered separately in the feasibility-based rule, no additional parameter is needed when using this rule, which can decrease the optimization difficulties [24, 25]. Moreover, although the nonfeasible solutions violate some constraints, they may also contain some information that is useful for finding good solutions . When this useful information is considered, the likelihood of finding high-quality solutions may be increased. This is one main reason why the nonfeasible solutions are considered in the feasibility-based rule.
Except the equality and inequality constraints, each design variable must satisfy its boundary constraints, as shown in (45). When any variable () cannot satisfy its boundary constraints, the saturation strategy is applied to modify as follows : where , , and have the same definitions as those in (45).
4.3. The SASPSO 2011-Based Framework for Solving COPs
Let denote the size of the swarm and let denote the maximum iteration number. The algorithmic scheme of applying SASPSO 2011 to solve COPs is illustrated in Algorithm 1.
5. Numerical Simulations and Analysis
To verify the proposed method, it is tested through 4 benchmark test functions and 2 real-world engineering problems against time-varying PSO (TVPSO) , constrained PSO (CPSO) , fine grained inertia weight PSO (FGIWPSO) , stability-based adaptive inertia weight PSO (SAIWPSO) , SPSO 2011 [22, 23], random PSO (RPSO) , and some well-established methods proposed in the literature. Note that the 4 benchmark test functions and 2 real-world engineering problems are listed in the Appendix.
In order to reduce the random discrepancy, a Monte Carlo experiment with 50 independent runs is conducted for each studied problem. After 50 independent runs, the best, mean, and worst results, as well as the standard deviation of each method, are examined and compared. In each run, the final solution of each method is obtained after 4000 iterations of 100 particles on MATLAB 2012B software on a Windows 8 personal computer with i3-2350 @ 2.30-GHz and 2 GB RAM. The simulation parameters for the different methods are shown in Table 1.
5.1. Simulations on 4 Benchmark Test Functions
This subsection shows the simulations results of all methods for the 4 benchmark test functions. The mathematical characteristics of the 4 benchmarks are shown in Table 2, where NV denotes the number of variables, LI and NI are, respectively, the numbers of linear and nonlinear inequality constraints, and LE and NE are, respectively, the numbers of linear and nonlinear equality constraints.
After performing 50 independent runs, the best, mean, worst, and standard deviation results of all methods for the four test functions are summarized in Tables 3–6, respectively. Note that, as shown in Tables 3–6, the statistical results of Krohling and Coelho  and Mezura-Montes and Coello Coello  are extracted from their corresponding references. In all cases, the best results with respect to the best, mean, and worst values and standard deviation are highlighted in boldface in these tables. Figure 7 displays the fitness curves of the best results of all PSO methods for the four test functions.
(a) Best fitness curves for
(b) Best fitness curves for
(c) Best fitness curves for
(d) Best fitness curves for
From Tables 3–6, it is evident that SASPSO 2011 and SAIWPSO are the first and second most efficient algorithms in terms of the average solution, which implies that SASPSO 2011 and SAIWPSO generally outperform the other algorithms in solving these 4 test functions. It can be also seen from these tables that the proposed algorithm can find the best solutions for the 4 benchmark test functions, and SAIWPSO can find the best worst solutions for , , and .
Moreover, it is clear from Tables 3–6 that SAIWPSO and SASPSO 2011 are the most stable and second most stable algorithms, since they can provide the least and second least standard deviations in solving the 4 benchmark test functions. The reason why our proposed algorithm and SAIWPSO are more robust than the other algorithms may be that these two algorithms are stability-based algorithms. By setting proper values of the three control parameters, the stability and convergence of these two algorithms can be guaranteed. It is notable from Tables 3–6 that the difference of the standard deviation between our proposed method and SAIWPSO is not significant for each test function.
Therefore, summarizing the simulation results shown in Tables 3–6, it can be concluded that the proposed method is highly competitive in solving the 4 benchmarks in terms of the solution quality. Moreover, the search robustness of the proposed method is comparable with that of SAIWPSO and outperforms those of the other methods.
5.2. Application on Two Real-World Engineering Problems
To further verify the proposed method, it is applied to solve two practical engineering problems: the tension compression spring design problem and the three-bar truss design problem. Note that the performance of the proposed algorithm is still compared with those of the aforementioned PSO algorithms and some other well-known methods proposed in the literature.
5.2.1. Tension Compression Spring Design Problem
The statistical results of different methods for the tension compression spring design problem are shown in Table 7. Note that, in this table, the statistical results of PSO-DE  and CDE  are extracted from their corresponding literature. Table 8 exhibits the details of the best solution searched by each method for this problem. The best fitness curves of different PSO algorithms are visualized in Figure 8.