Opposition-Based Barebones Particle Swarm for Constrained Nonlinear Optimization Problems

. This paper presents a modiﬁed barebones particle swarm optimization (cid:2) OBPSO (cid:3) to solve constrained nonlinear optimization problems. The proposed approach OBPSO combines barebones particle swarm optimization (cid:2) BPSO (cid:3) and opposition-based learning (cid:2) OBL (cid:3) to improve the quality of solutions. A novel boundary search strategy is used to approach the boundary between the feasible and infeasible search region. Moreover, an adaptive penalty method is employed to handle constraints. To verify the performance of OBPSO, a set of well-known constrained benchmark functions is used in the experiments. Simulation results show that our approach achieves a promising performance.


Introduction
Many engineering problems can be converted to constrained optimization problems.The aim of constrained optimization is to find a feasible solution with minimized cost In this paper, we only consider minimization problems .A general constrained minimization optimization problem can be defined as follows.
Minimize f x 1.1 subject to g j x ≤ 0, j 1, 2, . . ., q, h j x 0, j q 1, 2, . . ., m, where g x is inequality constraint, h x is the equality constraint, m is the number of constraints, q is the number of inequality constraints, and m − q is the number of equality constraints.
Particle swarm optimization PSO is a population-based stochastic search algorithm developed by Kennedy and Eberhart 1 .Although PSO shares many similarities with evolutionary algorithms EAs , the standard PSO does not use evolution operators such as crossover and mutation.For PSO's simple concept, easy implementation, and effectiveness, it has been widely applied to many optimization areas.
Although PSO has shown a good performance over many optimization problems, it does not work well when solving complex problems.Especially for constrained optimization problems, the standard PSO could hardly search promising solutions.The possible reason is that constrained optimization problems are usually multimodal and having some constraints.PSO could easily fall into local minima and hardly search feasible solutions.To enhance the performance of PSO on constrained optimization problems, many improved PSO variants have been proposed in the past several years.
Meng et al. 2 used a quantum-inspired PSO QPSO to solve constrained economic load dispatch.The QPSO shows stronger search ability and quicker convergence speed, not only because of the introduction of quantum computing theory, but also due to twonovel strategies: self-adaptive probability selection and chaotic sequences mutation.Coelho 3 presented a novel quantum-behaved PSO QPSO to solve constrained engineering design problems.The proposed approach embedded a Gaussian mutation into QPSO to prevent premature convergence to local minima.Sun et al. 4 proposed an improved vector PSO IVPSO based on multidimensional search, in which a simple constraint-preserving method is used to handle constraints.Liu et al. 5 used a hybrid PSO called PSO-DE to solve constrained numerical and engineering optimization problems.The PSO-DE integrates PSO with differential evolution DE to obtain a good performance.Venter and Haftka 6 proposed a new method to solve constrained optimization problems.The constrained, single objective optimization problem is converted into an unconstrained, biobjective optimization problem that is solved using a multiobjective PSO algorithm.Lu and Chen 7 presented an enhanced PSO by employing a dynamic inertia weight to avoid premature convergence.The inertia weight of every individual is dynamically controlled by the Euclidean distance between individual and the global best individual.Daneshyari and Yen 8 proposed a cultural-based constrained PSO to incorporate the information of the objective function and constraint violation.The archived information facilitates communication among swarms in the population space and assists in selecting the leading particles in three different levels: personal, swarm, and global levels.
There have been many modifications to the original PSO algorithm to improve the efficiency and robustness of the search.Although these modified PSO variants have shown good search abilities, their performance greatly depends on the control parameters in the velocity updating model, such as inertia weight w and acceleration coefficients c 1 and c 2 .Recently, a parameter-free PSO, known barebones PSO BPSO 9 , used Gaussian normal distribution to update the particles in the population.It does not involve inertia weight, acceleration coefficients, and velocity.Its performance has been found to be competitive, and a number of BPSO algorithms have been proposed in the past several years.Omran et al. 10 incorporate the idea of BPSO into DE.Krohling and Mendel 11 employed a jump strategy in BPSO to avoid premature convergence.Motivated by the idea of BPSO, this paper presents an improved BPSO, namely OBPSO, to solve constrained nonlinear optimization problems.In OBPSO, opposition-based learning OBL concept 12 is used for population initialization and generation jumping.To verify the performance of OBPSO, a set of wellknown constrained benchmark problems are used in the experiments.Results obtained by the proposed OBPSO are compared with those in the literature and discussed.
The rest of the paper is organized as follows.In Section 2, the standard PSO and barebones PSO are briefly introduced.Section 3 describes our proposed approach.Section 4 presents experimental simulations, results, and discussions.Finally, the work is concluded in Section 5.

Belief Descriptions of PSO and Barebones PSO
In traditional PSO, a member in the swarm, called a particle, represents a potential solution in the D-dimensional search space.Each particle has two vectors: velocity and position.It is attracted by its previous best particle pbest and the global best particle gbest during the evolutionary process.The velocity v ij and position x ij of the jth dimension of the ith particle are updated according to 2.1 13 : where i 1, 2, . . ., N is the particle's index, N is the population size, X i x i1 , x i2 , . . ., x iD is the position of the ith particle; V i v i1 , v i2 , . . ., v iD represents the velocity of the ith particle; the pbest i pbest i1 , pbest i2 , . . ., pbest iD is the best previous position yielding the best fitness value for the ith particle; gbest gbest 1 , gbest 2 , . . ., gbest D is the global best particle found by all particles so far.The parameter w, called inertia factor, which is used to balance the global and local search abilities of particles 13 , rand1 ij and rand2 ij are two random numbers generated independently within the range of 0, 1 , c 1 and c 2 are two learning factors which control the influence of the social and cognitive components, and t 1, 2, . . .indicates the iteration number.
Recently, Kennedy 9 developed the barebones PSO BPSO .This new version of PSO eliminates the velocity term, and the position is updated as follows.
where x ij t 1 is the position of the ith particle in the population, and G represents a Gaussian distribution with mean gbest j t pbest ij t /2 and standard deviation |gbest j t −pbest ij t |.Note that the particle positions are sampled by the above Gaussian distribution.The BPSO facilitates initial exploration, due to large deviation initially, pbest will be far from the gbest .As the number of generation increases, the deviation approaches to zero, by focussing on exploitation of the pbest and gbest 14 .

Opposition-Based Learning
Opposition-based learning OBL developed by Tizhoosh 12 is a new concept in computational intelligence.It has been proven to be an effective concept to enhance various optimization approaches 15-17 .When evaluating a solution x to a given problem, simultaneously computing its opposite solution will provide another chance for finding a candidate solution which is closer to the global optimum.
Let X x 1 , x 2 , . . ., x D be a candidate solution in a D-dimensional space, where 3.1 By staying within variables' interval static boundaries, we would jump outside of the already shrunken search space and the knowledge of the current converged search space would be lost.Hence, we calculate opposite particles by using dynamically updated interval boundaries a j t , b j t as follows 15 .
x * ij a j t b j t − x ij , 3.2 where x ij is the jth position element of the ith particle in the population, x * ij is the opposite particle of x ij , a j t and b j t are the minimum and maximum values of the jth dimension in current search space, respectively, rand a j t , b j t are random numbers within a j t , b j t , x min , x max is the box-constraint of the problem, and N is the population size, and t 1, 2, . .., indicates the generations.

Adaptive Constraint Handling
To handle the constraints in solving constrained optimization problems, this paper employs an adaptive penalty method APM which was early considered in 18-20 .It aims to help users avoid manually defining the coefficients of penalty functions.In the APM, each constraint of the candidate solutions is monitored.If a constraint seems to be more difficult to satisfy, then a larger penalty coefficient is added.
For each candidate solution, its jth constraint violation cv j x is computed as follows: cv j x h j x min 0, g j x , 3.5 where h j x is the jth equality constraint, and g j x is the jth inequality constraint.

Mathematical Problems in Engineering 5
The fitness value of candidate solution is defined by where m is the number of constraints, k j is a penalty coefficient for each constraint, and f x is defined by where f x is the average objective function values in the current swarm, and it is computed as where N is the population size.
For the penalty coefficient k j , it determines the scaled factor of the jth constraint.Every generation, the k j is adaptively adjusted as follows. 3.9 where cv l x is the violation of the lth constraint averaged over the current swarm, and it is computed by where cv ij x is the violation of ith particle on the jth constraint.

Boundary Search Strategy
For constrained optimization problems, the solution search space can be divided into two parts: feasible space and infeasible space.In some cases, the global optimum is located at the boundaries of feasible space.It is difficult to find this kind of solutions.Because many algorithms can hardly judge the boundaries of feasible space.To tackle this problem, this paper employs a boundary search strategy as follows.

Comparison of OBPSO with Similar PSO Algorithms
In this section, we compare the performance of OBPSO with standard PSO, barebones PSO BPSO , and OBPSO without boundary search strategy OBPSO-1 .To have a fair comparison, the same settings are used for common parameters.The population size N is set to 40.For PSO, The inertia weight w is set to 0.72984.The acceleration coefficients c 1 and c 2 are set to 1.49618.The maximum velocity V max was set to the half range of the search space for each dimension.For OBPSO-1 and OBPSO, the probability of opposition p o is set to 0.3.For each test functions, both OBPSO and PSO stop running when the number of iterations reaches to 1,000.
Table 2 presents average results of the four PSO algorithms over 30 runs, where Mean represents the mean best function values.As seen, PSO outperforms BPSO on only one problem G05.BPSO achieves better results than PSO on 7 problems.Both of them can find the global optimum on 5 problems.It demonstrates that the barebones PSO is better than standard PSO for these problems.
For the comparison of OBPSO with BPSO, both of them obtain the same results on 6 problems.For the rest 7 problems, OBPSO performs better than BPSO.It demonstrates that the opposition-based learning is helpful to improve the quality of solutions.
To verify the effects of the boundary search strategy, we compare the performance of OBPSO with OBPSO-1.For the OBPSO-1 algorithm, it does not use the proposed boundary search strategy.As seen, OBPSO outperforms OBPSO-1 on 6 problems, while they obtain the same results for the rest 7 problems.These results demonstrate the effectiveness of the boundary search strategy.
Figure 2 shows the evolutionary processes on four representative problems.It can be seen that OBPSO converges faster than other 3 PSO algorithms.The OBPSO-1 shows faster convergence rate than PSO and BPSO.This confirms that the opposition-based learning is beneficial for accelerating the evolution 15 .

Comparison of OBPSO with Other State-of-the-Art PSO Variants
In this section, we compare the performance of OBPSO with three other PSO variants on the test suite.The involved algorithms and parameter settings are listed as follows.
iii Self-adaptive velocity PSO SAVPSO 22 .iv Our approach OBPSO.The parameter settings of NVPSO are described in 4 .For RVPSO and SAVPSO, their parameter settings are given in 22 .For OBPSO, we use the same parameter values as described in the previous section.For each test functions, all algorithms stop running when the number of iterations reaches to the maximum value 1,000.
Table 3 presents average results of NVPSO, RVPSO, SAVPSO, and OBPSO over 30 runs, where Mean represents the mean best function values.The comparison results among OBPSO and other algorithms are summarized as w/t/l in the last row of the table, which means that OBPSO wins in w functions, ties in t functions, and loses in l functions, compared with its competitors.
From the results of Table 3, OBPSO outperforms NVPSO on 7 problems, while NVPSO only achieves better results on a single problem.For the rest 5 problems, both OBPSO and NVPSO can find the global optimum.OBPSO performs better than RVPSO on 8 problems, while both of them obtain the same results for the rest 5 problems.For the comparison of SAVPSO and OBPSO, both of them achieve the same results on 6 problems.For the rest 7 problems, OBPSO wins 6, while SAVPSO wins only 1.

Conclusion
This paper proposes a modified barebones particle swarm optimization to solve constrained nonlinear optimization problems.The proposed approach is called OBPSO which employs two novel strategies including opposition-based learning and boundary search.Compared to other improved PSO variants, OBPSO is almost a parameter-free algorithm except for the probability of opposition .Moreover, an adaptive penalty method is used to handle constraints.Experimental verifications on a set of constrained benchmark functions show that OBPSO achieves a promising performance compared to four other PSO variants.The parameter p o may affect the performance of OBPSO.To determine the best choice of p o , different values of p o will be investigated.This will be conducted in the future work.

Table 1 :
Summary of main characteristics of benchmark problems, where F means the feasible region, S is the whole search region, NE represents nonlinear equality, NI indicates nonlinear inequality, LI means linear inequality, and α is the number of active constraints at optimum.

Table 3 :
Comparison results of OBPSO with other three PSO algorithms, where "w/t/l" means that OBPSO wins in w functions, ties in t functions, and loses in l functions, compared with its competitors.The best results among the four algorithms are shown in bold.