Complex Optimization Problems Using Highly Efficient Particle Swarm Optimizer

Many engineering problems are the complex optimization problems with the large numbers of global andlocal optima. Due to its complexity, general particle swarm optimization method inclines towards stagnation phenomena in the later stage of evolution, which leads to premature convergence. Therefore, a highly efficient particle swarm optimizer is proposed in this paper, which employ the dynamic transitionstrategy ofinertia factor, search space boundary andsearchvelocitythresholdbased on individual cognitionin each cycle to plan large-scale space global search and refined local search as a whole according to the fitness change of swarm in optimization process of the engineering problems, and to improve convergence precision, avoid premature problem, economize computational expenses, and obtain global optimum. Several complex benchmark functions are used to testify the new algorithm and the results showed clearly the revised algorithm can rapidly converge at high quality solutions.


Introduction
As a newly developed population-based computational intelligence algorithm, Particle Swarm Optimization (PSO) was originated as a simulation of simplified social model of birds in a flock [1]- [4].The PSO algorithm has less parameters, easy implementation, fast convergence speed and other characteristics, is widely used in many fields,such as solving combinatorial optimization, fuzzy control, neural network training, etc.But, the PSO algorithm with other algorithms is also easy to fall into local optimumin fast convergence process, affecting the convergence precision, so how to overcome premature convergence, and improve the accuracy of convergence is always a hot and difficult problem in the research field [5]- [11].
To avoidthe premature problem and speed up the convergence process, thereare many approaches suggested by researchers.According to the research results published in recent years, the improvement of PSO algorithm mainly includes adjusting algorithm parameters, the improvement of topological structure, and mixed with other algorithm, etc [6]- [12].The purpose of improvement strategiesis to balance the global search ability and local search ability of particles, so as to improve the performance of the algorithm.
In this paper, we modified the traditional PSO (TPSO) algorithm with the dynamic transition strategy ofinertia factor, search space boundary andsearchvelocitythresholdbased on individual cognitionin each cycle,whichcan balance the global search ability and local search ability of particles, and has an excellent search performance to lead the search direction in early convergence stage of search process.Experimental results on several complexbenchmark functions demonstrate that this is a verypromisingway to improve the solution quality and rate of success significantly in optimizing complex engineering problems.
Section 2 gives some background knowledge of the PSO algorithm.In section 3, the proposed method and the experimental design are described in detail, and correlative results are given in section 4. Finally, the discussions are drawn in section 5.

Back Ground
In 1995, the particle swarm optimizer (PSO) is a populationbasedalgorithm that wasinvented by James Kennedy and Russell Eberhart,which was inspired by the social behaviorof animals such as fish schooling and bird flocking.Similar to other population-based  ISSN: 1693-6930 TELKOMNIKA Vol. 12, No. 4, December 2014: 1023 -1030 1024 algorithms, suchas evolutionary algorithms, PSO can solve a variety ofdifficult optimization problems but has shown a fasterconvergence rate than other evolutionary algorithms onsome problems.Anotheradvantage of PSO is that it has very few parameters toadjust, which makes it particularly easy to implement [1].In PSO, each potential solution is a "bird" in the search space, which is called "particle".Each particle has a fitness value evaluated by the objective function, and flies over the solution space with a velocity by following the current global best particle and its individual best position.With the directions of best particles, all particles of the swarm can eventually land on the best solution.
The foundation of PSO is based on the hypothesisthat social sharing of information among conspecificsoffers an evolutionary advantage.In the original PSO formula, particle i is denoted as X i =(x i1 ,x i2 ,...,x iD ), which represents a potential solution to a problem in D-dimensional space.Each particle maintains a memory of its previous best position Pbest, and a velocity along each dimension, represented as V i =(v i1 ,v i2 ,...,v iD ).At each iteration, the position of the particle with the best fitness in the search space, designated as g, and the P vector of the current particle are combined to adjust the velocity along each dimension, and that velocity is then used to compute a new position for the particle.
In TPSO, the velocity and position of particle i at (t+1)th iteration are updated as follows: Constants c1 and c2 determine the relative influence of the social and cognition components (learning rates), which often both are set to the same value to give each component equal weight; r 1 and r 2 are random numbers uniformly distributed in the interval [0,1].A constant, v max , was used to limit the velocities of the particles.The parameter w, which was introduced as an inertia factor, can dynamically adjust the velocity over time, gradually focusing the PSO into a local search [5].
To speed up the convergence process and avoid the premature problem, Shi proposed the PSO with linearly decrease factor method (LDWPSO) [4], [5].Suppose w max is the maximum of inertia factor, w min is the minimum of inertia factor, run is the current iterations, run max is the total iterations.The inertia factor is formulated as:

A Highly Efficient Particle Swarm Optimizer (HEPSO)
Due to thecomplexity of a great deal global and local optima,TPSO isrevised as HEPSO by fourdynamic strategies to adapt complex optimizationproblems.

Dynamic Harmonization Inertia Factor w
First of all, the larger w can enhance global search abilities of PSO, so to explore largescale search space and rapidly locate the approximate position of global optimum, the smaller w can enhance local search abilities of PSO, particles slow down, deploy refined local search, and obtain global optimum.Secondly, the more difficult the optimization problems are, the more fortified the global search abilities need, once located the approximate position of global optimum, the refined local search will further be strengthen to get global optimum [7]- [12].Therefore,the wcan harmonize global search and local searchautomatically, avoid premature convergenceand to rapidly gain global optimum.
According to the conclusions above, a new inertia factor decline curve ⑷for PSO is constructed, demonstrated in Figure 1:

3.3.Dynamic Transformation Search Space Boundary Strategy
In search process, all particles gather gradually to the current best region, the algorithmis propitious to quicken convergencebecause of the reducedsearch space, but, the global optima may be lost [7]- [10].In most cases, the global optima may be hidden somewhere in the gathering area nearby, and the effective search areafound is not easy.To solve theproblem, the improved algorithm not only reduces the search space to quicken convergence, but also avoids the premature problem, especially in complex optimizationproblems.Thus

Dynamic Transformation Search Velocity ThresholdStrategy
Many published works based on parameters selection principles pointed out, velocity threshold [v max (i),v min (i)] of a particleaffects the convergence precision and speed of algorithm strongly [9]- [11].Largev max (i)increases the search region, enhancing global search capability, as well as small v max (i) decreases the search region, adjusting search direction of each particle frequency.Thus, adynamic transformation search velocitythresholdstrategy is designed based on individual cognition.Thev max and v min are the threshold of the swarm in the k iterations, the computed equation is defined as: According to the above methods, TPSO is modified as HEPSO, whichhas the excellent search performance to optimize complex problems.The flow of the HEPSOalgorithm is as follows: Step1.Set algorithm parameters; Step2.Randomly initialize the speed and position of each particle; Step3.Evaluate the fitness of each particle and determine the initial values of the individual and global best positions: t id p and t gd p ; Step4.Update velocity and position using (1), ( 2) and (4); Step5.Evaluate the fitness and determine the current values of the individual and global best positions: t id p and t gd p ; Step6.Detect the gbest i , gbest i+1 and gbest i+k , to dynamically transformw,search space boundary and velocity threshold using ( 5), ( 6)and ( 7); Step7.Randomly initialize the speed and position after the k iterations; Step8.Loop to Step 4 and repeat until a given maximum iteration number is attained or the convergence criterion is satisfied.

Computational Experiments 4.1.Testing Functions
To test the HEPSO and compare it with other techniques in the literature, we adopt large variety of benchmark functions [8]- [16], among which most functions are multimodal, abnormal or computational time consuming, and can hardly get favorable results by current optimization algorithm.Due to limited space, we only select four representative functions optimization results to list in the paper.

4.3.Experimental Results
The testing functions is run50 times based onTPSO, LDWPSO and HEPSO, the comparison of statistical results of 20-1000 dimensions functions are shown in table 1-2, respectively.In addition, the datum of literature [12]

Conclusion
The experimental results of Table 1-2 can deduce that the effectiveness of the HEPSOalgorithm based on individual cognitionis validated,which guide particles to search in the more effective areathrough dynamic adjustmentthe search space, provide stable convergence, resulting in higher success rate and accuracy of convergence.The algorithm runs classical PSO only, so to keeps its simple and easy characteristic.
The experimental results of Figure 3-6 show that the HEPSOalgorithm has excellent search performance, especially complex engineering problems.As the dimensions of the functions grow fleetly, the increase of the average convergence steps is slow, so the algorithm has rapid convergence speed and can avoid premature.In addition, it can easily be applied to large and more complex practical multi-objective optimization problems.
Figure1.Dynamic harmonization w curve Figure 2. Dynamic transformation w curves , a dynamic transformation search spaceboundary strategy is designed based on individual cognition.Assume that a particle flight in the current boundary [b max (i),b min (i)], the algorithmreduce the search boundary if the current optimum is better, otherwise, expand search boundary in next iteration, and in the same breath, randomly initialize the speed and position of each particle after the k iterations.Theb max and b min are the boundary of the swarm in the k iteration.The computed equation is defined as: