Evolutionary algorithms (EAs) were shown to be effective for complex constrained optimization problems. However, inflexible explorationexploitation and improper penalty in EAs with penalty function would lead to losing the global optimum nearby or on the constrained boundary. To determine an appropriate penalty coefficient is also difficult in most studies. In this paper, we propose a bidirectional dynamic diversity evolutionary algorithm (BiDDEA) with multiagents guiding explorationexploitation through local extrema to the global optimum in suitable steps. In BiDDEA potential advantage is detected by three kinds of agents. The scale and the density of agents will change dynamically according to the emerging of potential optimal area, which play an important role of flexible explorationexploitation. Meanwhile, a novel double optimum estimation strategy with objective fitness and penalty fitness is suggested to compute, respectively, the dominance trend of agents in feasible region and forbidden region. This bidirectional evolving with multiagents can not only effectively avoid the problem of determining penalty coefficient but also quickly converge to the global optimum nearby or on the constrained boundary. By examining the rapidity and veracity of BiDDEA across benchmark functions, the proposed method is shown to be effective.
A great deal of engineering design problems can be formulated as constrained nonconvex optimization problems which are difficult to be solved with classic mathematical theory. Since swarm intelligence (SI) [
Genetic algorithm (GA) is a search heuristic which mimics the process of natural evolution by parallel computing. It became popular through the work of John Holland in the early 1970s, and particularly his book
Usually the fitness function, especially around the constrained boundary where there is a global optimum, becomes much more complex because of the mixed constraints and penalty functions. Overpenalty on the forbidden agents which is nearby the global optimum will lose important samples with better guiding function. Underpenalty on the forbidden agents which is far away from the global optimum will weaken strength of converging toward the global optimum. Then inflexible explorationexploitation will probably miss the global optimum and lead to illconvergence. In this paper, we propose a bidirectional dynamic diversity evolutionary algorithm (BiDDEA) with multiagents guiding adaptive explorationexploitation to the global optimum. In BiDDEA potential advantage is detected by three kinds of agents. The scale and the density of agents will change dynamically according to the emerging of potential optimal area, which play an important role of flexible explorationexploitation. Meanwhile, a novel double optimum estimation strategy with objective fitness and penalty fitness is suggested to compute, respectively, the dominance trend of agents in feasible region and forbidden region. Penalty fitness will guide agents in forbidden region to nearby constrained boundary with optimal objective fitness, and objective fitness will guide agents in feasible region to global optimum. This bidirectional evolving with multiagents can not only effectively avoid the problem of determining penalty coefficient but also quickly converge to the global optimum nearby or on the constrained boundary. Unlike other penalty methods, the penalty fitness suggested in this paper is not added to the objective function as a penalty term but constitutes the BiDDEA as a evaluation term to select optimal forbidden agents independently, which will make a convergence towards the global optimum from another direction in the forbidden region.
This paper is organized as follows. Section
Although penalty function method is one of the most common methods to solve constrained optimization problems, the design of penalty item should be fit for the algorithm with which we will deal with practical optimization problems. In order to avoid the complication caused by adding penalty term to the objective function, objective fitness (
The general constrained problem formulation that is also called the primal problem can be stated as follows:
Penalty fitness
Three types of agents, partition agents (PAs), basic agents (BAs), and creative agents (CAs), are combined together in BiDDEA to carry out adaptive explorationexploitation according to the guiding of
The evolutionary process of BiDDEA is mainly carried out by the rebirth of new agents around the senior ones. one PA will generate some BAs in a local region, and one BA will generate some CAs in a local region around the BA. Then some CAs with higher potential superiority will be selected as the next BAs. BAs alternate with CAs, which is also affected by PAs. The focus in BiDDEA is no longer on the position updating of previous agents but on the density distribution of newborn agents. The density of newborn agents is controlled by regulating the range and the scale of newborn agents according to the feedback of sampling information such as
The range of newborn agents is determined by two items in BiDDEA. The first one is narrowing range with
When newborn CAs are distributed, the sampling information will also affect the property, range, of PAs and the changing of Pas’ property will also affect the property, range, of CAs’. This process is called selfcorrecting in range which can be described as follows:
The scale of newborn agents is also determined by two items. The first one is the scale which is the same with their parent, which means the child agents will inherit
When newborn CAs are distributed, the sampling information will also affect the property, scale, of PAs and the changing of Pas’ property will also affect the property, scale, of CAs’. This process is called selfcorrecting in scale which can be described as follows:
Swarm scale ceil (SSC) is also introduced to prevent agents from excessive generating. Considering the forbidden agents will not be the global optimal one, the scale floor of forbidden agents will not be confined by SSF but limited to one or more. Thus, the scale of agents will be corrected in the following form:
The equation set (
BiDDEA mainly consists of bidirectional information processing suggested in Section
to divide searching space into a number of partitions with partition agents (PAs) and initialize the properties of each PA;
to distribute basic agents (BAs) in each partition according to the properties of PA;
to divide BAs into two types with the constrained conditions and, respectively, compute their objective fitness, penalty fitness, and fitness growth according to formulas (
to estimate the potential advantage of BAs both in feasible region and forbidden region according to formulas (
to assign the attributes of BAs according to formulas (
to distribute creative agents (CAs) around each BA according to the range and scale of the BA;
to divide CAs into two types with the constrained conditions and, respectively, compute their objective fitness, penalty fitness, and fitness growth according to formulas (
to estimate the potential advantage of CAs both in feasible region and forbidden region according to formulas (
to assign the attributes of CAs and PAs according to formulas (
to correct the scale of each agents according to formulas (
to select better CAs as next BAs according to the scale of PA in each partition;
to judge whether the breaking conditions are met. If so, then it continue to next step; otherwise it jumps to step (3);
to select the best BA as present global optimal agents and translate it into a PA;
to judge whether the breaking conditions are met. If so, then it continue to next step; otherwise it jumps to step (1);
output the results.
Obviously, all the agents that are divided into two types in BiDDEA will evolve along two directions with the dynamic diversity evolution method under the guidance of bidirectional information analysis proposed in this paper.
The performance of BiDDEA is evaluated by comparing its rapidity, accuracy, and universality with those of PSO algorithm and GA. To be meaningful, some benchmark functions that are commonly used in randomized populationbased algorithms testing are selected as objective functions. This gives a fair comparison being as far as possible free from biases that favor one style of algorithm over another. These functions include Rastrigin [
Benchmark functions.
For GA, the initial parameters are set to their default values as described below:
population size: 100;
probability of mutation: 0.05.
For PSO, the initial parameters are set to their default values as described below:
population size: 40;
inertia weight (
For BiDDEA, the initial parameters are set to their default values as described below:
partition dividing: from
learning factor (
SSF:
SSC:
outside loop number: from
inner loop number: from
This section compares the rapidity of BiDDEA with the results obtained by GA (genetic algorithm) and PSO (Particle swarm optimization) when they are applied to the six benchmark functions mentioned above. Sampling time is used to represent the optimization time logically. It refers to the cumulative time spent by every sample to calculate their fitness according to the coordinates. It can eliminate the program editor, environment, and other external factors affecting the evaluation of the optimization speed, which allows us to compare the rate of these algorithms more explicitly. An algorithm with different program design or in different computer environments will have a different performance. For example, a good programmer can implement an algorithm process with streamlined code to improve operational efficiency, but it is difficult to reflect its advantages when running in CPU with low frequency. In the experimental comparison, we would probably get wrong result due to these external factors. So, we use sampling time as reference to analyze their speed characteristics of the most original. Since the time used for each sample is almost equal, we use the number of samples to replace sampling time in this paper.
Figures
Comparison of speed across Rastrigin.
Comparison of speed across Rosenbrock.
Comparison of speed across Griewank.
Comparison of speed across NeedleinaHaystack.
Comparison of speed across Shubert.
Comparison of speed across Schaffer.
High accuracy and global superiority are the key indicators of good algorithm in quality evaluation. From Tables
Comparison of optimal results across Rastrigin.
Function  Times  GA  PSO  PLDDEA  BiDDEA 

Rastrigin  1  0.1833  0  0.0000  0.0000 
2  0.0093  0  0.0000  0.0000  
3  0.0136  0  0.0000  0.0000  
4  0.0216  0  0.0000  0.0000  
5  0.0411  0  0.0000  0.0000  
6  0.0008  0  0.0000  0.0000  
7  0.0025  0  0.0000  0.0000  
8  0.0747  0  0.0000  0.0000  
9  0.0450  0  0.0000  0.0000  
10  0.0394  0  0.0000  0.0000  
Mean  0.0431  0  0.0000  0.0000 
Comparison of optimal results across Rosenbrock.
Function  Times  GA  PSO  PLDDEA  BiDDEA 

Rosenbrock  1  1.1084  0.9903  0.9903  0.9903 
2  1.1828  0.9903  0.9903  0.9903  
3  1.1587  0.9903  0.9904  0.9903  
4  1.1089  0.9903  0.9903  0.9903  
5  1.1362  0.9903  0.9903  0.9903  
6  1.1643  0.9903  0.9903  0.9903  
7  1.0680  0.9903  0.9903  0.9903  
8  1.1026  0.9903  0.9903  0.9903  
9  1.1626  0.9903  0.9903  0.9903  
10  1.1017  0.9903  0.9903  0.9903  
Mean  1.0219  0.9903  0.9903  0.9903 
Comparison of optimal results across Griewank.
Function  Times  GA  PSO  PLDDEA  BiDDEA 

Griewank  1  0.0024  0  0  0 
2  0.0000  0  0  0  
3  0.0076  0.0074  0  0  
4  0.0107  0.0074  0  0  
5  0.0138  0.0099  0  0  
6  0.0094  0.0074  0  0  
7  0.0099  0.0074  0  0  
8  0.0055  0.0099  0  0  
9  0.0008  0.0074  0  0  
10  0.0096  0.0074  0  0  
Mean  0.0070  0.0064  0  0 
Comparison of optimal results across Needleinahaystack.
Function  Times  GA  PSO  PLDDEA  BiDDEA 

Needle 
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Mean 




Comparison of optimal results across Shubert.
Function  Times  GA  PSO  LPDDEA  BiDDEA 

Shubert  1 




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Comparison of optimal results across Schaffer.
Function  Times  GA  PSO  LPDDEA  BiDDEA 

Schaffer  1  0.0051  0.0000  0.0000  0.0000 
2  0.0011  0  0.0000  0.0000  
3  0.0030  0  0.0000  0.0000  
4  0.0027  0.0000  0.0000  0.0000  
5  0.0020  0.0000  0.0000  0.0000  
6  0.0006  0.0097  0.0000  0.0000  
7  0.0086  0.0097  0.0000  0.0000  
8  0.0001  0.0000  0.0000  0.0000  
9  0.0070  0.0000  0.0000  0.0000  
10  0.0024  0.0000  0.0000  0.0000  
Mean  0.0033  0.0019  0.0000  0.0000 
To avoid falling into local minimum and find better optimum or constrained boundary, a good algorithm should have the ability to explore the unknown feasible region. As a better cases analysis, we now consider the distribution of all the samples through the whole optimization process, as shown in Figure
Samples distribution across benchmark functions.
In addition, Figure
Experimental results have shown notably that BiDDEA with bidirection evolutions is more successful in solving nonconvex constrained optimization problems. The wonderful performance profits from the bidirection information processing and the dynamic diversity evolution carried out by multiagents consisting of PAs, Bas, and CAs both in feasible region and forbidden region. adaptive explorationexploitation in feasible region makes BiDDEA find out most of the local optimum which include the global optimum. It indicates that the suggested algorithm has the ability of discovering the character of function distribution, which provides robust evidence for the global searching ability of BiDDEA. Another evolution direction from the forbidden region is detected specially by agents which are distributed in infeasible region inevitably. The mechanism of BiDDEA does not penalize these infeasible agents with large penalty to eliminate their guidance, but to make full use of their guidance to the constrained boundary. When a feasible agent is generated by a forbidden agent, it will get much higher fitness growth according to its present fitness. Thus, more and more samples will be distributed on the boundary with higher fitness as shown in Figure
BiDDEA is not sensitive to initial population in most cases. From the mechanism analysis of BiDDEA, we can find that sample size in each iteration is regulated according to the feedback sampling information. Even though the setting population is different from the need greatly in the initial phase, BiDDEA will adjust the population scale automatically to a needed degree through a few iterations. It is worth noting that some special kinds initial partitions are able to improve efficiency of optimization algorithm, and these kinds of initial partitions are called sensitive initial conditions. How to determine the sensitive initial conditions will also be one of future research aspects for improving the performance of BiDDEA in practical application.
To let BiDDEA adapt to diversified and comprehensive constrained environment easily is another important goal of this paper. Although the six benchmark functions modified with the constrained condition can not give full instructions of BiDDEA’s generalization capability, the samples distribution has provided robust evidence for the predominance of bidirection dynamic diversity evolution. Many more experiments and practices are needed for verifying the validity of this optimization method. In fact, to find exactly the types for which an optimization method fit will be more important and useful than to improve hardly the generalization capability of the method. In future study, surveying and concluding the best parameters of BiDDEA to match different types of optimization problems will be done to expand the scope in which BiDDEA can give better results. There are also other special application fields in which BiDDEA can be attempted to apply, such as discrete optimization problems [
This work was supported by the National Natural Science Foundation of China under Grant no. 61074020 and the Fundamental Research Funds for the Central Universities.