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If you write a scientific paper describing research that made substantive use of this program, it is your obligation as a scientist to a mention the fashion in which this software was used in the Methods section; b mention the algorithm in the References section. The appropriate citation is:. Springer, Berlin, Germany, Moreover, as a personal note, I would appreciate it if you would email manuel. The so-called uniform sampling refers to the same probability of getting arbitrary solution.
It is also the reason why the uniform sampling can provide quantitative reference.
But, for some problems, it is difficult to achieve uniform sampling, and thus it will not be able to get OO ruler. In addition, the price of getting OO ruler for huge solution space is very high. These two problems limit the application of OO ruler in solution evaluation.
However, the introduction of ordinal performance has great inspiration for the research of solution quality evaluation for SI. In this section, we take traveling salesman problem TSP as an example to describe experimental analysis method of solution quality evaluation.
For SI, the feature of the algorithm itself determines that the sampling method in the search space is not uniform. Especially by the partial reinforcement effect, it makes the algorithm more and more concentrated in certain regions.
So it is not suitable for evaluating method directly using OO ruler. In addition, the algorithm produces a large number of feasible solutions. The feasible solution contains the search characteristics of some algorithms and the distribution of the solution space.
To obtain the hidden information and its rational utilization through some analysis methods, we need to do some research. It plays an important role in the research of qualtiy evaluation and improving the algorithm performance. Based on the above analysis, this paper presents a general framework of the quality evaluation method for SI.
The framework contains three procedures. First, to get some internal approximate uniform subclass, using cluster method, the solution samples corresponding to selected subset of OO were homogeneous processing.
In operations research and computer science it is common practice to evaluate the performance of optimization algorithms on the basis of computational results, . Parameter tuning is considered as a relatively new tool in method design and analysis, and it leads to the question of adapt- ability of optimization algorithms.
Second, discrete probability distribution solution samples of each subclass and the scale relationship of the subclass are estimated in the fitness space. Based on the characteristics of the subclass, the presupposition ratio of the good enough set is distributed to each subclass.
Last, alignment probability is calculated according to the model of solution quality evaluation, so as to complete the evaluation of the solution quality.
According to the characteristics of discrete space, uniform clustering of samples is that obtaining probability of solution is approximating same. Compared with the continuous space, clustering is very different from discrete space. General discrete spatial distance features are defined with the question, and not as the continuous space as a distance to define general way. This makes clustering method based on grid no longer applicable, which is used in continuous space such as density clustering and clustering method based on grid.
And the huge solution sample set also limits the use of some special clustering method. Therefore, we need to design a suitable and efficient clustering algorithm based on demand.
Approximate sampling probability is the purpose of clustering. The approximate sampling probability here refers to the neighbor characteristics including the distance and number of nearest neighbors consistent approximation. A feasible method for TSP is to calculate the distance between all solution samples. Then clustering is done according to the nearest neighbor statistical feature of each sample distance.
But it is only applicable to the small size of the solution sample. Another possible method is that the clustering centers are selected from the best solutions. The distance is calculated between each feasible solution and the cluster center. Then the solution samples are clustered according to the distance.
The calculation complexity of this algorithm is low. It is more suitable for clustering large scale solution samples. In the next section, we use this clustering method. The solution alignment probability is calculated using a priori ratio of the good enough set the ration between the good enough set and search space in OO. The ratio of each kind of the good enough sets is needed to know after clustering. The prior ratio requires decomposing prior ratio of each class. This decomposition has a certain relationship with each class distribution of samples and the class size. Therefore, the distribution characteristics of solution in the fitness value, as well as proportional relation of class size, are needed to estimate.
Estimation of distribution of solution in the fitness value is problem of one-dimensional distribution sequence estimation. The purpose of distribution estimation is to obtain the good enough set distribution. If the fitness value is arranged according to the order from small to large, ordered performance curve OPC can be obtained.
For the minimization problem, the good enough set is in the first half of the OPC. To obtain a true estimation of the good enough set, you need to consider the types of OPC. The original search space after clustering is divided into approximate uniform partition. Search space , the good enough set , and selected set of each partition and search space , good enough set , and selected set of the original search space have the following correspondence in the collection and base: Since the probability of any feasible solution pumped into each subclass is the same, for a sampling result has.
In this paper, we only concern the selected set whether has at least one solution in good enough set. So we can draw the following conclusions: The main steps to get the evaluation method by the above analysis are described in Algorithm 1. The main steps of assessment method. In this section, we take the Hopfield city problem, which is also used in [ 17 ], as the example to demonstrate our experimental analysis method.
The best path is or. We use two groups of experimental simulation to demonstrate effectiveness of proposed method, where is alignment probability. Statistics value represents the alignment probability by our methods. Computational value is the alignment probability, and the error represents the difference of two alignment probabilities.
Alignment probability is a measure of whether optimal solution belongs to the good enough set. It is a probability value. Therefore, studying this value has little significance in one experiment. It is needed to do many experiments to study the statistical laws. So, each kind of experiment independently does times. If the optimal of time belongs to the good enough set, let ; otherwise. Let be statistical frequency. Then, for times experiment, we have.
From 5 , the following can be seen, when tends to infinity: In general, we only need to compute the value which may be tested experimentally. Let be the alignment probability in an experiment by the evaluation method; is average value of. Let be the absolute value of error of and ; that is,. In the following experiments, we are using as the standard evaluation index. Vanderpooten Computational Optimization and Applications, 56 1: Improvements on bicriteria pairwise sequence alignment: Matias Bioinformatics, 29 8: On local search for bi-objective knapsack problems A.
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