An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics
Abstract
:1. Introduction
 Inspired by the work in [13], an improvement is proposed to the binarization technique that uses transfer functions, developed in [14], with the objective of metaheuristics which were defined to function in continuous spaces, efficiently solve COPs. This article includes the Knearest neighbor technique to improve the diversification and intensification properties of a specific metaheuristic. Unlike in [13], in which the perturbation operator is integrated with the kmeans clustering technique, in this article the perturbation operator is integrated with transfer functions, and these functions perform the binarization of the continuous metaheuristics. For this work, the Cuckoo Search (CS) metaheuristic was used. This algorithm was chosen due to its ease in parameter tuning, in addition to the existence of basic theoretical models of convergence.
 Unlike in [13], in which the multidimensional knapsack problem was tackled, this article addresses the set covering problem (SCP). This combinatorial problem has been widely studied and, because of that, instances of different difficulties are available which facilitate our analysis. In this work, we have chosen to use large size instances in order to adequately evaluate the contribution of the KNNperturbation operator.
 For a suitable evaluation of our KNN perturbation operator, we first use a parameter estimation methodology proposed in [15] with the goal to find the best metaheuristic configurations. Later, experiments are carried out to get insight into the contribution of the KNN operator. Finally, our hybrid algorithm is compared to the stateoftheart general binarization methods. The numerical results show that our proposal achieves highly competitive results.
2. Related Work
2.1. Hybridizing Metaheuristics with Machine Learning
 The first front is to apply ML at the level of the problem to be solved. This first front considers obtaining expert knowledge about the characteristics of the data of the problem under consideration. The problem model can be reformulated or it can be broken down to achieve a more efficient and effective solution. The good knowledge of the characteristics allows to design efficient metaheuristics and to understand the behavior of the algorithms. The benefits are varied, ranging from allowing the selection of an appropriate metaheuristic for each type of problem [25] to the possibility of finding an appropriate configuration of parameters [26]. For the knowledge of the ML characteristics it has several methods that can be used: neural networks [27], Bayesian networks [28], regression trees [26], support vector regression [29], Gaussian process [30], ridge regression [25], and random forest [30].
 The second front is to apply ML at the level of the components of metaheuristics. On this front, ML can be used to find suitable search components (or to find a good configuration of parameter values the latter in sin is an optimization problem. ML can be used to find good initial solutions which allows to improve the quality of the solutions and reduce processing costs since currently the initial solutions are randomly generated and of not very good quality [31]. Another participation of ML is in the design of the search operators: constructive, unary, binary, indirect, intensification, and diversification. Furthermore, ML can be present in the important task of finding a good configuration of parameters as this activity has direct impact on the performance of the algorithm [32]. Usually, the assignment of parameters is done by applying the technique of trial and error, which undoubtedly causes a loss of resources, especially time [33]. The number of parameters can vary between one metaheuristic and another, which makes experience an important factor.In general there are two major groups of parameters: those where the values are given before the execution of the algorithm known as static or offline parameters and there are also the parameters where the values are assigned during the execution of the algorithm also known as online or dynamic parameter setting. In the case of the offline parameters, the following ML methodologies can be used: unsupervised learning, supervised learning, and surrogatebased optimization. For the case of sustainable wall design, the kmeans unsupervised learning technique was used in [34] to allow algorithms that work naturally in continuous spaces to solve a combinatorial wall design problem. In the allocation of resources, the dbscan technique was used in [35], to solve the multidimensional knapsack problem. For the case of online parameter value assignment, where parameters are changed during the execution of the algorithm, the knowledge obtained during the search can serve as information to dynamically change the values of the parameters during its execution using ML methodologies, as they are Sequential learning approach, Classification/regression approach and Clustering approach. In [36], an algorithm has been proposed in order to carry out an intelligent initiation of algorithms based on populations. In this article, clustering techniques are used for the initiation. The results indicated that the proposed intelligent sampling has a significant impact, as it improves the performance of the algorithms with which it has been integrated. The integration of the knearest neighbors technique with a quantum cuckoo search algorithm was proposed in [37]. In this case, the proposed hybrid algorithm was applied to the multidimensional knapsack problem. The results showed that the hybrid algorithm is more robust than the original version.
 A third front is the choice of the best algorithm within a portfolio arranged for a certain problem. There are several metaheuristics to solve complex problems which may have common characteristics among them. For this reason, we can think of selecting an adequate metaheuristic to solve a problem with certain characteristics. We know that there is not a single metaheuristic that can solve a wide variety of problems, so the use of an alternative is to select from a portfolio of algorithms [38]. ML is a good tool for an adequate selection of the algorithm [39]. In this case, we can distinguish between offline learning where information is gathered from a series of instances in a previous form with the purpose of replicating new instances considering three approaches: classification, regression, and clustering. On the other hand, there is the online approach, which has the potential to be adaptive. Additionally, there are some hybrid approaches [40]. In [41], a cooperative strategy was implemented with the aim of land mine detection. The results of the strategy show good detection precision and robustness to environmental changes and data sets.
2.2. Related Binarization Work
 Transfer FunctionBinarization. This twostep binarization technique is widely used due to its low implementation cost. In the first step, transfer functions (TF) are used which produce values between 0 and 1 and then in the second step convert these values into binary using rules that allow to leave as value 0 or 1.There are two groups of transfer functions which are associated with the form of the function, which can be either S or V. A TF takes values of ${\mathbb{R}}^{n}$ and generates transition probability values of ${\left[0.1\right]}^{n}$. These were used to allow PSO to work with binary problems by relating the speed of particles to a transition probability. In PSO a particle is a solution which in each iteration has a position and velocity which is given by the difference of position between iterations. On the other hand, there are several rules to convert these values to binary among these are Complement, Static probability, Elitist, Elitist Roulette, or Monte Carlo.
 Angle ModulationRule. This binary technique has as a first step the use of Angle Modulation was used for phase and frequency modulation of the signal in the telecommunications industry [46]. It belongs to the family of fourparameter trigonometric functions by which it controls the frequency and displacement of the trigonometric function.$${g}_{i}\left({x}_{j}\right)=sin(2\pi ({x}_{j}{a}_{i}){b}_{i}cos\left(2\pi ({x}_{j}{a}_{i}){c}_{i}\right))+{d}_{i}$$In PSO, binary heuristic optimization applied to a set of reference functions was used for the first time [47].Consider an ndimensional binary problem, and let $X=({x}_{1},{x}_{2},...{x}_{n})$ be a solution. First of all, we define a fourdimensional search space. In this space, each dimension corresponds to a coefficient of Equation (1). As a first stage, using the fourdimensional space, we get a function. Specifically, from every tuple $({a}_{i},{b}_{i},{c}_{i},{d}_{i})$ in this space, we get a ${g}_{i}$. This ${g}_{i}$ corresponds to a trigonometric function.In the second stage, binarization, for each element ${x}_{j}$, the rule (2) is applied and getting an ndimensional binary solution.$${b}_{ij}=\left(\right)open="\{"\; close>\begin{array}{cc}1& \mathrm{if}{g}_{i}\left({x}_{j}\right)\ge 0\\ 0& \mathrm{otherwise}\end{array}$$Then, for each initial 4dimensional solution $({a}_{i},{b}_{i},{c}_{i},{d}_{i})$, we obtain a binary ndimensional solution $({b}_{i1},{b}_{i2},...,{b}_{in})$ that is a feasible solution of our nbinary problem. In [48], the authors successfully applied to network reconfiguration problems multiuser detection in multicarrier wireless broadband system [49], the antenna position problem [50], and Nqueens problems [51].
 Quantum binary. In reviewing the areas of quantum and evolutionary computing we can distinguish three types of algorithms [52]:
 Quantum evolutionary algorithms: In these methods, EC algorithms are used in order to apply them in quantum computing.
 Evolutionarybased quantum algorithms: The objective of these methods is to automate the generation of new quantum algorithms. This automation is done using evolutionary algorithms.
 Quantuminspired evolutionary algorithms: This category adapts concepts obtained from quantum computing in order to strengthen the EC algorithms.
The quantum binary approach is an evolutionary algorithm that adapts the concepts of qbits and overlap used in quantum computing applied to traditional computers.The position of a feasible solution is given by $X=({x}_{1},{x}_{2},...,{x}_{n})$ and a quantum bit vector q$Q=[{Q}_{1},{Q}_{2},...,{Q}_{n}]$ where in this approach the probability of change Q is the probability that ${x}_{j}$ takes the value 1. For each dimension, a random number between [0.1] is generated and compared to ${Q}_{j}$: if $rand<{Q}_{j}$, then ${x}_{j}=1$; otherwise, ${x}_{j}=0$. The updating mechanism of the Q vector is specific to each metaheuristic.
3. The Set Covering Problem
4. The Binary KNN Perturbed Algorithm
4.1. Initialization Operator
Algorithm 1 Init Operator 

4.2. KNN Perturbation Analysis Module
Algorithm 2 KNN perturbation analysis module 

4.3. KNN Perturbation Operator
Algorithm 3 KNN Perturbation operator 

4.4. Transfer Function Operator
4.5. Repair Operator
Algorithm 4 Repair Operator 

4.6. Heuristic Function
Algorithm 5 Heuristic function 

5. Numerical Results
5.1. Parameter Settings
 The percentage deviation of the best value resulting in ten runs compared with the bestknown value:$$bSolution=1\frac{KBestValBestVal}{KBestVal}$$
 The percentage deviation of the worst value resulting in ten runs compared to the bestknown value:$$wSolution=1\frac{KBestValWorstVal}{KBestVal}$$
 The average percentage deviation value resulting in ten runs compared with the bestknown value:$$aSolution=1\frac{KBestValAverageVal}{KBestVal}$$
 The convergence time in each experiment is standardized using Equation (16).$$nTime=1\frac{AvgConvTimeminTime}{maxTimeminTime}$$
5.2. Perturbation Operator Analysis
5.3. Comparisons
 KNN perturbation in the 4 types of problems outperforms BBH and BCS. These techniques use transfer functions as a method of binarization, the same methods used by KNN perturbation.
 KNN perturbation outperforms dbscanCS only on instance G. In all other instances, dbscanCS performs better. dbscanCS uses a binarization mechanism based on dbscan which adapts iteration to iteration. However, the difference is not statistically significant.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Parameters  Description  Value  Range 

$\nu $  Perturbation operator coefficient  25%  [20, 25, 30] 
N  Number of Nest  20  [20, 30,40] 
K  Neighbours for the perturbation  15  [10, 15,20] 
$\gamma $  Step Length  0.01  0.01 
$\kappa $  Levy distribution parameter  1.5  1.5 
Iterations  Maximum iterations  1000  [800, 900, 1000] 
Instance  Best Known  $\mathit{R}\mathit{a}\mathit{n}\mathit{d}$.25 (Best)  $\mathit{R}\mathit{a}\mathit{n}\mathit{d}$.50 (Best)  Non Perturbed (Best)  KNN Perturbed (Best)  $\mathit{R}\mathit{a}\mathit{n}\mathit{d}$.25 (Avg)  $\mathit{R}\mathit{a}\mathit{n}\mathit{d}$.50 (avg)  Non Perturbed (Avg)  KNN (Avg) 

E.1  29  30  30  31  29  30.6  30.9  31.8  29.4 
E.2  30  31  31  31  30  31.8  32.1  32.1  30.2 
E.3  27  28  29  29  27  28.7  29.5  29.7  27.6 
E.4  28  29  29  29  28  29.6  29.5  29.8  28.7 
E.5  28  28  28  28  28  28.7  28.8  29.1  28.4 
F.1  14  15  15  16  14  15.8  15.7  16.4  14.6 
F.2  15  15  15  15  15  15.7  16.1  15.9  15.2 
F.3  14  16  15  16  14  16.5  15.9  16.7  14.7 
F.4  14  15  16  16  14  15.4  16.5  16.7  14.8 
F.5  13  15  15  15  13  15.3  15.8  15.6  13.9 
G.1  176  179  180  182  176  180.1  181.2  183.2  177.2 
G.2  154  158  158  160  155  159.7  159.4  161.1  156.6 
G.3  166  171  172  171  168  172.4  173.2  172.3  169.2 
G.4  168  171  171  171  170  171.9  172.1  171.9  170.2 
G.5  168  171  171  172  168  172.1  172.0  173.2  168.6 
H.1  63  65  65  66  64  66.1  66.4  66.8  64.6 
H.2  63  65  65  65  64  65.8  65.9  66.1  64.8 
H.3  59  62  63  63  60  62.4  63.8  64.2  60.7 
H.4  58  60  60  60  59  61.3  61.2  61.1  59.4 
H.5  55  58  58  58  55  59.2  59.4  59.4  55.2 
Average  67.10  69.10  69.30  69.80  67.55  69.96  70.27  70.66  68.20 
Wilcoxon pvalue  1.2 × ${10}^{4}$  1.1 × ${10}^{5}$  2.7 × ${10}^{6}$ 
Instance  Best Known  dbScanCS (Best)  BBH (Best)  BCS (Best)  KNN (Perturbed) (Best)  dbScanCS (Avg)  BBH (Avg)  BCS (Avg)  KNN (Perturbed) (Avg)  Time (s) 

E.1  29  29  29  29  29  29.0  30.0  30.0  29.4  17.6 
E.2  30  30  31  31  30  30.1  31.0  32.0  30.2  19.1 
E.3  27  27  28  28  27  27.5  28.0  29.0  27.6  18.6 
E.4  28  28  29  30  28  28.1  29.0  31.0  28.7  20.1 
E.5  28  28  28  28  28  28.3  28.0  30.0  28.4  18.2 
F.1  14  14  14  14  14  14.1  15.0  14.0  14.6  17.8 
F.2  15  15  15  15  15  15.4  16.0  17.0  15.2  17.9 
F.3  14  14  16  15  14  14.4  16.0  16.0  14.7  19.4 
F.4  14  14  15  15  14  14.4  16.0  15.0  14.8  19.8 
F.5  13  14  14  14  13  13.4  15.0  15.0  13.9  18.4 
G.1  176  176  179  176  176  176.8  181.0  177.0  177.2  116.2 
G.2  154  157  158  156  155  156.8  160.0  157.0  156.6  114.7 
G.3  166  169  169  169  168  168.9  169.0  170.0  169.2  113.1 
G.4  168  169  170  170  170  170.1  171.0  171.0  170.2  110.6 
G.5  168  169  170  170  168  169.6  169.1  171.0  168.6  117.9 
H.1  63  64  66  64  64  64.5  67.0  64.0  64.6  104.2 
H.2  63  64  67  64  64  64.3  68.0  64.0  64.8  107.5 
H.3  59  60  65  61  60  60.6  65.0  63.0  60.7  95.6 
H.4  58  59  63  59  59  59.8  64.0  60.0  59.4  102.1 
H.5  55  55  62  56  55  55.2  62.0  57.0  55.2  92.1 
Average  67.1  67.8  69.4  68.2  67.6  68.1  70.0  69.2  68.2  63.05 
Wilcoxon pvalue  0.157  0.0005  0.0017  0.14  0.0001  0.0011 
Problems  Algorithm  Avg Gap  Gap Ratio 

E instances  dbscanCS  0.2  0.43 
BBH  0.8  4.0  
BCS  2.0  10  
KNNperturbation  0.46  1.00  
F instances  dbscanCS  0.4  0.625 
BBH  1.6  2.5  
BCS  1.4  2.19  
KNNperturbation  0.64  1.00  
G instances  dbscanCS  2.04  1.04 
BBH  3.62  1.85  
BCS  2.8  1.43  
KNNperturbation  1.96  1.00  
H instances  dbscanCS  1.28  0.96 
BBH  5.6  4.18  
BCS  2.0  1.50  
KNNperturbation  1.34  1.00 
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García, J.; Astorga, G.; Yepes, V. An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics. Mathematics 2021, 9, 225. https://doi.org/10.3390/math9030225
García J, Astorga G, Yepes V. An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics. Mathematics. 2021; 9(3):225. https://doi.org/10.3390/math9030225
Chicago/Turabian StyleGarcía, José, Gino Astorga, and Víctor Yepes. 2021. "An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics" Mathematics 9, no. 3: 225. https://doi.org/10.3390/math9030225