A, A Pavlenko

A, A Shigina

E, A Chzhan

V, A Kukartsev

V, S Tynchenko

V, V Kukartsev

English

Siberian Federal University Digital Repository

      Comparison of methods for start points initializing of a non-parametric optimization algorithm A A Pavlenko1, V V Kukartsev1,2, V S Tynchenko1,2, A A Shigina2, E A Chzhan2 and  V A Kukartsev2  1Reshetnev Siberian State University of Science and Technology, 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation 2 Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation  E-mail: vadimond@mail.ru  Abstract. The problem of global optimization arises in various fields of science and technology, and several different ways of solving it have been proposed. The results of the study of the effectiveness of the non-parametric global optimization algorithm are presented. A comparative analysis of this algorithm is presented. performance analysis of the algorithm based on the Ackley, Rastrigin, Shekel, Griewank and Rosenbrock function. In addition, studies were carried out for the three initial points of the distribution algorithms: the sequence LP, the sequence UDC, the uniform random distribution. thus, the best way to initialize the initial points of the non-parametric optimization algorithm on these test functions was identified. According to the research results, the effective parameters of the genetic algorithm were established.   1. Introduction The nonparametric global optimization algorithm [1] differs from the others in that it is the most universal global optimization algorithm. So, it doesn't matter for the algorithm what the object of optimization is and how complex the function is, which describes the object. The object has input and output parameters. We must find an extremum by certain criteria. The parameters of this algorithm are of a high value in order to reach the extremum more efficiently. The research was conducted on Ackley, Rastrigin, Shekel, Griewank and Rosenbrock function [2].  LP  sequence [3], UDC sequence, uniform random scatter are very interesting and effective scatter algorithms of initial points. Recent research in this area has been carried out in works [4]. These studies were applied to specific practical problems. The goal was not to average these parameters, but to test a complex type of a test function on a large number of practical tasks [5]. LP sequences is the algorithm of scattering of points based on the Marshal’s matrix of irreducible polynomials. UDC sequence is an algorithm of absolutely uniform distribution of points across all coordinates in multidimensional space [6, 7] regardless of the number of scattered points [8, 9]. Uniform random variation is a stochastic algorithm of scatter of points using a normal distribution (RND). The paper uses the best parameters for the nonparametric global optimization algorithm:  • The Nuclear function is hyperbolic with the parameter Q= 4. • Parameter gamma = 2. • Parameter P = 10.       2. Experimental study Figure 1 shows the flowchart of the nonparametric optimization algorithm [1].   Figure 1. Flowchart of a nonparametric optimization algorithm.  The optimal, in absolute value, parameters of the nonparametric algorithm (N – the number of initial scattered points, IM - the Initialization method, e – accuracy of finding the extremum, F1(x), F2(x) – test functions, Steps – steps of the algorithm to find the extremum with a given accuracy – e, RES, % is the percentage of finding the extremum at a given number of initial points) were determined, for each test function optimized for each point of view (Table 1).  Researches were conducted on test functions: Ackley, Rastrigin, Shekel and Griewank functions. The number of re-runs of the algorithm is 100.        Table 1. Choosing the most effective initialization method. The experiments were carried out with an accuracy of 0.0001. The dependence of the algorithm steps on the number of initial points and the dependence of the percentage of finding the extremum on the number of initial points are studied using LP spread, UDC spread and uniform scattering and random variation (RND). Dependence of algorithm steps on the number of starting points is presented in Figure 2.  N SI e=0,01 e=0,0001 F1(x) F2(x) F1(x) F2(x) Steps Res, % Steps Res, % Steps Res, % Steps Res, %           10 LP 14 0 19 100 32 0 34 100 RND 19 62 20 91 29 35 29 94 UDC 37 98 32 86 43 58 38 100 20 LP 7 0 11 100 7 0 8 0 RND 16 72 15 92 22 37 22 100 UDC 17 100 16 92 34 56 22 100 30 LP 10 100 7 100 9 100 8 100 RND 13 70 12 100 17 54 18 94 UDC 12 100 12 100 22 68 20 100 40 LP 8 100 7 100 9 100 8 100 RND 11 78 1 100 17 57 16 100 UDC 10 100 12 94 16 70 15 100 50 LP 8 100 7 100 9 100 20 100 RND 10 82 10 97 16 54 15 100 UDC 10 100 10 100 15 76 14 100 60 LP 8 100 7 100 9 100 21 100 RND 10 85 10 100 15 60 14 94 UDC 9 100 10 96 14 72 14 100 70 LP 8 100 7 100 9 100 19 100 RND 10 90 10 98 15 54 14 100 UDC 9 100 10 92 14 80 14 100 80 LP 8 100 7 100 19 100 19 100 RND 10 97 9 100 15 52 14 100 UDC 9 100 9 100 14 85 13 100 90 LP 8 100 7 100 19 100 20 100 RND 9 95 9 100 13 78 14 100 UDC 8 100 8 100 13 94 13 100 100 LP 6 100 6 100 15 100 15 100 RND 9 98 9 100 13 65 14 100 UDC 8 100 8 100 13 96 12 100         Figure 2. Dependence of algorithm steps on the number of starting points.  Dependence of the percentage of finding the extremum on the number of starting points is presented in Figure 3.   Figure 3. Dependence of the percentage of finding the extremum on the number of starting points.   3. Results According to table 1 and figures 1 and 2 we can say that the best way of initialization is based on the LP sequence both in terms of convergence and in terms of efficiency. Considering convergence initialization method based on UDC scatter is worse than initialization method based on uniform random spread only when the number of points is less than 40. In all other cases in terms of convergence and in all cases in terms of efficiency, the initialization method based on the UDC spread exceeds the initialization method based on a uniform random spread, but does not exceed the initialization method based on the LP spread. In the course of the work done, the LP-sequence generator was implemented; qualitative comparisons of the methods of scattering points were conducted. Initialization methods for starting points have been implemented. A non-parametric optimization algorithm was implemented; various initialization methods were compared using a non-parametric optimization algorithm, which showed       the best initialization method, that is LP scatter, UDC scatter was worse and uniform random scatter was the worst of all. 4. Conclusion The nonparametric algorithm of global optimization was analyzed. The research was conducted on Ackley, Rastrigin, Shekel, Griewank and Rosenbrock functions. The studies were carried out for three algorithms of initial point spread: LP sequence, UDC sequence, uniform random spread.  As a result of researches the best way of initialization of initial points was revealed. The dependence of the algorithm steps on the number of initial points were studied as well as the dependence of the percentage of finding the extremum on the number of initial points. 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Comparison of methods for start points initializing of a non-parametric optimization algorithm

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Comparison of methods for start points initializing of a non-parametric optimization algorithm

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