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TWO ALGORITHMS FOR GLOBAL OPTIMIZATION OF ONE-VARIABLE FUNCTIONS BASED ON THE SMALLEST ESTIMATE DISTANCES BETWEEN EXTREMES AND THEIR NUMBER

V. A. Kodnyanko

2020
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Radìoelektronika, Ìnformatika, Upravlìnnâ
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Contex. Making managerial decisions is often associated with solving one-dimensional global optimization problems. The most important property of global optimization methods is their speed, which is determined by the number of calls to the objective function in the optimization process. Objective. Development of high-performance algorithms global for optimizing the function of one variable, based on conditions that allow you to bring the problem to a form that opens up the practical possibility
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... of obtaining a solution with a given accuracy. Method. Two algorithms of conditional global optimization of a function of one variable are considered. The first is based on estimating the smallest distance between neighboring local extrema and allows you to find the global minimum of the goal function and, if necessary, all its local extrema. The second is suitable for finding the global minimum of a function if the number of local extrema in the uncertainty interval is known in advance. Both algorithms are based on segmentation methods of the initial uncertainty segment. The local extremum on a segment is determined by three or four points. An approach is proposed that, in most cases, allows localization of the extremum at three points, which provides savings in the calculation of digital filters, thereby contributing to an increase in the speed of the algorithm. Results. The results of solving optimization problems and data on the effectiveness of the proposed algorithms are presented. A comparative analysis of the speed of the developed algorithms and well-known algorithms is carried out on the example of solving test problems used in world practice to assess the effectiveness of global optimization algorithms. Examples of the practical use of algorithms are given. The analysis of the data obtained showed that according to the number of calls to the objective function, the algorithms in the sequential computing mode work several times faster than modern high-speed algorithms with which they were compared. Conclusions. The data presented indicate the efficiency and high speed of the proposed algorithms. Their speed will be even higher if the stated ideas of algorithmization are extended to parallel computations. This suggests that the proposed algorithms can find practical application in the global optimization of functions of the considered classes of problems. KEYWORDS: function of one variable, local minimum of a function, global minimum of a function, global optimization, Brent's method, algorithm performance. ABBREVIATIONS TF -target function; EPPefficiency of a parallel process. NOMENCLATURE f(x) is the objective function; G = [a, b] is segment of uncertainty; ε is the accuracy of determining the global extremum; d min is the smallest distance between the extremums of the CF; n is the number of segments on the segment G; h is the length of the segment; k min is the smallest number of local extrema on the segment G; k is the number of TF calls in optimization process.

doi:10.15588/1607-3274-2020-2-4
fatcat:jd37czizqneztl47dajd5z26bi