New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

Approximations of Fuzzy Numbers by Using r-s Piecewise Linear Fuzzy Numbers Based on Weighted Metric

Using simple fuzzy numbers to approximate general fuzzy numbers is an important research aspect of fuzzy number theory and application. The existing results in this field are basically based on the unweighted metric to establish the best approximation method for solving general fuzzy numbers. In order to obtain more objective and reasonable best approximation, in this paper, we use the weighted distance as the evaluation standard to establish a method to solve the best approximation of general fuzzy numbers. Firstly, the conceptions of I-nearest r-s piecewise linear approximation (in short, PLA) and the II-nearest r-s piecewise linear approximation (in short, PLA) are introduced for a general fuzzy number. Then, most importantly, taking weighted metric as a criterion, we obtain a group of formulas to get the I-nearest r-s PLA and the II-nearest r-s PLA. Finally, we also present specific examples to show the effectiveness and usability of the methods proposed in this paper.

Odd Mathieu Functions Application to Synthesize a Multi-element Radiator Flat-topped Radiation Pattern

The paper studies synthesizing capabilities of a flat-topped radiation pattern when using the expansion of the target radiation pattern into a series in terms of odd Mathieu functions. As parameters for comparing the target and synthesized radiation patterns, we used a main-lobe width at a level of -1 dB and an irregularity of the top of the main-lobe of the radiation pattern. The sector-shaped radiation pattern has been synthesized for linear radiators of various lengths. The convergence of the coefficients of the Mathieu series in the synthesis of the sector-shaped radiation pattern has been estimated. It is shown that the use of piecewise-linear approximation of the target radiation pattern in the synthesis using a series expansion into odd Mathieu functions allows us to improve the quality of the radiation pattern formed.The task that involved finding the amplitude-phase distribution for a linear emitter with a length of 3λ, 4λ and 5λ (λ is operation wavelength) for a target radiation pattern was solved. The target amplitude distribution has the following electrical characteristics: the main-lobe width is 37.5° at a level of -1 dB and the side lobe level (SLL) is -20 dB. The synthesis procedure was performed for two cases. In the first case, the target radiation pattern is represented by a piecewise constant function with a given width. In the second case, the target pattern was specified using piecewise linear approximation of the top and slopes of the main lobe.Comparison of the radiation patterns obtained shows that in the first case, the main-lobe width of the radiation pattern at a level of -1 dB is 34°, the SLL varies from -15.6 to -17 dB, and the irregularity of the main-lobe top of the radiation pattern lies within 0.9 ... 1.2 dB. In the second case, the main-lobe width of the antenna radiation pattern at a level of -1 dB is 36.5°, the SLL is -17.5 dB, and the irregularity of the main-lobe top is 0.4 dB at most. When used, the considered under consideration enables us to obtain both the synthesized patterns for linear radiators of various lengths, and the corresponding amplitude-phase distributions and coefficients of the Mathieu series. An estimate of the convergence of the Mathieu series shows that the use of linear approximation of the target radiation pattern in some cases allows up to 2.7-fold increase in acceleration of the convergence of the Mathieu series. The accuracy of reproducing the sector-shaped pattern by the synthesis method using the expansion into odd Mathieu functions gives good results when synthesizing the amplitude-phase distribution for the linear radiators with an electric length of 5λ or more.

MATHEMATICAL MODEL OF A CORRELATION RECEIVER WITH A PIECEWISE LINEAR APPROXIMATION OF THE DECISION FUNCTION OF THE THRESHOLD DEVICE

Разработана математическая модель двухканального корреляционного приемника радиосигналов с кусочно-линейной аппроксимацией, решающей функции порогового устройства. Приемник рассчитан на прием наиболее часто встречающихся на практике сигналов со случайной начальной фазой в условиях белого гауссовского шума. В синтезированной математической модели применяется кусочно-линейная аппроксимация решающей функции порогового устройства. Проведен сравнительный анализ характеристик обнаружения радиосигналов со случайной начальной фазой от отношения сигнал/шум, посчитанных с использованием разработанной математической модели корреляционного приемника с кусочно-линейной решающей функциeй порогового устройства. Представлены полученные в результате математического моделирования процесса функционирования корреляционного приемника при обнаружении сигналов со случайной начальной фазой в условиях шума зависимости вероятности правильного обнаружения от отношения сигнал/шум. Показано, что результаты имитационного моделирования согласуются с теоретическими расчетами. Выявлено, что представление решающих функций пороговых устройств в классической теории обнаружения сигналов в виде идеализированных (оптимальных), которые не учитывают их нелинейность, приводят либо к увеличению вероятности ложной тревоги, либо к уменьшению вероятности правильного обнаружения, что приводит к ошибкам первого рода We developed a mathematical model of a two-channel correlation receiver of radio signals with piecewise linear approximation of the decision function of the threshold device. The receiver is designed to receive the most commonly encountered signals in practice with a random initial phase in a white Gaussian noise environment. In the synthesized mathematical model, a piecewise linear approximation of the decision function of the threshold device is used. We carried out a comparative analysis of the characteristics of detecting radio signals with a random initial phase from the signal-to-noise ratio, calculated using the developed mathematical model of a correlation receiver with a piecewise linear decision function of the threshold device and known. The paper presents the dependences of the probability of correct detection on the signal-to-noise ratio obtained as a result of mathematical modeling of the process of functioning of the correlation receiver when detecting signals with a random initial phase under noise conditions. We show that the results of simulation are consistent with theoretical calculations. We found that the representation of the decision functions of threshold devices in the classical theory of signal detection in the form of idealized (optimal) ones, which do not take into account their nonlinearity, lead either to an increase in the probability of a false alarm, or to a decrease in the probability of correct detection, which leads to errors of the first kind