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z+XO7n{XO-w$#n06)D>XeYadP5@x=*)A@TwX050dowZ?U-oiybK6KA5KAj1rQE^R z>`KU?K6JQ153MxXS9o?X-%JLzpA{lAm7d(vx;LF)qVvPXR27?kZkx8quU#vB_z>4v zUl<~(!n)**-f4B-Y#bw2c>Ei@-=@9r!Ucgu5#u;m3v{cFc zrj5n(KW64bXJ^ujQy$fpxU?mb?D$zlNi^@IVhOlA=zUC3+_h_SUhMD&B{^}z_QHK` zIeisn!h)ta}PPjeIq9iLRQ_~M%{!mnN8wI= zYiz9MOXgf*1XoWQcyqAolC#iY=+;H1Ji;G#hw0IcEn~)$q5wS$6iqRtMkTH<7l;8q zoji|fx_gFdfEs7z6PkISs;MNvF*2|9Gp~|Lb>}\\ + + \hfill \break + \hfill \break + \hfill \break + \hfill \break + \large{Лабораторная работа №1}\\ + \large{по дисциплине}\\ + \large{<<Генетические алгоритмы>>}\\ + \large{Вариант 18}\\ + + % \hfill \break + \hfill \break + \end{center} + + \small{ + \begin{tabular}{lrrl} + \!\!\!Студент, & \hspace{2cm} & & \\ + \!\!\!группы 5130201/20101 & \hspace{2cm} & \underline{\hspace{3cm}} &Тищенко А. А. \\\\ + \!\!\!Преподаватель & \hspace{2cm} & \underline{\hspace{3cm}} & Большаков А. А. \\\\ + &&\hspace{4cm} + \end{tabular} + \begin{flushright} + <<\underline{\hspace{1cm}}>>\underline{\hspace{2.5cm}} 2025г. + \end{flushright} + } + + \hfill \break + % \hfill \break + \begin{center} \small{Санкт-Петербург, 2025} \end{center} + \thispagestyle{empty} % выключаем отображение номера для этой страницы + + % КОНЕЦ ТИТУЛЬНОГО ЛИСТА + \newpage + + \tableofcontents + + \newpage + \section {Постановка задачи} + В данной работе были поставлены следующие задачи: + + \begin{itemize} + \item Изучить теоретический материал; + \item Ознакомиться с вариантами кодирования хромосомы; + \item Рассмотреть способы выполнения операторов репродукции, + кроссинговера и мутации; + \item Выполнить индивидуальное задание на любом языке высокого + уровня с необходимыми комментариями и выводами + \end{itemize} + + \textbf{Индивидуальное задание вариант 18:} + + \textbf{Дано:} Функция одной переменной $f(x) = \frac{\sin(x)}{x^2}$; промежуток нахождения решения $x \in [3.1, 20.0]$. + + \vspace{0.3cm} + \textbf{Требуется:} + + \begin{enumerate} + \item Разработать простой генетический алгоритм для нахождения минимума данной функции в заданном промежутке; + \item Исследовать зависимость времени поиска, числа поколений (генераций) и точности нахождения решения от числа особей в популяции и вероятности кроссинговера и мутации; + \item Вывести на экран график функции с указанием найденного экстремума для каждого поколения; + \item Сравнить найденное решение с действительным. + \end{enumerate} + + \textbf{Ограничения:} + \begin{enumerate} + \item $ \min f(x) = \frac{\sin(x)}{x^2} \approx -0.04957 $ + \item Точность решения составляет 3 знака после запятой. + \end{enumerate} + + + \newpage + \section{Теоретические сведения} + + Генетиче ские алгоритмы (ГА) используют принципы и терминологию, заимствованные у биологической науки – генетики. В ГА каждая особь представляет потенциальное решение некоторой + проблемы. В классическом ГА особь кодируется строкой двоичных символов – хромосомой, каждый + бит которой называется геном. Множество особей – потенциальных решений составляет популяцию. Поиск (суб)оптимального решения проблемы выполняется в процессе эволюции популяции + - последовательного преобразования одного конечного множества решений в другое с помощью + генетических операторов репродукции, кроссинговера и мутации. + + Предварительно простой ГА случайным образом генерирует начальную популяцию стрингов + (хромосом). Затем алгоритм генерирует следующее поколение (популяцию), с помощью трех основных генетических операторов: + + \begin{enumerate} + \item Оператор репродукции (ОР); + \item Оператор скрещивания (кроссинговера, ОК); + \item Оператор мутации (ОМ). + \end{enumerate} + + ГА работает до тех пор, пока не будет выполнено заданное количество поколений (итераций) + процесса эволюции или на некоторой генерации будет получено заданное качество или вследствие + преждевременной сходимости при попадании в некоторый локальный оптимум. На Рис.~\ref{fig:alg} представлен простой генетический алгоритм. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.9\linewidth]{img/alg.png} + \caption{Простой генетический алгоритм.} + \label{fig:alg} + \end{figure} + + \newpage + \subsection{Основная терминология в генетических алгоритмах} + + \textbf{Ген} -- элементарный код в хромосоме $s_i$, называемый также знаком или детектором + (в классическом ГА $s_i = 0, 1$). + + \textbf{Хромосома} -- упорядоченная последовательность генов в виде закодированной структуры + данных $S = (s_1, s_2, \ldots, s_n)$, определяющая решение (в простейшем случае двоичная + последовательность -- стринг, где $s_i = 0, 1$). + + \textbf{Локус} -- местоположение (позиция, номер бита) данного гена в хромосоме. + + \textbf{Аллель} -- значение, которое принимает данный ген (например, 0 или 1). + + \textbf{Особь} -- одно потенциальное решение задачи (представляемое хромосомой). + + \textbf{Популяция} -- множество особей (хромосом), представляющих потенциальные решения. + + \textbf{Поколение} -- текущая популяция ГА на данной итерации алгоритма. + + \textbf{Генотип} -- набор хромосом данной особи. В популяции могут использоваться как отдельные + хромосомы, так и целые генотипы. + + \textbf{Генофонд} -- множество всех возможных генотипов. + + \textbf{Фенотип} -- набор значений, соответствующий данному генотипу. Это декодированное множество + параметров задачи (например, десятичное значение $x$, соответствующее двоичному коду). + + \textbf{Размер популяции $N$} -- число особей в популяции. + + \textbf{Число поколений} -- количество итераций, в течение которых производится поиск. + + \textbf{Селекция} -- совокупность правил, определяющих выживание особей на основе значений целевой функции. + + \textbf{Эволюция популяции} -- чередование поколений, в которых хромосомы изменяют свои признаки, + чтобы каждая новая популяция лучше приспосабливалась к среде. + + \textbf{Фитнесс-функция} -- функция полезности, определяющая меру приспособленности особи. + В задачах оптимизации она совпадает с целевой функцией или описывает близость к оптимальному решению. + + \subsection{Генетические операторы} + + \subsubsection{Оператор репродукции} + + Репродукция -- процесс копирования хромосом в промежуточную популяцию для дальнейшего + ``размножения'' в соответствии со значениями фитнесс-функции. В данной работе рассматривается метод колеса рулетки. Каждой хромосоме соответствует сектор, пропорциональный значению фитнесс-функции. + Хромосомы с большим значением имеют больше шансов попасть в следующее поколение. + + \subsubsection{Оператор скрещивания (кроссинговера)} + + Одноточечный кроссинговер выполняется следующим образом: + \begin{enumerate} + \item Из промежуточной популяции выбираются две хромосомы (родители). + \item Определяется случайная точка скрещивания $k \in [1, n-1]$, где $n$ -- длина хромосомы. + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\linewidth]{img/cross1.png} + \label{fig:cross1} + \end{figure} + \item Две новые хромосомы (потомки) формируются путём обмена подстрок после точки $k$. + \begin{figure}[h!] + \centering + \includegraphics[width=0.35\linewidth]{img/cross2.png} + \label{fig:cross2} + \end{figure} + \end{enumerate} + + \subsubsection{Оператор мутации} + + Мутация применяется с малой вероятностью $P_M \approx 0.001$: + \begin{enumerate} + \item В хромосоме $A = a_1a_2 \ldots a_n$ выбирается случайная позиция $k$. + \item Ген $a_k$ инвертируется: $a_k' = \lnot a_k$. + \end{enumerate} + + + + \newpage + \section*{Заключение} + \addcontentsline{toc}{section}{Заключение} + + В ходе первой лабораторной работы: + + \begin{enumerate} + \item Был изучен теоретический материал, основная терминология ГА, генетические операторы, + использующиеся в простых ГА; + \item Реализована программа на языке Python для нахождения минимума заданной функции; + \item Проведено исследование зависимости времени выполнения программы и поколения от мощности популяции и коэффициентов кроссинговера и мутации. + \end{enumerate} + +\newpage +\section*{Список литературы} +\addcontentsline{toc}{section}{Список литературы} + +\vspace{-1.5cm} +\begin{thebibliography}{0} + \bibitem{vostrov} + Методические указания по выполнению лабораторных работ к курсу «Генетические алгоритмы», 119 стр. +\end{thebibliography} + +\end{document} \ No newline at end of file