From c15867f02730ed62b882c99606955cc40d73fc67 Mon Sep 17 00:00:00 2001 From: Arity-T Date: Wed, 8 Oct 2025 17:30:39 +0300 Subject: [PATCH] =?UTF-8?q?Lab2=20=D0=BF=D0=BE=D0=BB=D0=BE=D0=B2=D0=B8?= =?UTF-8?q?=D0=BD=D0=B0=20=D0=BE=D1=82=D1=87=D1=91=D1=82=D0=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- lab2/main.py | 3 +- lab2/report/.gitignore | 6 + lab2/report/img/alg.png | Bin 0 -> 176421 bytes lab2/report/img/arithmetic_crossover.png | Bin 0 -> 26120 bytes lab2/report/img/blx_crossover.png | Bin 0 -> 13881 bytes lab2/report/img/geometric_crossover.png | Bin 0 -> 27270 bytes lab2/report/img/results/generation_001.png | Bin 0 -> 573005 bytes lab2/report/img/results/generation_002.png | Bin 0 -> 567574 bytes lab2/report/img/results/generation_003.png | Bin 0 -> 565543 bytes lab2/report/img/results/generation_005.png | Bin 0 -> 563280 bytes lab2/report/img/results/generation_007.png | Bin 0 -> 562726 bytes lab2/report/img/results/generation_010.png | Bin 0 -> 557137 bytes lab2/report/img/results/generation_013.png | Bin 0 -> 555833 bytes lab2/report/img/sbx_crossover.png | Bin 0 -> 58462 bytes lab2/report/report.tex | 602 +++++++++++++++++++++ 15 files changed, 610 insertions(+), 1 deletion(-) create mode 100644 lab2/report/.gitignore create mode 100644 lab2/report/img/alg.png create mode 100644 lab2/report/img/arithmetic_crossover.png create mode 100644 lab2/report/img/blx_crossover.png create mode 100644 lab2/report/img/geometric_crossover.png create mode 100644 lab2/report/img/results/generation_001.png create mode 100644 lab2/report/img/results/generation_002.png create mode 100644 lab2/report/img/results/generation_003.png create mode 100644 lab2/report/img/results/generation_005.png create mode 100644 lab2/report/img/results/generation_007.png create mode 100644 lab2/report/img/results/generation_010.png create mode 100644 lab2/report/img/results/generation_013.png create mode 100644 lab2/report/img/sbx_crossover.png create mode 100644 lab2/report/report.tex diff --git a/lab2/main.py b/lab2/main.py index e2c3cd7..fc1ebd6 100644 --- a/lab2/main.py +++ b/lab2/main.py @@ -17,7 +17,8 @@ config = GARunConfig( pm=0.01, minimize=True, seed=17, - save_generations=[1, 2, 3, 5, 7, 10, 15, 20, 50, 100], + fitness_avg_threshold=0.05, + save_generations=[1, 2, 3, 5, 7, 10, 15], ) result = genetic_algorithm(config) diff --git a/lab2/report/.gitignore b/lab2/report/.gitignore new file mode 100644 index 0000000..38887f8 --- /dev/null +++ b/lab2/report/.gitignore @@ -0,0 +1,6 @@ +* + +!**/ +!.gitignore +!report.tex +!img/**/*.png \ No newline at end of file diff --git a/lab2/report/img/alg.png b/lab2/report/img/alg.png new file mode 100644 index 0000000000000000000000000000000000000000..f55b76ff0756e7634882b5c55ca1bb42d4065eb4 GIT binary patch literal 176421 zcma&O2{hN=+CJP|niQoBr2%EG%ySwDp(yi|%$dnNG-#A$4k3gjgk&a^j0s6HXG${9 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footskip=10mm]{geometry} +\usepackage{ragged2e} %для растягивания по ширине +\usepackage{setspace} %для межстрочно го интервала +\usepackage{moreverb} %для работы с листингами +\usepackage{indentfirst} % для абзацного отступа +\usepackage{moreverb} %для печати в листинге исходного кода программ +\usepackage{pdfpages} %для вставки других pdf файлов +\usepackage{tikz} +\usepackage{graphicx} +\usepackage{afterpage} +\usepackage{longtable} +\usepackage{float} + + + +% \usepackage[paper=A4,DIV=12]{typearea} +\usepackage{pdflscape} +% \usepackage{lscape} + +\usepackage{array} +\usepackage{multirow} + +\renewcommand\verbatimtabsize{4\relax} +\renewcommand\listingoffset{0.2em} %отступ от номеров строк в листинге +\renewcommand{\arraystretch}{1.4} % изменяю высоту строки в таблице +\usepackage[font=small, singlelinecheck=false, justification=centering, format=plain, labelsep=period]{caption} %для настройки заголовка таблицы +\usepackage{listings} %листинги +\usepackage{xcolor} % цвета +\usepackage{hyperref}% для гиперссылок +\usepackage{enumitem} %для перечислений + +\newcommand{\specialcell}[2][l]{\begin{tabular}[#1]{@{}l@{}}#2\end{tabular}} + + +\setlist[enumerate,itemize]{leftmargin=1.2cm} %отступ в перечислениях + +\hypersetup{colorlinks, + allcolors=[RGB]{010 090 200}} %красивые гиперссылки (не красные) + +% подгружаемые языки — подробнее в документации listings (это всё для листингов) +\lstloadlanguages{ SQL} +% включаем кириллицу и добавляем кое−какие опции +\lstset{tabsize=2, + breaklines, + basicstyle=\footnotesize, + columns=fullflexible, + flexiblecolumns, + numbers=left, + numberstyle={\footnotesize}, + keywordstyle=\color{blue}, + inputencoding=cp1251, + extendedchars=true +} +\lstdefinelanguage{MyC}{ + language=SQL, +% ndkeywordstyle=\color{darkgray}\bfseries, +% identifierstyle=\color{black}, +% morecomment=[n]{/**}{*/}, +% commentstyle=\color{blue}\ttfamily, +% stringstyle=\color{red}\ttfamily, +% morestring=[b]", +% showstringspaces=false, +% morecomment=[l][\color{gray}]{//}, + keepspaces=true, + escapechar=\%, + texcl=true +} + +\textheight=24cm % высота текста +\textwidth=16cm % ширина текста +\oddsidemargin=0pt % отступ от левого края +\topmargin=-1.5cm % отступ от верхнего края +\parindent=24pt % абзацный отступ +\parskip=5pt % интервал между абзацами +\tolerance=2000 % терпимость к "жидким" строкам +\flushbottom % выравнивание высоты страниц + + +% Настройка листингов +\lstset{ + language=python, + extendedchars=\true, + inputencoding=utf8, + keepspaces=true, + % captionpos=b, % подписи листингов снизу +} + +\begin{document} % начало документа + + + + % НАЧАЛО ТИТУЛЬНОГО ЛИСТА + \begin{center} + \hfill \break + \hfill \break + \normalsize{МИНИСТЕРСТВО НАУКИ И ВЫСШЕГО ОБРАЗОВАНИЯ РОССИЙСКОЙ ФЕДЕРАЦИИ\\ + федеральное государственное автономное образовательное учреждение высшего образования «Санкт-Петербургский политехнический университет Петра Великого»\\[10pt]} + \normalsize{Институт компьютерных наук и кибербезопасности}\\[10pt] + \normalsize{Высшая школа технологий искусственного интеллекта}\\[10pt] + \normalsize{Направление: 02.03.01 <<Математика и компьютерные науки>>}\\ + + \hfill \break + \hfill \break + \hfill \break + \hfill \break + \large{Лабораторная работа №2}\\ + \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 Axis parallel hyper-ellipsoid function. + + Общая формула для n-мерного случая: + $$f(\mathbf{x}) = \sum_{i=1}^{n} i \cdot x_i^2$$ + где $\mathbf{x} = (x_1, x_2, \ldots, x_n)$, область определения $x_i \in [-5.12, 5.12]$ для всех $i = 1, \ldots, n$. + + Для двумерного случая (n=2): + $$f(x, y) = 1 \cdot x^2 + 2 \cdot y^2 = x^2 + 2y^2$$ + область нахождения решения $x \in [-5.12, 5.12], y \in [-5.12, 5.12]$. + + Глобальный минимум: $f(\mathbf{x}) = 0$ в точке $x_i = 0$ для всех $i = 1, \ldots, n$. Для двумерного случая: $\min f(x, y) = f(0, 0) = 0$. + + \vspace{0.3cm} + \textbf{Требуется:} + + \begin{enumerate} + \item Создать программу, использующую генетический алгоритм для нахождения минимума данной функции; + \item Для n=2 вывести на экран график функции с указанием найденного экстремума и точек популяции. Предусмотреть возможность пошагового просмотра процесса поиска решения; + \item Исследовать зависимость времени поиска, числа поколений (генераций), точности нахождения решения от основных параметров генетического алгоритма: числа особей в популяции, вероятности кроссинговера и мутации; + \item Повторить процесс поиска решения для n=3, сравнить результаты и скорость работы программы. + \end{enumerate} + + + \newpage + \section{Теоретические сведения} + + Генетические 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. Однако при работе с оптимизационными задачами в непрерывных пространствах вполне естественно представлять гены напрямую вещественными числами. В этом случае хромосома есть вектор вещественных чисел (real-coded алгоритмы). Их точность определяется исключительно разрядной сеткой ЭВМ. Длина хромосомы совпадает с длиной вектора-решения оптимизационной задачи, каждый ген отвечает за одну переменную. Генотип объекта становится идентичным его фенотипу. + + 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. Может быть представлена как двоичная + последовательность (где $s_i = 0, 1$) или как вектор вещественных чисел (real-coded представление). + + \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{Операторы кроссинговера для real-coded алгоритмов} + + Оператор скрещивания непрерывного ГА (кроссовер) порождает одного или нескольких потомков от двух хромосом. Требуется из двух векторов вещественных чисел получить новые векторы по определённым законам. Большинство real-coded алгоритмов генерируют новые векторы в окрестности родительских пар. + + Пусть $C_1=(c_{11},c_{21},\ldots,c_{n1})$ и $C_2=(c_{12},c_{22},\ldots,c_{n2})$ -- две хромосомы, выбранные оператором селекции для скрещивания. + + \textbf{Арифметический кроссовер (arithmetical crossover):} создаются два потомка $H_1=(h_{11},\ldots,h_{n1})$, $H_2=(h_{12},\ldots,h_{n2})$, где: + $$h_{k1}=w \cdot c_{k1}+(1-w) \cdot c_{k2}$$ + $$h_{k2}=w \cdot c_{k2}+(1-w) \cdot c_{k1}$$ + где $k=1,\ldots,n$, $w$ -- весовой коэффициент из интервала $[0;1]$. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\linewidth]{img/arithmetic_crossover.png} + \caption{Арифметический кроссовер} + \label{fig:arithmetic_crossover} + \end{figure} + + \textbf{Геометрический кроссовер (geometrical crossover):} создаются два потомка $H_1=(h_{11},\ldots,h_{n1})$, $H_2=(h_{12},\ldots,h_{n2})$, где: + $$h_{k1}=(c_{k1})^w \cdot (c_{k2})^{(1-w)}$$ + $$h_{k2}=(c_{k2})^w \cdot (c_{k1})^{(1-w)}$$ + где $w$ -- случайное число из интервала $[0;1]$. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\linewidth]{img/geometric_crossover.png} + \caption{Геометрический кроссовер} + \label{fig:geometric_crossover} + \end{figure} + + \textbf{Смешанный кроссовер (BLX-alpha crossover):} генерируется один потомок $H=(h_1,\ldots,h_k,\ldots,h_n)$, где $h_k$ -- случайное число из интервала $[c_{min}-I \cdot \alpha, c_{max}+I \cdot \alpha]$, $c_{min}=\min(c_{k1},c_{k2})$, $c_{max}=\max(c_{k1},c_{k2})$, $I=c_{max}-c_{min}$. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\linewidth]{img/blx_crossover.png} + \caption{Смешанный кроссовер} + \label{fig:blx_crossover} + \end{figure} + + \textbf{SBX кроссовер (Simulated Binary Crossover):} кроссовер, имитирующий двоичный, разработанный в 1995 году исследовательской группой под руководством K. Deb'а. Моделирует принципы работы двоичного оператора скрещивания, сохраняя важное свойство -- среднее значение функции приспособленности остаётся неизменным у родителей и их потомков. + + Создаются два потомка $H_k=(h_{1k}, \ldots, h_{jk}, \ldots, h_{nk})$, $k=1,2$, где: + $$h_{j1} = 0.5[(1+\beta_k)c_{j1} + (1-\beta_k)c_{j2}]$$ + $$h_{j2} = 0.5[(1-\beta_k)c_{j1} + (1+\beta_k)c_{j2}]$$ + + где $\beta_k \geq 0$ -- число, полученное по формуле: + $$\beta_k = \begin{cases} + (2u)^{\frac{1}{n+1}}, & \text{при } u \leq 0.5 \\ + \left(\frac{1}{2(1-u)}\right)^{\frac{1}{n+1}}, & \text{при } u > 0.5 + \end{cases}$$ + + где $u \in (0,1)$ -- случайное число, распределённое по равномерному закону, $n \in [2,5]$ -- параметр кроссовера. Увеличение $n$ повышает вероятность появления потомка в окрестности родителей. + + \begin{figure}[h!] + \centering + \includegraphics[width=0.7\linewidth]{img/sbx_crossover.png} + \caption{SBX кроссовер} + \label{fig:sbx_crossover} + \end{figure} + + \subsubsection{Операторы мутации для real-coded алгоритмов} + + В качестве оператора мутации наибольшее распространение получили: случайная и неравномерная мутация. + + \textbf{Случайная мутация (random mutation):} ген, подлежащий изменению, принимает случайное значение из интервала своего изменения. + + \textbf{Неравномерная мутация (non-uniform mutation):} из особи случайно выбирается точка $c_k$ с разрешёнными пределами изменения $[c_{kl}, c_{kr}]$. Точка меняется на: + $$c_k' = \begin{cases} + c_k + \Delta(t, c_{kr} - c_k), & \text{при } a = 1 \\ + c_k - \Delta(t, c_k - c_{kl}), & \text{при } a = 0 + \end{cases}$$ + + где $a$ -- случайно выбранное направление изменения, $\Delta(t, y)$ -- функция, возвращающая случайную величину в пределах $[0, y]$ таким образом, что при увеличении $t$ среднее возвращаемое значение уменьшается: + $$\Delta(t, y) = y \cdot r \cdot \left(1 - \frac{t}{T}\right)^b$$ + + где $r$ -- случайная величина на интервале $[0, 1]$, $t$ -- текущая эпоха работы генетического алгоритма, $T$ -- общее разрешённое число эпох алгоритма, $b$ -- задаваемый пользователем параметр, определяющий степень зависимости от числа эпох. + + \newpage + \section{Особенности реализации} + В рамках работы создана мини-библиотека \texttt{gen.py} для экспериментов с real-coded + генетическим алгоритмом для многомерных функций. Второй модуль + \texttt{expirements.py} организует серийные эксперименты (перебор параметров, + форматирование и сохранение результатов). + + \begin{itemize} + \item \textbf{Кодирование особей}: каждая хромосома представлена как \texttt{np.ndarray} вещественных чисел (\texttt{Chromosome = NDArray[np.float64]}). Длина хромосомы соответствует размерности задачи оптимизации. Популяция -- список хромосом (\texttt{Population = list[Chromosome]}). Инициализация случайными векторами в заданном диапазоне: + \begin{itemize} + \item \texttt{initialize\_population(pop\_size: int, x\_min: Chromosome, x\_max:}\\ \texttt{Chromosome) -> Population} + \end{itemize} + \item \textbf{Фитнесс и минимум/максимум}: целевая функция принимает хромосому (вектор) и возвращает скалярное значение фитнесса. Для режима минимизации используется внутреннее преобразование при селекции (сдвиг на минимальное значение), что позволяет применять рулетку при отрицательных значениях: + \begin{itemize} + \item \texttt{eval\_population(population: Population, fitness\_func: FitnessFn) -> Fitnesses} + \item Логика режима минимизации в \texttt{genetic\_algorithm(config: GARunConfig) -> GARunResult} + \end{itemize} + \item \textbf{Селекция (рулетка)}: вероятности нормируются после сдвига на минимальное значение в поколении (устойчиво к отрицательным фитнессам). Функция: + \texttt{reproduction(population: Population, fitnesses: Fitnesses) -> Population}. + \item \textbf{Кроссинговер}: реализованы арифметический и геометрический кроссоверы для real-coded алгоритмов. Кроссинговер выполняется попарно по перемешанной популяции с вероятностью $p_c$. Функции: + \begin{itemize} + \item \texttt{arithmetical\_crossover\_fn(p1: Chromosome, p2: Chromosome, w: float) -> tuple[Chromosome, Chromosome]} + \item \texttt{geometrical\_crossover\_fn(p1: Chromosome, p2: Chromosome, w: float) -> tuple[Chromosome, Chromosome]} + \item \texttt{crossover(population: Population, pc: float, crossover\_fn: CrossoverFn) -> Population} + \end{itemize} + \item \textbf{Мутация}: случайная мутация -- с вероятностью $p_m$ на хромосому изменяется один случайно выбранный ген на случайное значение из допустимого диапазона. Функции: + \begin{itemize} + \item \texttt{build\_random\_mutation\_fn(x\_min: Chromosome, x\_max: Chromosome) -> MutationFn} + \item \texttt{mutation(population: Population, pm: float, mutation\_fn: MutationFn) -> Population} + \end{itemize} + + \item \textbf{Критерий остановки}: поддерживается критерий по среднему значению фитнесс-функции в популяции и максимальному количеству поколений. Хранится история всех поколений. Проверка выполняется в функции: + + \texttt{genetic\_algorithm(config: GARunConfig) -> GARunResult}. + \item \textbf{Визуализация}: для двумерных функций реализованы 3D-графики поверхности и 2D-контурные графики с отображением популяций. Функции: + \begin{itemize} + \item \texttt{plot\_fitness\_surface(fitness\_func: FitnessFn, x\_min: Chromosome, x\_max: Chromosome, ax: Axes3D)} + \item \texttt{plot\_fitness\_contour(fitness\_func: FitnessFn, x\_min: Chromosome, x\_max: Chromosome, ax: Axes)} + \item \texttt{save\_generation(generation: Generation, history: list[Generation], config: GARunConfig)} + \end{itemize} + \item \textbf{Измерение времени}: длительность вычислений возвращается в миллисекундах как часть \texttt{GARunResult.time\_ms}. + \item \textbf{Файловая организация}: результаты экспериментов сохраняются иерархически в структуре \texttt{experiments/N/} с таблицами результатов в формате CSV. Задействованные функции: + \begin{itemize} + \item \texttt{clear\_results\_directory(results\_dir: str) -> None} + \item \texttt{run\_single\_experiment(pop\_size: int, pc: float, pm: float) -> tuple[float, float, float, float]} + \item \texttt{run\_experiments\_for\_population(pop\_size: int) -> PrettyTable} + \end{itemize} + \end{itemize} + + В модуле \texttt{expirements.py} задаётся целевая функция axis parallel hyper-ellipsoid: $f(x, y) = x^2 + 2y^2$ и параметры экспериментов. + Серийные запуски и сохранение результатов реализованы в функциях \texttt{run\_single\_experiment}, \texttt{run\_experiments\_for\_population} и \texttt{main}. + + \newpage + \section{Результаты работы} + На Рис.~\ref{fig:gen1}--\ref{fig:lastgen} представлены результаты работы генетического алгоритма со следующими параметрами: + \begin{itemize} + \item $N = 25$ -- размер популяции. + \item $p_c = 0.5$ -- вероятность кроссинговера. + \item $p_m = 0.01$ -- вероятность мутации. + \item $0.05$ -- минимальное среднее значение фитнесс функции по популяции для остановки алгоритма. Глобальный минимум функции равен $f(0, 0) = 0$. + \item Использован арифметический кроссовер для real-coded хромосом. + \end{itemize} + + С каждым поколением точность найденного минимума становится выше. Популяция постепенно сходится к глобальному минимуму в точке $(0, 0)$. На графиках показаны 2D-контурный график (a) и 3D-поверхность целевой функции с точками популяции текущего поколения (b) и (c). + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_001.png} + \caption{График целевой функции и популяции поколения №1} + \label{fig:gen1} + \end{figure} + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_002.png} + \caption{График целевой функции и популяции поколения №2} + \label{fig:gen2} + \end{figure} + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_003.png} + \caption{График целевой функции и популяции поколения №3} + \label{fig:gen3} + \end{figure} + + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_005.png} + \caption{График целевой функции и популяции поколения №5} + \label{fig:gen5} + \end{figure} + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_007.png} + \caption{График целевой функции и популяции поколения №7} + \label{fig:gen7} + \end{figure} + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_010.png} + \caption{График целевой функции и популяции поколения №10} + \label{fig:gen10} + \end{figure} + + \begin{figure}[h!] + \centering + \includegraphics[width=1\linewidth]{img/results/generation_013.png} + \caption{График целевой функции и популяции поколения №13} + \label{fig:lastgen} + \end{figure} + + + \newpage + \phantom{text} + \newpage + \phantom{text} + + \newpage + \section{Исследование реализации} + \subsection{Проведение измерений} + В рамках лабораторной работы необходимо было исследовать зависимость времени выполнения задачи и количества поколений от популяции и вероятностей кроссинговера и мутации хромосомы + + Для исследования были выбраны следующие значения параметров: + \begin{itemize} + \item $N = 10, 25, 50, 100$ -- размер популяции. + \item $p_c = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8$ -- вероятность кроссинговера. + \item $p_m = 0.001, 0.01, 0.05, 0.1, 0.2$ -- вероятность мутации. + \end{itemize} + + Результаты измерений представлены в таблицах \ref{tab:pc_pm_results}--\ref{tab:pc_pm_results4}. В ячейках указано усредненное время в миллисекундах нахождения минимума функции. В скобках указано усредненное количество поколений, за которое было найдено решение. Каждый эксперимент запускался 30 раз. Если в ячейке стоит прочерк, то это означает, что решение не было найдено за 200 поколений. Лучшее значение по времени выполнения для каждого размера популяции выделено жирным шрифтом. + + \newcolumntype{Y}{>{\centering\arraybackslash}X} + + \begin{table}[h!] + \centering + \small + \caption{Результаты для $N = 10$} + \begin{tabularx}{\linewidth}{l *{5}{Y}} + \toprule + $\mathbf{P_c \;\backslash\; P_m}$ & \textbf{0.001} & \textbf{0.010} & \textbf{0.050} & \textbf{0.100} & \textbf{0.200} \\ + \midrule + \textbf{0.3} & 11.5 (167) & 8.4 (123) & 5.4 (78) & 4.9 (71) & 3.3 (48) \\ + \textbf{0.4} & 10.1 (144) & 7.1 (104) & 6.3 (92) & 4.7 (67) & 4.7 (67) \\ + \textbf{0.5} & 11.4 (168) & 7.7 (112) & 5.4 (79) & 6.1 (83) & \textbf{3.1 (44)} \\ + \textbf{0.6} & 11.0 (160) & 6.7 (97) & 4.9 (70) & 4.7 (67) & 5.3 (74) \\ + \textbf{0.7} & 12.1 (174) & 9.3 (135) & 3.7 (52) & 4.7 (67) & 6.5 (92) \\ + \textbf{0.8} & 8.7 (126) & 8.3 (119) & 3.9 (57) & 7.9 (113)& 4.4 (61) \\ + \bottomrule + \end{tabularx} + \label{tab:pc_pm_results} + \end{table} + + \begin{table}[h!] + \centering + \small + \caption{Результаты для $N = 25$} + \begin{tabularx}{\linewidth}{l *{5}{Y}} + \toprule + $\mathbf{P_c \;\backslash\; P_m}$ & \textbf{0.001} & \textbf{0.010} & \textbf{0.050} & \textbf{0.100} & \textbf{0.200} \\ + \midrule + \textbf{0.3} & 14.7 (111) & 8.2 (62) & 4.9 (37) & 4.7 (35) & 8.7 (63) \\ + \textbf{0.4} & 12.8 (95) & 7.3 (54) & 4.7 (35) & 4.3 (32) & 8.2 (61) \\ + \textbf{0.5} & 10.5 (78) & 5.4 (40) & \textbf{2.2 (16)} & 5.5 (40) & 12.1 (89) \\ + \textbf{0.6} & 14.0 (103) & 6.5 (48) & 3.4 (25) & 4.0 (30) & 14.0 (87) \\ + \textbf{0.7} & 11.5 (84) & 6.2 (46) & 3.0 (22) & 3.2 (24) & 11.6 (83) \\ + \textbf{0.8} & 9.2 (64) & 5.8 (41) & 2.5 (18) & 3.0 (22) & 11.2 (78) \\ + \bottomrule + \end{tabularx} + \label{tab:pc_pm_results2} + \end{table} + + \begin{table}[h!] + \centering + \small + \caption{Результаты для $N = 50$} + \begin{tabularx}{\linewidth}{l *{5}{Y}} + \toprule + $\mathbf{P_c \;\backslash\; P_m}$ & \textbf{0.001} & \textbf{0.010} & \textbf{0.050} & \textbf{0.100} & \textbf{0.200} \\ + \midrule + \textbf{0.3} & 6.1 (26) & 5.2 (22) & 6.3 (26) & 11.6 (48) & 40.2 (147) \\ + \textbf{0.4} & 6.1 (26) & 4.5 (19) & 5.2 (22) & 9.8 (40) & 37.2 (149) \\ + \textbf{0.5} & 10.5 (44) & 4.9 (20) & 7.6 (28) & 17.1 (65) & 36.2 (144) \\ + \textbf{0.6} & 7.5 (31) & \textbf{4.6 (19)} & 5.6 (23) & 18.8 (76) & 42.0 (158) \\ + \textbf{0.7} & 7.6 (31) & 4.7 (20) & 7.6 (31) & 13.9 (55) & 34.3 (136) \\ + \textbf{0.8} & 10.8 (44) & 5.0 (21) & 6.1 (24) & 13.9 (56) & 36.5 (145) \\ + \bottomrule + \end{tabularx} + \label{tab:pc_pm_results3} + \end{table} + \begin{table}[h!] + \centering + \small + \caption{Результаты для $N = 100$} + \begin{tabularx}{\linewidth}{l *{5}{Y}} + \toprule + $\mathbf{P_c \;\backslash\; P_m}$ & \textbf{0.001} & \textbf{0.010} & \textbf{0.050} & \textbf{0.100} & \textbf{0.200} \\ + \midrule + \textbf{0.3} & \textbf{7.6 (16)} & 9.5 (21) & 29.0 (60) & 62.9 (128) & -- \\ + \textbf{0.4} & 8.0 (17) & 9.6 (21) & 31.5 (68) & 56.6 (120) & -- \\ + \textbf{0.5} & 9.1 (20) & 9.2 (20) & 22.6 (48) & 59.4 (124) & -- \\ + \textbf{0.6} & 17.8 (38) & 12.3 (26) & 30.0 (64) & 61.1 (128) & 95.3 (196) \\ + \textbf{0.7} & 10.0 (22) & 14.3 (31) & 30.3 (64) & 49.1 (103) & -- \\ + \textbf{0.8} & 16.4 (34) & 12.1 (25) & 31.4 (64) & 54.9 (114) & -- \\ + \bottomrule + \end{tabularx} + \label{tab:pc_pm_results4} + \end{table} + + \newpage + \phantom{text} + \newpage + \phantom{text} + \subsection{Анализ результатов} + + Ключевые наблюдения: + \begin{itemize} + \item При небольших популяциях ($N=10$) повышение $p_m$ ускоряет поиск; наилучшее время при $p_c=0.5$, $p_m=0.2$ (3.1 мс, 44 пок.). + \item Для $N=25$ оптимум при умеренной мутации $p_m\in[0.05,0.10]$; минимум времени при $p_c=0.5$, $p_m=0.05$ (2.2 мс, 16 пок.) — лучшее среди всех экспериментов. + \item Для $N=50$ лучшее время при $p_c=0.6$, $p_m=0.01$ (4.6 мс, 19 пок.). Слишком большая мутация ($p_m=0.2$) резко ухудшает результаты. + \item Для $N=100$ оптимальны низкие $p_m$; лучший результат при $p_c=0.3$, $p_m=0.001$ (7.6 мс, 16 пок.). При $p_m=0.2$ решение часто не находится за 200 поколений. + \item Рост $N$ не гарантирует ускорения: число поколений может уменьшаться, но суммарное время часто растёт из-за большей стоимости одной итерации. + \end{itemize} + + Практические выводы: + \begin{itemize} + \item Для умеренных затрат времени и стабильной сходимости разумно выбирать $N\approx25\text{--}50$, $p_c\approx0.5\text{--}0.6$, $p_m\approx0.01\text{--}0.05$. + \item Оптимальное $p_m$ снижается с ростом $N$: при малых популяциях полезна более агрессивная мутация, при больших — слабая. + \item Слишком большие значения $p_m$ и $p_c$ могут разрушать хорошие решения и ухудшать сходимость; стоит избегать $p_m\ge 0.2$ и высоких $p_c$ при больших $N$. + \end{itemize} + + \newpage + \section{Ответ на контрольный вопрос} + + \textbf{Вопрос}: Какую роль в ГА играет оператор репродукции (ОР)? + + \textbf{Ответ}: Оператор репродукции (ОР) в ГА играет роль селекции. Он выбирает наиболее приспособленных особей для дальнейшего участия в скрещивании и мутации. Это позволяет сохранить наиболее приспособленные особи и постепенно улучшить популяцию. + + + \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

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