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[Перевод] Раскрываем секреты роя: оптимизация на Python с помощью PSO

Начну с небольшой шутки:

"Знаете ли вы, что до изобретения часов людям приходилось активно ходить повсюду и спрашивать время?"

Этот незамысловатый анекдот иллюстрирует важную концепцию: информация, доступная одному члену группы, может распространяться среди остальных. Эта идея имеет глубокий смысл и находит применение во многих областях.

0c2348834f8a0369dd43f144429f37aa.png

Рассмотрим самоорганизующиеся системы в природе, например, стаи птиц или рыб. Представим такую систему как совокупность частиц, где каждая особь – это отдельная частица. Можно предположить, что движение каждой частицы в пространстве определяется двумя основными факторами:

Индивидуально оптимальная позиция: то, что особь считает наилучшим для себя.

И глобально оптимальная позиция: определяемая коллективным взаимодействием частиц, своего рода "инструкция", получаемая особью от "лидера группы".

В связи с этим возникает естественный вопрос: что считать "оптимальным" в природе? Что является наилучшим для отдельной особи и для всей группы? Не являясь биологом, я не могу дать ответы на эти вопросы. Однако, наблюдая за подобным поведением в природе, мы можем разработать эффективный алгоритм оптимизации. Другими словами, определив критерии "оптимальности", мы можем применить этот эволюционный подход для оптимизации заданной функции.

Данный алгоритм известен как оптимизация роем частиц (Particle Swarm Optimization, PSO). Возможно, это звучит несколько сложно. Что подразумевается под "оптимизацией"? Какова роль математики в этом процессе? Что именно оптимизируется? В статье я постараюсь подробно разъяснить все эти моменты. Более того, мы применим ООП на Python для создания собственного класса ParticleSwarmOptimizer(). И таким образом, мы пройдем путь от теоретических основ PSO до их практической реализации.

Итак, приступим! Желаю приятного чтения.

0. Введение в понятие "оптимизация"

Если вас заинтересовал метод "оптимизации роем частиц", вероятно, вы уже знакомы с общим понятием оптимизации. Однако, возможно, термин "рой частиц" для вас нов. Поэтому, чтобы не перегружать вас лишней информацией, я не буду подробно останавливаться на базовых понятиях оптимизации.

Давайте все же уточним постановку задачи. Как мы уже упомянули, что с математической точки зрения подразумевается под "наилучшим решением для роя"?

С математической точки зрения, "птицы роя" (или "частицы") представляют собой точки в K-мерном пространстве. В случае реального физического пространства, нашей "областью поиска" может быть привычное нам трехмерное пространство, однако в общем случае размерность может быть значительно больше. Важно отметить, что данный метод применим не только к примеру с птицами: мы можем использовать его для нахождения минимума или максимума любой заданной функции.

Рассмотрим пример: представим, что цена дома определяется автоматически с помощью алгоритма (хотя полностью полагаться на алгоритм в таком вопросе было бы крайне рискованно):

76457d0073bdf80b705ede25c601460f.png

Первоначально может быть не совсем понятно, что подразумевается под "Домом". На практике мы имеем дело со списком характеристик, таких как местоположение, размер и т.д.

bb95b30f66e2509f07e88503cab87753.png

Теперь картина становится яснее. Каждый дом можно представить как точку в многомерном пространстве признаков. Иными словами, каждый дом – это точка в k-мерном пространстве, где k – это количество характеристик (местоположение, размер и т.д.). Процесс преобразования этих характеристик в цену дома можно представить в виде "черного ящика":

7c4283bd06df8533f3188031e0e0021a.png

Ключевой вопрос, который мы ставим перед собой:

"Какими значениями признаков (x) должен обладать дом, чтобы его стоимость была минимальной?"

Именно в этом и заключается суть нашей задачи оптимизации.

df2613658adaeab9a23d6850c384e741.png

В конечном счете, метод оптимизации роем частиц поможет нам найти координаты xmin, то есть те значения признаков, которые обеспечат минимальную стоимость. Далее мы подробно рассмотрим, как это работает.

1. Знакомство с оптимизацией роем частиц

Прежде чем углубиться в детали, рекомендую ознакомиться с оригинальной статьей. Это одна из наиболее доступных научных статей, с которыми мне приходилось сталкиваться: она написана простым и понятным языком, но при этом информативна и увлекательна.

Итак, в чем заключается основная идея метода? Как уже упоминалось, мы создаем набор частиц, которые взаимодействуют и обмениваются информацией, чтобы найти глобальный минимум целевой функции. Если провести аналогию, это похоже на сериал "Очень странные дела", где герои объединяются и используют все доступные средства коммуникации, чтобы победить общего врага.

С точки зрения реализации, мы начинаем с заданного количества частиц (num_particles). Каждой частице присваивается случайное начальное положение в пространстве признаков. Для каждого положения рассчитывается значение целевой функции L. Например, частица 1 будет иметь координаты x1 и значение функции L(x1). Это нулевая итерация, после которой наша система готова к эволюции.

Как происходит эта эволюция? С помощью параметра, называемого "скоростью". Для каждой частицы x, каждого измерения (i) и каждой итерации (j) мы определяем скорость следующим образом:

de7e7adf2b080a1fb574614de0b887c3.png

Или, в более компактной векторной форме:

08d612e9c5ecd7c707b409caa7c7d42d.png

Теперь давайте разберемся, как определяется вектор скорости v. Именно здесь проявляется вся элегантность метода оптимизации роя частиц. Вектор v состоит из трех основных компонентов:

6544da6c92ae799ad09c5285447990af.png

Рассмотрим каждый компонент подробнее:

vinertial: Этот компонент представляет собой "память" о предыдущей скорости частицы. Термин заимствован из физики, где инерциальная система отсчета – это система, в которой тело, не подверженное воздействию внешних сил, сохраняет состояние покоя или равномерного прямолинейного движения.

622019351aedc690925bbd9e495f18a4.png

где kinertial – это константа.

vcognitive: Этот компонент представляет собой скорость, предлагаемую самой частицей (отсюда и название "когнитивный"). Он направляет частицу в сторону ее лучшего известного положения, то есть туда, где значение целевой функции было минимальным для этой конкретной частицы. Для добавления случайности в процесс поиска мы также используем случайное число r1:

0cd1482289da2b22c0ca3b294e6403ff.png

kcognitive – это также константа. xbest– это координаты лучшего известного положения для частицы x, то есть координаты, где значение целевой функции было минимальным для этой частицы.

vsocial: Этот компонент отвечает за учет информации от всех остальных частиц роя. Он направляет частицу в сторону глобально лучшего положения, то есть туда, где значение целевой функции было минимальным для всего роя.

25a8a09fd1c5f48594b5697ed9ef89ee.png

ksocial – это константа, r2– случайное число, а xglobal best – координаты точки в пространстве признаков, где целевая функция достигает своего глобального минимума (наименьшего значения для всего роя).

Этот подход, на мой взгляд, действительно довольно компактен. Давайте подведем итог:

  • Мы начинаем с создания заданного количества частиц и случайного размещения их в пространстве признаков.

  • Частицы перемещаются в пространстве признаков, основываясь на своих векторах скорости.

  • Скорость каждой частицы определяется тремя составляющими: инерцией, когнитивной и социальной составляющими.

  • Мы повторяем шаги движения и расчета скорости на протяжении заданного количества итераций (num_iteration).

  • По окончании итераций мы выбираем частицу, которая достигла наилучшего результата (минимального значения целевой функции).

На этом теоретическая часть завершена. Давайте перейдем к практике и напишем код на Python!

2. Реализация PSO на практике

Для реализации алгоритма PSO мы создадим два класса. Первый класс будет описывать отдельную частицу роя, а второй, используя первый как основу, будет отвечать за создание и управление всем роем.

Для начала создадим файл constants.py. В этом файле мы будем хранить все параметры по умолчанию, такие как границы области поиска для частиц, количество итераций алгоритма и значения коэффициентов k (инерции, когнитивной и социальной составляющих).

Как изменить эти параметры? Для этого необходимо создать файл pso_config.json в формате JSON и поместить его в папку config. В этом файле вы сможете задать собственные значения для параметров. Если файл pso_config.json отсутствует, скрипт, который мы напишем для классов, создаст его автоматически, используя параметры по умолчанию.

NUM_PARTICLES = 30
NUM_ITERATIONS = 100
INERTIA = 0.5
COGNITIVE = 2 
SOCIAL = 2
BOUNDS = [[-10,10],[-10,10]]
CONFIG_FILENAME = 'config/pso_config.json'
DEFAULT_CONFIG_NAMES = ["num_particles","num_iterations","inertia","cognitive","social","bounds"]
DEFAULT_PARTICLE = [NUM_PARTICLES, NUM_ITERATIONS,INERTIA, COGNITIVE, SOCIAL, BOUNDS]

Теперь все, что мы обсуждали, реализовано в коде, который находится в файле particle_swarm_optimizer.py.

import numpy as np 
import json 
from constants import * # Импортируем константы из файла constants.py
import os
import matplotlib.pyplot as plt
import matplotlib.animation as animation
import numpy as np


class Particle:
    def __init__(self, bounds):
        # Инициализируем позицию частицы случайным образом с учетом границ
        self.position = np.array([np.random.uniform(low, high) for low, high in bounds])
        # Инициализируем скорость частицы нулями
        self.velocity = np.zeros_like(self.position)
        # Инициализируем лучшую позицию частицы текущей позицией
        self.best_position = np.copy(self.position)
        # Инициализируем лучшее значение целевой функции бесконечностью
        self.best_value = float('inf')

    def update_velocity(self, global_best_position, inertia, cognitive, social):
        # Генерируем случайные числа для когнитивной и социальной составляющих скорости
        r1 = np.random.random(len(self.position))
        r2 = np.random.random(len(self.position))

        # Вычисляем когнитивную составляющую скорости
        cognitive_velocity = cognitive * r1 * (self.best_position - self.position)
        # Вычисляем социальную составляющую скорости
        social_velocity = social * r2 * (global_best_position - self.position)
        # Обновляем скорость частицы
        self.velocity = inertia * self.velocity + cognitive_velocity + social_velocity

    def update_position(self, bounds):
        # Обновляем позицию частицы
        self.position += self.velocity
        # Проверяем, не вышла ли частица за границы области поиска
        for i in range(len(self.position)):
            if self.position[i] < bounds[i][0]:
                self.position[i] = bounds[i][0]
            if self.position[i] > bounds[i][1]:
                self.position[i] = bounds[i][1]


class ParticleSwarmOptimizer:
    def __init__(self, particle_class, objective_function, config_filename = None):
        
        self.particle_class = particle_class  # Класс частицы, который будет использоваться
        if config_filename is None:
            self.config = self.load_or_create_config()
        else:
            self.config = self.load_or_create_config(config_filename)
        self.objective_function = objective_function # Целевая функция, которую нужно оптимизировать
        self.bounds = self.config['bounds'] # Границы области поиска
        self.num_particles = self.config['num_particles'] # Количество частиц в рое
        self.num_iterations = self.config['num_iterations'] # Количество итераций алгоритма
        self.inertia = self.config['inertia'] # Коэффициент инерции
        self.cognitive = self.config['cognitive'] # Коэффициент когнитивной составляющей
        self.social = self.config['social'] # Коэффициент социальной составляющей
        # Создаем рой частиц
        self.swarm = [self.particle_class(self.bounds) for _ in range(self.num_particles)]
        # Инициализируем глобально лучшую позицию и значение
        self.global_best_position = np.copy(self.swarm[0].position)
        self.global_best_value = float('inf')


    def load_or_create_config(self,filename=CONFIG_FILENAME):
        if os.path.exists(filename):
            # Если файл конфигурации существует, загружаем его
            with open(filename, 'r') as f:
                config = json.load(f)
                print("Loaded configuration from", filename)
        else:
            # Если файл конфигурации не найден, создаем его с значениями по умолчанию
            with open(filename, 'w') as f:
                default_config = {DEFAULT_CONFIG_NAMES[i]:DEFAULT_PARTICLE[i] for i in range(len(DEFAULT_PARTICLE))}
                json.dump(default_config, f, indent=4)
                config = default_config
                print(f"Configuration file not found. Created default config at {filename}")
        return config

        
    def optimize(self, save_iterations=False, print_iterations=False):
        # Инициализируем списки для хранения лучших значений и позиций на каждой итерации, если требуется
        if save_iterations:
            best_values_per_iteration = np.zeros(self.num_iterations)
            best_positions_per_iteration = np.zeros((self.num_iterations, len(self.bounds)))
    
        for iteration in range(self.num_iterations):
            for particle in self.swarm:
                # Вычисляем значение целевой функции для текущей позиции частицы
                value = self.objective_function(particle.position)
                
                # Обновляем личный лучший результат частицы
                if value < particle.best_value:
                    particle.best_value = value
                    particle.best_position = np.copy(particle.position)
    
                # Обновляем глобально лучший результат
                if value < self.global_best_value:
                    self.global_best_value = value
                    self.global_best_position = np.copy(particle.position)
    
            # Сохраняем текущее глобально лучшее значение и позицию для этой итерации, если требуется
            if save_iterations:
                best_values_per_iteration[iteration] = self.global_best_value
                best_positions_per_iteration[iteration] = np.copy(self.global_best_position)
    
            # Обновляем скорость и позицию каждой частицы
            for particle in self.swarm:
                particle.update_velocity(self.global_best_position, self.inertia, self.cognitive, self.social)
                particle.update_position(self.bounds)
    
            # Выводим информацию о текущей итерации, если требуется
            if print_iterations:
                print(f"Iteration {iteration+1}/{self.num_iterations}, Global Best Value: {self.global_best_value}")
        
        # Возвращаем лучшие значения и позиции для каждой итерации, если save_iterations=True
        if save_iterations:
            self.best_values_per_iteration = best_values_per_iteration
            self.best_positions_per_iteration = best_positions_per_iteration
            return self.global_best_position, self.global_best_value, self.best_values_per_iteration, self.best_positions_per_iteration
        else:
            return self.global_best_position, self.global_best_value, None, None



    
    def plot_pso_convergence(self, animated=False, gif_name="pso_convergence.gif"):
        """
        Строит график сходимости алгоритма PSO с контурным графиком целевой функции.
        
        Параметры:
        - objective_function: целевая функция для построения контурного графика.
        - animated: если True, создает анимированный GIF, показывающий процесс сходимости с движением частиц.
        - gif_name: имя GIF-файла для сохранения, если animated=True.
        """
        objective_function = self.objective_function
        iteration_values = self.best_values_per_iteration
        iteration_positions = self.best_positions_per_iteration
        
        # Создаем сетку точек для вычисления целевой функции для построения контурного графика
        x = np.linspace(self.bounds[0][0], self.bounds[0][1], 100)
        y = np.linspace(self.bounds[1][0], self.bounds[1][1], 100)
        X, Y = np.meshgrid(x, y)
        Z = np.array([[objective_function([x_val, y_val]) for x_val in x] for y_val in y])
        
        if animated:
            # Создаем анимированный GIF с движением частиц и глобально лучшей позицией
            fig, ax = plt.subplots()
    
            # Строим контурный график целевой функции
            contour = ax.contourf(X, Y, Z, levels=50, cmap='viridis')
            plt.colorbar(contour, ax=ax)
    
            # Инициализируем точечный график для позиций частиц
            particles = ax.scatter([], [], c='blue', label="Particles", s=50)
            global_best = ax.scatter([], [], c='red', label="Global Best", s=100, marker='x')
    
            ax.set_title("PSO Particle Movement and Convergence")
            ax.set_xlabel("X Position")
            ax.set_ylabel("Y Position")
    
            def init():
                particles.set_offsets([])
                global_best.set_offsets([])
                return particles, global_best
    
            def update(frame):
                # Обновляем позиции частиц (в этом случае только глобально лучшую позицию)
                particle_positions = iteration_positions[frame]
                particles.set_offsets(particle_positions)
                
                # Обновляем глобально лучшую позицию (красный крестик)
                global_best_position = iteration_positions[frame]
                global_best.set_offsets(global_best_position)
    
                return particles, global_best
    
            ani = animation.FuncAnimation(fig, update, frames=len(iteration_positions), init_func=init, blit=True)
    
            # Сохраняем анимацию как GIF
            ani.save(gif_name, writer='imagemagick', fps=5)
            print(f"Animation saved as {gif_name}")
    
        else:
            # Статический график с контуром и сходимостью глобально лучшего значения по итерациям
            fig, ax = plt.subplots()
    
            # Строим контурный график целевой функции
            contour = ax.contourf(X, Y, Z, levels=50, cmap='viridis')
            plt.colorbar(contour, ax=ax)
    
            # Строим график глобально лучших значений по итерациям
            plt.plot(iteration_values, marker='o', linestyle='-', color='b')
            plt.title("PSO Convergence Plot")
            plt.xlabel("Iterations")
            plt.ylabel("Global Best Value")
            plt.grid(True)
            plt.show()

Давайте кратко рассмотрим структуру реализованного кода:

Класс Particle():

Этот класс отвечает за создание и описание отдельных частиц роя. Каждая частица инициализируется случайным положением в пространстве признаков. Класс также содержит методы update_velocity() и update_position(), которые обновляют скорость и положение частицы на каждой итерации алгоритма в соответствии с описанными выше правилами.

Класс ParticleSwarmOptimizer():

Этот класс отвечает за создание и управление всем роем частиц. При создании объекта этого класса необходимо передать два аргумента:

  1. Particle: Используется для создания объектов-частиц. Вы можете модифицировать или расширить класс Particle(), чтобы изменить поведение алгоритма или добавить новые функции.

  2. objective_function: Это функция, которую мы стремимся оптимизировать (найти ее минимум). Она представляет собой "черный ящик", который принимает на вход координаты частицы в пространстве признаков и возвращает значение целевой функции в этой точке.

Основная логика алгоритма PSO реализована в классе ParticleSwarmOptimizer(). Метод optimize() запускает процесс оптимизации. Вы можете управлять выводом информации во время оптимизации с помощью параметров print_iterations (выводить ли информацию о каждой итерации) и save_iterations (сохранять ли историю лучших значений и позиций на каждой итерации). Метод plot_pso_convergence() отвечает за визуализацию процесса сходимости алгоритма. Если установить параметр animated=True, метод создаст GIF-анимацию, наглядно демонстрирующую движение частиц и изменение положения глобального минимума на каждой итерации (вскоре мы увидим пример).

3. Практические примеры

Мы проделали всю необходимую подготовительную работу и теперь можем запустить алгоритм PSO всего несколькими строками кода.

Для начала рассмотрим простую квадратичную функцию:

7e8774531ca20a36dcefabca2a27aaa1.png
import numpy as np
import json
import os
from particle_swarm_optimizer import * 

def quadratic_objective_function(x):
    return x[0]**2 + x[1]**2 + 50  # Простая квадратичная функция
pso = ParticleSwarmOptimizer(Particle,quadratic_objective_function)
best_position, best_value, iteration_values,iteration_positions = pso.optimize(save_iterations=True,print_iterations=False)
print("Best Position:", best_position)
print("Best Value:", best_value)
pso.plot_pso_convergence(animated=True)

Вот гиф, которую мы получим:

7bbc07c66aea3799c56e30fe319e9063.gif

Алгоритм PSO отлично справляется даже с достаточно сложными целевыми функциями, например:

5194ed100ae2bbaff268a6d6123c507f.png

Запустив тот же код, что и в предыдущем примере, мы получим:

def complex_objective_function(x):
    # Комбинация квадратичной функции, синусоиды и экспоненциального затухания
    term1 = (x[0]**2 + x[1]**2)  # Квадратичная функция
    term2 = 10 * np.sin(x[0]) * np.sin(4*x[1])  # Синусоида, вносящая нелинейность
    term3 = np.exp(-0.1 * (x[0]**2 + x[1]**2))  # Экспоненциальное затухание для создания локальных минимумов
    return term1 - term2 + 5 * term3 + 20  # 20 просто сдвигает всю поверхность вверх

# Создаем объект оптимизатора PSO
pso = ParticleSwarmOptimizer(Particle,complex_objective_function)
# Запускаем оптимизацию и сохраняем значения на каждой итерации
best_position, best_value, iteration_values,iteration_positions = pso.optimize(save_iterations=True,print_iterations=False)
# Выводим результаты
print("Best Position:", best_position)
print("Best Value:", best_value)
# Строим график сходимости с анимацией
pso.plot_pso_convergence(animated=True)

Наша гифка:

696e4a3ec06aa816a48c4f22dc751886.gif

Мы убедились, что алгоритм PSO не только эффективно находит минимум для функций с одной ярко выраженной точкой минимума, но и способен находить глобальный минимум (по крайней мере, мы можем предположить, что это именно глобальный минимум, так как аналитически это сложно проверить) даже для сложных функций с множеством локальных минимумов.

4. Выводы

Благодарю вас за то, что уделили время прочтению этой статьи и вместе со мной погрузились в увлекательный мир оптимизации с помощью PSO. Надеюсь, мне удалось доступно и понятно изложить основные идеи алгоритма.

В рамках статьи мы:

  • Рассмотрели подход к решению задач оптимизации, основанный на анализе информации, получаемой от множества частиц (агентов).

  • Проиллюстрировали задачу "оптимизации" на простом и наглядном примере поиска минимальной цены дома.

  • Ознакомились с основными принципами оптимизации роем частиц (PSO) и ее ключевой идеей.

  • Изучили математические основы алгоритма PSO и проследили, как он эволюционирует от начального случайного состояния роя, состоящего из num_particles частиц, к нахождению наилучшей оценки глобального минимума целевой функции.

  • Реализовали алгоритм PSO с нуля, используя ООП на Python.

  • Протестировали нашу реализацию на простой квадратичной функции и на более сложной функции с множеством локальных минимумов. В обоих случаях алгоритм продемонстрировал высокую эффективность и точность!

Источник

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  • 15.10.25 18:07 crypto

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  • 15.10.25 21:52 harristhomas7376

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  • 15.10.25 21:52 harristhomas7376

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  • 17.10.25 20:17 tyleradams

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  • 17.10.25 20:20 lindseyvonn

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  • 17.10.25 20:22 richardcharles

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  • 17.10.25 21:42 marcushenderson624

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  • 17.10.25 21:42 marcushenderson624

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  • 17.10.25 21:42 marcushenderson624

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  • 21.10.25 08:39 debby131

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  • 21.10.25 11:45 harristhomas7376

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  • 21.10.25 11:45 harristhomas7376

    "In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 22.10.25 04:48 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 22.10.25 07:36 donnacollier

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  • 22.10.25 11:59 elizabethrush89

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  • 22.10.25 11:59 elizabethrush89

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  • 22.10.25 16:31 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 23.10.25 01:52 marcushenderson624

    Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 23.10.25 01:52 marcushenderson624

    Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 23.10.25 15:47 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 23.10.25 21:43 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 23.10.25 21:43 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.10.25 06:08 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 24.10.25 06:40 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.10.25 06:40 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.10.25 06:54 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 24.10.25 18:18 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.10.25 18:18 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 25.10.25 00:49 tyleradams

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  • 25.10.25 00:50 stevekalfman

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  • 25.10.25 05:05 victoriabenny463

    A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)

  • 25.10.25 10:13 wendytaylor015

    My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 25.10.25 10:13 wendytaylor015

    My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 01:03 Christopherbelle

    A terrible financial shock hit me hard. I lost $860,000 in a fake crypto scheme. Things looked good until I tried to cash out my earnings. Suddenly, my account vanished into thin air. I told the police many times, but help never showed up. Despair washed over me; I felt totally beaten. Then, I found Sylvester Bryant Intelligence. Right away, they were skilled and honest about the whole thing. They looked deep into my situation. They tracked where the stolen funds went. They fought hard for me every step of the way. To my great surprise, they got my money back. I thought that was impossible. I owe them so much for their hard work and truthfulness. Sylvester Bryant Intelligence restored my peace. If scams took your crypto, contact them now. Contact Info: Yt7cracker@gmail . com WhatsApp: ‪+1 512 577 7957. or +44 7428 662701.

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