【算法】模拟退火

一、引言

        模拟退火算法（Simulated Annealing, SA）是一种启发式搜索算法，它通过模拟物理中的退火过程来解决优化问题。这种算法能够跳出局部最优解，寻找全局最优解，特别适用于解决复杂的优化问题。

二、算法原理

        模拟退火算法的核心原理是模拟物理中的退火过程，将问题的解状态视为物理系统的状态，目标函数值视为系统的能量。算法从初始温度开始，通过随机扰动当前解产生新解，并根据Metropolis准则决定是否接受新解。随着温度的逐渐降低，系统逐渐趋于稳定，最终在低温下达到全局最优或近似最优解。

        模拟退火算法（Simulated Annealing, SA）也是一种基于概率的优化算法，灵感来源于金属退火过程。金属退火是一个加热和缓慢冷却的过程，目的是提高材料的强度和硬度。模拟退火算法试图从一个初始解出发，逐步搜索其他可能解，以找到全局最优解。

其工作原理如下：

• 随机选择初始解和初始温度。
• 利用当前温度对解进行扰动，产生新解。
• 计算新解与旧解的目标函数值：
  • 如果新解更优，则接受新解。
  • 如果新解不优，则以一定概率接受新解（即存在“跳出局部最优”的可能性），概率依赖于当前温度和解的优劣。
• 随着每次迭代，逐渐降低温度。
• 直到达到终止条件（如达到最大迭代次数或温度降至某个阈值）。

三、数据结构

模拟退火算法主要涉及以下数据结构：

• 解空间：表示所有可能解的集合。
• 当前解：当前迭代中正在考虑的解。
• 新解：通过随机扰动当前解产生的新解。
• 温度参数：控制算法搜索过程的冷却速度。

四、算法使用场景

模拟退火算法适用于多种优化问题，包括但不限于：

• 调度问题：在满足约束条件下，优化任务的执行顺序。
• 神经网络权重优化：调整神经网络的权重和偏置，以提高模型性能,超参数优化等。
• 组合优化问题：如旅行商问题（TSP）、背包问题等。

• N-Queens 问题：寻找 N 皇后问题的解。
• 约束满足问题：如图着色问题。

五、算法实现

• 初始化：选择初始解，并设置初始温度。
• 迭代：在当前解的邻域内生成一个新解。
• 接受准则：如果新解比当前解好，则接受新解；如果新解较差，则根据概率接受新解（这个概率随着温度的降低而减少）。
• 降温：逐步降低温度，使得接受较差解的概率逐渐减小。
• 终止条件：当温度降低到指定值或迭代次数达到上限时，算法停止。

import mathimport randomdef simulated_annealing(initial_state, temperature, cooling_rate, max_iterations, objective_function):current_state = initial_statecurrent_energy = objective_function(current_state)for i in range(max_iterations):# 生成新解new_state = perturb(current_state) # perturb是自定义的新解生成函数new_energy = objective_function(new_state)# 计算接受概率if new_energy < current_energy:current_state = new_statecurrent_energy = new_energyelse:acceptance_probability = math.exp((current_energy - new_energy) / temperature)if random.random() < acceptance_probability:current_state = new_statecurrent_energy = new_energy# 降温temperature *= cooling_ratereturn current_statedef perturb(state):# 这里定义扰动操作，比如随机交换两个元素new_state = state[:] # 复制当前状态i, j = random.sample(range(len(state)), 2)new_state[i], new_state[j] = new_state[j], new_state[i]return new_statedef objective_function(state):# 计算目标函数值，示例计算归并值return sum(state)# 示例使用initial_state = [5, 3, 1, 7, 2, 4, 6]temperature = 1000cooling_rate = 0.95max_iterations = 1000best_state = simulated_annealing(initial_state, temperature, cooling_rate, max_iterations, objective_function)print(\"Best state:\", best_state)print(\"Objective value:\", objective_function(best_state))

六、其他同类算法对比

• 遗传算法：基于自然选择和遗传机制，适合大规模复杂的优化问题，相比之下计算开销大。
• 粒子群优化：通过群体智能进行搜索，适合多维和非线性问题，通常收敛速度较快。
• 蚁群算法：通过模拟蚂蚁觅食行为进行优化，适合图论中的路径问题。

七、多语言代码实现

Java

import java.util.Random;public class SimulatedAnnealing {private static double objectiveFunction(int[] state) {double sum = 0;for (int i : state) {sum += i;}return sum;}private static int[] perturb(int[] state) {Random rand = new Random();int[] newState = state.clone();int i = rand.nextInt(state.length);int j = rand.nextInt(state.length);// 交换元素int temp = newState[i];newState[i] = newState[j];newState[j] = temp;return newState;}public static int[] simulatedAnnealing(int[] initialState, double temperature, double coolingRate, int maxIterations) {int[] currentState = initialState;double currentEnergy = objectiveFunction(currentState);for (int i = 0; i < maxIterations; i++) {int[] newState = perturb(currentState);double newEnergy = objectiveFunction(newState);if (newEnergy < currentEnergy) {currentState = newState;currentEnergy = newEnergy;} else {double acceptanceProbability = Math.exp((currentEnergy - newEnergy) / temperature);if (Math.random() < acceptanceProbability) {currentState = newState;currentEnergy = newEnergy;}}// 降温temperature *= coolingRate;}return currentState;}public static void main(String[] args) {int[] initialState = {5, 3, 1, 7, 2, 4, 6};double temperature = 1000.0;double coolingRate = 0.95;int maxIterations = 1000;int[] bestState = simulatedAnnealing(initialState, temperature, coolingRate, maxIterations);System.out.println(\"Best state: \" + java.util.Arrays.toString(bestState));System.out.println(\"Objective value: \" + objectiveFunction(bestState));}}

C++

#include #include #include #include #include double objectiveFunction(const std::vector& state) {return std::accumulate(state.begin(), state.end(), 0.0);}std::vector perturb(const std::vector& state) {std::vector newState = state;std::swap(newState[rand() % state.size(), rand() % state.size()]);return newState;}std::vector simulatedAnnealing(std::vector initialState, double temperature, double coolingRate, int maxIterations) {std::vector currentState = initialState;double currentEnergy = objectiveFunction(currentState);for (int i = 0; i < maxIterations; i++) {auto newState = perturb(currentState);double newEnergy = objectiveFunction(newState);if (newEnergy < currentEnergy) {currentState = newState;currentEnergy = newEnergy;} else {double acceptanceProbability = exp((currentEnergy - newEnergy) / temperature);if ((static_cast(rand()) / RAND_MAX) < acceptanceProbability) {currentState = newState;currentEnergy = newEnergy;}}temperature *= coolingRate;}return currentState;}int main() {std::vector initialState = {5, 3, 1, 7, 2, 4, 6};double temperature = 1000.0;double coolingRate = 0.95;int maxIterations = 1000;auto bestState = simulatedAnnealing(initialState, temperature, coolingRate, maxIterations);std::cout << \"Best state: \";for (const auto& val : bestState) {std::cout << val << \" \";}std::cout << \"\\nObjective value: \" << objectiveFunction(bestState) << std::endl;return 0;}

Python

import mathimport randomclass SimulatedAnnealing: def __init__(self, cooling_rate=0.99): self.temperature = 1000 self.cooling_rate = cooling_rate def find_solution(self, distances): current = self.initialize_solution(len(distances)) best = current.copy() while self.temperature > 1: next = self.perturb_solution(current) delta = self.calculate_cost(distances, next) - self.calculate_cost(distances, current) if self.accept(delta): current = next if self.is_better(current, best):  best = current.copy() self.temperature *= self.cooling_rate return best def accept(self, delta): return math.exp(-delta / self.temperature) > random.random()

Go

package mainimport (\"fmt\"\"math\"\"math/rand\"\"time\")func objectiveFunction(state []int) float64 {sum := 0for _, v := range state {sum += v}return float64(sum)}func perturb(state []int) []int {newState := make([]int, len(state))copy(newState, state)i, j := rand.Intn(len(state)), rand.Intn(len(state))newState[i], newState[j] = newState[j], newState[i]return newState}func simulatedAnnealing(initialState []int, temperature, coolingRate float64, maxIterations int) []int {currentState := initialStatecurrentEnergy := objectiveFunction(currentState)for i := 0; i < maxIterations; i++ {newState := perturb(currentState)newEnergy := objectiveFunction(newState)if newEnergy < currentEnergy {currentState = newStatecurrentEnergy = newEnergy} else {acceptanceProbability := math.Exp((currentEnergy - newEnergy) / temperature)if rand.Float64() < acceptanceProbability {currentState = newStatecurrentEnergy = newEnergy}}temperature *= coolingRate}return currentState}func main() {rand.Seed(time.Now().UnixNano())initialState := []int{5, 3, 1, 7, 2, 4, 6}temperature := 1000.0coolingRate := 0.95maxIterations := 1000bestState := simulatedAnnealing(initialState, temperature, coolingRate, maxIterations)fmt.Println(\"Best state:\", bestState)fmt.Println(\"Objective value:\", objectiveFunction(bestState))}

八、实际服务应用场景的代码框架

        在物流配送系统中，模拟退火算法可以用于优化配送路径，减少配送时间和成本。系统会根据实时交通数据、配送点位置等信息，不断调整配送路径，以达到最优配送效果。

        为一个无线网络调度问题应用模拟退火算法，整个代码框架示例：

wireless_network_optimization/├── main.py # 主程序入口├── optimization.py # 模拟退火算法实现├── network.py # 网络相关数据结构及功能实现└── utils.py # 其他辅助函数

main.py 实现

from optimization import SimulatedAnnealingfrom network import Networkdef main():network = Network()initial_configuration = network.get_initial_configuration()best_configuration = SimulatedAnnealing.run(initial_configuration,temperature=1000,cooling_rate=0.95,max_iterations=1000,objective_function=network.objective_function)print(\"最优配置:\", best_configuration)print(\"最优目标值:\", network.objective_function(best_configuration))if __name__ == \"__main__\":main()

optimization.py 实现

import mathimport randomclass SimulatedAnnealing:@staticmethoddef run(initial_state, temperature, cooling_rate, max_iterations, objective_function):current_state = initial_statecurrent_energy = objective_function(current_state)for i in range(max_iterations):new_state = SimulatedAnnealing.perturb(current_state)new_energy = objective_function(new_state)if new_energy < current_energy:current_state = new_statecurrent_energy = new_energyelse:acceptance_probability = math.exp((current_energy - new_energy) / temperature)if random.random() < acceptance_probability:current_state = new_statecurrent_energy = new_energytemperature *= cooling_ratereturn current_state@staticmethoddef perturb(state):# 定义扰动逻辑pass

network.py 实现

class Network:def __init__(self):# 初始化网络相关参数passdef get_initial_configuration(self):# 获取初始配置passdef objective_function(self, configuration):# 计算目标函数pass

utils.py 实现

# 辅助函数，可能包含数据处理等def load_data(file_path):# 加载数据passdef save_results(results, file_path):# 保存结果pass

        模拟退火算法（Simulated Annealing, SA）是一种启发式全局优化算法，灵感来源于固体退火原理。在冶金学中，退火是将金属加热到一定温度，再缓慢冷却以消除内部应力，使金属结构达到稳定状态。在优化问题中，模拟退火算法通过接受一定概率的“坏解”（即解质量下降的情况），以跳出局部最优，最终逼近全局最优解。

        模拟退火算法是一种强大并且灵活的优化算法，适合多种应用场景。