【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 math
import random

def simulated_annealing(initial_state, temperature, cooling_rate, max_iterations, objective_function):
current_state = initial_state
current_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_state
current_energy = new_energy
else:
acceptance_probability = math.exp((current_energy - new_energy) / temperature)
if random.random() < acceptance_probability:
current_state = new_state
current_energy = new_energy

# 降温
temperature *= cooling_rate

return current_state

def 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_state

def 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 = 1000
cooling_rate = 0.95
max_iterations = 1000
best_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 <iostream>
#include <vector>
#include <cmath>
#include <random>
#include <algorithm>

double objectiveFunction(const std::vector<int>& state) {
return std::accumulate(state.begin(), state.end(), 0.0);
}

std::vector<int> perturb(const std::vector<int>& state) {
std::vector<int> newState = state;
std::swap(newState[rand() % state.size(), rand() % state.size()]);
return newState;
}

std::vector<int> simulatedAnnealing(std::vector<int> initialState, double temperature, double coolingRate, int maxIterations) {
std::vector<int> 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<double>(rand()) / RAND_MAX) < acceptanceProbability) {
currentState = newState;
currentEnergy = newEnergy;
}
}

temperature *= coolingRate;
}

return currentState;
}

int main() {
std::vector<int> 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 math
import random

class 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 main

import (
"fmt"
"math"
"math/rand"
"time"
)

func objectiveFunction(state []int) float64 {
sum := 0
for _, 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 := initialState
currentEnergy := objectiveFunction(currentState)

for i := 0; i < maxIterations; i++ {
newState := perturb(currentState)
newEnergy := objectiveFunction(newState)

if newEnergy < currentEnergy {
currentState = newState
currentEnergy = newEnergy
} else {
acceptanceProbability := math.Exp((currentEnergy - newEnergy) / temperature)
if rand.Float64() < acceptanceProbability {
currentState = newState
currentEnergy = newEnergy
}
}
temperature *= coolingRate
}

return currentState
}

func main() {
rand.Seed(time.Now().UnixNano())
initialState := []int{5, 3, 1, 7, 2, 4, 6}
temperature := 1000.0
coolingRate := 0.95
maxIterations := 1000

bestState := 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 SimulatedAnnealing
from network import Network

def 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 math
import random

class SimulatedAnnealing:

@staticmethod
def run(initial_state, temperature, cooling_rate, max_iterations, objective_function):
current_state = initial_state
current_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_state
current_energy = new_energy
else:
acceptance_probability = math.exp((current_energy - new_energy) / temperature)
if random.random() < acceptance_probability:
current_state = new_state
current_energy = new_energy

temperature *= cooling_rate

return current_state

@staticmethod
def 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
pass

def get_initial_configuration(self):
# 获取初始配置
pass

def 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
pass

def 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。