2024/8/25周报

摘要

多目标优化（Multi-Objective Optimization, MOO）是优化领域的一个分支，它处理的是同时优化多个相互冲突的目标函数的问题。在实际应用中，很少有决策问题只涉及单一目标，通常需要在多个目标之间找到平衡点。例如，在工程设计中可能需要同时考虑成本、性能和可靠性等因素。本周进一步学习一些多目标优化算法的常见类型。

Abstract

Multi-objective optimization (MOO) is a branch of optimization field, which deals with the problem of simultaneously optimizing multiple conflicting objective functions. In practical application, few decision-making problems only involve a single goal, and it is usually necessary to find a balance between multiple goals. For example, factors such as cost, performance and reliability may need to be considered simultaneously in engineering design. Learn more about some common types of multi-objective optimization algorithms this week.

多目标优化算法

多目标优化的基本概念

1. 目标函数：每个目标函数衡量了某个特定方面的性能或质量。
2. 解空间：所有可能的解决方案组成的集合。
3. 可行解：满足所有约束条件的解。
4. Pareto 优势：如果一个解在至少一个目标上优于另一个解，并且在其他目标上不劣于该解，则称这个解对另一个解具有 Pareto 优势。
5. Pareto 最优解：没有其他解能够对该解产生 Pareto 优势，这样的解称为 Pareto 最优解。
6. Pareto 前沿：所有 Pareto 最优解在目标空间中的投影构成的集合。

实现步骤

1. 初始化种群：随机生成初始解集。
2. 评估：计算每个个体的目标函数值。
3. 选择：根据某种策略选择下一代的父代个体。
4. 遗传操作：执行交叉和变异操作以产生新个体。
5. 更新：根据 Pareto 优势更新种群。
6. 终止条件：达到预定迭代次数或其他终止标准后停止。

多目标优化算法常见类型及其实现

多目标优化算法的目标是找到一组 Pareto 最优解，而不是单一的最佳解。常见的多目标优化算法包括但不限于以下几种：

NSGA-II (Non-dominated Sorting Genetic Algorithm II)

• 使用快速非支配排序来评价种群中的个体。
• 利用拥挤距离保持种群多样性。
• 通过遗传操作（如交叉和变异）生成新的后代。

代码如下：

import random
import numpy as np
from deap import base, creator, tools, algorithms

# 定义问题和解决方案的类型
creator.create("FitnessMin", base.Fitness, weights=(-1.0, -1.0))
creator.create("Individual", list, fitness=creator.FitnessMin)

# 初始化工具箱
toolbox = base.Toolbox()

# 注册属性初始化器
toolbox.register("attr_float", random.random)
toolbox.register("individual", tools.initRepeat, creator.Individual, toolbox.attr_float, n=2)
toolbox.register("population", tools.initRepeat, list, toolbox.individual)

# 注册评估函数
def evaluateZDT1(individual):
    f1 = individual[0]
    g = 1 + 9 * sum(individual[1:]) / (len(individual) - 1)
    f2 = g * (1 - (f1 / g) ** 0.5)
    return f1, f2

toolbox.register("evaluate", evaluateZDT1)

# 注册遗传操作
toolbox.register("mate", tools.cxSimulatedBinaryBounded, eta=20.0, low=0, up=1)
toolbox.register("mutate", tools.mutPolynomialBounded, eta=20.0, low=0, up=1, indpb=1.0/2)
toolbox.register("select", tools.selNSGA2)

# 主程序
def main():
    # 初始化种群
    pop = toolbox.population(n=90)
    hof = tools.ParetoFront()
    stats = tools.Statistics(lambda ind: ind.fitness.values)
    stats.register("avg", np.mean, axis=0)
    stats.register("std", np.std, axis=0)
    stats.register("min", np.min, axis=0)
    stats.register("max", np.max, axis=0)

    # 进化过程
    pop, logbook = algorithms.eaMuPlusLambda(pop, toolbox, mu=len(pop), lambda_=len(pop),
                                             cxpb=0.9, mutpb=0.1, ngen=20,
                                             stats=stats, halloffame=hof, verbose=True)

    return pop, logbook, hof

if __name__ == "__main__":
    pop, log, hof = main()

这段代码的主要组成部分如下：

1.定义问题和解决方案的类型：
creator.create(“FitnessMin”, base.Fitness, weights=(-1.0, -1.0)) 定义了一个最小化多目标问题。
creator.create(“Individual”, list, fitness=creator.FitnessMin) 创建了一个列表类型的个体，其中包含一个FitnessMin实例作为其fitness属性。

2.注册属性初始化器：
toolbox.register(“attr_float”, random.random) 定义了一个随机浮点数生成器。
toolbox.register(“individual”, tools.initRepeat, creator.Individual, toolbox.attr_float, n=2) 注册了个体初始化器，每个个体有两个基因。
toolbox.register(“population”, tools.initRepeat, list, toolbox.individual) 注册了种群初始化器。

3.评估函数：
evaluateZDT1 是一个示例评估函数，用于评估ZDT1测试问题的个体。该函数返回两个目标值。

4.遗传操作：
cxSimulatedBinaryBounded 和 mutPolynomialBounded 分别用于执行模拟二进制交叉和多项式变异。
selNSGA2 用于执行快速非支配排序的选择操作。

5.主程序：
初始化种群和统计信息。
执行进化算法 eaMuPlusLambda。
输出最终的种群、日志和Pareto前沿。
结果如下：

SPEA2 (Strength Pareto Evolutionary Algorithm 2)

• 引入个体适应度的概念，根据支配关系和密度估计进行排序。
• 通过存档机制保持历史最优解。

代码如下：

import random
import numpy as np
from deap import base, creator, tools, algorithms

# 定义问题和解决方案的类型
creator.create("FitnessMin", base.Fitness, weights=(-1.0, -1.0))
creator.create("Individual", list, fitness=creator.FitnessMin)

# 初始化工具箱
toolbox = base.Toolbox()

# 注册属性初始化器
toolbox.register("attr_float", random.random)
toolbox.register("individual", tools.initRepeat, creator.Individual, toolbox.attr_float, n=2)
toolbox.register("population", tools.initRepeat, list, toolbox.individual)

# 注册评估函数
def evaluateZDT1(individual):
    f1 = individual[0]
    g = 1 + 9 * sum(individual[1:]) / (len(individual) - 1)
    f2 = g * (1 - (f1 / g) ** 0.5)
    return f1, f2

toolbox.register("evaluate", evaluateZDT1)

# 注册遗传操作
toolbox.register("mate", tools.cxSimulatedBinaryBounded, eta=20.0, low=0, up=1)
toolbox.register("mutate", tools.mutPolynomialBounded, eta=20.0, low=0, up=1, indpb=1.0/2)

# SPEA2的选择函数
def selSPEA2(individuals, k):
    if len(individuals) < k:
        return individuals[:]
    
    # 计算支配关系
    dominance = tools.SPEADomination()
    dominated_counts = [sum(dominance(i, j) for j in individuals) for i in individuals]

    # 计算邻近度
    k_nearest = tools.knn(20)
    distances = [k_nearest(i, individuals) for i in individuals]

    # 创建存档
    archive = tools.ParetoFront()
    archive.update(individuals)

    # 选择个体
    selected = []
    while len(selected) < k:
        candidates = [i for i in range(len(individuals)) if i not in selected]
        if not candidates:
            break
        
        # 选择支配数量最少的个体
        min_dominated_count = min(dominated_counts[i] for i in candidates)
        candidates = [i for i in candidates if dominated_counts[i] == min_dominated_count]

        # 如果有多个个体，则选择距离最大的个体
        if len(candidates) > 1:
            max_distance = max(distances[i] for i in candidates)
            candidates = [i for i in candidates if distances[i] == max_distance]

        selected.append(candidates[0])
        
        # 更新支配计数和邻近度
        for i in candidates:
            dominated_counts[i] = sum(dominance(i, j) for j in individuals if j not in selected)
            distances[i] = k_nearest(i, [j for j in individuals if j not in selected])

    return [individuals[i] for i in selected], archive

# 主程序
def main():
    # 初始化种群
    pop = toolbox.population(n=90)
    stats = tools.Statistics(lambda ind: ind.fitness.values)
    stats.register("avg", np.mean, axis=0)
    stats.register("std", np.std, axis=0)
    stats.register("min", np.min, axis=0)
    stats.register("max", np.max, axis=0)

    # 进化过程
    hof = tools.ParetoFront()
    logbook = tools.Logbook()
    record = stats.compile(pop)
    logbook.record(gen=0, evals=len(pop), **record)
    print(logbook.stream)

    for gen in range(1, 21):
        offspring = algorithms.varOr(pop, toolbox, lambda_=len(pop), cxpb=0.9, mutpb=0.1)
        invalid_ind = [ind for ind in offspring if not ind.fitness.valid]
        fitnesses = toolbox.map(toolbox.evaluate, invalid_ind)
        for ind, fit in zip(invalid_ind, fitnesses):
            ind.fitness.values = fit

        # 选择操作
        pop, archive = selSPEA2(offspring + pop, len(pop))

        record = stats.compile(pop)
        logbook.record(gen=gen, evals=len(pop), **record)
        print(logbook.stream)

    return pop, logbook, hof

if __name__ == "__main__":
    pop, log, hof = main()

MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition)

• 将多目标问题分解为多个单目标子问题。
• 每个个体负责优化一个子问题。
• 使用局部搜索策略提高优化效率。

代码如下：

import random
import numpy as np
from deap import base, creator, tools, algorithms

# 定义问题和解决方案的类型
creator.create("FitnessMin", base.Fitness, weights=(-1.0, -1.0))
creator.create("Individual", list, fitness=creator.FitnessMin)

# 初始化工具箱
toolbox = base.Toolbox()

# 注册属性初始化器
toolbox.register("attr_float", random.random)
toolbox.register("individual", tools.initRepeat, creator.Individual, toolbox.attr_float, n=2)
toolbox.register("population", tools.initRepeat, list, toolbox.individual)

# 注册评估函数
def evaluateZDT1(individual):
    f1 = individual[0]
    g = 1 + 9 * sum(individual[1:]) / (len(individual) - 1)
    f2 = g * (1 - (f1 / g) ** 0.5)
    return f1, f2

toolbox.register("evaluate", evaluateZDT1)

# 注册遗传操作
toolbox.register("mate", tools.cxSimulatedBinaryBounded, eta=20.0, low=0, up=1)
toolbox.register("mutate", tools.mutPolynomialBounded, eta=20.0, low=0, up=1, indpb=1.0/2)

# 分解多目标问题
def decompose_objectives(objective_values, weight_vector):
    weighted_sum = sum(w * obj for w, obj in zip(weight_vector, objective_values))
    return weighted_sum

# MOEA/D的选择函数
def selMOEAD(individuals, k, weights):
    if len(individuals) < k:
        return individuals[:]
    
    # 计算每个个体对应的子问题权重
    for ind in individuals:
        ind.f_weights = weights[ind.id]

    # 创建存档
    archive = tools.ParetoFront()
    archive.update(individuals)

    # 选择个体
    selected = []
    while len(selected) < k:
        candidates = [i for i in range(len(individuals)) if i not in selected]
        if not candidates:
            break

        # 选择支配数量最少的个体
        min_dominated_count = min(ind.fitness.values[0] for ind in individuals if ind.id not in selected)
        candidates = [ind for ind in individuals if ind.fitness.values[0] == min_dominated_count and ind.id not in selected]

        # 如果有多个个体，则随机选择一个
        if len(candidates) > 1:
            selected.append(random.choice(candidates))
        else:
            selected.append(candidates[0].id)

        # 更新支配计数
        for i in candidates:
            i.fitness.values = decompose_objectives(i.fitness.values, i.f_weights)

    return [individuals[i] for i in selected], archive

# 主程序
def main():
    # 初始化种群
    pop = toolbox.population(n=90)
    weights = tools.uniform_reference_points(nobj=2, p=len(pop))
    for i, ind in enumerate(pop):
        ind.id = i
        ind.fitness.values = decompose_objectives(toolbox.evaluate(ind), weights[i])
    
    # 进化过程
    hof = tools.ParetoFront()
    logbook = tools.Logbook()
    record = tools.Statistics(lambda ind: ind.fitness.values)
    record.register("avg", np.mean, axis=0)
    record.register("std", np.std, axis=0)
    record.register("min", np.min, axis=0)
    record.register("max", np.max, axis=0)

    record = record.compile(pop)
    logbook.record(gen=0, evals=len(pop), **record)
    print(logbook.stream)

    for gen in range(1, 21):
        offspring = algorithms.varOr(pop, toolbox, lambda_=len(pop), cxpb=0.9, mutpb=0.1)
        invalid_ind = [ind for ind in offspring if not ind.fitness.valid]
        fitnesses = toolbox.map(toolbox.evaluate, invalid_ind)
        for ind, fit in zip(invalid_ind, fitnesses):
            ind.fitness.values = decompose_objectives(fit, weights[ind.id])

        # 选择操作
        pop, archive = selMOEAD(offspring + pop, len(pop), weights)

        record = record.compile(pop)
        logbook.record(gen=gen, evals=len(pop), **record)
        print(logbook.stream)

    return pop, logbook, hof

if __name__ == "__main__":
    pop, log, hof = main()

ε-MOEA (Epsilon-Dominance Based Multi-Objective Evolutionary Algorithm)

• 引入 ε-支配关系，允许一定程度的目标函数值差距。
• 可以更好地控制 Pareto 前沿的分布。

代码如下：

import random
import numpy as np
from deap import base, creator, tools, algorithms

# 定义问题和解决方案的类型
creator.create("FitnessMin", base.Fitness, weights=(-1.0, -1.0))
creator.create("Individual", list, fitness=creator.FitnessMin)

# 初始化工具箱
toolbox = base.Toolbox()

# 注册属性初始化器
toolbox.register("attr_float", random.random)
toolbox.register("individual", tools.initRepeat, creator.Individual, toolbox.attr_float, n=2)
toolbox.register("population", tools.initRepeat, list, toolbox.individual)

# 注册评估函数
def evaluateZDT1(individual):
    f1 = individual[0]
    g = 1 + 9 * sum(individual[1:]) / (len(individual) - 1)
    f2 = g * (1 - (f1 / g) ** 0.5)
    return f1, f2

toolbox.register("evaluate", evaluateZDT1)

# 注册遗传操作
toolbox.register("mate", tools.cxSimulatedBinaryBounded, eta=20.0, low=0, up=1)
toolbox.register("mutate", tools.mutPolynomialBounded, eta=20.0, low=0, up=1, indpb=1.0/2)

# ε-支配关系
def epsilon_dominates(a, b, epsilons):
    dominates = True
    for ai, bi, eps in zip(a, b, epsilons):
        if ai > bi + eps:
            return False
        elif ai < bi - eps:
            dominates = False
    return dominates

# ε-MOEA的选择函数
def selEPSMOEA(individuals, k, epsilons):
    if len(individuals) < k:
        return individuals[:]
    
    # 创建存档
    archive = tools.ParetoFront()
    archive.update(individuals)

    # 选择个体
    selected = []
    while len(selected) < k:
        candidates = [i for i in range(len(individuals)) if i not in selected]
        if not candidates:
            break

        # 选择支配数量最少的个体
        min_dominated_count = min(sum(epsilon_dominates(ind.fitness.values, other.fitness.values, epsilons)
                                      for other in individuals if other.id != ind.id)
                                  for ind in individuals if ind.id not in selected)
        candidates = [ind for ind in individuals if ind.fitness.values[0] == min_dominated_count and ind.id not in selected]

        # 如果有多个个体，则随机选择一个
        if len(candidates) > 1:
            selected.append(random.choice(candidates))
        else:
            selected.append(candidates[0].id)

    return [individuals[i] for i in selected], archive

# 主程序
def main():
    # 初始化种群
    pop = toolbox.population(n=90)
    epsilons = [0.05, 0.05]  # 目标函数的ε值

    # 进化过程
    hof = tools.ParetoFront()
    logbook = tools.Logbook()
    record = tools.Statistics(lambda ind: ind.fitness.values)
    record.register("avg", np.mean, axis=0)
    record.register("std", np.std, axis=0)
    record.register("min", np.min, axis=0)
    record.register("max", np.max, axis=0)

    record = record.compile(pop)
    logbook.record(gen=0, evals=len(pop), **record)
    print(logbook.stream)

    for gen in range(1, 21):
        offspring = algorithms.varOr(pop, toolbox, lambda_=len(pop), cxpb=0.9, mutpb=0.1)
        invalid_ind = [ind for ind in offspring if not ind.fitness.valid]
        fitnesses = toolbox.map(toolbox.evaluate, invalid_ind)
        for ind, fit in zip(invalid_ind, fitnesses):
            ind.fitness.values = fit

        # 选择操作
        pop, archive = selEPSMOEA(offspring + pop, len(pop), epsilons)

        record = record.compile(pop)
        logbook.record(gen=gen, evals=len(pop), **record)
        print(logbook.stream)

    return pop, logbook, hof

if __name__ == "__main__":
    pop, log, hof = main()

PESA-II (Performance-Scalarizing Evolutionary Algorithm)

• 使用标量化方法将多目标问题转换为一系列单目标问题。
• 通过动态调整权重向量来探索 Pareto 前沿的不同区域。

代码如下：

import random
import numpy as np
from deap import base, creator, tools, algorithms

# 定义问题和解决方案的类型
creator.create("FitnessMin", base.Fitness, weights=(-1.0,))
creator.create("Individual", list, fitness=creator.FitnessMin)

# 初始化工具箱
toolbox = base.Toolbox()

# 注册属性初始化器
toolbox.register("attr_float", random.random)
toolbox.register("individual", tools.initRepeat, creator.Individual, toolbox.attr_float, n=2)
toolbox.register("population", tools.initRepeat, list, toolbox.individual)

# 注册评估函数
def evaluateZDT1(individual, weights):
    f1 = individual[0]
    g = 1 + 9 * sum(individual[1:]) / (len(individual) - 1)
    f2 = g * (1 - (f1 / g) ** 0.5)
    scalarized_value = weights[0]*f1 + weights[1]*f2
    return scalarized_value,

toolbox.register("evaluate", evaluateZDT1)

# 注册遗传操作
toolbox.register("mate", tools.cxSimulatedBinaryBounded, eta=20.0, low=0, up=1)
toolbox.register("mutate", tools.mutPolynomialBounded, eta=20.0, low=0, up=1, indpb=1.0/2)

# PESA-II的选择函数
def selPESA2(individuals, k, weights):
    if len(individuals) < k:
        return individuals[:]
    
    # 计算所有个体的适应度
    fitnesses = [toolbox.evaluate(ind, weights) for ind in individuals]
    
    # 按照适应度排序
    sorted_individuals = [ind for _, ind in sorted(zip(fitnesses, individuals))]
    
    # 选择前k个个体
    return sorted_individuals[:k]

# 主程序
def main():
    # 初始化种群
    pop = toolbox.population(n=90)
    num_weights = 2  # 两个目标
    num_weights_per_generation = 5  # 每代使用的权重数量
    
    # 进化过程
    hof = tools.ParetoFront()
    logbook = tools.Logbook()
    record = tools.Statistics(lambda ind: ind.fitness.values)
    record.register("avg", np.mean, axis=0)
    record.register("std", np.std, axis=0)
    record.register("min", np.min, axis=0)
    record.register("max", np.max, axis=0)

    record = record.compile(pop)
    logbook.record(gen=0, evals=len(pop), **record)
    print(logbook.stream)

    for gen in range(1, 21):
        # 生成权重向量
        weights = generate_weights(num_weights, num_weights_per_generation)
        
        offspring = algorithms.varOr(pop, toolbox, lambda_=len(pop), cxpb=0.9, mutpb=0.1)
        invalid_ind = [ind for ind in offspring if not ind.fitness.valid]
        fitnesses = toolbox.map(lambda ind: toolbox.evaluate(ind, weights), invalid_ind)
        for ind, fit in zip(invalid_ind, fitnesses):
            ind.fitness.values = fit

        # 选择操作
        pop = selPESA2(offspring + pop, len(pop), weights)

        record = record.compile(pop)
        logbook.record(gen=gen, evals=len(pop), **record)
        print(logbook.stream)

    return pop, logbook, hof

# 生成权重向量
def generate_weights(num_objectives, num_weights):
    weights = []
    for _ in range(num_weights):
        w = [random.random() for _ in range(num_objectives)]
        w_sum = sum(w)
        w = [wi / w_sum for wi in w]  # 归一化
        weights.append(w)
    return weights

if __name__ == "__main__":
    pop, log, hof = main()

这些算法通常用于解决复杂问题，比如设计优化、资源分配等问题。每种算法都有其特点和适用场景，选择哪种算法取决于具体问题的需求。

一些常见的dos命令

“DOS命令”通常指的是在MS-DOS（Microsoft Disk Operating System）或者兼容的命令行界面（如Windows命令提示符cmd.exe）中使用的命令。

更换盘符： d：
查看当前目录下的文件及文件夹：dir
进入文件夹： cd 文件夹的名字
返回上一级目录：cd …
创建文件夹：mkdir/md 文件夹名
删除文件夹：rd 文件夹的名字
清屏：cls
退出： exit

总结

本周进一步学习了一些多目标优化算法的常见类型，下周将继续学习项目申报书中提到的动态优化GA模型等内容。