一种求解鲁棒优化问题的多目标进化方法

鲁棒优化问题（Robust Optimization Problem,ROP）是进化算法（Evolutionary Algorithms,EAs）研究的重要方面之一,对于许多实际工程优化问题,通常需要得到鲁棒最优解。利用多目标优化中的Pareto思想优化ROP的鲁棒性和最优性,将ROP转化为一个两目标的优化问题,一个目标为解的鲁棒性,一个目标为解的最优性。针对ROP与多目标优化的特点,利用动态加权思想,设计一种求解ROP的多目标进化算法。通过测试函数的实验仿真,验证了该方法的有效性。     

一类新型动态多目标鲁棒进化优化方法

传统动态多目标优化问题（Dynamic multi-objective optimization problems,DMOPs）的求解方法,通常需要在新环境下,通过重新激发寻优过程,获得适应该环境的Pareto最优解.这可能导致较高的计算代价和资源成本,甚至无法在有限时间内执行该优化解.由此,提出一类寻找动态鲁棒Pareto最优解集的进化优化方法.动态鲁棒Pareto解集是指某一时刻下的Pareto较优解可以以一定稳定性阈值,逼近未来多个连续动态环境下的真实前沿,从而直接作为这些环境下的Pareto解集,以减小计算代价.为合理度量Pareto解的环境适应性,给出了时间鲁棒性和性能鲁棒性定义,并将其转化为两类鲁棒优化模型.引入基于分解的多目标进化优化方法和无惩罚约束处理方法,构建了动态多目标分解鲁棒进化优化方法.特别是基于移动平均预测模型实现了未来动态环境下适应值的多维时间序列预测.基于提出的两类新型性能评价测度,针对8个典型动态测试函数的仿真实验,结果表明该方法得到满足决策者精度要求,且具有较长平均生存时间的动态鲁棒Pareto最优解.     

多目标进化算法搜索鲁棒最优解效率研究

鲁棒最优解是进化计算研究的重要方面,同时也是研究难点。多目标进化算法搜索鲁棒最优解时,通常要用蒙特卡罗积分（MCI）近似估计有效目标函数（EOF）,而已有求解方法近似精度不高,使得算法搜索鲁棒最优解的性能较差。提出用拟蒙特卡罗方法（Q-MC）来估计有效目标函数方法,其所引入的Q-MC方法——Korobov点阵能更精确地估计EOF。实验结果表明,与现有的原始蒙特卡罗方法（C-MC）相比,拟蒙特卡罗方法（Q-MC）可以较大地提高多目标进化算法搜索鲁棒最优解的效率。     

多目标进化算法鲁棒性实验研究*

多目标进化算法的研究主要集中于搜寻全局最优解。在现实中,环境不是一成不变的,需找到抗干扰能力强的鲁棒解。多目标鲁棒最优化问题的研究较少,主要归结于环境的不确定性和缺乏合适的测试函数。针对不同特性测试函数,通过实验检验了在不同干扰下算法的性能变化情况。实验结果表明,存在干扰的情况下,原来的测试函数不再适用,需构造鲁棒测试函数。     

基于拟蒙特卡罗方法的进化算法搜索鲁棒最优解的性能提高研究

鲁棒最优解在工程应用中具有十分重要的意义,它是进化计算的重要研究内容,也是研究难点．进化算法搜索鲁棒最优解时,通常使用蒙特卡罗积分(MCI)近似估计有效目标函数(EOF),但由于现有的原始蒙特卡罗方法(C-MC)近似精度不高,导致进化算法搜索鲁棒最优解的性能较差．文中提出用拟蒙特卡罗方法(Q-MC)估计有效目标函数,通过大量的数值实验,结果表明,与C-MC相比,文中所引入的Q-MC 方法、SQRT序列、SOBOL序列和Korobov点阵能更精确估计EOF,进而较大提高进化算法搜索鲁棒最优解的性能．     

一种改进的鲁棒多目标优化方法

针对在解决某些复杂多目标优化问题过程中,所得到的Pareto最优解易受设计参数或环境参数扰动的影响,引入了鲁棒的概念并提出一种改进的鲁棒多目标优化方法,它利用了经典的基于适应度函数期望和方差方法各自的优势,有效地将两种方法结合在一起。为了实现该方法,给出一种基于粒子群优化算法的多目标优化算法。仿真实例结果表明,所给出的方法能够得到更为鲁棒的Pareto最优解。     

一种鲁棒故障检测与反馈控制的最优集成设计方法

研究线性不确定系统的反馈控制器与鲁棒故障检测滤波器集成设计问题.基于新提出 的性能指标函数,将鲁棒故障检测滤波器设计问题归结为最优化问题,通过求解Riccati方程可 得到鲁棒故障检测滤波器设计问题的最优解.在共用同一状态观测器的情况下,将反馈控制器 和鲁棒故障检测滤波器的集成设计问题归结为两目标优化问题,解决了同时满足闭环控制系统 设计要求和故障诊断系统鲁棒性能的最优集成设计问题.简例验证了提出算法的有效性.     

多配送中心危险货物配送路径鲁棒优化

针对危险货物配送路径对不确定因素敏感度较高的问题,提出了鲁棒性可调的多配送中心危险货物配送路径鲁棒优化方法。首先,以最小化运输风险和最小化运输成本为目标,根据Bertsimas鲁棒离散优化理论,建立鲁棒优化模型;然后,在改进型强度Pareto进化算法（SPEA2）的基础上设计一种三段式编码的多目标遗传算法进行求解,在遗传操作中对不同染色体段分别采用不同的交叉和变异操作,有效避免了种群进化过程中不可行解的产生;最后,以庆阳市西峰区部分路网为例进行实证研究,并将配送方案落实到运输过程的路段中,形成具体的运输路径。研究结果表明：在多配送中心下,运用该鲁棒优化模型及算法,能快速得到具有较好鲁棒性的危险货物配送路径。     

高精度RBP-模糊推理复合学习系统

该文提出了高精度RBP-模糊推理复合学习系统.系统主要由基于鲁棒估计的鲁棒BP 学习环节和基于混合合成推理的模糊推理环节构成.该学习系统的主要特点是可由鲁棒BP 算法和min-max,max-min模糊推理算法简单地实现.最后通过在目标跟踪问题中应用结 果,表示了该算法的高精度和鲁棒性.     

Flow-shop系统的鲁棒性与最优鲁棒控制

使用极大代数模块网络法,研究Flow-shop离散事件动态系统(DEDS)在没有缓冲 器下的无阻塞鲁棒性及最优鲁棒控制.提出了系统鲁棒性的Kharitonov-like判据;证明了最 优鲁棒控制是一类线性状态反馈;还讨论了鲁棒控制系统的周期稳定性和鲁棒极点对摄动参 数的敏感性.     

Robust optimization using Bayesian optimization algorithm: Early detection of non-robust solutions

Probabilistic robustness evaluation is a promising approach to evolutionary robust optimization; however, high computational time arises. In this paper, we apply this approach to the Bayesian optimization algorithm (BOA) with a view to improving its computational time. To this end, we analyze the Bayesian networks constructed in BOA in order to extract the patterns of non-robust solutions. In each generation, the solutions that match the extracted patterns are detected and then discarded from the process of evaluation; therefore, the computational time in discovering the robust solutions decreases. The experimental results demonstrate that our proposed method reduces computational time, while increasing the robustness of solutions.     

Evolutionary robustness analysis for multi-objective optimization: benchmark problems

This paper presents a new approach to robustness analysis in multi-objective optimization problems aimed at obtaining the most robust Pareto front solutions and distributing the solutions along the most robust regions of the optimal Pareto set. A new set of test problems accounting for the different types of robustness cases is presented in this study. Non-dominated solutions are classified according to their degree of robustness and are distributed along the Pareto front according to specific algorithm parameter values. Verification of the proposed method is carried out using the developed test problems and artificial and real world benchmark test problems present in the literature.     

Introducing robustness in multi-objective optimization

In optimization studies including multi-objective optimization, the main focus is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the so-called global best solutions, particularly when these solutions are quite sensitive to the variable perturbations which cannot be avoided in practice. In such cases, practitioners are interested in finding the robust solutions which are less sensitive to small perturbations in variables. Although robust optimization is dealt with in detail in single-objective evolutionary optimization studies, in this paper, we present two different robust multi-objective optimization procedures, where the emphasis is to find a robust frontier, instead of the global Pareto-optimal frontier in a problem. The first procedure is a straightforward extension of a technique used for single-objective optimization and the second procedure is a more practical approach enabling a user to set the extent of robustness desired in a problem. To demonstrate the differences between global and robust multi-objective optimization principles and the differences between the two robust optimization procedures suggested here, we develop a number of constrained and unconstrained test problems having two and three objectives and show simulation results using an evolutionary multi-objective optimization (EMO) algorithm. Finally, we also apply both robust optimization methodologies to an engineering design problem.     

Periodical switching between related goals for improving evolvability to a fixed goal in multi-objective problems

Evolutionary computation plays a principal role in intelligent design automation. Evolutionary approaches have discovered novel and patentable designs. Nonetheless, evolutionary techniques may lead to designs that lack robustness. This critical issue is strongly connected to the concept of evolvability. In nature, highly evolvable species tend to be found in rapidly changing environments. Such species can be considered robust against environmental changes. Consequently, to create robust engineering designs it could be beneficial to use variable, rather than fixed, fitness criteria. In this paper, we study the performance of an evolutionary programming algorithm with periodical switching between goals, which are selected randomly from a set of related goals. It is shown by a dual-objective filter optimization example that altering goals may improve evolvability to a fixed goal by enhancing the dynamics of solution population, and guiding the search to areas where improved solutions are likely to be found. Our reference algorithm with a single goal is able to find solutions with competitive fitness, but these solutions are results of premature convergence, because they are poorly evolvable. By using the same algorithm with switching goals, we can extend the productive search length considerably; both the fitness and robustness of such designs are improved.     

柔性系统的最小时间鲁棒时滞滤波器设计

本文提出了一种最小时间鲁棒时滞滤波器控制,来提高柔性系统的时间最优控制的 鲁棒性．这种最小时间鲁棒时滞滤波器控制方法与其扩展系统传统的时间最优控制是一致的 ,这种等价性提供了验证最小时间鲁棒时滞滤波器最优性的理论方法,即所求得的解是否满 足Pontryagin 最小值原理．仿真结果进一步表明了这种最小时间鲁棒时滞滤波器的优越性 ．     

Robust optimization with convex model considering bounded constraints on performance variation

In robust optimization, double-looped structures are often adopted where the outer loop is used to seek for the optimal design and the optimization performed in the inner loop is for the robustness assessment of the candidate solutions. However, the double-looped techniques usually will lead to a significant increase in computational efforts. Therefore, in this paper, a new robustness index is developed to handle bounded constraints on performance variation where no optimization run is required for the robustness evaluation work in the inner loop. The computation of this new index is based on the sensitivity Jacobian matrix of the system performances with respect to the uncertainties and it can quantitatively measure the maximal allowable magnitude of system variations. By introducing this index, the robust design problem can be reformulated as a deterministic optimization with robustness indices requirements. Two numerical examples are tested to show the effectiveness and efficiency of the proposed approach, whose solutions and computational efforts are compared to those from a double-looped approach proposed in previous literature.     

记忆增强的动态多目标分解进化算法

现实世界中的一些多目标优化问题经常受动态环境影响而不断发生变化,要求优化算法不断地及时跟踪时变的Pareto 最优解集.提出了一种记忆增强的动态多目标分解进化算法.将动态多目标优化问题分解为若干个动态单目标优化子问题并同时优化这些子问题,以便快速逼近Pareto 最优解集.给出了一个改进的环境变化检测算子,以便更好地检测环境变化.设计了一种基于子问题的串式记忆方法,利用过去类似环境下搜索到的最优解来有效地响应新的环境变化.在8 个标准的测试问题上,将新算法与其他3 种记忆增强的动态进化多目标优化算法进行了实验比较.结果表明,新算法比其他3 种算法具有更快的运行速度、更强的记忆能力与鲁棒性能,并且新算法所获得的解集还具有更好的收敛性与分布性.     

Robust optimization to secure urban bulk water supply against extreme drought and uncertain climate change

Urban bulk water systems supply water with high reliability and, in the event of extreme drought, must avoid catastrophic economic and social collapse. In view of the deep uncertainty about future climate change, it is vital that robust solutions be found that secure urban bulk water systems against extreme drought. To tackle this challenge an approach was developed integrating: 1) a stochastic model of multi-site streamflow conditioned on future climate change scenarios; 2) Monte Carlo simulation of the urban bulk water system incorporated into a robust optimization framework and solved using a multi-objective evolutionary algorithm; and 3) a comprehensive decision space including operating rules, investment in new sources and source substitution and a drought contingency plan with multiple actions with increasingly severe economic and social impact. A case study demonstrated the feasibility of this approach for a complex urban bulk water supply system. The primary objective was to minimize the expected present worth cost arising from infrastructure investment, system operation and the social cost of “normal” and emergency restrictions. By introducing a second objective which minimizes either the difference in present worth cost between the driest and wettest future climate change scenarios or the present worth cost for driest climate scenario, the trade-off between efficiency and robustness was identified. The results show that a significant change in investment and operating strategy can occur when the decision maker expresses a stronger preference for robustness and that this depends on the adopted robustness measure. Moreover, solutions are not only impacted by the degree of uncertainty about future climate change but also by the stress imposed on the system and the range of available options.