机会受到限制的优化问题允许建模问题,其中涉及随机组件的约束仅应以较小的概率侵犯。进化算法已应用于这种情况,并证明可以实现高质量的结果。在本文中,我们有助于对进化算法的理论理解,以进行偶然的优化。我们研究独立且正态分布的随机组件的场景。考虑到简单的单对象(1+1)〜EA,我们表明,施加额外的统一约束已经导致局部最佳选择,对于非常有限的场景和指数优化时间。因此,我们引入了问题的多目标公式,该公式可以摆脱预期成本及其差异。我们表明,在使用此公式时,多目标进化算法是非常有效的,并获得一组解决方案,该解决方案包含最佳解决方案,以适用于施加在约束上的任何可能的置信度。此外,我们证明这种方法还可以用于计算一组最佳解决方案,以限制最小跨越树问题。为了在多目标配方中呈指数指数的折衷,我们提出并分析了改进的凸多目标方法。关于NP-固定随机最小重量占主导地位问题的实例的实验研究证实了多目标和改进的凸多目标方法的益处。
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Evolutionary algorithms (EAs) are a kind of nature-inspired general-purpose optimization algorithm, and have shown empirically good performance in solving various real-word optimization problems. During the past two decades, promising results on the running time analysis (one essential theoretical aspect) of EAs have been obtained, while most of them focused on isolated combinatorial optimization problems, which do not reflect the general-purpose nature of EAs. To provide a general theoretical explanation of the behavior of EAs, it is desirable to study their performance on general classes of combinatorial optimization problems. To the best of our knowledge, the only result towards this direction is the provably good approximation guarantees of EAs for the problem class of maximizing monotone submodular functions with matroid constraints. The aim of this work is to contribute to this line of research. Considering that many combinatorial optimization problems involve non-monotone or non-submodular objective functions, we study the general problem classes, maximizing submodular functions with/without a size constraint and maximizing monotone approximately submodular functions with a size constraint. We prove that a simple multi-objective EA called GSEMO-C can generally achieve good approximation guarantees in polynomial expected running time.
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Evolutionary algorithms (EAs) have found many successful real-world applications, where the optimization problems are often subject to a wide range of uncertainties. To understand the practical behaviors of EAs theoretically, there are a series of efforts devoted to analyzing the running time of EAs for optimization under uncertainties. Existing studies mainly focus on noisy and dynamic optimization, while another common type of uncertain optimization, i.e., robust optimization, has been rarely touched. In this paper, we analyze the expected running time of the (1+1)-EA solving robust linear optimization problems (i.e., linear problems under robust scenarios) with a cardinality constraint $k$. Two common robust scenarios, i.e., deletion-robust and worst-case, are considered. Particularly, we derive tight ranges of the robust parameter $d$ or budget $k$ allowing the (1+1)-EA to find an optimal solution in polynomial running time, which disclose the potential of EAs for robust optimization.
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在进化计算中使用非豁免主义时的一个希望是放弃当前最佳解决方案的能力,艾滋病们离开本地最佳效果。为了提高我们对这种机制的理解,我们对基本的非精英进化算法(EA),$(\ mu,\ lambda)$ ea进行严格的运行时分析,在最基本的基准函数上,具有本地最佳的基本基准函数跳跃功能。我们证明,对于参数和问题的所有合理值,$(\ mu,\ lambda)$ ~ea的预期运行时间除了下订单条款之外,至少与其Elitist对应的预期运行时间,$(\ mu + \ lambda)$〜ea(我们对跳转功能进行第一个运行时分析以允许此比较)。因此,$(\ mu,\ lambda)$ ~ea将本地最优方式留给劣质解决方案的能力不会导致运行时优势。我们补充了这个下限的下限,即对于参数的广泛范围,与我们的下限不同,与下顺序不同。这是一个在多模态问题上的非精英算法的第一个运行时结果,除了下订单术语。
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线性函数在进化算法的运行时分析中起关键作用,研究为分析进化计算方法提供了广泛的新见解和技术。通过对可分离功能的研究和进化算法的优化行为以及来自机会约束优化领域的目标函数的优化行为,我们研究了两个转换线性函数的加权总和的目标函数类别。我们的结果表明,(1+1)EA的突变速率取决于功能的重叠位数,在预期时间O(n log n)中为这些函数获得了最佳解决方案,从而推广了一个众所周知的。线性函数的结果范围更广泛。
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非主导的分类遗传算法II(NSGA-II)是现实应用中最强烈使用的多目标进化算法(MOEA)。然而,与几个通过数学手段分析的几个简单的MOES相反,到目前为止,NSGA-II也不存在这种研究。在这项工作中,我们表明,数学运行时分析也可用于NSGA-II。结果,我们证明,由于持续因素大于帕累托前方大小的人口大小,具有两个经典突变算子的NSGA-II和三种不同的选择父母的方式满足与Semo和GSEMO相同的渐近运行时保证基本ineminmax和Lotz基准函数的算法。但是,如果人口大小仅等于帕累托前面的大小,那么NSGA-II就无法有效地计算完整的帕累托前部(对于指数迭代,人口总是错过帕累托前部的恒定分数) 。我们的实验证实了上述研究结果。
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The Makespan Scheduling problem is an extensively studied NP-hard problem, and its simplest version looks for an allocation approach for a set of jobs with deterministic processing times to two identical machines such that the makespan is minimized. However, in real life scenarios, the actual processing time of each job may be stochastic around the expected value with a variance, under the influence of external factors, and the actual processing times of these jobs may be correlated with covariances. Thus within this paper, we propose a chance-constrained version of the Makespan Scheduling problem and investigate the theoretical performance of the classical Randomized Local Search and (1+1) EA for it. More specifically, we first study two variants of the Chance-constrained Makespan Scheduling problem and their computational complexities, then separately analyze the expected runtime of the two algorithms to obtain an optimal solution or almost optimal solution to the instances of the two variants. In addition, we investigate the experimental performance of the two algorithms for the two variants.
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大多数进化算法具有多个参数,它们的值大大影响性能。由于参数的常复相互作用,将这些值设置为特定问题(参数调整)是一个具有挑战性的任务。当最佳参数值在算法运行期间最佳参数值发生显着变化时,此任务变得更加复杂。然后是必要的动态参数选择(参数控制)。在这项工作中,我们提出了一个懒惰但有效的解决方案,即从一个适当缩放的幂律分布中随机地选择所有参数值(在那里这是有意义的)。为了展示这种方法的有效性,我们使用以这种方式选择的所有三个参数执行$(1 +(\ lambda,\ lambda))$遗传算法的运行时分析。我们展示该算法一方面可以模仿像$(1 + 1)$ EA这样的简单山羊,给出了onemax,领导者或最小生成树等问题的相同渐近运行时。另一方面,该算法对跳跃功能也非常有效,其中最佳静态参数与优化简单问题所需的静态参数非常不同。我们证明了具有可比性的性能保证,有时比静态参数所知的最佳性能更好。我们通过严格的实证研究来补充我们的理论结果,证实了渐近运行时期结果的建议。
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Clustering is a fundamental problem in many areas, which aims to partition a given data set into groups based on some distance measure, such that the data points in the same group are similar while that in different groups are dissimilar. Due to its importance and NP-hardness, a lot of methods have been proposed, among which evolutionary algorithms are a class of popular ones. Evolutionary clustering has found many successful applications, but all the results are empirical, lacking theoretical support. This paper fills this gap by proving that the approximation performance of the GSEMO (a simple multi-objective evolutionary algorithm) for solving the three popular formulations of clustering, i.e., $k$-center, $k$-median and $k$-means, can be theoretically guaranteed. Furthermore, we prove that evolutionary clustering can have theoretical guarantees even when considering fairness, which tries to avoid algorithmic bias, and has recently been an important research topic in machine learning.
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The NSGA-II is one of the most prominent algorithms to solve multi-objective optimization problems. Despite numerous successful applications, several studies have shown that the NSGA-II is less effective for larger numbers of objectives. In this work, we use mathematical runtime analyses to rigorously demonstrate and quantify this phenomenon. We show that even on the simple OneMinMax benchmark, where every solution is Pareto optimal, the NSGA-II also with large population sizes cannot compute the full Pareto front (objective vectors of all Pareto optima) in sub-exponential time when the number of objectives is at least three. Our proofs suggest that the reason for this unexpected behavior lies in the fact that in the computation of the crowding distance, the different objectives are regarded independently. This is not a problem for two objectives, where any sorting of a pair-wise incomparable set of solutions according to one objective is also such a sorting according to the other objective (in the inverse order).
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最近,已经进行了多目标进化优化器NSGA-II的第一个数学运行时分析(AAAI 2022,GECCO 2022(出现),ARXIV 2022)。我们通过对由两个多模式目标组成的基准问题进行该算法的第一个运行时分析继续进行这一研究。我们证明,如果人口尺寸$ n $至少是帕累托阵线的四倍,那么NSGA-II具有四种不同方法的NSGA-II选择父母,并且位于Bit Wise突变将优化OnejumpzeroJump基准,其跳高尺寸〜$ 2 \ le lek \ le n/4 $ in Time $ o(n n^k)$。当使用快速突变(最近提出的重型突变操作员)时,此保证将提高$ k^{\ omega(k)} $。总体而言,这项工作表明,NSGA-II至少与全球SEMO算法有关OnejumpZeroJump问题的局部优势。
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Evolutionary algorithms (EAs) are general-purpose optimization algorithms, inspired by natural evolution. Recent theoretical studies have shown that EAs can achieve good approximation guarantees for solving the problem classes of submodular optimization, which have a wide range of applications, such as maximum coverage, sparse regression, influence maximization, document summarization and sensor placement, just to name a few. Though they have provided some theoretical explanation for the general-purpose nature of EAs, the considered submodular objective functions are defined only over sets or multisets. To complement this line of research, this paper studies the problem class of maximizing monotone submodular functions over sequences, where the objective function depends on the order of items. We prove that for each kind of previously studied monotone submodular objective functions over sequences, i.e., prefix monotone submodular functions, weakly monotone and strongly submodular functions, and DAG monotone submodular functions, a simple multi-objective EA, i.e., GSEMO, can always reach or improve the best known approximation guarantee after running polynomial time in expectation. Note that these best-known approximation guarantees can be obtained only by different greedy-style algorithms before. Empirical studies on various applications, e.g., accomplishing tasks, maximizing information gain, search-and-tracking and recommender systems, show the excellent performance of the GSEMO.
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最近,已经进行了NSGA-II的第一个数学运行时分析,这是最常见的多目标进化算法(Zheng,Liu,Doerr(AAAI 2022))。继续这一研究方向,我们证明了NSGA-II在使用交叉时,渐近渐近地测试了OneJumpZeroJump基准测试。这是NSGA-II首次证明这种交叉的优势。我们的论点可以转移到单目标优化。然后,他们证明,跨界可以以不同的方式加速$(\ MU+1)$遗传算法,并且比以前更为明显。我们的实验证实了交叉的附加值,并表明观察到的加速度甚至比我们的证明所能保证的要大。
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跳跃功能是随机搜索启发式理论中的{最多研究的非单峰基准,特别是进化算法(EA)。他们对我们的理解显着改善了EASE逃离当地最优的理解。然而,他们的特殊结构 - 离开本地最佳的结构只能直接跳到全球最优 - 引发代表性这种结果的问题。出于这个原因,我们提出了一个扩展的$ \ textsc {jump} _ {k,\ delta} $ jump函数,其中包含宽度$ \ delta $的低适合度vally以距离$ k $从全局最佳v $开始。我们证明了几个以前的结果延伸到这一更普遍的类:对于所有{$ k \ le \ frac {n ^ {1/3}} {\ ln {n}} $}和$ \ delta <k $,最佳$(1 + 1)$〜EA的突变率是$ \ FRAC {\ delta} $,并且快速$(1 + 1)$〜EA运行比经典$(1 + 1)$更快〜ea在$ \ delta $中的一个超级指数。但是,我们还观察到一些已知结果不概括:随机本地搜索算法具有停滞检测,其比$ \ textsc的$ k $ k $ k $ k $ k $ k $ k $ x $ \ textsc {跳} _K $,在某些$ \ textsc {jump} _ {k,\ delta} $实例上以$ n $的因子多项式慢。计算地,新类允许使用更宽的健身谷的实验,特别是当它们远离全球最佳时。
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在处理机器人技术,游戏和组合优化等领域的问题时,质量多样性(QD)算法已被证明非常成功。它们的目的是最大程度地提高基本问题所谓行为空间不同区域的解决方案的质量。在本文中,我们应用QD范式来模拟背包问题上的动态编程行为,并提供对QD算法的第一个运行时分析。我们证明他们能够在预期的伪多项式时间内计算最佳解决方案,并揭示导致完全多项式随机近似方案(FPRAS)的参数设置。我们的实验研究根据在行为空间中构建的解决方案以及获得最佳解决方案所需的运行时评估了经典基准集的不同方法。
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进化算法(EAS)是通用优化仪,其具有父母和后代群体的大小或突变率。众所周知,EAS的性能可能在这些参数上急剧上依赖。最近的理论研究表明,自调节参数控制机制在算法运行期间调整参数的调节参数可以在离散问题上可被显着优于最佳静态参数。然而,大多数这些研究有关的Elitist EAB,我们没有明确的答案,以及是否可以申请非Elitist EA。我们研究了一个最着名的参数控制机制,第五个成功规则,控制后代人口尺寸$ \ lambda $ \ llambda $ ea。众所周知,$(1,\ lambda)$ ea有一个尖锐的阈值,关于$ \ lambda $的选择,其中基准函数的预期运行时间onemax从多项式变为指数时间。因此,目前尚不清楚参数控制机制是否能够找到和维护$ \ lambda $的合适值。对于OneMax,我们表明答案是至关重要的,这取决于成功率$ s $(即一+ 1)美元成功规则)。我们证明,如果成功率适当小,则自我调整$(1,\ Lambda)$ EA优化ONEMAX以美元(n)$预期的几代人和$ O(n \ log n)$预期评估任何一元无偏见的黑匣子算法最好的运行时。一个小的成功率至关重要:我们还表明,如果成功率太大,则该算法对onemax具有指数运行计划和具有相似特征的其他功能。
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In noisy evolutionary optimization, sampling is a common strategy to deal with noise. By the sampling strategy, the fitness of a solution is evaluated multiple times (called \emph{sample size}) independently, and its true fitness is then approximated by the average of these evaluations. Most previous studies on sampling are empirical, and the few theoretical studies mainly showed the effectiveness of sampling with a sufficiently large sample size. In this paper, we theoretically examine what strategies can work when sampling with any fixed sample size fails. By constructing a family of artificial noisy examples, we prove that sampling is always ineffective, while using parent or offspring populations can be helpful on some examples. We also construct an artificial noisy example to show that when using neither sampling nor populations is effective, a tailored adaptive sampling (i.e., sampling with an adaptive sample size) strategy can work. These findings may enhance our understanding of sampling to some extent, but future work is required to validate them in natural situations.
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$(1 +(\ lambda,\ lambda))$遗传算法是一种较年轻的进化算法,试图从劣质解决方案中获利。关于单峰的健身功能的严格运行时分析表明它确实可以比古典进化算法更快,但在这些简单的问题上,收益只有中等。在这项工作中,我们在多模式问题类中进行了该算法的第一个运行时分析,跳跃功能基准。我们展示了使用正确的参数,\ ollga优化任何跳跃尺寸$ 2 \ Le K \ Le N / 4 $的任何跳跃功能,在预期的时间$ O(n ^ {(k + 1)/ 2} e ^ {o( k)}} k ^ { - k / 2}),它显着且已经持续了〜$ k $优于基于标准的突变的算法与他们的$ \ theta(n ^ k)$运行时与它们的标准交叉的算法$ \ tilde {o}(n ^ {k-1})$运行时保证。对于离开局部跳跃功能的局部最佳的孤立问题,我们确定了导致$(n / k)^ {k / 2} e ^ {\ theta(k)} $的运行时间的最佳参数。这表明有关如何设置\ ollga的参数的一般建议,这可能会缓解该算法的进一步使用。
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由于NSGA-II的种群动态更为复杂,因此该算法的现有运行时保证都没有伴随着非平凡的下限。通过对NSGA-II人口动态的首次数学理解,即通过估计具有一定客观价值的个体的预期数量,我们证明具有合适人口大小的NSGA-II需要$ \ omega(nn \ log) n)$函数评估,以找到Oneminmax问题的帕累托正面和$ \ omega(nn^k)$评估,$ jumpzerojump问题与跳跃尺寸$ k $。这些界限在渐近上(即,它们匹配先前显示的上限),并表明这里的NSGA-II甚至在平行运行时(迭代次数)中也没有从较大的人口大小中的利润。对于OneJumpZeroJump问题,当使用相同的排序用于计算两个目标的拥挤距离贡献时,我们甚至获得了一个紧张的运行时估计,其中包括领导常数。
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In many real-world optimization problems, the objective function evaluation is subject to noise, and we cannot obtain the exact objective value. Evolutionary algorithms (EAs), a type of general-purpose randomized optimization algorithm, have been shown to be able to solve noisy optimization problems well. However, previous theoretical analyses of EAs mainly focused on noise-free optimization, which makes the theoretical understanding largely insufficient for the noisy case. Meanwhile, the few existing theoretical studies under noise often considered the one-bit noise model, which flips a randomly chosen bit of a solution before evaluation; while in many realistic applications, several bits of a solution can be changed simultaneously. In this paper, we study a natural extension of one-bit noise, the bit-wise noise model, which independently flips each bit of a solution with some probability. We analyze the running time of the (1+1)-EA solving OneMax and LeadingOnes under bit-wise noise for the first time, and derive the ranges of the noise level for polynomial and super-polynomial running time bounds. The analysis on LeadingOnes under bit-wise noise can be easily transferred to one-bit noise, and improves the previously known results. Since our analysis discloses that the (1+1)-EA can be efficient only under low noise levels, we also study whether the sampling strategy can bring robustness to noise. We prove that using sampling can significantly increase the largest noise level allowing a polynomial running time, that is, sampling is robust to noise.
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