Evolutionary algorithms (EAs) are population-based general-purpose optimization algorithms, and have been successfully applied in various real-world optimization tasks. However, previous theoretical studies often employ EAs with only a parent or offspring population and focus on specific problems. Furthermore, they often only show upper bounds on the running time, while lower bounds are also necessary to get a complete understanding of an algorithm. In this paper, we analyze the running time of the ($μ$+$λ$)-EA (a general population-based EA with mutation only) on the class of pseudo-Boolean functions with a unique global optimum. By applying the recently proposed switch analysis approach, we prove the lower bound $Ω(n \ln n+ μ+ λn\ln\ln n/ \ln n)$ for the first time. Particularly on the two widely-studied problems, OneMax and LeadingOnes, the derived lower bound discloses that the ($μ$+$λ$)-EA will be strictly slower than the (1+1)-EA when the population size $μ$ or $λ$ is above a moderate order. Our results imply that the increase of population size, while usually desired in practice, bears the risk of increasing the lower bound of the running time and thus should be carefully considered.
In the present paper we describe new heuristic technique, which can be applied to the optimization of pseudo-Boolean functions including Black-Box functions. This technique is based on a simple procedure which consists in transition from the optimization problem over Boolean hypercube to the optimization problem of auxiliary function in a specially constructed metric space. It is shown that there is a natural connection between the points of the original Boolean hypercube and points from the new metric space. For the Boolean hypercube with fixed dimension it is possible to construct a number o…
With this paper, we contribute to the understanding of ant colony optimization (ACO) algorithms by formally analyzing their runtime behavior. We study simple MAX-MIN ant systems on the class of linear pseudo-Boolean functions defined on binary strings of length 'n'. Our investigations point out how the progress according to function values is stored in pheromone. We provide a general upper bound of O((n^3 \log n)/ ρ) for two ACO variants on all linear functions, where (ρ) determines the pheromone update strength. Furthermore, we show improved bounds for two well-known linear pseudo-Boolean fun…
Finding Boolean functions suitable for cryptographic primitives is a complex combinatorial optimization problem, since they must satisfy several properties to resist cryptanalytic attacks, and the space is very large, which grows super exponentially with the number of input variables. Recent research has focused on the study of Boolean functions that satisfy properties on restricted sets of inputs due to their importance in the development of the FLIP stream cipher. In this paper, we consider one such property, perfect balancedness, and investigate the use of Genetic Programming (GP) and Genet…
NegaBent, No Regrets: Evolving Spectrally Flat Boolean Functions
University of Paris 8（巴黎第八大学）；University of Bergen（卑尔根大学）；Faculty of Electrical Engineering and Computing, University of Zagreb（萨格勒布大学电气工程和计算学院）；University of Twente（特温特大学）；Radboud University（拉德堡德大学）
University of Málaga（马拉加大学）；Colorado State University（科罗拉多州立大学）；University of São Paulo（圣保罗大学）
Local search algorithms and iterated local search algorithms are a basic technique. Local search can be a stand along search methods, but it can also be hybridized with evolutionary algorithms. Recently, it has been shown that it is possible to identify improving moves in Hamming neighborhoods for k-bounded pseudo-Boolean optimization problems in constant time. This means that local search does not need to enumerate neighborhoods to find improving moves. It also means that evolutionary algorithms do not need to use random mutation as a operator, except perhaps as a way to escape local optima. …
Finding balanced, highly nonlinear Boolean functions is a difficult problem where it is not known what nonlinearity values are possible to be reached in general. At the same time, evolutionary computation is successfully used to evolve specific Boolean function instances, but the approach cannot easily scale for larger Boolean function sizes. Indeed, while evolving smaller Boolean functions is almost trivial, larger sizes become increasingly difficult, and evolutionary algorithms perform suboptimally. In this work, we ask whether genetic programming (GP) can evolve constructions resulting in b…
Monotone but Exciting: On Evolving Monotone Boolean Functions with High Nonlinearity
University of Paris 8（巴黎第八大学）；University of Bergen（卑尔根大学）；Faculty of Electrical Engineering and Computing, University of Zagreb（萨格勒布大学电气工程和计算学院）；Semantics, Cybersecurity and Services Group, University of Twente（特文特大学语义、网络安全与服务组）；Digital Security Group, Radboud University（拉波德国立大学数字安全组）
The binary value function, or BinVal, has appeared in several studies in theory of evolutionary computation as one of the extreme examples of linear pseudo-Boolean functions. Its unbiased black-box complexity was previously shown to be at most $\lceil \log_2 n \rceil + 2$, where $n$ is the problem size. We augment it with an upper bound of $\log_2 n + 2.42141558 - o(1)$, which is more precise for many values of $n$. We also present a lower bound of $\log_2 n + 1.1186406 - o(1)$. Additionally, we prove that BinVal is an easiest function among all unimodal pseudo-Boolean functions at least for u…
Runtime Analysis of Evolutionary Algorithms for Multi-party Multi-objective Optimization
School of Artificial Intelligence, Nanjing University of Information Science and Technology（人工智能学院，南京信息工程大学）；Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies, Institute of Cyberspace Security, School of Computer Science and Technology, Harbin Institute of Technology（新型安全智能技术广东省重点实验室，网络空间安全学院，计算机科学与技术学院，哈尔滨工业大学）
IDEM Enough? Evolving Highly Nonlinear Idempotent Boolean Functions
University of Paris 8（巴黎第八大学）；University of Bergen（卑尔根大学）；Faculty of Electrical Engineering and Computing, University of Zagreb（萨格勒布大学电气工程和计算学院）；University of Twente（特文特大学）；Digital Security Group, Radboud University（拉德堡德大学数字安全组）
We present a new method for analyzing the running time of parallel evolutionary algorithms with spatially structured populations. Based on the fitness-level method, it yields upper bounds on the expected parallel running time. This allows to rigorously estimate the speedup gained by parallelization. Tailored results are given for common migration topologies: ring graphs, torus graphs, hypercubes, and the complete graph. Example applications for pseudo-Boolean optimization show that our method is easy to apply and that it gives powerful results. In our examples the possible speedup increases wi…
Selection Hyper-heuristics Can Automatically Adjust the Learning Period to Optimally Solve Pseudo-Boolean Problems
Laboratoire d’Informatique (LIX), CNRS, École Polytechnique, Institut Polytechnique de Paris（信息实验室（LIX），法国国家科学研究中心，巴黎高等理工学院，巴黎理工学院）；Department of Computer Science and Engineering, Southern University of Science and Technology（计算机科学与工程系，南方科技大学）；School of Computing & Communications, Lancaster University Leipzig（计算与通信学院，莱斯特大学莱比锡分校）
Laboratoire d’Informatique (LIX)（计算机科学实验室（LIX））；CNRS（法国国家科学研究中心）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）
While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the \textsc{LeadingO…
One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a boolean function $f:\{0,1\}^n \to {\mathbb R}$. The algorithm starts with a random search point $ξ\in \{0,1\}^n$, and in each round it flips each bit of $ξ$ with probability $c/n$ independently at random, where $c>0$ is a fixed constant. The thus created offspring $ξ'$ replaces $ξ$ if and only if $f(ξ') \ge f(ξ)$. The analysis of the runtime of this simple algorithm on monotone and on linear functions turned out to be highly non-trivial. In this paper we review known …
Drift analysis has become a powerful tool to prove bounds on the runtime of randomized search heuristics. It allows, for example, fairly simple proofs for the classical problem how the (1+1) Evolutionary Algorithm (EA) optimizes an arbitrary pseudo-Boolean linear function. The key idea of drift analysis is to measure the progress via another pseudo-Boolean function (called drift function) and use deeper results from probability theory to derive from this a good bound for the runtime of the EA. Surprisingly, all these results manage to use the same drift function for all linear objective functi…
Hillclimbing is an essential part of any optimization algorithm. An important benchmark for hillclimbing algorithms on pseudo-Boolean functions $f: \{0,1\}^n \to \mathbb{R}$ are (strictly) montone functions, on which a surprising number of hillclimbers fail to be efficient. For example, the $(1+1)$-Evolutionary Algorithm is a standard hillclimber which flips each bit independently with probability $c/n$ in each round. Perhaps surprisingly, this algorithm shows a phase transition: it optimizes any monotone pseudo-boolean function in quasilinear time if $c<1$, but there are monotone functions fo…
