Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives
School of Computer Science and Technology（计算机科学与技术学院）；International Research Institute for Artificial Intelligence（国际人工智能研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Southern University of Science and Technology（南方科技大学）；Laboratoire d’Informatique (LIX)（信息学实验室）；École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Institut Polytechnique de Paris（巴黎理工学院）
Weijie Zheng, Benjamin Doerr
备注：
Extended version of a paper that appeared in the proceedings of AAAI 2021. 49 pages. Latest version only fixed two bugs in the references
The theoretical understanding of MOEAs is lagging far behind their success in practice. In particular, previous theory work considers mostly easy problems that are composed of unimodal objectives.
As a first step towards a deeper understanding of how evolutionary algorithms solve multimodal multiobjective problems, we propose the OJZJ problem, a bi-objective problem composed of two objectives isomorphic to the classic jump function benchmark. We prove that SEMO with probability one does not compute the full Pareto front, regardless of the runtime. In contrast, for all problem sizes $n$ and all jump sizes ${k \in [4..\frac n2 - 1]}$, the global SEMO (GSEMO) covers the Pareto front in an expected number of $Θ((n-2k)n^{k})$ iterations. For $k = o(n)$, we also show the tighter bound $\frac 32 e n^{k+1} \pm o(n^{k+1})$, which might be the first runtime bound for an MOEA that is tight apart from lower-order terms. We also combine the GSEMO with two approaches that showed advantages in single-objective multimodal problems. When using the GSEMO with a heavy-tailed mutation operator, the expected runtime improves by a factor of at least $k^{Ω(k)}$. When adapting the recent stagnation-detection strategy of Rajabi and Witt (2022) to the GSEMO, the expected runtime also improves by a factor of at least $k^{Ω(k)}$ and surpasses the heavy-tailed GSEMO by a small polynomial factor in $k$. Via an experimental analysis, we show that these asymptotic differences are visible already for small problem sizes: A factor-$5$ speed-up from heavy-tailed mutation and a factor-$10$ speed-up from stagnation detection can be observed already for jump size~$4$ and problem sizes between $10$ and $50$. Overall, our results show that the ideas recently developed to aid single-objective evolutionary algorithms to cope with local optima can be effectively employed also in multiobjective optimization.
Sun Yat-sen University（中山大学）；Guangdong University of Finance and Economics（广东财经大学）
Learning-based heuristics for solving combinatorial optimization problems has recently attracted much academic attention. While most of the existing works only consider the single objective problem with simple constraints, many real-world problems have the multiobjective perspective and contain a rich set of constraints. This paper proposes a multiobjective deep reinforcement learning with evolutionary learning algorithm for a typical complex problem called the multiobjective vehicle routing problem with time windows (MO-VRPTW). In the proposed algorithm, the decomposition strategy is applied …
University of Tsukuba（筑波大学）；University of Stirling（斯特林大学）
Understanding the search dynamics of multiobjective evolutionary algorithms (MOEAs) is still an open problem. This paper extends a recent network-based tool, search trajectory networks (STNs), to model the behavior of MOEAs. Our approach uses the idea of decomposition, where a multiobjective problem is transformed into several single-objective problems. We show that STNs can be used to model and distinguish the search behavior of two popular multiobjective algorithms, MOEA/D and NSGA-II, using 10 continuous benchmark problems with 2 and 3 objectives. Our findings suggest that we can improve ou…
Clustering-based Transfer Learning for Dynamic Multimodal MultiObjective Evolutionary Algorithm
School of Automation and Electrical Engineering, Zhongyuan University of Technology（中原文理大学自动化与电气工程学院）；School of Electrical Engineering, Zhengzhou University（郑州大学电气工程学院）
Evolutionary Biparty Multiobjective UAV Path Planning: Problems and Empirical Comparisons
Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies, School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen（广东省新型安全智能技术重点实验室，哈尔滨工业大学计算机科学与技术学院，深圳）；Peng Cheng Laboratory（鹏城实验室）；School of Artificial Intelligence/School of Future Technology, Nanjing University of Information Science and Technology（人工智能学院/未来技术学院，南京信息科学技术大学）
Shenzhen University（深圳大学）；The Hong Kong Polytechnic University（香港理工大学）
Recent decades have witnessed great advancements in multiobjective evolutionary algorithms (MOEAs) for multiobjective optimization problems (MOPs). However, these progressively improved MOEAs have not necessarily been equipped with scalable and learnable problem-solving strategies for new and grand challenges brought by the scaling-up MOPs with continuously increasing complexity from diverse aspects, mainly including expensive cost of function evaluations, many objectives, large-scale search space, time-varying environments, and multi-task. Under different scenarios, divergent thinking is requ…
In this paper we systematically study the importance, i.e., the influence on performance, of the main design elements that differentiate scalarizing functions-based multiobjective evolutionary algorithms (MOEAs). This class of MOEAs includes Multiobjecitve Genetic Local Search (MOGLS) and Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D) and proved to be very successful in multiple computational experiments and practical applications. The two algorithms share the same common structure and differ only in two main aspects. Using three different multiobjective combinatorial op…
Central South University（中南大学）；Hunan University（湖南大学）；South China University of Technology（华南理工大学）
The goal of constrained multiobjective evolutionary optimization is to obtain a set of well-converged and welldistributed feasible solutions. To complete this goal, there should be a tradeoff among feasibility, diversity, and convergence. However, it is nontrivial to balance these three elements simultaneously by using a single tradeoff model since the importance of each element varies in different evolutionary phases. As an alternative, we adapt different tradeoff models in different phases and propose a novel algorithm called ATM-R. In the infeasible phase, ATM-R takes the tradeoff between d…
Multi-objective optimisation is regarded as one of the most promising ways for dealing with constrained optimisation problems in evolutionary optimisation. This paper presents a theoretical investigation of a multi-objective optimisation evolutionary algorithm for solving the 0-1 knapsack problem. Two initialisation methods are considered in the algorithm: local search initialisation and greedy search initialisation. Then the solution quality of the algorithm is analysed in terms of the approximation ratio.
A hybrid framework combining the branch and bound method with multiobjective evolutionary algorithms is proposed for nonconvex multiobjective optimization. The hybridization exploits the complementary character of the two optimization strategies. A multiobjective evolutionary algorithm is intended for inducing tight lower and upper bounds during the branch and bound procedure. Tight bounds such as the ones derived in this way can reduce the number of subproblems that have to be solved. The branch and bound method guarantees the global convergence of the framework and improves the search capabi…
In this report, we suggest nine test problems for multi-task multi-objective optimization (MTMOO), each of which consists of two multiobjective optimization tasks that need to be solved simultaneously. The relationship between tasks varies between different test problems, which would be helpful to have a comprehensive evaluation of the MO-MFO algorithms. It is expected that the proposed test problems will germinate progress the field of the MTMOO research.
Multiobjective Evolutionary Component Effect on Algorithm behavior
IRIT - CNRS UMR5505, University of Toulouse（图卢兹大学IRIT（法国国家科学研究中心UMR5505））；University of Brasilia（巴西利亚大学）；University of Stirling（斯特林大学）；University of Tsukuba（筑波大学）
In evolutionary algorithms, a preselection operator aims to select the promising offspring solutions from a candidate offspring set. It is usually based on the estimated or real objective values of the candidate offspring solutions. In a sense, the preselection can be treated as a classification procedure, which classifies the candidate offspring solutions into promising ones and unpromising ones. Following this idea, we propose a classification based preselection (CPS) strategy for evolutionary multiobjective optimization. When applying classification based preselection, an evolutionary algor…
Subset selection is an important component in evolutionary multiobjective optimization (EMO) algorithms. Clustering, as a classic method to group similar data points together, has been used for subset selection in some fields. However, clustering-based methods have not been evaluated in the context of subset selection from solution sets obtained by EMO algorithms. In this paper, we first review some classic clustering algorithms. We also point out that another popular subset selection method, i.e., inverted generational distance (IGD)-based subset selection, can be viewed as clustering. Then, …
Multiobjective Evolutionary Algorithms based on Decomposition (MOEA/D) represent a widely used class of population-based metaheuristics for the solution of multicriteria optimization problems. We introduce the MOEADr package, which offers many of these variants as instantiations of a component-oriented framework. This approach contributes for easier reproducibility of existing MOEA/D variants from the literature, as well as for faster development and testing of new composite algorithms. The package offers an standardized, modular implementation of MOEA/D based on this framework, which was desi…
The present survey provides the state-of-the-art of research, copiously devoted to Evolutionary Approach (EAs) for clustering exemplified with a diversity of evolutionary computations. The Survey provides a nomenclature that highlights some aspects that are very important in the context of evolutionary data clustering. The paper missions the clustering trade-offs branched out with wide-ranging Multi Objective Evolutionary Approaches (MOEAs) methods. Finally, this study addresses the potential challenges of MOEA design and data clustering, along with conclusions and recommendations for novice a…
The paper analyzes the scalability of multiobjective estimation of distribution algorithms (MOEDAs) on a class of boundedly-difficult additively-separable multiobjective optimization problems. The paper illustrates that even if the linkage is correctly identified, massive multimodality of the search problems can easily overwhelm the nicher and lead to exponential scale-up. Facetwise models are subsequently used to propose a growth rate of the number of differing substructures between the two objectives to avoid the niching method from being overwhelmed and lead to polynomial scalability of MOE…
We propose the cone epsilon-dominance approach to improve convergence and diversity in multiobjective evolutionary algorithms (MOEAs). A cone-eps-MOEA is presented and compared with MOEAs based on the standard Pareto relation (NSGA-II, NSGA-II*, SPEA2, and a clustered NSGA-II) and on the epsilon-dominance (eps-MOEA). The comparison is performed both in terms of computational complexity and on four performance indicators selected to quantify the quality of the final results obtained by each algorithm: the convergence, diversity, hypervolume, and coverage of many sets metrics. Sixteen well-known…
Population-based evolutionary algorithms have great potential to handle multiobjective optimisation problems. However, these algorithms depends largely on problem characteristics, and there is a need to improve their performance for a wider range of problems. References, which are often specified by the decision maker's preference in different forms, are a very effective method to improve the performance of algorithms but have not been fully explored in literature. This paper proposes a novel framework for effective use of references to strengthen algorithms. This framework considers reference…
Evolutionary algorithms (EAs) have been well acknowledged as a promising paradigm for solving optimisation problems with multiple conflicting objectives in the sense that they are able to locate a set of diverse approximations of Pareto optimal solutions in a single run. EAs drive the search for approximated solutions through maintaining a diverse population of solutions and by recombining promising solutions selected from the population. Combining machine learning techniques has shown great potentials since the intrinsic structure of the Pareto optimal solutions of an multiobjective optimisat…
EVITA, standing for Evolutionary Inventory and Transportation Algorithm, is a two-level methodology designed to address the Inventory and Transportation Problem (ITP) in retail chains. The top level uses an evolutionary algorithm to obtain delivery patterns for each shop on a weekly basis so as to minimise the inventory costs, while the bottom level solves the Vehicle Routing Problem (VRP) for every day in order to obtain the minimum transport costs associated to a particular set of patterns. The aim of this paper is to investigate whether a multiobjective approach to this problem can yield an…
National University of Defense Technology（国防科技大学）；Hunan Key Laboratory of Multi-energy System Intelligent Interconnection Technology（多能系统智能互联技术湖南省重点实验室）；Xiangjiang Laboratory（湘江实验室）；Tsinghua University（清华大学）
A Novel Immune Algorithm for Multiparty Multiobjective Optimization
Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies（广东省新型安全智能技术重点实验室）；School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳）计算机科学与技术学院）；Peng Cheng Laboratory（鹏城实验室）；School of Artificial Intelligence, Nanjing University of Information Science and Technology（南京信息工程大学人工智能学院）；School of Computer Science and Engineering, Southern University of Science and Technology（南方科技大学计算机科学与工程系）
The source number identification is an essential step in direction-of-arrival (DOA) estimation. Existing methods may provide a wrong source number due to inferior statistical properties (in low SNR or limited snapshots) or modeling errors (caused by relaxing sparse penalties), especially in impulsive noise. To address this issue, we propose a novel idea of simultaneous source number identification and DOA estimation. We formulate a multiobjective off-grid DOA estimation model to realize this idea, by which the source number can be automatically identified together with DOA estimation. In parti…
Variable division and optimization (D\&O) is a frequently utilized algorithm design paradigm in Evolutionary Algorithms (EAs). A D\&O EA divides a variable into partial variables and then optimize them respectively. A complicated problem is thus divided into simple subtasks. For example, a variable of portfolio problem can be divided into two partial variables, i.e. the selection of assets and the allocation of capital. Thereby, we optimize these two partial variables respectively. There is no formal discussion about how are the partial variables iteratively optimized and why can it work for b…
Many real-world optimization problems such as engineering design can be eventually modeled as the corresponding multiobjective optimization problems (MOPs) which must be solved to obtain approximate Pareto optimal fronts. Multiobjective evolutionary algorithm based on decomposition (MOEA/D) has been regarded as a significantly promising approach for solving MOPs. Recent studies have shown that MOEA/D with uniform weight vectors is well-suited to MOPs with regular Pareto optimal fronts, but its performance in terms of diversity usually deteriorates when solving MOPs with irregular Pareto optima…
This paper describes a scalable algorithm for solving multiobjective decomposable problems by combining the hierarchical Bayesian optimization algorithm (hBOA) with the nondominated sorting genetic algorithm (NSGA-II) and clustering in the objective space. It is first argued that for good scalability, clustering or some other form of niching in the objective space is necessary and the size of each niche should be approximately equal. Multiobjective hBOA (mohBOA) is then described that combines hBOA, NSGA-II and clustering in the objective space. The algorithm mohBOA differs from the multiobjec…
Multi-task multi-constraint differential evolution with elite-guided knowledge transfer for coal mine integrated energy system dispatching
School of Information and Control Engineering, China University of Mining and Technology（中国矿业大学信息与控制工程学院）；School of Artificial Intelligence and Computer Science, Jiangnan University（江南大学人工智能与计算机学院）；School of Information Science and Technology, Qingdao University of Science and Technology（青岛科技大学信息科学与技术学院）
Deep neural networks have been found vulnerable to adversarial attacks, thus raising potentially concerns in security-sensitive contexts. To address this problem, recent research has investigated the adversarial robustness of deep neural networks from the architectural point of view. However, searching for architectures of deep neural networks is computationally expensive, particularly when coupled with adversarial training process. To meet the above challenge, this paper proposes a bi-fidelity multiobjective neural architecture search approach. First, we formulate the NAS problem for enhancin…
In parallel and distributed environments, generational evolutionary algorithms often do not exploit the full potential of the computation system since they have to wait until the entire population is evaluated before starting selection procedures. Steady-state algorithms are often seen as a solution to this problem, since fitness evaluation can be done by multiple threads in an asynchronous way. However, if the algorithm updates its state in a complicated way, the threads will eventually have to wait until this update finishes. State update procedures that are computationally expensive are com…
Balancing common disease treatment and epidemic control is a key objective of medical supplies procurement in hospitals during a pandemic such as COVID-19. This problem can be formulated as a bi-objective optimization problem for simultaneously optimizing the effects of common disease treatment and epidemic control. However, due to the large number of supplies, difficulties in evaluating the effects, and the strict budget constraint, it is difficult for existing evolutionary multiobjective algorithms to efficiently approximate the Pareto front of the problem. In this paper, we present an appro…
Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；University of Science and Technology of China（中国科学技术大学）；Southern University of Science and Technology（南方科技大学）
In the field of evolutionary multiobjective optimization, the decision maker (DM) concerns conflicting objectives. In the real-world applications, there usually exist more than one DM and each DM concerns parts of these objectives. Multiparty multiobjective optimization problems (MPMOPs) are proposed to depict the MOP with multiple decision makers involved, where each party concerns about certain some objectives of all. However, in the evolutionary computation field, there is not much attention paid on MPMOPs. This paper constructs a series of MPMOPs based on distance minimization problems (DM…
Technical indicators use graphic representations of data sets by applying various mathematical formulas to financial time series of prices. These formulas comprise a set of rules and parameters whose values are not necessarily known and depend on many factors: the market in which it operates, the size of the time window, and others. This paper focuses on the real-time optimization of the parameters applied for analyzing time series of data. In particular, we optimize the parameters of technical and financial indicators and propose other applications, such as glucose time series. We propose the…
Multiobjective Evolutionary Pruning of Deep Neural Networks with Transfer Learning for improving their Performance and Robustness
University of Granada（格拉纳达大学）；Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI)（安达卢西亚数据科学与计算智能研究所（DaSCI））；TECNALIA（泰克纳利亚）；Basque Research & Technology Alliance (BRTA)（巴斯克研究与技术联盟（BRTA））；University of the Basque Country (UPV/EHU)（巴斯克大学（UPV/EHU））
In this paper we propose a novel methodology to assist in identifying vulnerabilities in a real-world complex heterogeneous industrial control systems (ICS) using two evolutionary multiobjective optimisation (EMO) algorithms, NSGA-II and SPEA2. Our approach is evaluated on a well known benchmark chemical plant simulator, the Tennessee Eastman (TE) process model. We identified vulnerabilities in individual components of the TE model and then made use of these to generate combinatorial attacks to damage the safety of the system, and to cause economic loss. Results were compared against random at…
Southern University of Science and Technology（南方科技大学）；City University of Hong Kong（香港城市大学）；University of Surrey（萨里大学）
Recently, increasing works have proposed to drive evolutionary algorithms using machine learning models. Usually, the performance of such model based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e. the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales, due to the curse of dimensionality. To address this issue, we propose a multi-objective evolutionary algorithm driven by the generative adversarial n…
Recently, the discretization of decision and objective spaces has been discussed in the literature. In some studies, it is shown that the decision space discretization improves the performance of evolutionary multi-objective optimization (EMO) algorithms on continuous multi-objective test problems. In other studies, it is shown that the objective space discretization improves the performance on combinatorial multi-objective problems. However, the effect of the simultaneous discretization of both spaces has not been examined in the literature. In this paper, we examine the effects of the decisi…
One of the major distinguishing features of the dynamic multiobjective optimization problems (DMOPs) is the optimization objectives will change over time, thus tracking the varying Pareto-optimal front becomes a challenge. One of the promising solutions is reusing the "experiences" to construct a prediction model via statistical machine learning approaches. However most of the existing methods ignore the non-independent and identically distributed nature of data used to construct the prediction model. In this paper, we propose an algorithmic framework, called Tr-DMOEA, which integrates transfe…
Semantics-Guided Automatic Tensorization for Multiobjective Evolutionary Algorithms: A Multi-Agent Framework
The Hong Kong Polytechnic University（香港理工大学）；The Hong Kong Polytechnic University Shenzhen Research Institute（香港理工大学深圳研究院）；The Hong Kong Polytechnic University-Daya Bay Technology and Innovation Research Institute（香港理工大学-大亚湾技术与创新研究院）
Proven Advantage of Multiobjective Evolutionary Algorithms for Problems with Different Degrees of Conflict
School of Computer Science and Technology, International Research Institute for Artificial Intelligence, Harbin Institute of Technology, Shenzhen, China（哈尔滨工业大学深圳，计算机科学与技术学院，人工智能国际研究院）
University of Tsukuba（筑波大学）；University of Brasilia（巴西利亚大学）；University of Stirling（斯特林大学）
The performance of multiobjective algorithms varies across problems, making it hard to develop new algorithms or apply existing ones to new problems. To simplify the development and application of new multiobjective algorithms, there has been an increasing interest in their automatic design from component parts. These automatically designed metaheuristics can outperform their human-developed counterparts. However, it is still uncertain what are the most influential components leading to their performance improvement. This study introduces a new methodology to investigate the effects of the fin…
Xidian University（西安电子科技大学）；Polytechnic Institute of Leiria（莱里亚理工学院）；Leiden University（莱顿大学）；Microsoft Research Asia（微软亚洲研究院）；Newcastle University（纽卡斯尔大学）
Finding a good classifier is a multiobjective optimization problem with different error rates and the costs to be minimized. The receiver operating characteristic is widely used in the machine learning community to analyze the performance of parametric classifiers or sets of Pareto optimal classifiers. In order to directly compare two sets of classifiers the area (or volume) under the convex hull can be used as a scalar indicator for the performance of a set of classifiers in receiver operating characteristic space. Recently, the convex hull based multiobjective genetic programming algorithm w…
Multiobjective evolutionary algorithms (MOEAs) have been successfully applied to a number of constrained optimization problems. Many of them adopt mutation and crossover operators from differential evolution. However, these operators do not explicitly utilise features of fitness landscapes. To improve the performance of algorithms, this paper aims at designing a search operator adapting to fitness landscapes. Through an observation, we find that principle component analysis (PCA) can be used to characterise fitness landscapes. Based on this finding, a new search operator, called PCA-projection…
The unit commitment (UC) problem is a nonlinear, high-dimensional, highly constrained, mixed-integer power system optimization problem and is generally solved in the literature considering minimizing the system operation cost as the only objective. However, due to increasing environmental concerns, the recent attention has shifted to incorporating emission in the problem formulation. In this paper, a multi-objective evolutionary algorithm based on decomposition (MOEA/D) is proposed to solve the UC problem as a multi-objective optimization problem considering minimizing cost and emission as the…
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. …
A Detection Region Method-Based Evolutionary Algorithm for Binary Constrained Multiobjective Optimization
National University of Defense Technology（国防科技大学）；Hunan Key Laboratory of Multi-energy System Intelligent Interconnection Technology（湖南省多能系统智能互联技术重点实验室）；Xiangjiang Laboratory（湘江实验室）；Tsinghua University（清华大学）
Evolutionary algorithms have been successfully applied to a variety of optimisation problems in stationary environments. However, many real world optimisation problems are set in dynamic environments where the success criteria shifts regularly. Population diversity affects algorithmic performance, particularly on multiobjective and dynamic problems. Diversity mechanisms are methods of altering evolutionary algorithms in a way that promotes the maintenance of population diversity. This project intends to measure and compare the performance effect a variety of diversity mechanisms have on an evo…
