Kalyanmoy Deb

Zhichao Lu, Ran Cheng, Yaochu Jin et al.

The ongoing advancements in network architecture design have led to remarkable achievements in deep learning across various challenging computer vision tasks. Meanwhile, the development of neural architecture search (NAS) has provided promising approaches to automating the design of network architectures for lower prediction error. Recently, the emerging application scenarios of deep learning have raised higher demands for network architectures considering multiple design criteria: number of parameters/floating-point operations, and inference latency, among others. From an optimization point of view, the NAS tasks involving multiple design criteria are intrinsically multiobjective optimization problems; hence, it is reasonable to adopt evolutionary multiobjective optimization (EMO) algorithms for tackling them. Nonetheless, there is still a clear gap confining the related research along this pathway: on the one hand, there is a lack of a general problem formulation of NAS tasks from an optimization point of view; on the other hand, there are challenges in conducting benchmark assessments of EMO algorithms on NAS tasks. To bridge the gap: (i) we formulate NAS tasks into general multi-objective optimization problems and analyze the complex characteristics from an optimization point of view; (ii) we present an end-to-end pipeline, dubbed $\texttt{EvoXBench}$, to generate benchmark test problems for EMO algorithms to run efficiently -- without the requirement of GPUs or Pytorch/Tensorflow; (iii) we instantiate two test suites comprehensively covering two datasets, seven search spaces, and three hardware devices, involving up to eight objectives. Based on the above, we validate the proposed test suites using six representative EMO algorithms and provide some empirical analyses. The code of $\texttt{EvoXBench}$ is available from $\href{https://github.com/EMI-Group/EvoXBench}{\rm{here}}$.

Shihua Huang, Zhichao Lu, Kalyanmoy Deb et al.

Efforts to improve the adversarial robustness of convolutional neural networks have primarily focused on developing more effective adversarial training methods. In contrast, little attention was devoted to analyzing the role of architectural elements (such as topology, depth, and width) on adversarial robustness. This paper seeks to bridge this gap and present a holistic study on the impact of architectural design on adversarial robustness. We focus on residual networks and consider architecture design at the block level, i.e., topology, kernel size, activation, and normalization, as well as at the network scaling level, i.e., depth and width of each block in the network. In both cases, we first derive insights through systematic ablative experiments. Then we design a robust residual block, dubbed RobustResBlock, and a compound scaling rule, dubbed RobustScaling, to distribute depth and width at the desired FLOP count. Finally, we combine RobustResBlock and RobustScaling and present a portfolio of adversarially robust residual networks, RobustResNets, spanning a broad spectrum of model capacities. Experimental validation across multiple datasets and adversarial attacks demonstrate that RobustResNets consistently outperform both the standard WRNs and other existing robust architectures, achieving state-of-the-art AutoAttack robust accuracy of 61.1% without additional data and 63.7% with 500K external data while being $2\times$ more compact in terms of parameters. Code is available at \url{ https://github.com/zhichao-lu/robust-residual-network}

Amir H Gandomi, Kalyanmoy Deb, Ronald C Averill et al.

To solve complex real-world problems, heuristics and concept-based approaches can be used in order to incorporate information into the problem. In this study, a concept-based approach called variable functioning Fx is introduced to reduce the optimization variables and narrow down the search space. In this method, the relationships among one or more subset of variables are defined with functions using information prior to optimization; thus, instead of modifying the variables in the search process, the function variables are optimized. By using problem structure analysis technique and engineering expert knowledge, the $Fx$ method is used to enhance the steel frame design optimization process as a complex real-world problem. The proposed approach is coupled with particle swarm optimization and differential evolution algorithms and used for three case studies. The algorithms are applied to optimize the case studies by considering the relationships among column cross-section areas. The results show that $Fx$ can significantly improve both the convergence rate and the final design of a frame structure, even if it is only used for seeding.

Stephen Kelly, Daniel S. Park, Xingyou Song et al.

Autonomous robots deployed in the real world will need control policies that rapidly adapt to environmental changes. To this end, we propose AutoRobotics-Zero (ARZ), a method based on AutoML-Zero that discovers zero-shot adaptable policies from scratch. In contrast to neural network adaptation policies, where only model parameters are optimized, ARZ can build control algorithms with the full expressive power of a linear register machine. We evolve modular policies that tune their model parameters and alter their inference algorithm on-the-fly to adapt to sudden environmental changes. We demonstrate our method on a realistic simulated quadruped robot, for which we evolve safe control policies that avoid falling when individual limbs suddenly break. This is a challenging task in which two popular neural network baselines fail. Finally, we conduct a detailed analysis of our method on a novel and challenging non-stationary control task dubbed Cataclysmic Cartpole. Results confirm our findings that ARZ is significantly more robust to sudden environmental changes and can build simple, interpretable control policies.

Julian Blank, Kalyanmoy Deb

Significant effort has been made to solve computationally expensive optimization problems in the past two decades, and various optimization methods incorporating surrogates into optimization have been proposed. Most research focuses on either exploiting the surrogate by defining a utility optimization problem or customizing an existing optimization method to use one or multiple approximation models. However, only a little attention has been paid to generic concepts applicable to different types of algorithms and optimization problems simultaneously. Thus this paper proposes a generalized probabilistic surrogate-assisted framework (GPSAF), applicable to a broad category of unconstrained and constrained, single- and multi-objective optimization algorithms. The idea is based on a surrogate assisting an existing optimization method. The assistance is based on two distinct phases, one facilitating exploration and another exploiting the surrogates. The exploration and exploitation of surrogates are automatically balanced by performing a probabilistic knockout tournament among different clusters of solutions. A study of multiple well-known population-based optimization algorithms is conducted with and without the proposed surrogate assistance on single- and multi-objective optimization problems with a maximum solution evaluation budget of 300 or less. The results indicate the effectiveness of applying GPSAF to an optimization algorithm and the competitiveness with other surrogate-assisted algorithms.

Devesh Shah, Anirudh Suresh, Alemayehu Admasu et al.

The evaluation of synthetic micro-structure images is an emerging problem as machine learning and materials science research have evolved together. Typical state of the art methods in evaluating synthetic images from generative models have relied on the Fréchet Inception Distance. However, this and other similar methods, are limited in the materials domain due to both the unique features that characterize physically accurate micro-structures and limited dataset sizes. In this study we evaluate a variety of methods on scanning electron microscope (SEM) images of graphene-reinforced polyurethane foams. The primary objective of this paper is to report our findings with regards to the shortcomings of existing methods so as to encourage the machine learning community to consider enhancements in metrics for assessing quality of synthetic images in the material science domain.

Zhichao Lu, Kalyanmoy Deb, Erik Goodman et al.

In this paper, we propose an efficient NAS algorithm for generating task-specific models that are competitive under multiple competing objectives. It comprises of two surrogates, one at the architecture level to improve sample efficiency and one at the weights level, through a supernet, to improve gradient descent training efficiency. On standard benchmark datasets (C10, C100, ImageNet), the resulting models, dubbed NSGANetV2, either match or outperform models from existing approaches with the search being orders of magnitude more sample efficient. Furthermore, we demonstrate the effectiveness and versatility of the proposed method on six diverse non-standard datasets, e.g. STL-10, Flowers102, Oxford Pets, FGVC Aircrafts etc. In all cases, NSGANetV2s improve the state-of-the-art (under mobile setting), suggesting that NAS can be a viable alternative to conventional transfer learning approaches in handling diverse scenarios such as small-scale or fine-grained datasets. Code is available at https://github.com/mikelzc1990/nsganetv2

Mohamed Wiem Mkaouer, Marouane Kessentini, Adnan Shaout et al.

Software systems nowadays are complex and difficult to maintain due to continuous changes and bad design choices. To handle the complexity of systems, software products are, in general, decomposed in terms of packages/modules containing classes that are dependent. However, it is challenging to automatically remodularize systems to improve their maintainability. The majority of existing remodularization work mainly satisfy one objective which is improving the structure of packages by optimizing coupling and cohesion. In addition, most of existing studies are limited to only few operation types such as move class and split packages. Many other objectives, such as the design semantics, reducing the number of changes and maximizing the consistency with development change history, are important to improve the quality of the software by remodularizing it. In this paper, we propose a novel many-objective search-based approach using NSGA-III. The process aims at finding the optimal remodularization solutions that improve the structure of packages, minimize the number of changes, preserve semantics coherence, and re-use the history of changes. We evaluate the efficiency of our approach using four different open-source systems and one automotive industry project, provided by our industrial partner, through a quantitative and qualitative study conducted with software engineers.

Zhichao Lu, Gautam Sreekumar, Erik Goodman et al.

Neural architecture search (NAS) has emerged as a promising avenue for automatically designing task-specific neural networks. Existing NAS approaches require one complete search for each deployment specification of hardware or objective. This is a computationally impractical endeavor given the potentially large number of application scenarios. In this paper, we propose Neural Architecture Transfer (NAT) to overcome this limitation. NAT is designed to efficiently generate task-specific custom models that are competitive under multiple conflicting objectives. To realize this goal we learn task-specific supernets from which specialized subnets can be sampled without any additional training. The key to our approach is an integrated online transfer learning and many-objective evolutionary search procedure. A pre-trained supernet is iteratively adapted while simultaneously searching for task-specific subnets. We demonstrate the efficacy of NAT on 11 benchmark image classification tasks ranging from large-scale multi-class to small-scale fine-grained datasets. In all cases, including ImageNet, NATNets improve upon the state-of-the-art under mobile settings ($\leq$ 600M Multiply-Adds). Surprisingly, small-scale fine-grained datasets benefit the most from NAT. At the same time, the architecture search and transfer is orders of magnitude more efficient than existing NAS methods. Overall, the experimental evaluation indicates that, across diverse image classification tasks and computational objectives, NAT is an appreciably more effective alternative to conventional transfer learning of fine-tuning weights of an existing network architecture learned on standard datasets. Code is available at https://github.com/human-analysis/neural-architecture-transfer

Zhichao Lu, Kalyanmoy Deb, Vishnu Naresh Boddeti

Convolutional neural networks have witnessed remarkable improvements in computational efficiency in recent years. A key driving force has been the idea of trading-off model expressivity and efficiency through a combination of $1\times 1$ and depth-wise separable convolutions in lieu of a standard convolutional layer. The price of the efficiency, however, is the sub-optimal flow of information across space and channels in the network. To overcome this limitation, we present MUXConv, a layer that is designed to increase the flow of information by progressively multiplexing channel and spatial information in the network, while mitigating computational complexity. Furthermore, to demonstrate the effectiveness of MUXConv, we integrate it within an efficient multi-objective evolutionary algorithm to search for the optimal model hyper-parameters while simultaneously optimizing accuracy, compactness, and computational efficiency. On ImageNet, the resulting models, dubbed MUXNets, match the performance (75.3% top-1 accuracy) and multiply-add operations (218M) of MobileNetV3 while being 1.6$\times$ more compact, and outperform other mobile models in all the three criteria. MUXNet also performs well under transfer learning and when adapted to object detection. On the ChestX-Ray 14 benchmark, its accuracy is comparable to the state-of-the-art while being $3.3\times$ more compact and $14\times$ more efficient. Similarly, detection on PASCAL VOC 2007 is 1.2% more accurate, 28% faster and 6% more compact compared to MobileNetV2. Code is available from https://github.com/human-analysis/MUXConv

Zhichao Lu, Ian Whalen, Yashesh Dhebar et al.

Early advancements in convolutional neural networks (CNNs) architectures are primarily driven by human expertise and by elaborate design processes. Recently, neural architecture search was proposed with the aim of automating the network design process and generating task-dependent architectures. While existing approaches have achieved competitive performance in image classification, they are not well suited to problems where the computational budget is limited for two reasons: (1) the obtained architectures are either solely optimized for classification performance, or only for one deployment scenario; (2) the search process requires vast computational resources in most approaches. To overcome these limitations, we propose an evolutionary algorithm for searching neural architectures under multiple objectives, such as classification performance and floating-point operations (FLOPs). The proposed method addresses the first shortcoming by populating a set of architectures to approximate the entire Pareto frontier through genetic operations that recombine and modify architectural components progressively. Our approach improves computational efficiency by carefully down-scaling the architectures during the search as well as reinforcing the patterns commonly shared among past successful architectures through Bayesian model learning. The integration of these two main contributions allows an efficient design of architectures that are competitive and in most cases outperform both manually and automatically designed architectures on benchmark image classification datasets: CIFAR, ImageNet, and human chest X-ray. The flexibility provided from simultaneously obtaining multiple architecture choices for different compute requirements further differentiates our approach from other methods in the literature. Code is available at https://github.com/mikelzc1990/nsganetv1

Amin Ibrahim, Azam Asilian Bidgoli, Shahryar Rahnamayan et al.

As the interest in multi- and many-objective optimization algorithms grows, the performance comparison of these algorithms becomes increasingly important. A large number of performance indicators for multi-objective optimization algorithms have been introduced, each of which evaluates these algorithms based on a certain aspect. Therefore, assessing the quality of multi-objective results using multiple indicators is essential to guarantee that the evaluation considers all quality perspectives. This paper proposes a novel multi-metric comparison method to rank the performance of multi-/ many-objective optimization algorithms based on a set of performance indicators. We utilize the Pareto optimality concept (i.e., non-dominated sorting algorithm) to create the rank levels of algorithms by simultaneously considering multiple performance indicators as criteria/objectives. As a result, four different techniques are proposed to rank algorithms based on their contribution at each Pareto level. This method allows researchers to utilize a set of existing/newly developed performance metrics to adequately assess/rank multi-/many-objective algorithms. The proposed methods are scalable and can accommodate in its comprehensive scheme any newly introduced metric. The method was applied to rank 10 competing algorithms in the 2018 CEC competition solving 15 many-objective test problems. The Pareto-optimal ranking was conducted based on 10 well-known multi-objective performance indicators and the results were compared to the final ranks reported by the competition, which were based on the inverted generational distance (IGD) and hypervolume indicator (HV) measures. The techniques suggested in this paper have broad applications in science and engineering, particularly in areas where multiple metrics are used for comparisons. Examples include machine learning and data mining.

Claudio Lucio do Val Lopes, Flávio Vinícius Cruzeiro Martins, Elizabeth Fialho Wanner et al.

Dominance move (DoM) is a binary quality indicator that can be used in multi-objective and many-objective optimization to compare two solution sets obtained from different algorithms. The DoM indicator can differentiate the sets for certain important features, such as convergence, spread, uniformity, and cardinality. DoM does not use any reference, and it has an intuitive and physical meaning, similar to the $ε$-indicator, and calculates the minimum total move of members of one set so that all elements in another set are to be dominated or identical to at least one member of the first set. Despite the aforementioned properties, DoM is hard to calculate, particularly in higher dimensions. There is an efficient and exact method to calculate it in a bi-objective case only. This work proposes a novel approach to calculate DoM using a mixed integer programming (MIP) approach, which can handle sets with three or more objectives and is shown to overcome the $ε$-indicator's information loss. Experiments, in the bi-objective space, are done to verify the model's correctness. Furthermore, other experiments, using 3, 5, 10, 15, 20, 25 and 30-objective problems are performed to show how the model behaves in higher-dimensional cases. Algorithms, such as IBEA, MOEA/D, NSGA-III, NSGA-II, and SPEA2 are used to generate the solution sets (however any other algorithms can also be used with the proposed MIP-DoM indicator). Further extensions are discussed to handle certain idiosyncrasies with some solution sets and also to improve the quality indicator and its use for other situations.

Sukrit Mittal, Dhish Kumar Saxena, Kalyanmoy Deb et al.

"Innovization" is a task of learning common relationships among some or all of the Pareto-optimal (PO) solutions in multi- and many-objective optimization problems. Recent studies have shown that a chronological sequence of non-dominated solutions obtained in consecutive iterations during an optimization run also possess salient patterns that can be used to learn problem features to help create new and improved solutions. In this paper, we propose a machine-learning- (ML-) assisted modelling approach that learns the modifications in design variables needed to advance population members towards the Pareto-optimal set. We then propose to use the resulting ML model as an additional innovized repair (IR2) operator to be applied on offspring solutions created by the usual genetic operators, as a novel mean of improving their convergence properties. In this paper, the well-known random forest (RF) method is used as the ML model and is integrated with various evolutionary multi- and many-objective optimization algorithms, including NSGA-II, NSGA-III, and MOEA/D. On several test problems ranging from two to five objectives, we demonstrate improvement in convergence behaviour using the proposed IR2-RF operator. Since the operator does not demand any additional solution evaluations, instead using the history of gradual and progressive improvements in solutions over generations, the proposed ML-based optimization opens up a new direction of optimization algorithm development with advances in AI and ML approaches.

Yashesh Dhebar, Kalyanmoy Deb, Subramanya Nageshrao et al.

Black-box AI induction methods such as deep reinforcement learning (DRL) are increasingly being used to find optimal policies for a given control task. Although policies represented using a black-box AI are capable of efficiently executing the underlying control task and achieving optimal closed-loop performance, the developed control rules are often complex and neither interpretable nor explainable. In this paper, we use a recently proposed nonlinear decision-tree (NLDT) approach to find a hierarchical set of control rules in an attempt to maximize the open-loop performance for approximating and explaining the pre-trained black-box DRL (oracle) agent using the labelled state-action dataset. Recent advances in nonlinear optimization approaches using evolutionary computation facilitates finding a hierarchical set of nonlinear control rules as a function of state variables using a computationally fast bilevel optimization procedure at each node of the proposed NLDT. Additionally, we propose a re-optimization procedure for enhancing closed-loop performance of an already derived NLDT. We evaluate our proposed methodologies (open and closed-loop NLDTs) on different control problems having multiple discrete actions. In all these problems our proposed approach is able to find relatively simple and interpretable rules involving one to four non-linear terms per rule, while simultaneously achieving on par closed-loop performance when compared to a trained black-box DRL agent. A post-processing approach for simplifying the NLDT is also suggested. The obtained results are inspiring as they suggest the replacement of complicated black-box DRL policies involving thousands of parameters (making them non-interpretable) with relatively simple interpretable policies. Results are encouraging and motivating to pursue further applications of proposed approach in solving more complex control tasks.

Yashesh Dhebar, Sparsh Gupta, Kalyanmoy Deb

Classification of datasets into two or more distinct classes is an important machine learning task. Many methods are able to classify binary classification tasks with a very high accuracy on test data, but cannot provide any easily interpretable explanation for users to have a deeper understanding of reasons for the split of data into two classes. In this paper, we highlight and evaluate a recently proposed nonlinear decision tree approach with a number of commonly used classification methods on a number of datasets involving a few to a large number of features. The study reveals key issues such as effect of classification on the method's parameter values, complexity of the classifier versus achieved accuracy, and interpretability of resulting classifiers.

Yashesh Dhebar, Kalyanmoy Deb

For supervised classification problems involving design, control, other practical purposes, users are not only interested in finding a highly accurate classifier, but they also demand that the obtained classifier be easily interpretable. While the definition of interpretability of a classifier can vary from case to case, here, by a humanly interpretable classifier we restrict it to be expressed in simplistic mathematical terms. As a novel approach, we represent a classifier as an assembly of simple mathematical rules using a non-linear decision tree (NLDT). Each conditional (non-terminal) node of the tree represents a non-linear mathematical rule (split-rule) involving features in order to partition the dataset in the given conditional node into two non-overlapping subsets. This partitioning is intended to minimize the impurity of the resulting child nodes. By restricting the structure of split-rule at each conditional node and depth of the decision tree, the interpretability of the classifier is assured. The non-linear split-rule at a given conditional node is obtained using an evolutionary bilevel optimization algorithm, in which while the upper-level focuses on arriving at an interpretable structure of the split-rule, the lower-level achieves the most appropriate weights (coefficients) of individual constituents of the rule to minimize the net impurity of two resulting child nodes. The performance of the proposed algorithm is demonstrated on a number of controlled test problems, existing benchmark problems, and industrial problems. Results on two to 500-feature problems are encouraging and open up further scopes of applying the proposed approach to more challenging and complex classification tasks.

Kyle Robert Harrison, Azam Asilian Bidgoli, Shahryar Rahnamayan et al.

In the context of optimization, visualization techniques can be useful for understanding the behaviour of optimization algorithms and can even provide a means to facilitate human interaction with an optimizer. Towards this goal, an image-based visualization framework, without dimension reduction, that visualizes the solutions to large-scale global optimization problems as images is proposed. In the proposed framework, the pixels visualize decision variables while the entire image represents the overall solution quality. This framework affords a number of benefits over existing visualization techniques including enhanced scalability (in terms of the number of decision variables), facilitation of standard image processing techniques, providing nearly infinite benchmark cases, and explicit alignment with human perception. Furthermore, image-based visualization can be used to visualize the optimization process in real-time, thereby allowing the user to ascertain characteristics of the search process as it is progressing. To the best of the authors' knowledge, this is the first realization of a dimension-preserving, scalable visualization framework that embeds the inherent relationship between decision space and objective space. The proposed framework is utilized with 10 different mapping schemes on an image-reconstruction problem that encompass continuous, discrete, binary, combinatorial, constrained, dynamic, and multi-objective optimization. The proposed framework is then demonstrated on arbitrary benchmark problems with known optima. Experimental results elucidate the flexibility and demonstrate how valuable information about the search process can be gathered via the proposed visualization framework.

Shaik Tanveer ul Huq, Vadlamani Ravi, Kalyanmoy Deb

Most optimization-based community detection approaches formulate the problem in a single or bi-objective framework. In this paper, we propose two variants of a three-objective formulation using a customized non-dominated sorting genetic algorithm III (NSGA-III) to find community structures in a network. In the first variant, named NSGA-III-KRM, we considered Kernel k means, Ratio cut, and Modularity, as the three objectives, whereas the second variant, named NSGA-III-CCM, considers Community score, Community fitness and Modularity, as three objective functions. Experiments are conducted on four benchmark network datasets. Comparison with state-of-the-art approaches along with decomposition-based multi-objective evolutionary algorithm variants (MOEA/D-KRM and MOEA/D-CCM) indicates that the proposed variants yield comparable or better results. This is particularly significant because the addition of the third objective does not worsen the results of the other two objectives. We also propose a simple method to rank the Pareto solutions so obtained by proposing a new measure, namely the ratio of the hyper-volume and inverted generational distance (IGD). The higher the ratio, the better is the Pareto set. This strategy is particularly useful in the absence of empirical attainment function in the multi-objective framework, where the number of objectives is more than two.

Jonatas B. C. Chagas, Julian Blank, Markus Wagner et al.

In this paper, we propose a method to solve a bi-objective variant of the well-studied Traveling Thief Problem (TTP). The TTP is a multi-component problem that combines two classic combinatorial problems: Traveling Salesman Problem (TSP) and Knapsack Problem (KP). We address the BI-TTP, a bi-objective version of the TTP, where the goal is to minimize the overall traveling time and to maximize the profit of the collected items. Our proposed method is based on a biased-random key genetic algorithm with customizations addressing problem-specific characteristics. We incorporate domain knowledge through a combination of near-optimal solutions of each subproblem in the initial population and use a custom repair operator to avoid the evaluation of infeasible solutions. The bi-objective aspect of the problem is addressed through an elite population extracted based on the non-dominated rank and crowding distance. Furthermore, we provide a comprehensive study showing the influence of each parameter on the performance. Finally, we discuss the results of the BI-TTP competitions at EMO-2019 and GECCO-2019 conferences where our method has won first and second places, respectively, thus proving its ability to find high-quality solutions consistently.

Tao Chen, Miqing Li, Ke Li et al.

Search-Based Software Engineering (SBSE) is a promising paradigm that exploits the computational search to optimize different processes when engineering complex software systems. Self-adaptive system (SAS) is one category of such complex systems that permits to optimize different functional and non-functional objectives/criteria under changing environments (e.g., requirements and workload), which involves problems that are subject to search. In this regard, over years, there has been a considerable amount of work that investigates SBSE for SASs. In this paper, we provide the first systematic and comprehensive survey exclusively on SBSE for SASs, covering papers in 27 venues from 7 repositories, which eventually leads to several key statistics from the most notable 74 primary studies in this particular field of research. Our results, surprisingly, have revealed five disappointments that are of utmost importance and can result in serve consequences but have been overwhelmingly ignored in existing studies. We provide theoretical and/or experimental evidence to justify our arguments against the disappointments, present suggestions, and highlight the promising research opportunities towards their mitigation. We also elaborate on three other emergent, but currently under-explored opportunities for future work on SBSE for SASs. By mitigating the disappointments revealed in this work, together with the highlighted opportunities, we hope to be able to excite a much more significant growth in this particular research direction.

Julian Blank, Kalyanmoy Deb

Python has become the programming language of choice for research and industry projects related to data science, machine learning, and deep learning. Since optimization is an inherent part of these research fields, more optimization related frameworks have arisen in the past few years. Only a few of them support optimization of multiple conflicting objectives at a time, but do not provide comprehensive tools for a complete multi-objective optimization task. To address this issue, we have developed pymoo, a multi-objective optimization framework in Python. We provide a guide to getting started with our framework by demonstrating the implementation of an exemplary constrained multi-objective optimization scenario. Moreover, we give a high-level overview of the architecture of pymoo to show its capabilities followed by an explanation of each module and its corresponding sub-modules. The implementations in our framework are customizable and algorithms can be modified/extended by supplying custom operators. Moreover, a variety of single, multi and many-objective test problems are provided and gradients can be retrieved by automatic differentiation out of the box. Also, pymoo addresses practical needs, such as the parallelization of function evaluations, methods to visualize low and high-dimensional spaces, and tools for multi-criteria decision making. For more information about pymoo, readers are encouraged to visit: https://pymoo.org

Ke Li, Minhui Liao, Kalyanmoy Deb et al.

The ultimate goal of multi-objective optimisation is to help a decision maker (DM) identify solution(s) of interest (SOI) achieving satisfactory trade-offs among multiple conflicting criteria. This can be realised by leveraging DM's preference information in evolutionary multi-objective optimisation (EMO). No consensus has been reached on the effectiveness brought by incorporating preference in EMO (either a priori or interactively) versus a posteriori decision making after a complete run of an EMO algorithm. Bearing this consideration in mind, this paper i) provides a pragmatic overview of the existing developments of preference-based EMO; and ii) conducts a series of experiments to investigate the effectiveness brought by preference incorporation in EMO for approximating various SOI. In particular, the DM's preference information is elicited as a reference point, which represents her/his aspirations for different objectives. Experimental results demonstrate that preference incorporation in EMO does not always lead to a desirable approximation of SOI if the DM's preference information is not well utilised, nor does the DM elicit invalid preference information, which is not uncommon when encountering a black-box system. To a certain extent, this issue can be remedied through an interactive preference elicitation. Last but not the least, we find that a preference-based EMO algorithm is able to be generalised to approximate the whole PF given an appropriate setup of preference information.

Zhichao Lu, Ian Whalen, Vishnu Boddeti et al.

This paper introduces NSGA-Net -- an evolutionary approach for neural architecture search (NAS). NSGA-Net is designed with three goals in mind: (1) a procedure considering multiple and conflicting objectives, (2) an efficient procedure balancing exploration and exploitation of the space of potential neural network architectures, and (3) a procedure finding a diverse set of trade-off network architectures achieved in a single run. NSGA-Net is a population-based search algorithm that explores a space of potential neural network architectures in three steps, namely, a population initialization step that is based on prior-knowledge from hand-crafted architectures, an exploration step comprising crossover and mutation of architectures, and finally an exploitation step that utilizes the hidden useful knowledge stored in the entire history of evaluated neural architectures in the form of a Bayesian Network. Experimental results suggest that combining the dual objectives of minimizing an error metric and computational complexity, as measured by FLOPs, allows NSGA-Net to find competitive neural architectures. Moreover, NSGA-Net achieves error rate on the CIFAR-10 dataset on par with other state-of-the-art NAS methods while using orders of magnitude less computational resources. These results are encouraging and shows the promise to further use of EC methods in various deep-learning paradigms.

Zhun Fan, Yugen You, Haodong Zheng et al.

A new kind of six degree-of-freedom teaching manipulator without actuators is designed, for recording and conveniently setting a trajectory of an industrial robot. The device requires good gravity balance and operating force performance to ensure being controlled easily and fluently. In this paper, we propose a process for modeling the manipulator and then the model is used to formulate a multi-objective optimization problem to optimize the design of the testing manipulator. Three objectives, including total mass of the device, gravity balancing and operating force performance are analyzed and defined. A popular non-dominated sorting genetic algorithm (NSGA-II-CDP) is used to solve the optimization problem. The obtained solutions all outperform the design of a human expert. To extract design knowledge, an innovization study is performed to establish meaningful implicit relationship between the objective space and the decision space, which can be reused by the designer in future design process.

Zhun Fan, Wenji Li, Xinye Cai et al.

This paper proposes a push and pull search (PPS) framework for solving constrained multi-objective optimization problems (CMOPs). To be more specific, the proposed PPS divides the search process into two different stages, including the push and pull search stages. In the push stage, a multi-objective evolutionary algorithm (MOEA) is adopted to explore the search space without considering any constraints, which can help to get across infeasible regions very fast and approach the unconstrained Pareto front. Furthermore, the landscape of CMOPs with constraints can be probed and estimated in the push stage, which can be utilized to conduct the parameters setting for constraint-handling approaches applied in the pull stage. Then, a constrained multi-objective evolutionary algorithm (CMOEA) equipped with an improved epsilon constraint-handling is applied to pull the infeasible individuals achieved in the push stage to the feasible and non-dominated regions. Compared with other CMOEAs, the proposed PPS method can more efficiently get across infeasible regions and converge to the feasible and non-dominated regions by applying push and pull search strategies at different stages. To evaluate the performance regarding convergence and diversity, a set of benchmark CMOPs is used to test the proposed PPS and compare with other five CMOEAs, including MOEA/D-CDP, MOEA/D-SR, C-MOEA/D, MOEA/D-Epsilon and MOEA/D-IEpsilon. The comprehensive experimental results demonstrate that the proposed PPS achieves significantly better or competitive performance than the other five CMOEAs on most of the benchmark set.

Ankur Sinha, Pekka Malo, Kalyanmoy Deb

Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community. Only limited work exists on bilevel problems using evolutionary computation techniques; however, recently there has been an increasing interest due to the proliferation of practical applications and the potential of evolutionary algorithms in tackling these problems. This paper provides a comprehensive review on bilevel optimization from the basic principles to solution strategies; both classical and evolutionary. A number of potential application problems are also discussed. To offer the readers insights on the prominent developments in the field of bilevel optimization, we have performed an automated text-analysis of an extended list of papers published on bilevel optimization to date. This paper should motivate evolutionary computation researchers to pay more attention to this practical yet challenging area.

Ke Li, Kalyanmoy Deb, Xin Yao

Most existing studies on evolutionary multi-objective optimization focus on approximating the whole Pareto-optimal front. Nevertheless, rather than the whole front, which demands for too many points (especially in a high-dimensional space), the decision maker might only interest in a partial region, called the region of interest. In this case, solutions outside this region can be noisy to the decision making procedure. Even worse, there is no guarantee that we can find the preferred solutions when tackling problems with complicated properties or a large number of objectives. In this paper, we develop a systematic way to incorporate the decision maker's preference information into the decomposition-based evolutionary multi-objective optimization methods. Generally speaking, our basic idea is a non-uniform mapping scheme by which the originally uniformly distributed reference points on a canonical simplex can be mapped to the new positions close to the aspiration level vector specified by the decision maker. By these means, we are able to steer the search process towards the region of interest either directly or in an interactive manner and also handle a large number of objectives. In the meanwhile, the boundary solutions can be approximated given the decision maker's requirements. Furthermore, the extent of the region of the interest is intuitively understandable and controllable in a closed form. Extensive experiments, both proof-of-principle and on a variety of problems with 3 to 10 objectives, fully demonstrate the effectiveness of our proposed method for approximating the preferred solutions in the region of interest.

Zhun Fan, Wenji Li, Xinye Cai et al.

Multi-objective evolutionary algorithms (MOEAs) have progressed significantly in recent decades, but most of them are designed to solve unconstrained multi-objective optimization problems. In fact, many real-world multi-objective problems contain a number of constraints. To promote research on constrained multi-objective optimization, we first propose a problem classification scheme with three primary types of difficulty, which reflect various types of challenges presented by real-world optimization problems, in order to characterize the constraint functions in constrained multi-objective optimization problems (CMOPs). These are feasibility-hardness, convergence-hardness and diversity-hardness. We then develop a general toolkit to construct difficulty-adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPs when the number of objectives is greater than three) with three types of parameterized constraint functions developed to capture the three proposed types of difficulty. Based on this toolkit, we suggest nine difficulty-adjustable and scalable CMOPs and nine CMaOPs. The experimental results reveal that mechanisms in MOEA/D-CDP may be more effective in solving convergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be more effective in solving DAS-CMOPs with simultaneous diversity-, feasibility- and convergence-hardness. Mechanisms in C-NSGA-III may be more effective in solving feasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective in solving CMaOPs with convergence-hardness. In addition, none of them can solve these problems efficiently, which stimulates us to continue to develop new CMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

Nikhil Padhye, Pulkit Mittal, Kalyanmoy Deb

Evolutionary Algorithms (EAs) are being routinely applied for a variety of optimization tasks, and real-parameter optimization in the presence of constraints is one such important area. During constrained optimization EAs often create solutions that fall outside the feasible region; hence a viable constraint- handling strategy is needed. This paper focuses on the class of constraint-handling strategies that repair infeasible solutions by bringing them back into the search space and explicitly preserve feasibility of the solutions. Several existing constraint-handling strategies are studied, and two new single parameter constraint-handling methodologies based on parent-centric and inverse parabolic probability (IP) distribution are proposed. The existing and newly proposed constraint-handling methods are first studied with PSO, DE, GAs, and simulation results on four scalable test-problems under different location settings of the optimum are presented. The newly proposed constraint-handling methods exhibit robustness in terms of performance and also succeed on search spaces comprising up-to 500 variables while locating the optimum within an error of 10$^{-10}$. The working principle of the IP based methods is also demonstrated on (i) some generic constrained optimization problems, and (ii) a classic `Weld' problem from structural design and mechanics. The successful performance of the proposed methods clearly exhibits their efficacy as a generic constrained-handling strategy for a wide range of applications.

Ankur Sinha, Pekka Malo, Kalyanmoy Deb

Bilevel optimization problems are a class of challenging optimization problems, which contain two levels of optimization tasks. In these problems, the optimal solutions to the lower level problem become possible feasible candidates to the upper level problem. Such a requirement makes the optimization problem difficult to solve, and has kept the researchers busy towards devising methodologies, which can efficiently handle the problem. Despite the efforts, there hardly exists any effective methodology, which is capable of handling a complex bilevel problem. In this paper, we introduce bilevel evolutionary algorithm based on quadratic approximations (BLEAQ) of optimal lower level variables with respect to the upper level variables. The approach is capable of handling bilevel problems with different kinds of complexities in relatively smaller number of function evaluations. Ideas from classical optimization have been hybridized with evolutionary methods to generate an efficient optimization algorithm for generic bilevel problems. The efficacy of the algorithm has been shown on two sets of test problems. The first set is a recently proposed SMD test set, which contains problems with controllable complexities, and the second set contains standard test problems collected from the literature. The proposed method has been evaluated against two benchmarks, and the performance gain is observed to be significant.