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Multi-objective immune algorithm (MOIA) is a heuristic algorithm based on artificial immune system model. Due to its characteristics of antibody clonal selection, automatic antigen recognition and immune memory in the immune system, artificial immune algorithm has become a research hotspot in the field of multi-objective optimization after the evolutionary algorithms. In this paper, most MOIAs can be classified into three main categories according to the type of problem solving, i.e., they are mostly designed to solve multi-objective optimization problems (MOPs), dynamic MOPs, and constrained MOPs. In this paper, a comprehensive survey is presented to summarize most existing MOIAs, in which their corresponding characteristics, principles and theoretical analyses are discussed in details. Moreover, the performance of MOIAs on solving various kinds of MOPs and many-objective optimization problems is also studied in our experimental comparisons. Finally, a brief conclusion is given to summarize the current drawbacks, challenges, and some future directions for MOIAs. [ABSTRACT FROM AUTHOR]

• An analysis of lexicographic optimization for feature selection and data classification. • Cooperative co-evolution for feature selection while the classifier hyperparameters are also being optimized. • Optimization of multiple objectives using a simple evolutionary algorithm. • A new lexicographic ranking methodology to compare the results of different feature selection methods. • Quite good results for real high-dimensional datasets. This paper starts with two hypotheses. The first one is that the simultaneous optimization of the hyperparameters regulating the classifier within a wrapper method, while the best subset of features is being determined, should improve the results with respect to those obtained with a pre-parameterized classifier. The second one is that solving these two problems can be formulated as a lexicographic optimization problem, allowing the use of a simple single-objective evolutionary algorithm to solve this multi-objective problem. The fitness function is of key importance for such wrapper methods. It is responsible for guiding the search towards potentially good solutions and it also consumes most of the runtime. Having these issues in mind, this paper also proposes a new lexicographic fitness function, designed to minimize the runtime of the algorithm and also to avoid over-fitting. Furthermore, the execution time and the quality of the results obtained by the wrapper procedure also depend on some algorithmic hyperparameters: the similarity thresholds used when comparing two different solutions lexicographically and the percentage of data samples used for validation during the training process. Thus, an experimental analysis has been carried out to find adequate values for these hyperparameters. Finally, the lexicographic cooperative co-evolutionary wrapper approach, using the new fitness function proposed in this paper, has been tested with several datasets belonging to the University of California, Irvine (UCI) repository and also with some real high-dimensional datasets, obtaining quite good results, compared to other state-of-the-art wrapper methods. The comparison has also been made lexicographically, with a new methodology proposed in this paper. [ABSTRACT FROM AUTHOR]

Combinatorial optimization problems (COPs) play an important role in both industrial production and theoretical research. As a classical class of optimization algorithms, evolutionary algorithms have shown good performance in COPs. However, as the size of the problem increases, it becomes difficult for evolutionary algorithms to produce satisfactory solutions within the specified time. In recent years, neural network algorithms have been applied to solve COPs and have shown superior effects. Inspired by this, this paper proposes a new algorithm based on graph structure for solving COPs. Through relaxation, COPs are transformed into their corresponding continuous optimization problems. The output state of our algorithm is converted to a discrete solution by approximation method. Compared with previous algorithms, our algorithm can not only solve COPs with general form such as binary quadratic programming (BQP), but also solve COPs with special form, which greatly expands the application scope of neural network algorithms. In addition, our algorithm can be embedded into other problem frameworks such as solving multi-objective combinatorial optimization problem (MOCOP) through scalarization method. A new penalty function is added to the objective function, which enables our algorithm to overcome the inconsistency between the optimal solution of the original problem and the corresponding continuous problem, so that our algorithm can perform well on various problems. Several numerical simulations are used to illustrate the effectiveness of our algorithm in solving COPs. Experimental results show that our algorithm can effectively solve COPs with general form and has a shorter solving time compared to traditional evolutionary algorithms. [ABSTRACT FROM AUTHOR]

Adaptive operator selection is an online method that automatically adjusts the application rate of different operators based on their actual performance. This paper proposes an adaptive operator selection paradigm based on dueling deep Q-network (DDQN), aiming to improve the training efficiency of the Q-network for solving multi-objective optimization problems. The Q-network is decomposed into state value and action advantage networks, allowing the agent to learn state values more frequently and accurately. A novel state space and reward mechanism are designed to adapt to the characteristics of evolutionary multi-objective optimization. Combining adaptive operator selection with a multi-objective evolutionary algorithm with adaptive weights (AdaW), an AdaW-DDQN algorithm with high adaptability is proposed. Experimental results on three complex benchmark suites demonstrate that the proposed adaptive operator selection paradigm significantly improves the performance of multi-objective optimization algorithms. The AdaW-DDQN algorithm provides an effective and efficient solution for solving multi-objective optimization problems. • An adaptive operator selection paradigm based on dueling deep Q-network is proposed. • Ten effective crossover operators are designed as a pool of candidate operators. • A new state space and reward mechanism are designed for the multi-objective optimization problem. • The proposed adaptive operator selection paradigm is embedded into MOEA/D with adaptive weights. • Experimental results show that AdaW-DDQN has strong generalization ability. [ABSTRACT FROM AUTHOR]

In multi-objective optimization, the Pareto-optimal solutions must be widely and uniformly distributed on the Pareto-optimal front. When this condition is satisfied, decision makers can choose satisfactory solutions. In this paper, gaps in the evolving nondominated front are filled by a multi-objective evolutionary algorithm using a uniformly evolving scheme (MOEA/UE), thus generating a uniform and wide Pareto-optimal front. The MOEA/UE first calculates the distance between the points on the nondominated front and identifies those with relatively wide gaps. At the widely spaced points, the objective and constraint functions are approximated by linear functions expressed as Taylor series in auxiliary linear programs. Finally, the MOEA/UE solves these auxiliary linear programs and retains the optimal or intermediate solutions as the potentially superior individuals that will be selected in the next generation of the population. Convergence toward potentially superior solutions is accelerated by a mutation operator (a basic differential evolution operator). In constrained multi-objective optimization problems, the MOEA/UE implements a new constraint-handling strategy that pulls infeasible solutions into or nearby the feasible domain. Experimental results illustrate that MOEA/UE obtains wider and more uniform Pareto-optimal fronts and better Pareto-optimal solutions. [ABSTRACT FROM AUTHOR]

The weapon-target assignment problem is a challenging optimization issue, but reliability is seldom considered in the majority of existing literature. To address the high-reliability weapon-target assignment problem, this paper integrates weapon reliability and mission reliability into a multi-objective optimization model (MOD) and presents an improved algorithm termed MOEA/D-iAM2M to the problem. This algorithm effectively combines the strengths of adaptive search space decomposition-based MOEA (MOEA/D-AM2M) and two-stage hybrid learning-based MOEA (HLMEA), resulting in a faster convergence rate and a more extensive distribution of the Pareto solution set. Furthermore, a reference point is incorporated into MOEA/D-iAM2M to facilitate the adaptive weight adjustment. Numerical experiments are carried out to confirm the effectiveness of the proposed MOEA/D-iAM2M. This research is significant in the field of optimization and has practical value in the defense industry. [ABSTRACT FROM AUTHOR]

Support vector machine (SVM) is a popular machine learning method with a solid theoretical foundation, and has shown promising performance on different classification problems. However, it suffers from an expensive training cost, which makes it not be very suitable for the application with a large-scale training set. To this end, as a data pre-processing technique, training set selection (TSS) for SVM has received much attention recently, since it can reduce the size of SVM training set without degrading the performance. In this paper, a subregion division based multi-objective evolutionary algorithm termed SDMOEA-TSS is proposed for SVM training set selection, where objective space is divided into several subregions for effectively searching good solutions. Specifically, in SDMOEA-TSS, a divided based initialization strategy is firstly suggested to initialize the population to locate in different regions of objective space. Then a subregion based evolutionary (including crossover, mutation and update) strategy is developed, which not only makes full use of individuals in each subregion for local search but also maintains the whole population's global search ability. Empirical studies on 21 public data sets demonstrate the superiority of the proposed algorithm over the state-of-the-arts in terms of both quality and diversity of the selected SVM training subset. [ABSTRACT FROM AUTHOR]

In software engineering, optimal feature selection for software product lines (SPLs) is an important and complicated task, involving simultaneous optimization of multiple competing objectives in large but highly constrained search spaces. A feature model is the standard representation of features of all possible products as well as the relationships among them for an SPL. Recently, various multi-objective evolutionary algorithms have been used to search for valid product configurations. However, the issue of the balance between correctness and diversity of solutions obtained in a reasonable time has been found very challenging for these algorithms. To tackle this problem, this paper proposes a novel aggregation-based dominance (ADO) for Pareto-based evolutionary algorithms to direct the search for high-quality solutions. Our method was tested on two widely used Pareto-based evolutionary algorithms: NSGA-II and SPEA2+SDE and validated on nine different SPLs with up to 10,000 features and two real-world SPLs with up to 7 objectives. Our experiments have shown the effectiveness and efficiency of both ADO-based NSGA-II and SPEA2+SDE: (1) Both algorithms could generate 100% valid solutions for all feature models. (2) The performance of both algorithms was improved as measured by the hypervolume metric in 7/9 and 8/9 feature models. (3) Even for the largest tested feature model with 10,000 features, it required under 40 s on a standard desktop to find 100% valid solutions in a single run of both algorithms. [ABSTRACT FROM AUTHOR]

Highlights • A succinct and efficient Opposition-Based Learning strategy is inherited in the algorithm. • A novel efficient strategy called Global Grid Ranking is proposed. • A charming multi-objective optimization algorithm called MOWOA is demonstrated. Abstract Nature-inspired computing has attracted a lot of research effort especially for addressing real-world multi-objective optimization problem (MOP). This paper proposes a new nature-inspired optimization algorithm which is named opposition-based multi-objective whale optimization algorithm with global grid ranking (MOWOA). The proposed approach utilizes several parts to enhance the performance in optimization. First, the efficient evolution process is inherited from the single objective whale optimization algorithm(WOA). Second, opposition-based learning(OBL) is applied into the algorithm. Meanwhile, a novel mechanism called global grid ranking(GGR) which is inspired by grid mechanism has been incorporated into the proposed algorithm. To show the significance of the proposed algorithm, MOWOA is tested on a diverse set of benchmark with a series of well-known evolutionary algorithms and the influence of each individual strategy is also verified through 14 benchmarks. Moreover, the new proposed algorithm is also applied to the simple data clustering problem and a real-world water optimization problem in China. The results demonstrate that MOWOA is not only an algorithm with well performance for bench-mark problems but also expected to have a more wide application in real-world engineering problems. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy set is a useful tool to handle the uncertainty in data. In order to deal with the uncertainty in an image and meanwhile overcome the sensitivity to image noise, a multi-objective evolutionary intuitionistic fuzzy clustering algorithm with multiple image spatial information (MOEIFC-MSI) is proposed to perform image segmentation in this paper. The important innovations of the method are listed as follows: (1) an intuitionistic fuzzy set of the image is first constructed by using a generalized fuzzy complement function; (2) this intuitionistic fuzzy set of the image is then utilized to compute fitness functions and a fuzzy evaluation index for selecting the optimal solution; (3) two kinds of complementary image spatial information are also introduced into the fitness functions and the fuzzy evaluation index to make the proposed method robust to image noise; (4) a real-coded variable string length technique is utilized to encode the cluster centers to automatically determine the number of clusters. Experimental results on synthetic, Berkeley and magnetic resonance (MR) images show that the proposed method outperforms state-of-the-art methods in noise robustness and segmentation performance. [ABSTRACT FROM AUTHOR]

This paper proposes ELMOEA/D, a surrogate-assisted MOEA, for solving costly multi-objective problems in small evaluation budgets. The proposed approach encompasses a state-of-the-art MOEA based on decomposition and Differential Evolution (MOEA/D-DE) assisted by Extreme Learning Machines (ELMs). ELMOEA/D is tested in instances from three well-known benchmarks (ZDT, DTLZ and WFG) with 5–60 decision variables, 2 and 5 objectives. The ELMOEA/D׳s performance is also analyzed on a real problem (Airfoil Shape Optimization). The impact of some ELMs parameters and an automatic model selection mechanism is investigated. The results obtained by ELMOEA/D are compared with those of two state-of-the-art surrogate approaches (MOEA/D-RBF and ParEGO) and a non-surrogate-based MOEA (MOEA/D). The ELMOEA/D variants are among the best results for most benchmark instances and for the real problem. [ABSTRACT FROM AUTHOR]

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) has attracted a great deal of attention and has obtained enormous success in the field of evolutionary multiobjective optimization. It converts a multiobjective optimization problem (MOP) into a set of scalar optimization subproblems and then uses the evolutionary algorithm (EA) to optimize these subproblems simultaneously. However, there is a great deal of randomness in MOEA/D. Researchers in the field of evolutionary algorithm, reinforcement learning and neural network have reported that the simultaneous consideration of randomness and opposition has an advantage over pure randomness. A new scheme, called opposition-based learning (OBL), has been proposed in the machine learning field. In this paper, OBL has been integrated into the framework of MOEA/D to accelerate its convergence speed. Hence, our proposed approach is called opposition-based learning MOEA/D (MOEA/D-OBL). Compared with MOEA/D, MOEA/D-OBL uses an opposition-based initial population and opposition-based learning strategy to generate offspring during the evolutionary process. It is compared with its parent algorithm MOEA/D on four representative kinds of MOPs and many-objective optimization problems. Experimental results indicate that MOEA/D-OBL outperforms or performs similar to MOEA/D. Moreover, the parameter sensitivity of generalization opposite point and the probable to use OBL is experimentally investigated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new multi-objective evolutionary algorithm-based ensemble optimizer coupled with neural network models for undertaking feature selection and classification problems. Specifically, the Modified micro Genetic Algorithm (MmGA) is used to form the ensemble optimizer. The aim of the MmGA-based ensemble optimizer is two-fold, i.e. to select a small number of input features for classification and to improve the classification performances of neural network models. To evaluate the effectiveness of the proposed system, a number of benchmark problems are first used, and the results are compared with those from other methods. The applicability of the proposed system to a human motion detection and classification task is then evaluated. The outcome positively demonstrates that the proposed MmGA-based ensemble optimizer is able to improve the classification performances of neural network models with a smaller number of input features. [Copyright &y& Elsevier]

Abstract: This paper describes a novel Bacterial Foraging Optimization (BFO) approach to multi-objective optimization, called Multi-objective Bacterial Foraging Optimization (MBFO). The objectives in the Multi-objective Bacterial Foraging Optimization are maintained by a fitness survive mechanism. Bacteria with the smaller health values have the better chance to survive. Meanwhile, the main goal of multi-objective optimization problems is to obtain a superior non-dominated front which is closed to the true Pareto front. With identification of such features, the idea of integration between health sorting approach and pareto dominance mechanism are developed to search for Pareto-optimal set of problems. Moreover, strategy keeping a certain unfeasible border solutions based on a given probability is considered to improve the diversity of individuals. In addition, two different performance metrics: Diversity and Generational Distance are introduced as well to evaluate multi-objective optimization problems. Compared to two other multi-objective optimization evolutionary algorithms MOPSO and NSGA-II, simulation results show that in most cases, the proposed MBFO is able to find a much better spread of solutions and convergence to the true Pareto-optimal front faster. It suggests that MBFO is very promising in dealing with ordinary multi-objective optimization problems. [Copyright &y& Elsevier]