Evolutionary Multiobjective Optimization Algorithms Research Articles

A repository of real-world datasets for data-driven evolutionary multiobjective optimization

Many real-world optimization applications have more than one objective, which are modeled as multiobjective optimization problems. Generally, those complex objective functions are approximated by expensive simulations rather than cheap analytic functions, which have been formulated as data-driven multiobjective optimization problems. The high computational costs of those problems pose great challenges to existing evolutionary multiobjective optimization algorithms. Unfortunately, there have not been any benchmark problems reflecting those challenges yet. Therefore, we carefully select seven benchmark multiobjective optimization problems from real-world applications, aiming to promote the research on data-driven evolutionary multiobjective optimization by suggesting a set of benchmark problems extracted from various real-world optimization applications.

A Framework to Handle Multimodal Multiobjective Optimization in Decomposition-Based Evolutionary Algorithms

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. While decomposition-based evolutionary algorithms have good performance for multi-objective optimization, they are likely to perform poorly for multi-modal multi-objective optimization due to the lack of mechanisms to maintain the solution space diversity. To address this issue, this paper proposes a framework to improve the performance of decomposition-based evolutionary algorithms for multi-modal multi-objective optimization. Our framework is based on three operations: assignment, deletion, and addition operations. One or more individuals can be assigned to the same subproblem to handle multiple equivalent solutions. In each iteration, a child is assigned to a subproblem based on its objective vector, i.e., its location in the objective space. The child is compared with its neighbors in the solution space assigned to the same subproblem. The performance of improved versions of six decomposition-based evolutionary algorithms by our framework is evaluated on various test problems regarding the number of objectives, decision variables, and equivalent Pareto optimal solution sets. Results show that the improved versions perform clearly better than their original algorithms.

Explicit Control of Implicit Parallelism in Decomposition-Based Evolutionary Many-Objective Optimization Algorithms [Research Frontier

Over the past two decades, Evolutionary Multi-objective Optimization (EMO) algorithms have demonstrated their ability to find and maintain multiple trade-off solutions in two and three-objective problems, making EMO as one of the most emergent and exciting fields of research and application within Computational Intelligence (CI) area. The main reason for EMO's success is that the population-based EMO operators are able to establish an implicit parallel search within an evolving population to find multiple Pareto-optimal regions of the search space parallelly. For many-objective optimization problems involving a largedimensional objective space, the extent of implicit parallelism is argued here to be too generic, compared to the same in a lower-dimensional objective space. Decomposition-based EMO algorithms - a recent trend in EMO literature - which divide the overall computing task into a number of sub-tasks of focusing within a region of the search space have found to be successful in solving many-objective problems. In this paper, we study the effect of explicit control of an algorithm's implicit parallelism mechanism for achieving an enhanced performance of decomposition-based EMO algorithms. We consider three decomposition-based many-objective evolutionary algorithms (EAs) - MOEA/D, MOEA/D-M2M, and NSGA-III - for this purpose. We also investigate another explicit control strategy of suitably choosing a normalization method of objectives for improving the performance of MOEA/D and MOEA/ D-M2M methods, and report much improved performance than their original counterparts. The principles of this study are valid for any population-based search and optimization algorithms and can be extended to improve the performance other single-objective EA, EMO, and other relevant CI methods.

An improved evolutionary multi-objective optimization algorithm based on decomposition

Multi-objective optimization (MOP) is a hot and difficult problem in the field of industrial production and scientific research, involving simultaneous optimization of multiple objectives. In this paper, an improved decomposition-based multi-objective evolutionary algorithm is proposed to solve MOP. Firstly, the Latin hypercube sampling method is used to generate the initial population in order to improve the diversity of the initial population and the probability of convergence to the global optimal solution. Secondly, the adaptive differential crossover operator is used to adjust the parameter settings adaptively for different problems, including the differential proportionality factor and crossover probability, so as to improve the performance of the algorithm. Finally, the reference point is set to coordinate origin, which can pull the population back to the real front faster and accelerate the convergence speed of the population. In a group of benchmark problems, ZDT and DTLZ test problems, the algorithm is widely compared with four most advanced algorithms. The experimental results show that the algorithm has better ability to solve MOP problems.

Multi-scenario microgrid optimization using an evolutionary multi-objective algorithm

Multi-scenario microgrid optimization arises regularly in real life. It refers to finding optimal scheduling strategies of a microgrid under multiple scenarios where each scenario corresponds to a specific working condition. For example, in an industrial park, there are often many users with different load demands. We need to efficiently find the optimal scheduling strategies for all users. The easiest way is to conduct the operation search for each user separately, which however, is obviously inefficient. Inspired by the underlying parallelism of evolutionary multi-objective optimization (EMO), this study proposes to optimize all scenarios simultaneously, i.e., finding the optimal scheduling strategies for all users in a single algorithm run. Specifically, the multi-scenario microgrid optimization problem is transformed into a bi-objective problem in which one objective is to minimize the number of scenarios and the other is to minimize the overall cost of the microgrid. The bi-objective problem is then solved by a typical EMO algorithm. The obtained Pareto optimal solutions correspond to the optimal scheduling strategies for different scenarios. Experimental results show that the proposed method is both effective and efficient, and can find more appropriate scheduling strategies than dealing with each scenario individually.

An R2 indicator and weight vector-based evolutionary algorithm for multi-objective optimization

A two-stage R2 indicator-based evolution algorithm (TS-R2EA) was proposed in the recent years. A good balance between convergence and diversity can be achieved, due to the R2 indicator and reference vector-guided selection strategy. However, TSR2-EA is sensitive to problem geometries. In order to address this issue, a weight vector-based selection strategy is introduced, and a weight vector adaptive strategy based on population partition is proposed. In the selection strategy, each candidate solution is ranked according to the scalarizing function values in the corresponding neighbor, and the candidate solutions with good performance can be selected. In the adaptive strategy, the population is partitioned by associating each individual with its closest weight vector, and the weight vectors with a worse performance are adjusted. Similar to TS-R2EA, these strategies are combined with the R2 indicator to solve multi-objective optimization problems. The performance of proposed algorithm has been validated and compared with four related algorithms on a variety of benchmark test problems. The experimental results have demonstrated that the proposed algorithm has high competition and is less sensitive to problem geometries.

Evolutionary Multiobjective Optimization With Robustness Enhancement

Uncertainty is an important feature abstracted from real-world applications. Multiobjective optimization problems (MOPs) with uncertainty can always be characterized as robust MOPs (RMOPs). Over recent years, multiobjective optimization evolutionary algorithms (EAs) have demonstrated the success in solving MOPs. However, most of them do not consider disturbance in the design. In order to handling the uncertainty in the optimization problem, we first give a thorough analysis of three important issues on robust optimization. Then, a novel EA called multiobjective optimization EA with robustness enhancement is developed, where the seamless integration of robustness and optimality is achieved by a proposed novel archive updating mechanism applied on the evolutionary process as well as the new robust optimal front building strategy designed to construct the final robust optimal front. Furthermore, the new designed archive updating mechanism makes the robust optimization process free of the enormous computational workload induced from sampling. The experimental results on a set of benchmark functions show the superiority of the proposed design in terms of both solutions' quality under the disturbance and computational efficiency in solving RMOPs.

Fast and insightful bi-objective optimization for prostate cancer treatment planning with high-dose-rate brachytherapy

Purpose:Prostate high-dose-rate brachytherapy (HDR-BT) planning involves determining the movement that a high-strength radiation stepping source travels through the patient’s body, such that the resulting radiation dose distribution sufficiently covers tumor volumes and safely spares nearby healthy organs from radiation risks. The Multi-Objective Real-Valued Gene-pool Optimal Mixing Evolutionary Algorithm (MO-RV-GOMEA) has been shown to be able to effectively handle this inherent bi-objective nature of HDR-BT planning. However, in clinical practice there is a very restricted planning time budget (often less than 1 h) for HDR-BT planning, and a considerable amount of running time needs to be spent before MO-RV-GOMEA finds a good trade-off front of treatment plans (about20–30 min on a single CPU core) with sufficiently accurate dose calculations, limiting the applicability of the approach in the clinic. To address this limitation, we propose an efficiency enhancement technique for MO-RV-GOMEA solving the bi-objective prostate HDR-BT planning problem. Methods:Dose-Volume (DV) indices are often used to assess the quality of HDR-BT plans. The accuracy of these indices depends on the number of dose calculation points at which radiation doses are computed. These are randomly uniformly sampled inside target volumes and organs at risk. In available HDR-BT planning optimization algorithms, the number of dose calculation points is fixed. The more points are used, the better the accuracy of the obtained results will be, but also the longer the algorithms need to be run. In this work, we introduce a so-called multi-resolution scheme that gradually increases the number of dose calculation points during the optimization run such that the running time can be substantially reduced without compromising on the accuracy of the obtained results. Results and conclusion:Experiments on a data set of 18 patient cases show that with the multi-resolution scheme, MO-RV-GOMEA can achieve a sufficiently good trade-off front of treatment plans after five minutes of running time on a single CPU core (4–6 times faster than the old approach with a fixed number of dose calculation points). When the optimization with the multi-resolution scheme is run on a quad-core machine, five minutes are enough to obtain trade-off fronts that are nearly as good as those obtained by running optimization with the old approach in one hour (i.e., 12 times faster). This leaves ample time to perform the selection of the preferred treatment plan from the trade-off front for the specific patient at hand. Furthermore, comparisons with real clinical treatment plans, which were manually made by experienced BT planners within 30–60 min, confirm that the plans obtained by our approach are superior in terms of DV indices. These results indicate that our proposed approach has the potential to be employed in clinical practice.

A Generator for Multiobjective Test Problems With Difficult-to-Approximate Pareto Front Boundaries

In some real-world applications, it has been found that the performance of multiobjective optimization evolutionary algorithms (MOEAs) may deteriorate when boundary solutions in the Pareto front (PF) are more difficult to approximate than others. Such a problem feature, referred to as difficult-to-approximate (DtA) PF boundaries, is seldom considered in existing multiobjective optimization test problems. To fill this gap and facilitate possible systematic studies, we introduce a new test problem generator. The proposed generator enables the design of test problems with controllable difficulties regarding the feature of DtA PF boundaries. Three representative MOEAs, NSGA-II, SMS-EMOA, and MOEA/D-DRA, are performed on a series of test problems created using the proposed generator. Experimental results indicate that all the three algorithms perform poorly on the new test problems. Meanwhile, a modified variant of MOEA/D-DRA, denoted as MOEA/D-DRA-UT, is validated to be more effective in dealing with these problems. Subsequently, it is concluded that the rational allocation of computational resources between different PF parts is crucial for MOEAs to handle the problems with DtA PF boundaries.

Precision animal feed formulation: An evolutionary multi-objective approach

Most livestock producers aim for optimal ways of feeding their animals. Conventional algorithms approach optimum feed formulation by minimizing feed costs while satisfying constraints related to nutritional requirements of the animal. The optimization process needs to be performed every time a nutritional requirement is changed due to the nonlinear relationship between the relaxation of the different nutritional requirements and the feed cost. Consequently, decision-making becomes a time-consuming trial and error process. In addition, the nonlinear relationship changes depending on the type of materials used, their nutritional compositions and costs as well as the animal’s nutritional requirements. Therefore, in this work, we formulated a multi-objective feed formulation problem comprising of two objects – a) minimizing feed cost and b) minimizing deviation from the specified requirements. The problem is solved using a population-based evolutionary multi-objective optimization algorithm (NSGA-II) that results in an optimal set of comprised solutions in a single run. The availability of the entire set of comprised solutions facilitates the understanding of the relationship between different nutritional requirements and cost, thus leading to a more efficient decision-making process. We demonstrated the applicability of the proposed method by performing experimental simulations on several cases of dairy and beef cattle feed formulation.

Thermal optimization of a kirigami-patterned wearable lithium-ion battery based on a novel design of composite phase change material

In this study, the thermal behavior of a stretchable lithium-ion battery following kirigami pattern is investigated and optimized online based on the finite element method. A novel distribution of expanded graphite/paraffin composite phase change material units, which is suitable for this unique battery, is proposed to reduce the temperature rise and improve the temperature uniformity at a high discharge rate of 5C. Before the optimization, a high dimensional modelling representation based global sensitivity analysis is performed to filter unimportant design variables to thermal management system, and a high quality metamodel is obtained by back propagation neural network. Instead of using trial-and-error method, the optimization is performed online by using different evolutionary multi-objective optimization algorithms, and the result obtained by strength Pareto evolutionary algorithm-II is shown to be the best. This optimal result shows that the thermal management system with optimized design exhibits great thermal performance. This study provides a guide for the design of battery thermal management of stretchable batteries.

An adaptive weight vector guided evolutionary algorithm for preference-based multi-objective optimization

Recently, multi-objective evolutionary algorithms (MOEAs) have been widely explored and applied to many real-world problems. Particularly, preference-based MOEAs are among the mostly studied. Several preference-based optimization algorithms have already been proposed in literature. However, most existing studies focus on how to locate the region of interest (ROI) and how to control the size of ROI, and overlook the relationship between preference information and distribution of the final solutions. Given that the distribution of the final solutions is also an important factor, in this paper, we propose a new preference-based MOEAs called MOEA/D-AWV using an adaptive weight vector generation strategy (AWV). The weight vectors are generated adaptively by the decision maker’s preference, and finally guide the solutions to converge to a preferred distribution. Solutions will converge to the reference point as close as possible with the AWV strategy, which will lead to the loss of diversity. In order to prevent the search process from being trapped at local optima, we propose an adaptive parameter tuning scheme (APT) to maintain diversity during the search process. In addition, since the distribution of weight vectors should adapt to the changes of decision makers’ preference, the APT scheme can help algorithm find desired results in different scenarios. Compared with five state-of-the-art preference-based MOEAs on 23 test instances, MOEA/D-AWV achieves the best performance. Especially, in many-objective optimization problems with high-dimensional objective space, our proposed MOEA/D-AWV still shows a competitive performance.

Efficient computation of expected hypervolume improvement using box decomposition algorithms

In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms. MOBGO utilizes Gaussian Process models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A commonly used criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far, it has been a challenge to calculate exact EHVI values efficiently. This paper proposes an efficient algorithm for the exact calculation of the EHVI for in a generic case. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities can be improved from previously O (n^2) and O(n^3), for two- and three-objective problems respectively, to varTheta (nlog n), which is asymptotically optimal. This article generalizes the scheme in higher dimensional cases by utilizing a new hyperbox decomposition technique, which is proposed by Dächert et al. (Eur J Oper Res 260(3):841–855, 2017). It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI).

Sensitivity Analysis of Evolutionary Algorithm for Software Reusability

Fast and competitive software industry demands rapid development using Component Based Software Development (CBSD). CBSD is dependent on the availability of the high-quality reusable component libraries. Recently, evolutionary multi-objective optimization algorithms have been used to identify sets of reusable software components from the source-code of Object Oriented (OO) software, using different quality indicators (e.g. cohesion, coupling, etc.). Sometimes, these used quality indicators are quite sensitive towards the small variations in their values, although they should not be. Therefore, this paper analyzes the sensitivity of the evolutionary technique for three quality indicators used during the identification: Frequent Usage Pattern (FUP), Semantic and evolutionary coupling. The sensitivity analysis is performed on three widely used open-source OO software. The experimentation is performed by mutating the system to different degrees. Results of the empirical analysis indicate that the semantic parameter is most sensitive and important. Ignoring this feature highly degrades the quality; FUP relation is uniformly sensitive and evolutionary relations's sensitivity is non-uniform.

On redundant coverage maximization in wireless visual sensor networks: Evolutionary algorithms for multi-objective optimization

Wireless visual sensor networks can provide valuable information for a variety of monitoring and control applications. Frequently, a set of targets must be covered by visual sensors, as such visual sensing redundancy is a desired condition specially when applications have availability requirements for multiple coverage perspectives. If visual sensors become rotatable, their sensing orientations can be adjusted to optimize coverage and redundancy, bringing different challenges as there may be different coverage optimization objectives. Actually, the specific issue of redundant coverage maximization is inherently a multi-objective problem, but usual approaches are not designed accordingly to compute visual sensing redundancy. This article proposes two different evolutionary algorithms that exploit the multi-objective nature of the redundant coverage maximization problem: a lexicographic ”a priori” algorithm and a NSGA-II ”a posteriori” algorithm. The performance of both algorithms are compared, using a previously proposed single-objective greedy-based algorithm as a reference. Numerical results outline the benefits of employing evolutionary algorithms for adjustments of sensors’ orientations, potentially benefiting deployment and management of wireless visual sensor networks for different monitoring scenarios.

A niching indicator-based multi-modal many-objective optimizer

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. Some evolutionary algorithms for multi-modal multi-objective optimization have been proposed in the literature. However, there is no efficient method for multi-modal many-objective optimization, where the number of objectives is more than three. To address this issue, this paper proposes a niching indicator-based multi-modal multi- and many-objective optimization algorithm. In the proposed method, the fitness calculation is performed among a child and its closest individuals in the solution space to maintain the diversity. The performance of the proposed method is evaluated on multi-modal multi-objective test problems with up to 15 objectives. Results show that the proposed method can handle a large number of objectives and find a good approximation of multiple equivalent Pareto optimal solutions. The results also show that the proposed method performs significantly better than eight multi-objective evolutionary algorithms.

Learning to Decompose: A Paradigm for Decomposition-Based Multiobjective Optimization

The decomposition-based evolutionary multiobjective optimization (EMO) algorithm has become an increasingly popular choice for a posteriori multiobjective optimization. However, recent studies have shown that their performance strongly depends on the Pareto front (PF) shapes. This can be attributed to the decomposition method, of which the reference points and subproblem formulation settings are not well adaptable to various problem characteristics. In this paper, we develop a learning-to-decompose (LTD) paradigm that adaptively sets the decomposition method by learning the characteristics of the estimated PF. Specifically, it consists of two interdependent parts, i.e., a learning module and an optimization module. Given the current nondominated solutions from the optimization module, the learning module periodically learns an analytical model of the estimated PF. Thereafter, useful information is extracted from the learned model to set the decomposition method for the optimization module: 1) reference points compliant with the PF shape and 2) subproblem formulations whose contours and search directions are appropriate for the current status. Accordingly, the optimization module, which can be any decomposition-based EMO algorithm in principle, decomposes the multiobjective optimization problem into a number of subproblems and optimizes them simultaneously. To validate our proposed LTD paradigm, we integrate it with two decomposition-based EMO algorithms, and compare them with four state-of-the-art algorithms on a series of benchmark problems with various PF shapes.

A Scalable Indicator-Based Evolutionary Algorithm for Large-Scale Multiobjective Optimization

The performance of traditional multiobjective evolutionary algorithms (MOEAs) often deteriorates rapidly as the number of decision variables increases. While some efforts were made to design new algorithms by adapting existing techniques to large-scale single-objective optimization to the MOEA context, the specific difficulties that may arise from large-scale multiobjective optimization have rarely been studied. In this paper, the exclusive challenges along with the increase of the number of variables of a multiobjective optimization problem (MOP) are examined empirically, and the popular benchmarks are categorized into three groups accordingly. Problems in the first category only require MOEAs to have stronger convergence, and can thus be mitigated using techniques employed in large-scale single-objective optimization. Problems that require MOEAs to have stronger diversification but ignore a correlation between position and distance functions are grouped as the second. The rest of the problems that pose a great challenge to the balance between diversification and convergence by considering a correlation between position and distance functions are grouped as the third. While existing large-scale MOEAs perform well on the problems in the first two categories, they suffer a significant loss when applied to those in the third category. To solve large-scale MOPs in this category, we have developed a novel indicator-based algorithm with an enhanced diversification mechanism. The proposed algorithm incorporates a new solution generator with an external archive, thus forcing the search toward different subregions of the Pareto front using a dual local search mechanism. The results obtained by applying the proposed algorithm to a wide variety of problems (108 instances in total) with up to 8192 variables demonstrate that it outperforms eight state-of-the-art approaches on the examined problems in the third category and show its advantage in the balance between diversification and convergence.

Dual-information-based evolution and dual-selection strategy in evolutionary multiobjective optimization

Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously in a collaborative manner in one run. The recently proposed stable matching (STM)-based selection is a variant of MOEA/D that achieves one-to-one STM between subproblems and solutions on the basis of mutual preferences. However, the STM has a high probability of matching a good convergence solution with a subproblem, which results in an imbalance between convergence and diversity of selection result. In this study, we propose a new variant of MOEA/D with dual-information and dual-selection (DS) strategy (MOEA/D-DIDS). Different from other evolutionary operations, we use an adaptive historical and neighboring information in generating new individuals to avoid local optima and accelerate convergence rate. In the selection operation, we use the adaptive limited STM ($$ \beta {\text{LSTM}} $$) strategy, where parameter β is adaptive in accordance with the evolutionary process, as a guideline to select a population from the mixed population that survives as the next parent population. In addition to $$ \beta {\text{LSTM}} $$, we use an STM to select competitive individuals as the members of the next mixed population. This DS strategy not only balances convergence and diversity but also holds the elite solutions. The effectiveness and competitiveness of MOEA/D-DIDS are validated and compared with several state-of-the-art evolutionary multiobjective optimization algorithms on benchmark problems.

A dynamic multi-objective optimization model with interactivity and uncertainty for real-time reservoir flood control operation

• A dynamic multi-objective model is developed for reservoir flood control. • The proposed model considers the environment uncertainty and the interactivity. • An algorithmic framework of optimizers for the proposed model is proposed. • Two change reaction strategies are developed for adapting the environment change. Despite the successes of both multi-objective optimization and uncertainty handling techniques in reservoir flood control operation, no work has been done yet on developing and investigating dynamic multi-objective optimization models for this problem. In this work, a dynamic multi-objective optimization model with interactivity and uncertainty was developed for the real-time reservoir flood control operation. Accordingly, a dynamic multi-objective optimization algorithmic framework with two newly designed change reaction strategies was proposed for solving the proposed dynamic model. Following the proposed algorithmic framework, any evolutionary multi-objective optimization algorithm can be converted into a dynamic optimizer. After investigating the difficulty variation of the proposed dynamic model, the effectiveness and robustness of the proposed algorithmic framework have been validated based on experiential studies on two typical floods of Ankang reservoir.