Dynamic Multi-objective Optimization Problems Research Articles

A weighted knowledge extraction strategy for dynamic multi-objective optimization

Multi-objective evolutionary algorithms suffer from performance degradation when solving dynamic multi-objective optimization problems (DMOPs) with a new conditional configuration from scratch, which motivates the research on knowledge extraction. However, most knowledge extraction strategies only focus on obtaining effective information from a single knowledge source, while ignoring the useful information from other knowledge sources with similar properties. Motivated by this, a weighted multi-source knowledge extraction strategy-based dynamic multiobjective evolutionary algorithm is proposed. First, a similarity criterion based on angle information is constructed to quantify similarity between different source domains and the target domain. Second, a knowledge extraction technique is developed to select a specific number of individuals from each source domain using a distance metric. Third, a generation strategy based on dynamic weighting mechanism is proposed, which generates a certain number of individuals and merges these individuals into the initial population within the new environment. Finally, the comprehensive experiments are conducted on public DMOP benchmarks and demonstrate the devised method significantly outperforms the state-of-the-art competing algorithms.

Dynamic multiobjective optimization via an improved r-dominance relation and a novel prediction approach

When decision makers are only interested in a portion of the Pareto optimal front (PF) in dynamic multiobjective optimization problems (DMOPs), dynamic multiobjective evolutionary algorithms (DMOEAs) need to search only for the PF portion of interest to decision makers. However, this is challenging for most existing DMOEAs, as they are designed to search the entire PF, overlooking the preferences of decision makers. Therefore, we present a novel dynamic multiobjective optimization algorithm based on decision makers’ preference information, involving an improved r-dominance relation and a response strategy. It focuses on searching for the preference solutions (the region of interest) according to the decision makers’ preference information in dynamic multiobjective optimization. The improved r-dominance relation adapts the angle to measure the closeness between the solution and the preference information, which solves the convergence problem of the r-dominance relation since the original r-dominance relation has difficulty converging to the true PF when the preference information is located in the feasible objective region. The prediction mechanism is based on the movement of the population’s special points; it helps the population make adjustments in its moving direction and step size towards the new PF when a change is detected. Experimental results show that the proposed algorithm is efficient for dynamic multiobjective optimization and is competitive compared to state-of-the-art methods.

Dynamic multi-objective optimization based on classification response of decision variables

In recent years, many dynamic multi-objective optimization algorithms (DMOAs) have been proposed to address dynamic multi-objective optimization problems (DMOPs). Most existing DMOAs treat all decision variables uniformly and respond to them in an identical manner. This paper proposes a dynamic multi-objective optimization algorithm based on the classification response of decision variables (CRDV-DMO). Firstly, CRDV-DMO categorizes the decision variables into convergence variables and diversity variables. Different decision variables adopt distinct response strategies. The response strategy of diversity variable (RSDV) uses Latin hypercube sampling to generate the diversity variables of the new environment. For each dimensional convergence variable, the response strategy of convergence variable (RSCV) first evaluates whether the basic center prediction strategy (CPS) yields positive feedback or negative feedback, further determining the predictability of that dimensional convergence variable. RSCV then decides to either use the basic CPS to generate the convergence variable for that dimension or to retain that dimensional convergence variable from the current environment, based on the predictability of that dimensional convergence variable. The proposed algorithm is extensively studied through comparison with several advanced DMOAs, demonstrating its effectiveness in dealing with the benchmark DMOPs and the parameter-tuning problem of the PID controller on a dynamic system.

Dynamic Multiobjective Optimization Based on Multi-Environment Knowledge Selection and Transfer

Background: Dynamic multiobjective optimization problems (DMOPs) involve multiple conflicting and time-varying objectives, and dynamic multiobjective algorithms (DMOAs) aim to find Pareto optima that are closer to the real one in the new environment as soon as possible. In particular, the introduction of transfer learning in DMOAs has led to good results in solving DMOPs. However, the selection of valuable historical knowledge and the mitigation of negative transfer remain important problems in existing transfer learning-based DMOAs. Method: A DMOA based on multi-environment knowledge selection and transfer (MST-DMOA) is proposed in this article. First, by clustering historical Pareto optima, some representative solutions that can reflect the main evolutionary information are selected as knowledge of the environment. Second, the similarity between the historical and current environments is evaluated, and then the knowledge of multiple similar environments is selected as valuable historical knowledge to construct the source domain. Third, solutions with high quality in the new environment are obtained to form the target domain, which can better help historical knowledge to adapt to the current environment, thus effectively alleviating negative transfer. Conclusions: We compare the proposed MST-DMOA with five state-of-the-art DMOAs on fourteen benchmark test problems, and the experimental results verify the excellent performance of MST-DMOA in solving DMOPs.

A Novel Dynamic Multiobjective Optimization Algorithm With Non-Inductive Transfer Learning Based on Multi-Strategy Adaptive Selection.

In this article, a novel multi-strategy adaptive selection-based dynamic multiobjective optimization algorithm (MSAS-DMOA) is proposed, which adopts the non-inductive transfer learning (TL) paradigm to solve dynamic multiobjective optimization problems (DMOPs). In particular, based on a scoring system that evaluates environmental changes, the source domain is adaptively constructed with several optional groups to enrich the knowledge. Along with a group of guide solutions, the importance of historical experiences is estimated via the kernel mean matching (KMM) method, which avoids designing strategies to label individuals. The proposed MSAS-DMOA is comprehensively evaluated on 14 DMOPs, and the results show an overwhelming performance improvement in terms of both convergence and diversity as compared with other four popular DMOAs. In addition, ablation studies are also conducted to validate the superiority of the applied strategies in MSAS-DMOA, which can effectively alleviate the negative transfer phenomenon. Without the conventional labeling procedure, the proposed method also yields satisfactory results, which can provide valuable reference for designing other evolutionary transfer optimization (ETO) algorithms.

Integrating machine learning with dynamic multi-objective optimization for real-time decision-making

Real-time decision-making in dynamic multi-objective optimization problems (DMOPs) is challenging due to constantly changing objectives and constraints. This paper integrates machine learning with Non-dominated Sorting Genetic Algorithm II (NSGA-II) to solve DMOPs and make real-time decisions. Learning-based methods have gained popularity for predicting solutions in new environments and capturing changing patterns in optimal solutions. However, existing approaches often struggle with training difficulty and reduced prediction accuracy due to irrelevant or redundant variables. Therefore, we introduce a new interdependent prediction (IDP) technique to identify correlations between variables and prediction targets and select significant variables for a predictive model. In this way, a better initial population is predicted. The IDP strategy is integrated within the dynamic NSGA-II, introducing a new algorithm called IDP-DNSGA-II. This integration facilitates rapid convergence, finding optimal or near-optimal solutions. The proposed method is evaluated against standard benchmarks, demonstrating superior performance in convergence speed and solution diversity with the changes in the problem environment. The IDP-DNSGA-II is validated through real-world optimization challenges in sustainable automobile production distribution in order-to-delivery systems to enhance environmental sustainability and operational efficiency. This study identifies the minimum frequency of change required in real-world problems to adequately track the optimal decision in real-time.

Learning to Guide Particle Search for Dynamic Multiobjective Optimization.

Dynamic multiobjective optimization problems (DMOPs) are characterized by multiple objectives that change over time in varying environments. More specifically, environmental changes can be described as various dynamics. However, it is difficult for existing dynamic multiobjective algorithms (DMOAs) to handle DMOPs due to their inability to learn in different environments to guide the search. Besides, solving DMOPs is typically an online task, requiring low computational cost of a DMOA. To address the above challenges, we propose a particle search guidance network (PSGN), capable of directing individuals' search actions, including learning target selection and acceleration coefficient control. PSGN can learn the actions that should be taken in each environment through rewarding or punishing the network by reinforcement learning. Thus, PSGN is capable of tackling DMOPs of various dynamics. Additionally, we efficiently adjust PSGN hidden nodes and update the output weights in an incremental learning way, enabling PSGN to direct particle search at a low computational cost. We compare the proposed PSGN with seven state-of-the-art algorithms, and the excellent performance of PSGN verifies that it can handle DMOPs of various dynamics in a computationally very efficient way.

The IGD-based prediction strategy for dynamic multi-objective optimization

In recent years, an increasing number of prediction-based strategies have shown promising results in handling dynamic multi-objective optimization problems (DMOPs), and prediction models are also considered to be very favorable. Nevertheless, some linear prediction models may not always be effective. In particular, when the motion direction trends of different individuals are not aligned, these models can yield inaccurate prediction results. Inverted generational distance (IGD) is a commonly used metric for evaluating the performance of algorithms. This paper proposes a prediction model based on the IGD metric. Specifically, we assume that the pareto optimal front (POF) of the population at the previous time step is the true POF, and the POF at the current time step is the approximate POF. We cluster the current population with reference to the euclidean distances from uniform points on the true POF to the current POF points, with slight overlap between adjacent clusters, enables a better tradeoff between convergence and diversity in the prediction process. We consider the movement directions of individuals within each cluster separately through different cluster distributions, while balancing the individual movement directions and the overall population movement direction by overlaying cluster coverage areas, thereby helping to avoid the clustered prediction population from getting trapped in local optima. Experimental results and comparisons with other algorithms demonstrate that this strategy exhibits strong competitiveness in handling DMOPs.

Fusion prediction strategy-based dynamic multi-objective sparrow search algorithm

Solving dynamic multi-objective optimization problems with time-varying Pareto front (PF) or Pareto set (PS) is a challenging task. Such problems require algorithms to react to environmental changes and efficiently track optimal solutions. For this purpose, a dynamic multi-objective sparrow search algorithm (SSA) with fusion prediction strategy, based on difference model and kernel extreme learning machine (DMOSSA-FPS), is proposed. Given the diversity of change characteristics, a single prediction model is insufficient. Therefore, based on the historical information of the population, a difference model and a kernel extreme learning machine are integrated for PS prediction. The former is used to predict the solutions of some individuals under approximate linear changes and the latter is employed for nonlinear predictions. In a new environment, the combined predictions increase the diversity of the initial population. Additionally, a new static optimizer is proposed, which combines decomposition- and dominance-based approaches to constitute a new individual screening mechanism. Then the optimization mode of SSA is introduced to enhance both algorithmic diversity and convergence rate. The experimental results on the DF test suite demonstrate that, compared with several other advanced algorithms, DMOSSA-FPS exhibits stronger convergence and robustness.

A new prediction-based evolutionary dynamic multiobjective optimization algorithm aided by Pareto optimal solution estimation strategy

Dynamic multiobjective optimization problems (DMOPs) typically involve multiple conflicting time-varying objectives that require optimization algorithms to quickly track the changing Pareto-optimal front (POF). To this end, several methods have been developed to predict new locations of moving Pareto-optimal solution set (POS) so that populations can be re-initialized around the predicted locations. In this paper, a dynamic multi-objective optimization algorithm based on a multi-directional difference model (MOEA/D-MDDM) is proposed. The multi-directional difference model predicts the initial population through the estimated populations developed by a designed POS estimation strategy. An adaptive crossover-rate approach is incorporated into the optimization process to cope with different POS structures. To investigate the performance of the proposed approach, MOEA/D-MDDM has been compared with six state-of-the-art dynamic multiobjective optimization evolutionary algorithms (DMOEAs) on 19 benchmark problems. The experimental results demonstrate that the proposed algorithm can effectively deal with DMOPs whose POS has a single-modality characteristic and continuous manifolds.

Hybrid response dynamic multi-objective optimization algorithm based on multi-arm bandit model

Dynamic multi-objective optimization is a relatively challenging problem within the field of multi-objective optimization. Nevertheless, these problems have significant real-world applications. The key to addressing dynamic multi-objective problems effectively is promptly tracking changes in the Pareto set (PS) and Pareto front (PF). Dynamic multi-objective optimization encompasses various types of problems, and a single-type response strategy proves effective for some specific scenarios. However, as problem complexity and diversity increase, a single-type response strategy often falls short in solving dynamic multi-objective optimization problems. To address this issue, this paper proposes a hybrid response dynamic multi-objective optimization algorithm. The suggested algorithm utilizes the multi-arm bandit model to adaptively adjust the proportion of different response strategies for each type of multi-objective optimization problem. Furthermore, it achieves rapid convergence through an enhanced two-stage MOEA/D. Experiments demonstrate the effectiveness of the strategies employed in the proposed algorithm and its competitiveness compared to other state-of-the-art algorithms.

A dynamic multi-objective evolutionary algorithm with variable stepsize and dual prediction strategies

The prediction strategy is a key method for solving dynamic multi-objective optimization problems (DMOPs), particularly the commonly used linear prediction strategy, which has an advantage in solving problems with regular changes. However, using the linear prediction strategy may have limited advantages in addressing problems with complex changes, as it may result in the loss of population diversity. To tackle this issue, this paper proposes a dynamic multi-objective optimization algorithm with variable stepsize and dual prediction strategies (VSDPS), which aims to maintain population diversity while making predictions. When an environmental change is detected, the variable stepsize is first calculated. The stepsize of the nondominated solutions is expressed by the centroid of the population, while the stepsize of the dominated solutions is determined by the centroids of the clustered subpopulations. Then, the dual prediction strategies combine an improved linear prediction strategy with a dynamic particle swarm prediction strategy to track the new Pareto-optimal front (PF) or Pareto-optimal set (PS). The improved linear prediction strategy aims to enhance the convergence of the population, while the dynamic particle swarm prediction strategy focuses on preserving the diversity of the population. There have also been some improvements made in the static optimization phase, which are advantageous for both population convergence and diversity. VSDPS is compared with six state-of-the-art dynamic multi-objective evolutionary algorithms (DMOEAs) on 28 test instances. The experimental results demonstrate that VSDPS outperforms the compared algorithms in most instances.

An online cooperative control strategy for enhancing microgrid participation into electricity market

This paper develops a cooperative control methodology for the online energy management of grid-connected microgrids. The main aim of this methodology is to actively manage the active power outputs of all dispatchable energy sources available within the microgrid as well as the power exchanged with the utility grid so as to match the total load demand at the minimum operating cost. This implies to solve a constrained multi-objective dynamic optimization problem aimed at minimizing the total microgrid operating costs and ensuring the real-time balance within the microgrid in compliance with its technical-operational constraints. Lyapunov's theorem using sensitivity theory is adopted to solve this optimization.To test the performances of the proposed control methodology, several computer simulations corresponding to different operating scenarios have been conducted on the PrInCE Lab experimental microgrid built at the Polytechnic University of Bari.

A Co-evolutionary Multi-population Evolutionary Algorithm for Dynamic Multiobjective Optimization

Dynamic multiobjective optimization problems (DMOPs) widely appear in various real-world applications and have attracted increasing attention worldwide. However, how to obtain both good population diversity and fast convergence speed to efficiently solve DMOPs are two challenging issues. Inspired by that the multiple populations for multiple objectives (MPMO) framework can provide algorithms with good population diversity and fast convergence speed, this paper proposes a new efficient algorithm called a co-evolutionary multi-population evolutionary algorithm (CMEA) based on the MPMO framework together with three novel strategies, which are helpful for solving DMOPs efficiently from two aspects. First, in the evolution control aspect, a convergence-based population evolution strategy is proposed to select the suitable population for executing the evolution in different generations, so as to accelerate the convergence speed of the algorithm. Second, in the dynamic control aspect, a multi-population-based dynamic detection strategy and a multi-population-based dynamic response strategy are proposed to help the algorithm maintain the population diversity, which are efficient for detecting and responding to the dynamic changes of environments. Integrating with the above strategies, the CMEA is proposed to solve the DMOP efficiently. The superiority of the proposed CMEA is validated in experiments on widely-used DMOP benchmark problems.

A dynamic multi-objective optimization evolutionary algorithm with adaptive boosting

Dynamic multi-objective optimization problems (DMOPs) are prevalent in the real world, where the challenge in solving DMOPs is how to track the time-varying Pareto-optimal front (PF) and Pareto-optimal set (PS) quickly and accurately. However, balancing convergence and diversity is challenging as a single strategy can only address a particular type of DMOP. To solve this issue, a dynamic multi-objective optimization evolutionary algorithm with adaptive boosting (AB-DMOEA) is proposed in this paper. In the AB-DMOEA, an adaptive boosting response mechanism will increase the weights of high-performing strategies, including those based on prediction, memory, and diversity, which have been improved and integrated into the mechanism to tackle various problems. Additionally, the dominated solutions reinforcement strategy optimizes the population to ensure the effective operation of the above mechanism. In static optimization, the static optimization boosting mechanism selects the appropriate static multi-objective optimizer for the current problem. AB-DMOEA is compared with the other seven state-of-the-art DMOEAs on 35 benchmark DMOPs. The comprehensive experimental results demonstrate that the overall performance of the AB-DMOEA is superior or comparable to that of the compared algorithms. The proposed AB-DMOEA is also successfully applied to the smart greenhouses problem.

A dynamic multi-objective evolutionary algorithm based on Mahalanobis distance and intra-cluster individual correlation rectification

Responding quickly and accurately to environmental changes is a challenge in addressing dynamic multi-objective optimization problems (DMOPs). Although many dynamic multi-objective evolutionary algorithms (DMOEAs) have demonstrated impressive performance, there is still room for improvement in accurately predicting population behavior. To address this limitation, this paper proposes a prediction strategy based on Mahalanobis distance and intra-cluster individual correlation rectification (MCIR) to deal with DMOPs. First, a manifold clustering method is used to partition the Pareto set of the population into subpopulations, in which individuals with similar movement trends are grouped into clusters. Second, the Mahalanobis distance is introduced to systematically measure the relationships between clusters in adjacent environments. Time series models are established for each cluster center to predict their positions in the neighboring environment. On this basis, the movement characteristics of individuals within clusters are further rectified by calculating the correlation between intra-cluster individuals and the cluster center, facilitating more accurate tracking of the changing Pareto set/Pareto front. Finally, Gaussian noise is introduced to ensure the diversity of new individuals. The effectiveness of the MCIR algorithm is demonstrated by comparing it with four DMOEAs using 18 test instances. Experimental results confirm that MCIR holds great promise in addressing DMOPs.

A Metaheuristic Framework with Experience Reuse for Dynamic Multi-Objective Big Data Optimization

Dynamic multi-objective big data optimization problems (DMBDOPs) are challenging because of the difficulty of dealing with large-scale decision variables and continuous problem changes. In contrast to classical multi-objective optimization problems, DMBDOPs are still not intensively explored by researchers in the optimization field. At the same time, there is lacking a software framework to provide algorithmic examples to solve DMBDOPs and categorize benchmarks for relevant studies. This paper presents a metaheuristic software framework for DMBDOPs to remedy these issues. The proposed framework has a lightweight architecture and a decoupled design between modules, ensuring that the framework is easy to use and has enough flexibility to be extended and modified. Specifically, the framework now integrates four basic dynamic metaheuristic algorithms, eight test suites of different types of optimization problems, as well as some performance indicators and data visualization tools. In addition, we have proposed an experience reuse method, speeding up the algorithm’s convergence. Moreover, we have implemented parallel computing with Apache Spark to enhance computing efficiency. In the experiments, algorithms integrated into the framework are tested on the test suites for DMBDOPs on an Apache Hadoop cluster with three nodes. The experience reuse method is compared to two restart strategies for dynamic metaheuristics.

Effective response strategies based on adaptive selection for dynamic multi-objective evolutionary optimization

Dynamic multiobjective optimization problems (DMOPs) change over time, which require Evolutionary algorithms (EA) track Pareto-optimal solution (PS) or/and Pareto-optimal front (PF) in a constantly change environment. Prediction-based algorithms are the most common method to solve DMOPs. However, a single elaborate predictor is not always suitable for extracting changing pattern of different DMOPs, and not to mention DMOPs with unpredictable changes. To overcome these limitations, a simple yet effective algorithm, response strategies based on adaptive selection (RSAS), are proposed in this paper. When a change occurs, RSAS provides diversified solutions by different proposed strategies, that is, a center-guided self-correcting prediction, an individual-based prediction, and a precision-controllable mutation. Based on the quality of their generated solution set, an adaptive selection mechanism can adjust the selection probability of these three strategies. Since RSAS consists of not merely two different prediction strategies but also a mutation strategy, which can be more responsive to predictable and unpredictable changes. To validate the performance of RSAS, DMOP benchmarks in CEC2018 with recommended configurations are adopted. Compared to four state-of-the-art algorithms, the experimental results show that RSAS is effective and efficient.

Dynamic multi-objective evolutionary algorithm based on decomposition with hybrid prediction

The proposed dynamic multi-objective evolutionary algorithm, DMOEA/D-HP, addresses temporal variations in both the Pareto Front (PF) and Pareto Set (PS) for dynamic multi-objective optimization problems (DMOPs). Utilizing a hybrid prediction approach, the algorithm adapts to the dynamic nature of the problem. The population is divided into three segments for prediction: individuals with a distance greater than a threshold in PS for central prediction, those with a distance less than a threshold in PS for differential evolutionary prediction, and the remaining individuals for cross-mutation to maintain diversity. To assess DMOEA/D-HP’s effectiveness, it is compared with three advanced algorithms in DMOP by using the DF test set. Experimental results demonstrate that DMOEA/D-HP outperforms in terms of distribution and convergence when solving DMOPs.

Scalable benchmarks and performance measures for dynamic multi-objective optimization

Dynamic multi-objective optimization problems (DMOPs) can be utilized to model certain real-world problems that have a dynamic nature. Algorithms for solving DMOPs can be evaluated and improved by comparing their performance on different benchmarks. However, some existing benchmarks for DMOPs have the limitation of non-uniform weights for decision variables. Additionally, dynamic many-objective optimization problems (DMaOPs) involve more than three objectives, but only a few existing benchmarks can be extended to accommodate DMaOPs. Furthermore, some existing performance measures for DMOPs may not effectively compare the relative performance differences between multiple algorithms or evaluate the search uniformity among different objectives. In this paper, we propose improvements to an existing benchmark for DMOPs by expanding the impact range of decision variables. Moreover, a benchmark framework that can be extended to accommodate DMaOPs is proposed, thus addressing a research gap between the optimization of DMOPs and DMaOPs. Additionally, a set of performance measures for DMOPs are proposed, which can evaluate the relative performance and search uniformity of multi-objective optimization algorithms. By comparing the performance of state-of-the-art and commonly used algorithms on test problems, we can gain a better understanding of the characteristics and strengths and weaknesses of the algorithms and test problems.