Dynamic Multi-objective Problems Research Articles

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 novel preference-driven evolutionary algorithm for dynamic multi-objective problems

Most studies in dynamic multi-objective optimization have predominantly focused on rapidly and accurately tracking changes in the Pareto optimal front (POF) and Pareto optimal set (POS) when the environment undergoes changes. However, there are real-world scenarios where it is necessary to simultaneously solve changing objective functions and satisfy the preference of Decision Makers (DMs). In particular, the DMs may be only interested in a partial region of the POF, known as the region of interest (ROI), rather than requiring the entire POF. To meet the challenge of simultaneously predicting a changing POF and/or POS and dynamic ROI, this paper proposes a new dynamic multi-objective evolutionary algorithm (DMOEAs) based on the preference. The proposed algorithm consists of three key components: an evolutionary direction adjustment strategy based on changing reference points to accommodate shifts in preferences, an angle-based search strategy for tracking the varying ROI, and a hybrid prediction strategy that combines linear prediction models and population manifold estimation within the ROI to ensure convergence and distribution in scenarios where preferences remain unchanged. Experimental studies conducted on 30 widely used benchmark problems in which it outperforms contrasting algorithms on 71% of test suits. Empirical results demonstrate the significant advantages of the proposed algorithm over existing state-of-the-art DMOEAs.

A novel population robustness-based switching response framework for solving dynamic multi-objective problems

In this paper, a novel population robustness-based switching response framework (PR-SRF) is proposed to develop effective dynamic multi-objective optimization algorithm (DMOA), which integrates different response strategies to comprehensively cope with the dynamic behaviors. In particular, the population robustness is described by the quantification of how severely the environmental changes affect current population, which is timely graded as three levels of weak, strong, and normal to enable the adaptive switch of three different responses of diversity-enhancement, diversity-maintenance, and the knowledge-transfer, respectively. In this way, associations between the adopted responses and the changing environments are successfully established, thereby facilitating more intelligent decision when handling the dynamic behaviors. According to the benchmark evaluation results, the proposed PR-SRF-DMOA yields better comprehensive performance than several other DMOAs with popular response strategies, and it also outperforms another three DMOAs with hybrid responses, which demonstrates the great competitiveness of our algorithm. In addition, ablation study proves that the proposed PR-SRF can sufficiently exploit the merits of different responses, which effectively alleviates the negative knowledge transfer in extremely fluctuating environments, thereby providing valuable references for the development of evolutionary transfer optimization (ETO) algorithms.

A learnable population filter for dynamic multi-objective optimization

In this paper, a novel learnable population filter is proposed to solve the dynamic multi-objective problems (DMOPs), whose main idea is to pick out potential valuable individuals in the changing situation via the trained filter to facilitate the population convergence in a new environment. In particular, by selecting training samples from three different optional pools, rich information in the previous searching experiences can be sufficiently used to cope with the dynamic behaviors in DMOPs, thereby handling the imperfect history data availability caused by the environmental uncertainty to some extents. Then, the screened population in the new environment, which is regarded to have reliable quality that benefits accelerating the evolution, is used to initialize the static optimizer. In addition, partial random solutions are employed as supplement to the initial population to further enhance the diversity. It is shown from the benchmark evaluations that the proposed method outperforms some other popular algorithms on both convergence and diversity by seamlessly combining the mainstream strategies in dealing with the dynamic behaviors, which is a competitive method that conforms to the novel evolutionary transfer optimization (ETO) trend. Moreover, effectiveness of the core components is validated through extensive ablation studies, which brings a new sight in developing the learnable DMOP solvers.

A dynamic multi-objective evolutionary algorithm using center and multi-direction prediction strategies.

Dynamic multi-objective optimization problems have been popular because of its extensive application. The difficulty of solving the problem focuses on the moving PS as well as PF dynamically. A large number of efficient strategies have been put forward to deal with such problems by speeding up convergence and keeping diversity. Prediction strategy is a common method which is widely used in dynamic optimization environment. However, how to increase the efficiency of prediction is always a key but difficult issue. In this paper, a new prediction model is designed by using the rank sums of individuals, and the position difference of individuals in the previous two adjacent environments is defined to identify the present change type. The proposed prediction strategy depends on environment change types. In order to show the effectiveness of the proposed algorithm, the comparison is carried out with five state-of-the-art approaches on 20 benchmark instances of dynamic multi-objective problems. The experimental results indicate the proposed algorithm can get good convergence and distribution in dynamic environments.

AugGKG: a grid-augmented geographic knowledge graph representation and spatio-temporal query model

ABSTRACTAs an emerging knowledge representation model in the domain of knowledge graphs, geographic knowledge graph can take full advantage of semantic, spatial and temporal information to facilitate answering spatio-temporal questions and completing relations. However, the representation of geographic knowledge graphs still has issues such as the difficulty of unified heterogeneous spatio-temporal data modelling, weak ability to answer spatio-temporal queries for dynamic multiobjective problems, and low efficiency of graph querying. This paper presents a grid-augmented geographic knowledge graph (AugGKG) based on the GeoSOT global subdivision grid model and time slice subgraph architecture. AugGKG discretely normalizes the spatio-temporal data of the graph, which involves five types of nodes and two types of relations. By using the geo-hidden layer of the graph and geocoding algebraic operations, the AugGKG can quickly answer complex multiobjective spatio-temporal queries and complete implicit spatio-temporal relations. Compared with existing geographic knowledge graphs (YAGO, GeoKG and GEKG), the comparative experiments verified the obvious advantages of AugGKG in terms of uniformity of accuracy, completeness, and efficiency. Hence, AugGKG is expected to be regarded as an innovative and robust geographic knowledge graph that can perform fast computation and relation completion for complex spatio-temporal queries in future geospatial question answering applications.

A novel control strategy for active battery thermal management systems based on dynamic programming and a genetic algorithm

An appropriate active battery thermal management system (ABTMS) control strategy should be able to extend battery life, improve battery capacity consistency and reduce system energy consumption; however, these three purposes are mutually conflicting. To address this conflict, this paper proposes representing the control of the ABTMS as a dynamic multiobjective control problem, with the battery state of health (SOH), battery SOH difference and system energy consumption as the optimization objectives and the airflow rate and refrigeration power as the control variables. Weight coefficients are proposed to couple the multiple objectives. Next, a combined control strategy comprising dynamic programming and a genetic algorithm is developed to obtain the optimal weight coefficient set and calculate the optimal path of the ABTMS that minimizes the total cost of the multiple objectives. Experimental results indicate that the combined strategy effectively balances the three optimization objectives to comprehensively optimize performance. Compared with the performance of a rule-based multiparameter control strategy, the proposed combined control strategy decreases the battery capacity attenuation rate by 3.7%, saves system energy consumption by 22.4%, and improves battery capacity consistency by 23.3%. These findings provide a basis for more efficient and intelligent control strategies for ABMTSs.

A dynamic multi-objective evolutionary algorithm based on Niche prediction strategy

In reality, many multi-objective optimization problems are dynamic. The Pareto optimal front (PF) or Pareto optimal solution (PS) of these dynamic multi-objective problems (DMOPs) changes as the environment change. Therefore, solving such problems requires an optimization algorithm that can quickly track the PF or PS after an environment change. Prediction-based response mechanism is a common method used to deal with environmental changes, which is commonly known as center point-based prediction. However, if the predicted direction of the center point is inaccurate, the predicted population will be biased towards one side. In this paper, we propose a niche prediction strategy based on center and boundary points (PCPB) to solve the dynamic multi-objective optimization problems, which consists of three steps. After environmental changes are detected, the first step is to divide the niche, dividing different individuals in the PS into different niche populations. The second step is to independently predict different niches, and select individuals with good convergence and distribution in the niche to predict the individuals that will produce the next generation. Finally, some different individuals are randomly generated in the next possible PS area to ensure the diversity of the population. To verify whether our proposed strategy is effective and competitive, PCPB was compared with five state-of-the-art strategies. The experimental results show that PCPB performed competitively in solving dynamic multi-objective optimization problems, which proves that our algorithm has good competitiveness.

A computation offloading algorithm based on multi-objective evolutionary optimization in mobile edge computing

For computation offloading problem (COP) in mobile edge computing (MEC), the energy consumption of terminal equipments(TEs) and the delay of mobile equipment applications are two optimization goals. In real life, terminal equipment is dynamic, and their number, mobility, and continuous changes in wireless channels will affect the balance between the mentioned energy consumption and delay. Different from available works, we model the COP in MEC as a dynamic multi-objective problem (DMOP) in this paper, and propose an improved dynamic multi-objective evolutionary optimization based on decomposition (DMOEA/D) to solve it, namely DMOEA/D-COPMEC. In the proposed algorithm, the environmental change is detected by a fixed detector, and whether the current change is similar to the historical change is determined. If so, the difference prediction is used to re-locate the population individual in the new MEC environment, otherwise, the memory-based strategy is used to response environmental change. In MOEA/D, an adaptive weight adjustment strategy based on chain segmentation (CS) is adopted to generate a set of uniformly distributed weight vectors. The simulation results show that the proposed algorithm can better balance the application delay and the terminal energy consumption if there is environment change. The solution set is closer to reality and better than the related algorithms.

Data-Driven Dynamic Multiobjective Optimal Control: An Aspiration-Satisfying Reinforcement Learning Approach.

This article presents an iterative data-driven algorithm for solving dynamic multiobjective (MO) optimal control problems arising in control of nonlinear continuous-time systems. It is first shown that the Hamiltonian functional corresponding to each objective can be leveraged to compare the performance of admissible policies. Hamiltonian inequalities are then used for which their satisfaction guarantees satisfying the objectives' aspirations. Relaxed Hamilton-Jacobi-Bellman (HJB) equations in terms of HJB inequalities are then solved in a dynamic constrained MO framework to find Pareto optimal solutions. Relation to satisficing (good enough) decision-making framework is shown. A sum-of-square (SOS)-based iterative algorithm is developed to solve the formulated aspiration-satisfying MO optimization. To obviate the requirement of complete knowledge of the system dynamics, a data-driven satisficing reinforcement learning approach is proposed to solve the SOS optimization problem in real time using only the information of the system trajectories measured during a time interval without having full knowledge of the system dynamics. Finally, two simulation examples are utilized to verify the analytical results of the proposed algorithm.

DPb-MOPSO: A Dynamic Pareto bi-level Multi-objective Particle Swarm Optimization Algorithm

Particle Swarm Optimization (PSO) system based on the distributed architecture over multiple sub-swarms is very efficient for static multi-objective optimization but has not been considered for solving dynamic multi-objective problems (DMOPs). Tracking the most effective solutions over time and ensuring good exploitation and exploration are the main challenges of solving DMOP. This study proposes a Dynamic Pareto bi-level Multi-Objective Particle Swarm Optimization (DPb-MOPSO) algorithm including two parallel optimization levels. At the first level, all solutions are managed in a single search space. When a dynamic change is successfully detected in the objective values, the Pareto ranking operator is used to enable multiple sub-swarm’ subdivisions and processing which drives the second level of enhanced exploitation. A dynamic handling strategy based on random detectors is used to track the changes in the objective function due to time-varying parameters. A response strategy consisting in reevaluating all unimproved solutions and replacing them with newly generated ones is also implemented. The DPb-MOPSO system is tested on DMOPs with different types of time-varying Pareto Optimal Set (POS) and Pareto Optimal Front (POF). Inverted generational distance (IGD), mean inverted generational distance (MIGD), hypervolume difference (HVD), Robust IGD (RIGD), and Robust General Distance (RGD) metrics are used to assess the DPb-MOPSO performance. Quantitative results are analyzed using Friedman’s analysis of variance, and the Wilcoxon sum ranks test, while the stability is analyzed using Lyapunov’s theorem. The DPb-MOPSO is more robust than several dynamic multi-objective evolutionary algorithms in solving 21 complex problems over a range of changes in both the POS and POF. On IGD and HVD, DPb-MOPSO can solve 8/13 and 8/13 of the 13 UDF and ZJZ functions with moderate changes. DPb-MOPSO can resolve 7/8 FDA and DMOP benchmarks with severe changes to the MIGD, and 6/8 with moderate changes. DPb-MOPSO assumes 7/8, 6/8, and 5/8 for solving FDA, and dMOP functions on IGD and 6/8, 5/8, and 5/8 on HVD metrics considering severe, moderate, and slight environmental changes respectively. Also, it is the winner for solving 8 DMOPs based on RIGD, and RGD metrics.

Solving dynamic multi-objective problems using polynomial fitting-based prediction algorithm

Recently, dynamic multi-objective optimization has received growing attention due to its popularity in real-world applications. Inspired by polynomial fitting, this paper proposes a polynomial fitting-based prediction algorithm (PFPA) and incorporates it into the model-based multi-objective estimation of distribution algorithm (RM-MEDA) for solving dynamic multi-objective optimization problems. When an environment change is detected, the main mission of PFPA is to predict high-quality search populations for tracking the moving Pareto-optimal set effectively. Firstly, the non-dominated solutions obtained in past environments are utilized to predict high-quality solutions based on a multi-step movement strategy. Secondly, a polynomial fitting-based strategy is designed to fit the distribution of variables according to the obtained search populations, and capture the relationship between variables in the new search environment. Thirdly, some effective search agents are generated for improving population convergence and diversity based on characteristics of variables. To evaluate the performance of the proposed algorithm, experimental results on a set of benchmark functions, with a variety of different dynamic characteristics and difficulties, and two classical dynamic engineering design problems show that PFPA is competitive with some state-of-the-art algorithms.

Fitness Landscape Analysis on Binary Dynamic Optimization Problems

Fitness Landscape Analysis (FLA) denotes the task of analyzing black-box optimization problems and capturing their characteristic features with the goal of providing additional information, that may help in algorithm selection, parametrization or guidance. Many real-world optimization tasks require dynamic on-going optimization and a plethora of meta-heuristic algorithms has been introduced for this task. However, most analysis focuses on static problems or dynamic optimization tasks without time-linkage, where the dynamic changes of the problem are independent of the decisions taken by the optimizer, but many real-world optimization problems display very heavy dependence on previous states and decisions. In this paper, the techniques of the static FLA are combined with dynamic and domain specific measures and applied to two dynamic problems. A time-linked dynamic OneMax problem and a dynamic multi-objective knapsack problem are presented and the impact of time-linkage on their FLA features is analyzed.

A novel predictive method based on key points for dynamic multi-objective optimization

Dynamic multi-objective problem is very difficult to be solved because of the variability of the objective function with time. To overcome the difficult caused by such variability, a predictive method utilizing some key points (including polar points and centroid points) is designed, which contains four critical steps. First, the whole population is automatically divided into multiple clusters, which will be used to preserve a good diversity in the process of population evolution. Second, the technique for order of preference by similarity to ideal solution (TOPSIS), a well-regarded multi-attribute decision making strategy, is exploited to improve its convergence speed further. Third, the polar point and centroid point in each cluster are utilized to obtain the initial population by using sequence predictive method when environmental changes are detected. Fourth, to accelerate the convergence speed, the quantitative value for each individual determined in the prediction process is also used in mating selection and environmental selection. The numerical results imply that the new method can deal with the change of environment effectively and track the Pareto optimal front (POF) quickly. Meanwhile, the comparison results with several selected state-of-the-art methods also show that the overall performance of the proposed method is the best on most benchmark problems.

Optimal Route Selection Decision-Making Based on Intelligent Network

Modern intelligent network provides a lot of convenience for the development of the society, especially for some practical NP-hard optimal problems. In this paper, we study a game case which is a problem of optimal strategy of desert crossing based on intelligent network and then propose an optimal route selection decision-making. We construct some dynamic multiobjective network problems and use heuristic algorithm to solve the two problems in four levels in the paper. The innovation of this paper lies in the optimal route selection strategy which considers different environment conditions and their own situations. The cross research between the practical engineering problems and mathematics is applied flexibly in this article. Moreover, the strategy given in this paper is a general direction rather than a specific result, which is also in line with the characteristics of diversity of the game optimal results.

Solving dynamic multi-objective problems with a new prediction-based optimization algorithm.

This paper proposes a new dynamic multi-objective optimization algorithm by integrating a new fitting-based prediction (FBP) mechanism with regularity model-based multi-objective estimation of distribution algorithm (RM-MEDA) for multi-objective optimization in changing environments. The prediction-based reaction mechanism aims to generate high-quality population when changes occur, which includes three subpopulations for tracking the moving Pareto-optimal set effectively. The first subpopulation is created by a simple linear prediction model with two different stepsizes. The second subpopulation consists of some new sampling individuals generated by the fitting-based prediction strategy. The third subpopulation is created by employing a recent sampling strategy, generating some effective search individuals for improving population convergence and diversity. Experimental results on a set of benchmark functions with a variety of different dynamic characteristics and difficulties illustrate that the proposed algorithm has competitive effectiveness compared with some state-of-the-art algorithms.

Time consistency for scalar multivariate risk measures

Abstract In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)).

Improved NSGA-III with Second-Order Difference Random Strategy for Dynamic Multi-Objective Optimization

Most real-world problems that have two or three objectives are dynamic, and the environment of the problems may change as time goes on. For the purpose of solving dynamic multi-objective problems better, two proposed strategies (second-order difference strategy and random strategy) were incorporated with NSGA-III, namely SDNSGA-III. When the environment changes in SDNSGA-III, the second-order difference strategy and random strategy are first used to improve the individuals in the next generation population, then NSGA-III is employed to optimize the individuals to obtain optimal solutions. Our experiments were conducted with two primary objectives. The first was to test the values of the metrics mean inverted generational distance (MIGD), mean generational distance (MGD), and mean hyper volume (MHV) on the test functions (Fun1 to Fun6) via the proposed algorithm and the four state-of-the-art algorithms. The second aim was to compare the metrics’ value of NSGA-III with single strategy and SDNSGA-III, proving the efficiency of the two strategies in SDNSGA-III. The comparative data obtained from the experiments demonstrate that SDNSGA-III has good convergence and diversity compared with four other evolutionary algorithms. What is more, the efficiency of second-order difference strategy and random strategy was also analyzed in this paper.

A decision variable classification-based cooperative coevolutionary algorithm for dynamic multiobjective optimization

This paper proposes a new decision variable classification-based cooperative coevolutionary algorithm, which uses the information of decision variable classification to guide the search process, for handling dynamic multiobjective problems. In particular, the decision variables are divided into two groups: convergence variables (CS) and diversity variables (DS), and different strategies are introduced to optimize these groups. Two kinds of subpopulations are used in the proposed algorithm, i.e., the subpopulations that represent DS and the subpopulations that represent CS. In the evolution process, the coevolution of DS and CS is carried out through genetic operators, and subpopulations of CS are gradually merged into DS, which is optimized in the global search space, based on an indicator to avoid becoming trapped in local optimum. Once a change is detected, a prediction method and a diversity introduction approach are adopted for these two kinds of variables to get a promising population with good diversity and convergence in the new environment. The proposed algorithm is tested on 16 benchmark dynamic multiobjective optimization problems, in comparison with state-of-the-art algorithms. Experimental results show that the proposed algorithm is very competitive for dynamic multiobjective optimization.

A heredity-based adaptive variation operator for reinitialization in dynamic multi-objective problems

A reinitialization approach is an effective way of generalizing a static multi-objective optimization method to a dynamic one. It is usually comprised of a prediction operator for predicting the approximate location(s) of the optimal solution(s) and a variation operator for enhancing the diversity of the reinitialized solution(s) after a change. While many recent studies have focused on prediction methods, the importance of the variation operator has usually been overlooked. This study systematically explores the effects of the accuracy of the prediction method employed as well as the frequency and severity of the change on the optimal strength of the variation used for reinitialization. Subsequently, it introduces an adaptive variation operator for dynamic multi-objective optimization which can learn the optimal variation strength on-the-fly. To develop this method, firstly, a heredity measure for evolutionary algorithms is formulated to quantify the contribution of each reinitialized solution to the optimization process by measuring the presence of its traits in the final population. Some carefully designed descriptive simulations are performed to explore the capability of the proposed method to learn the optimal variation strength and its sensitivity to the change severity, initial variation strength, and accuracy of the employed prediction method. Finally, the performance of this variation operator on 42 dynamic multi-objective test problems is compared with those of five other popular ones, with numerical comparisons revealing its superior learning capability.