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The search region in multiobjective optimization problems is a part of the objective space where nondominated points could lie. It plays an important role in the generation of the nondominated set of multiobjective combinatorial optimization (MOCO) problems. In this paper, we establish the representation of the search region by half-open polyblocks (a variant concept of “polyblock” in monotonic optimization) and propose a new procedure for updating the search region. We also study the impact of stack policies to the new procedure and the existing methods that update the search region. Stack policies are then analyzed, pointing out their performance effectiveness by means of the results of rich computational experiments on finding the whole set of nondominated points of MOCO problems.

This paper analyzes the convergence rates of the Frank–Wolfe method for solving convex constrained multiobjective optimization. We establish improved convergence rates under different assumptions on the objective function, the feasible set, and the localization of the limit point of the sequence generated by the method. Notably, we demonstrate that the method can achieve linear convergence rates in terms of a merit function whenever the objectives are strongly convex and the limit point is in the relative interior of the feasible set, or when the feasible set is strongly convex and it does not contain an unconstrained weak Pareto point. Moreover, improved sublinear convergence rates can also be obtained in other scenarios where the feasible set is uniformly convex. Additionally, we explore enhanced convergence rates with respect to an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.

In this paper, we propose and analyze an away-step Frank–Wolfe algorithm designed for solving multiobjective optimization problems over polytopes. We prove that each limit point of the sequence generated by the algorithm is a weak Pareto optimal solution. Furthermore, under additional conditions, we show linear convergence of the whole sequence to a Pareto optimal solution. Numerical examples illustrate a promising performance of the proposed algorithm in problems where the multiobjective Frank–Wolfe convergence rate is only sublinear.

Improving current efficiency and reducing energy consumption are two important technical goals of the electrolytic aluminum process (EAP). However, because the process involves complex noise characteristics (i.e., unknown types, redundant distributions and variable forms), it is very difficult to accurately develop a multiobjective prediction model. To overcome this problem, in this paper, a novel framework of multiobjective incremental learning based on a multi-source filter neural network (MSFNN) is presented. The proposed framework first presents a “multi-source filter” (MSF) technique that utilizes the mean and variance in the unscented Kalman filter (UKF) to guide the importance function of the particle filter (PF) based on a density kernel estimation method. Then, the MSF is embedded in the mutated neural network to adjust weights in real time. Third, weights are calculated and normalized by a modified importance function, which is the basis for further optimizing a secondary sampling based on sampling importance resampling (SIR). Finally, the incremental learning model with two objectives (i.e., process power consumption and current efficiency) based on the MSFNN in the EAP is established. The presented framework has been verified by the real-world EAP and some closely related methods. All test results indicate that the MSFNN’s relative prediction errors of the above two objectives are controlled within 0.51% and 0.38%, respectively and prove that MSFNN has significant competitive advantages over other recent filtering network models. Successfully establishment of the proposed framework provides a model foundation for multiobjective optimization problems in the EAP.

Network robustness optimization is crucial for enhancing the resilience of industrial networks and social systems against malicious attacks. Existing studies typically evaluate the robustness by simulating the sequential removal of nodes or edges and recording the residual connectivity at each step. However, the attack simulation is computationally expensive and becomes impractical for large-scale networks. Therefore, this paper proposes a multiobjective evolutionary algorithm assisted by a graph isomorphism network (GIN)-based surrogate model to efficiently optimize network robustness. First, the robustness optimization task is formulated as a multiobjective problem that simultaneously considers network robustness against attacks and the structural modification cost. Then, a GIN-based surrogate model is constructed to approximate the robustness, replacing the expensive simulation assessments. Finally, the multiobjective evolutionary algorithm is employed to explore promising network structures guided by the surrogate model, which is continuously updated via online learning to improve both prediction accuracy and optimization performance. Experimental results in various synthetic and real-world networks demonstrate that the proposed algorithm reduces the computational cost of the robustness evaluation by about 65% while achieving comparable or even superior robustness optimization performance compared with those of baseline algorithms. These results indicate that the proposed method is practical and scalable and can be applied to enhance the robustness of industrial and social networks.

Economic dispatch (ED) problems, especially in multi-area power networks, have been challenging concerns for power system operators for several decades. In this paper, we introduce a novel approach for solving the multiobjective multi-area dynamic ED (MADED) problem in the presence of practical constraints such as valve-point effect (VPE), prohibited operating zone (POZ), multi-fuel operation (MFO), and ramp rate (RR) limitations. Different objective functions including energy not supplied (ENS), generation costs, and emissions are investigated. The reliability objective, which has been less studied in economic dispatch area, distinguishes the proposed study from other studies. A compromise has been made from economic and reliability points of view. The MADED problem in the power system is inherently a complex and nonlinear problem, considering the operational constraint increments and the intricacy of the problem. Hence, the modified grasshopper optimization (MGO) algorithm based on a chaos mechanism is presented to prevent being trapped in local optima. The proposed method is tested on two systems including a 10 unit, 3-zone test system and a 40-unit 3-zone test system, and then, the outcomes are compared with those of other evolutionary techniques such as gray wolf optimization (GWO) and modified honey bee mating optimization (MHBMO). The simulation results demonstrate that the suggested strategy is successful in resolving both single-objective and multiobjective MADED problems.

Vehicle routing problems (VRPs) are challenging problems. Many variants of the VRP have been proposed. However, few studies on VRP have combined robustness and just-in-time (JIT) requirements with uncertainty. To solve the problem, this paper proposes the just-in-time-based robust multiobjective vehicle routing problem with time windows (JIT-RMOVRPTW) for the assembly workshop. Based on the conflict between uncertain time and JIT requirements, a JIT strategy was proposed. To measure the robustness of the solution, a metric was designed as the objective. Afterwards, a two-stage nondominated sorting ant colony algorithm with deep reinforcement learning (NSACOWDRL) was proposed. In stage I, ACO combines with NSGA-III to obtain the Pareto frontier. Based on the model, a pheromone update strategy and a transfer probability formula were designed. DDQN was introduced as a local search algorithm which trains networks through Pareto solutions to participate in probabilistic selection and nondominated sorting. In stage II, the Pareto frontier was quantified in feasibility by Monte Carlo simulation, and tested by diversity-robust selection based on uniformly distributed weights in the solution space to select robust Pareto solutions that take diversity into account. The effectiveness of NSACOWDRL was demonstrated through comparative experiments with other algorithms on instances. The impact of JIT strategy is analyzed and the effect of networks on the NSACOWDRL is further discussed.

In this paper, a proximal bundle-based method for solving nonsmooth nonconvex constrained multiobjective optimization problems with inexact information is proposed and analyzed. In this method, each objective function is treated individually without employing any scalarization. Using the improvement function, we transform the problem into an unconstrained one. At each iteration, by the proximal bundle method, a piecewise linear model is built and by solving a convex subproblem, a new candidate iterate is obtained. For locally Lipschitz objective and constraint functions, we study the problem of computing an approximate substationary point (a substationary point), when only inexact (exact) information about the functions and subgradient values are accessible. At the end, some numerical experiments are provided to illustrate the effectiveness of the method and certify the theoretical results.

This study proposes a mathematical model of an intelligent multi-objective home energy management (HEM) scheme with the integration of small-scale renewable energy sources. The main aim of the proposed model is to handle the residential load demand in a smart way to minimise both the consumer's energy bill and the system peak demand simultaneously. To generate the best compromise solution of the proposed multi-objective problem, a cooperative game theory approach is used in this study on the basis of super-criterion and a Pareto optimal solution concept. In the cooperative game process, each HEM objective is assigned as a player and every player tries to maximise their own payoff. Bargaining model in the form of super criterion is considered in this game approach. Finally, all players can get win–win nature with collective negotiations. Generally, HEM method deals with various controllable devices having distinct operating characteristics. Because of this, the proposed HEM problem is modelled as a mixed-integer problem. Consequently, a mixed-integer non-linear programming is applied in this game process to maximise the super-criterion. To show the effectiveness of the proposed model, different case studies and various scenarios are carried out.

One of the important issues in the planning stage of active distribution networks (ADNs) is the optimal design of microgrids (MGs). The design, as a multi-MG system, is comprehensively investigated in this study. In this way, the allocation of energy storage systems (ESSs) and partitioning of ADN are simultaneously performed in order to minimise the cost and maximise the self-adequacy and the reliability considering the uncertainty of load and renewable energy resources. In this study, two approaches are considered. In approach I, the cost, reliability and self-adequacy objectives are taken into account whereas, in approach II, a new probabilistic index representing the ratio of load to storage capacity is also added to mentioned objectives. The proposed multi-objective problem is solved with non-dominated sorting genetic algorithm-II (NSGA-II) as a well-known algorithm based on a probabilistic approach using the Monte-Carlo simulation method (MCSM) and in each approach, several Pareto optimal solutions are evaluated. To simulate and validate the effectiveness of the proposed method, two benchmark distribution networks (the 33-bus and the 119-bus) are used.