Constrained Multi-objective Optimization Research Articles

Multi-Objective Constrained Optimization Model and Molten Iron Allocation Application Based on Hybrid Archimedes Optimization Algorithm

The challenge of distributing molten iron involves the optimal allocation of blast furnace output to various steelmaking furnaces, considering the blast furnace’s production capacity and the steelmaking converter’s consumption capacity. The primary objective is to prioritize the distribution from the blast furnace to achieve a balance between iron and steel production while ensuring that the volume of hot metal within the system remains within a safe range. To address this, a constrained multi-objective nonlinear programming model is abstracted. A linear weighting method combines multiple objectives into a single objective function, while the Lagrange multiplier method addresses constraints. The proposed hybrid Archimedes optimization algorithm effectively solves this problem, demonstrating significant improvements in time efficiency and precision compared to existing methods.

ATM-R: An Adaptive Tradeoff Model With Reference Points for Constrained Multiobjective Evolutionary Optimization.

The goal of constrained multiobjective evolutionary optimization is to obtain a set of well-converged and well-distributed feasible solutions. To achieve this goal, a delicate tradeoff must be struck among feasibility, diversity, and convergence. However, balancing these three elements simultaneously through a single tradeoff model is nontrivial, mainly because the significance of each element varies in different evolutionary phases. As an alternative approach, we adapt distinct tradeoff models in various phases and introduce a novel algorithm named adaptive tradeoff model with reference points (ATM-R). In the infeasible phase, ATM-R takes the tradeoff between diversity and feasibility into account, aiming to move the population toward feasible regions from diverse search directions. In the semi-feasible phase, ATM-R promotes the transition from "the tradeoff between feasibility and diversity" to "the tradeoff between diversity and convergence." This transition is instrumental in discovering an adequate number of feasible regions and accelerating the search for feasible Pareto optima in succession. In the feasible phase, ATM-R places an emphasis on balancing diversity and convergence to obtain a set of feasible solutions that are both well-converged and well-distributed. It is worth noting that the merits of reference points are leveraged in ATM-R to accomplish these tradeoff models. Also, in ATM-R, a multiphase mating selection strategy is developed to generate promising solutions beneficial to different evolutionary phases. Systemic experiments on a diverse set of benchmark test functions and real-world problems demonstrate that ATM-R is effective. When compared to eight state-of-the-art constrained multiobjective optimization evolutionary algorithms, ATM-R consistently demonstrates its competitive performance.

A parallel variable-fidelity algorithm for efficient constrained multi-objective aerodynamic design optimization

Modern aerodynamic design optimization aims to discover optimal configurations using computational fluid dynamics under complex flow conditions, which is a typical expensive multi-objective optimization problem. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) combined with efficient global optimization is a promising method but requires enhanced efficiency and faces limitations in its application to multi-objective aerodynamic design optimization (MOADO). To address the issues, an efficient parallel MOEA/D assisted with variable-fidelity optimization (VFO) is proposed for solving MOADO, called the MOEA/D-VFO algorithm. Variable-fidelity surrogates are built for objectives and constraints, achieving higher accuracy using fewer high-fidelity samples and a great number of low-fidelity samples. By retaining more good candidates, the sub-optimization problems defined by decomposing original objectives are capable of discovering more favorable samples using MOEA/D, which prompts optimization convergence. A constraint-handling strategy is developed by incorporating the probability of feasibility functions in the sub-optimizations. The selection of new samples for parallel evaluation is improved by filtering out poor candidates and selecting effective promising samples, which improves the feasibility and diversity of solved Pareto solutions. A Pareto front (PF) can be efficiently found in a single optimization run. The proposed approach is demonstrated by four analytical test functions and verified by two aerodynamic design optimizations of airfoils with and without constraints, respectively. The results indicate that the MOEA/D-VFO approach can greatly improve optimization efficiency and obtain the PF satisfying constraints within an affordable computational budget.

Evolutionary Constrained Multiobjective Optimization: Scalable High-Dimensional Constraint Benchmarks and Algorithm

Evolutionary constrained multiobjective optimization has received extensive attention and research in the past two decades, and a lot of benchmarks have been proposed to test the effect of the constrained multiobjective evolutionary algorithms (CMOEAs). Specially, the constraint functions are highly correlated with the objective values, which makes the features of constraints too monotonic and differ from the properties of the real-world problems. Accordingly, previous CMOEAs cannot solve real-world problems well, which generally involve decision space constraints with multi-modal/non-linear features. Therefore, we propose a new benchmark framework and design a suite of new test functions with scalable high-dimensional decision space constraints. To be specific, different high-dimensional constraint functions and mixed linkages in variables are considered to be close to realistic features. In this framework, several parameter interfaces are provided, so that users can easily adjust the parameters to obtain the variant functions and test the generalization performance of the algorithms. Different types of existing CMOEAs are employed to test the use of the proposed test functions, and the results show that they are easy to fall into local feasible regions. Therefore, we improve one evolutionary multitasking-based CMOEA to better handle these problems, in which a new search algorithm is designed to enhance the search abilities of populations. Compared with the existing CMOEAs, the proposed CMOEA presents better performance.

Constraints Separation Based Evolutionary Multitasking for Constrained Multi-Objective Optimization Problems

Constraints Separation Based Evolutionary Multitasking for Constrained Multi-Objective Optimization Problems

Two-stage coevolutionary constrained multi-objective optimization algorithm for solving optimal power flow problems with wind power and FACTS devices

As a large amount of wind energy is integrated into the grid, the randomness it brings poses a challenge to modern power systems. The application of Flexible AC Transmission Systems (FACTS) in the grid is becoming more and more common, and it is necessary to consider how to choose suitable equipment in the appropriate locations. In this paper, a multi-objective optimal power flow (MOOPF) model with wind farms and FACTS devices is established. The Weibull probability density function is used to establish the wind speed model, and the cost problem brought by wind power is considered. The locations and ratings of thyristor-controlled series compensators, thyristor-controlled phase shifters, and static VAR compensators are added to the system as control variables. In addition, the constraints on the prohibited operating areas of thermal power generators and the valve point effect are also considered. Coevolutionary constrained multi-objective optimization algorithm (CCMO) is an advanced technology, and this paper improves it and names it two-stage coevolutionary constrained multi-objective optimization algorithm (TSCCMO). The proposed algorithm uses the constraint violation value as an additional objective function in the sub-population environmental selection process, and integrates a neighborhood selection strategy into the mating selection process. The population evolution process is divided into two stages, in the first stage the two populations cooperate weakly, and in the second stage the two populations will have strong cooperation. TSCCMO is used to solve this complex constrained MOOPF problem, and its results are compared and analyzed with CCMO, NSGA–II–CDP, C3M, and PPS. The comprehensive performance of TSCCMO is the best among the 6 cases.

A Multiobjective Optimization Algorithm for Fluid Catalytic Cracking Process with Constraints and Dynamic Environments

The optimization problems in a fluid catalytic cracking process with dynamic constraints and conflicting objectives are challenging due to the complicated constraints and dynamic environments. The decision variables need to be reoptimized to obtain the best objectives when dynamic environments arise. To solve these problems, we established a mathematical model and proposed a dynamic constrained multiobjective optimization evolution algorithm for the fluid catalytic cracking process. In this algorithm, we design an offspring generation strategy based on minimax solutions, which can explore more feasible regions and converge quickly. Additionally, a dynamic response strategy based on population feasibility is proposed to improve the feasible and infeasible solutions by different perturbations, respectively. To verify the effectiveness of the algorithm, we test the algorithm on ten instances based on the hypervolume metric. Experimental results show that the proposed algorithm is highly competitive with several state-of-the-art competitors.

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 federated GAN network-based evolutionary constrained optimization approach to integrated coal mine energy system

The integrated coal mine energy system (ICMES) is a kind of system with multiple scenarios, variables and parameters, which belongs to dynamic constrained multi-objective optimization problem (DCMOP). One of the challenges in solving ICMES lies in searching for feasible solutions when the frequency of changes is quick. To solve the above mentioned issues, this paper proposes a federated GAN network-based evolutionary constrained optimization for ICMES (FGECO). Firstly, multiple GAN networks are utilized in the framework of federated learning (FL) to estimate the distribution of feasible regions that satisfy each constraint. Following that, they are fused to realize the intersection of all feasible regions, and generate one feasible region that can meet all constrained requirements. Subsequently, an initial population with guidance of evolution is generated based on the proposed shared GAN network model. Finally, FGECO is compared with four popular dynamic constrained multi-objective evolutionary algorithms (DCMOEAs) on ICMES. Experimental results indicate its superiority.

Planning trajectories for connected and automated vehicle platoon on curved roads: A two-dimensional cooperative approach

This paper presents a cooperative two-dimensional trajectory planning algorithm for connected and automated vehicle (CAV) platoons. Specifically, the proposed algorithm generates two-dimensional optimal trajectories for CAVs with car-following relationships cooperatively within a complex road geometry. By extending the simplified Newell’s car-following model, we propose a two-dimensional Newell’s car-following model as an equilibrium car-following policy for CAVs. Based on this, a multi-objective constrained optimization is systematically formulated under a Cartesian coordinate. Due to the constraint’s complexity, a new solving algorithm based on the rapid random tree (RRT) technique is proposed. To test the effectiveness of our proposed models and algorithm, numerical simulation experiments with a real-world road geometry are conducted. Results indicate that our proposed method is able to generate trajectories for CAV platoons which are close to the equilibrium condition with smooth controls, while avoiding road obstacles. We further extend the definition of a one-dimensional car-following control string stability to a two-dimensional case. By this definition, we find that the proposed method can achieve empirical two-dimensional string stability, ensuring that both lateral and longitudinal disturbances are attenuated through vehicular strings.

Dynamic constrained multi-objective optimization algorithm based on co-evolution and diversity enhancement

Dynamic constrained multi-objective optimization problems (DCMOPs) involve objectives, constraints, and parameters that change over time. This kind of problem presents a greater challenge for evolutionary algorithms because it requires the population to quickly track the changing pareto-optimal set (PS) under constrained conditions while maintaining the feasibility and good distribution of the population. To address these challenges, this paper proposes a dynamic constrained multi-objective optimization algorithm based on co-evolution and diversity enhancement (CEDE), in which we have made improvements to both the static optimization and dynamic response parts, innovatively utilizing the valuable information latent in the optimization process to help the population evolve more comprehensively. The static optimization involves the co-evolution of three populations, through which their mutual synergy can more comprehensively identify potential true PS and provide more useful historical information for dynamic response. Additionally, to prevent the elimination of potentially valuable infeasible individuals (i.e., individuals that are not dominated by feasible individuals) due to pareto domination, we employ an archive set to store and update these individuals. When the environment changes, to effectively enhance population diversity under complex dynamic constraints and help the population to respond quickly to changes, we propose a diversity enhancement strategy, which includes a diversity maintenance strategy and a center point-based exploration strategy. This strategy effectively enhances population diversity in complex and changing environments, helping the population respond quickly to changes. The effectiveness of the algorithm is validated through two test sets. The experimental results show that CEDE can effectively use valuable information to cope with complex dynamic constraint environments. Compared with several of the most advanced algorithms, it is superior in 94% of the test problems, demonstrating strong competitiveness in handling DCMOPs.

Constrained multiobjective optimization design for ordinary shovel attachment of hydraulic excavator based on evolutionary algorithm

Shovel attachment is one of the most commonly used working attachments in hydraulic excavators. Its mechanical design remains a challenging optimization problem. To tackle this issue, the optimizing model of ordinary shovel attachment is firstly established which contains twenty-three design variables, six objective functions, and sixty-three constraints. Then, an improved decomposition-based constrained many-objective evolution algorithm is proposed, which integrates advanced push & pull search constraint handling technique and differential evolution operator to deal with the optimization problems of shovel attachment. Finally, the proposed algorithm has demonstrated its superior optimization ability for designing shovel attachments through the case studies of a 70-ton face-shovel hydraulic excavator, outperforming seventeen state-of-the-art constrained multiobjective evolutionary algorithms. Furthermore, the first quantitative comparison between ordinary shovel attachment and TriPower shovel attachment under the same main machine is presented by the optimized results. The result demonstrates the complexity of the optimal design for shovel attachment and the effectiveness of multiobjective evolutionary algorithm.

A staged diversity enhancement method for constrained multiobjective evolutionary optimization

Optimizing the convergence and diversity of solutions simultaneously under constraints is a challenge in solving constrained multiobjective optimization problems. In existing multiobjective optimization algorithms, general diversity maintenance mechanisms have difficulty determining all optimal solutions in discrete feasible regions. This paper proposes a staged constrained multiobjective optimization algorithm with a diversity enhancement method (SDEM), which can explore potential discrete feasible regions by retaining well-distributed offspring. Specifically, after solutions have converged to optimal feasible regions by niching-based constraint dominance in the early stage, the SDEM improves the diversity of solutions through a proposed diversity enhancement dominance principle in the mid-term. Finally, the optimize objective functions and constraints of all solutions are optimized under constraint dominance to balance convergence, diversity, and feasibility during the three stages. Experiments on four well-known test suites and six real-world case studies demonstrate that the SDEM is competitive with or comparable to seven state-of-the-art constrained multiobjective evolutionary algorithms.

Data-driven modeling and multi-objective optimization of a continuous kraft pulping digester

Understanding wood characteristics is essential for enhancing the economic sustainability of the kraft process. In a recent work, we included wood characteristics in the phenomenological modeling of digesters. The developed model allows the optimization of the process. Here, we propose a constrained multi-objective optimization approach to minimize the residual lignin content while maximizing the carbohydrate content in the cellulose pulp, using the Non-dominated Sorting Genetic Algorithm II. In order to reduce computational costs during the optimization, we developed a surrogate model, based on simulated data, to predict key performance indicators. We evaluated two machine learning techniques: Multilayer Perceptron (MLP) and decision tree-based methods, considering single and multiple outputs. The combination between the multiple-output approach and MLP performed well and resulted in a more simplified surrogate model. The results for the simulation and the surrogate model were similar, and the computational cost was approximately 50 times lower for the surrogate model. An uncertainty was associated with the multiplicative factor of the delignification rate as a way to observe how its estimate interferes with the Pareto curve. The present work presents contributions to data-driven modeling and multi-objective optimization of digesters, allowing us to determine the importance of wood characteristics in operation.

Optimal design of a hinge-spring-friction device for enhancing wind induced structural response of onshore wind turbines

Wind turbines are subjected to fluctuating wind loads leading to considerable forces and structural fatigue to the towers, ultimately affecting costs, performances, and lifespan, for the tower as well as for the foundation. To mitigate these issues, both in terms of peak and fatigue structural demand, this paper presents the development and optimization of a Hinge-Spring-Friction Device (HSFD) designed for onshore wind turbines. A decoupled numerical model of the wind turbine system incorporating the HSFD is first established. The wind load is modelled by means of the open-source software QBlade© accounting for different wind conditions. These loads are then applied to a FEM structural model of the wind turbine developed in Simulink© and optimized for computational efficiency. The optimal design parameters (strength and stiffness of the frictional and elastic part, respectively) of the HSFD are determined through a multi-objective constrained optimization algorithm, minimizing the peak base moment and total damage fatigue. The proposed framework is then applied to a NREL 5 MW wind turbine to provide an applicative example. The results show that the optimized HSFD can significantly reduce the fatigue damage and the base moment demand to the tower, so providing a really promising solution for the effective design of wind turbines as well as for the repowering of existing plants.

A dual-population auxiliary multiobjective coevolutionary algorithm for constrained multiobjective optimization problems

The key to solving constrained multiobjective optimization problems (CMOPs) lies in maintaining the feasibility, convergence, and diversity of the population. In recent years, various constraint handling techniques (CHTs) and strategies have been proposed to enhance the performance of constrained multiobjective evolutionary algorithms (CMOEAs). However, most of these algorithms face difficulties in dealing with problems that have large infeasible regions and discontinuous small feasible regions, as they have trouble crossing large infeasible regions while simultaneously maintaining the convergence and diversity of the population. To tackle this issue, this paper proposes a dual-population auxiliary coevolutionary algorithm with an enhanced operator, denoted as DAEAEO. Auxiliary population 1 employs an improved ϵ-constraint handling technique to provide high-quality feasible solutions for the main population. Auxiliary population 2 adopts the non-dominated sorting method to provide favorable objective information for the main population to help it cross the infeasible region. In addition, to further improve diversity, each population adopts an enhanced operator and a genetic operator to generate offspring, respectively. Finally, knowledge transfer between offspring is realized. Compared to six state-of-the-art CMOEAs on DASCMOPs, LIR-CMOPs, DOC test suites, and two real-world problems, the proposed DAEAEO achieved superior performance, especially for CMOPs with large infeasible regions and discontinuous small feasible regions.

DpEA: A dual-population evolutionary algorithm for dynamic constrained multiobjective optimization

Dynamic Constrained Multiobjective Optimization Problems (DCMOPs) are very difficult to solve because both of the objectives and constraints may change over time. The existing approaches for solving DCMOPs mainly develop dynamic response techniques and constraint handling techniques. But they do not focus on the search capability of the static optimizer in each environment, which ignores the intrinsic requirement of quickly locating Pareto-optimal Front (PF) in each environment when solving DCMOPs. To this end, this paper proposes a dual-population evolutionary algorithm for solving DCMOPs, called as DpEA, which maintains a population without considering constraints (called UP) for exploration and a population with considering constraints (called CP) for exploitation in each environment. In each iteration of a new environment, UP firstly adopts a stratified mutation strategy (SMS) and a dominated solution repairment strategy (DSR) to enhance the exploration ability of finding promising regions where the PF may reside. SMS uses solutions from different nondominated fronts to generate offspring, while DSR repairs the single-optimal variables of the dominated solutions by sampling from the distribution of those variables of nondominated solutions. Secondly, this paper uses an adaptive offspring ratio adjustment strategy to control the offspring number generated by UP and CP according to the normalized Hausdorff distance between nondominated solution sets from the two latest generations of UP. This strategy is helpful to balance the intensity between exploration and exploitation and thereby ensures efficient search. Experimental results on CEC 2023 DCF test suite show that DpEA has a superior performance over six state-of-the-art algorithms.

Knowledge-embedded constrained multiobjective evolutionary algorithm based on structural network control principles for personalized drug targets recognition in cancer

The structural network control principle for identifying personalized drug targets (SNCPDTs) is a kind of constrained multiobjective optimization (CMO) problem with NP-hard features, which makes traditional mathematical methods difficult to adopt. Therefore, this study designs a knowledge-embedded multitasking constrained multiobjective evolutionary algorithm (KMCEA) to solve the SNCPDTs by mining relevant knowledge. Specifically, the relationships between two optimization objectives (minimizing the number of driver nodes and maximizing prior-known drug-target information) and constraints (guaranteeing network control) are analyzed from the perspective of CMO. We find that two objectives are difficult to optimize; thus two single-objective auxiliary tasks are created to optimize two objectives respectively, so as to maintain diversity along the Pareto front. Furthermore, we find that two optimization objectives have a complex reverse relation and a simple positive relation with constraints, respectively; thus, a population initialization method and a local auxiliary task are designed for two single-objective auxiliary tasks, respectively, so as to improve the performance of the algorithm on two objective functions. Finally, KMCEA is used to solve two kinds of models with three kinds of datasets. Compared with various methods, KMCEA can not only effectively discover clinical combinatorial drugs but also better solve the SNCPDTs regarding convergence and diversity.

Multipopulation Evolution-Based Dynamic Constrained Multiobjective Optimization Under Diverse Changing Environments

Dynamic constrained multiobjective optimization involves irregular changes in the distribution of the true Pareto-optimal fronts, drastic changes in the feasible region caused by constraints, and the movement directions and magnitudes of the optimal distance variables due to diverse changing environments. To solve these problems, we propose a multi-population evolution based dynamic constrained multiobjective optimization algorithm. In this algorithm, we design a tribe classification operator to divide the population into different tribes according to a feasibility check and the objective values, which is beneficial for driving the population toward the feasible region and Pareto-optimal fronts. Meanwhile, a population selection strategy is proposed to identify promising solutions from tribes and exploit them to update the population. The optimal values of the distance variables vary differently with dynamic environments, thus, we design a dynamic response strategy for solutions in different tribes that estimates their distances to approach the Pareto-optimal fronts and regenerates a promising population when detecting environmental changes. In addition, a scalable generator is designed to simulate diverse movement directions and magnitudes of the optimal distance variables in real-world problems under dynamic environments, obtaining a set of improved test problems. Experimental results show the effectiveness of test problems, and the proposed algorithm is impressively competitive with several chosen state-of-the-art competitors.

A surrogate-assisted a priori multiobjective evolutionary algorithm for constrained multiobjective optimization problems

We consider multiobjective optimization problems with at least one computationally expensive constraint function and propose a novel surrogate-assisted evolutionary algorithm that can incorporate preference information given a priori. We employ Kriging models to approximate expensive objective and constraint functions, enabling us to introduce a new selection strategy that emphasizes the generation of feasible solutions throughout the optimization process. In our innovative model management, we perform expensive function evaluations to identify feasible solutions that best reflect the decision maker’s preferences provided before the process. To assess the performance of our proposed algorithm, we utilize two distinct parameterless performance indicators and compare them against existing algorithms from the literature using various real-world engineering and benchmark problems. Furthermore, we assemble new algorithms to analyze the effects of the selection strategy and the model management on the performance of the proposed algorithm. The results show that in most cases, our algorithm has a better performance than the assembled algorithms, especially when there is a restricted budget for expensive function evaluations.