Constrained Multi-objective Optimization Research Articles

A dynamic dual-population co-evolution multi-objective evolutionary algorithm for constrained multi-objective optimization problems

Many multi-objective evolutionary algorithms are proposed to handle constrained multi-objective optimization problems. Nevertheless, they often fail to appropriately balance feasibility, convergence and diversity of the population. This paper proposed a dynamic dual-population co-evolution multi-objective evolutionary algorithm (DDCMEA) to solve this issue. In DDCMEA, a dynamic dual-population co-evolution strategy is employed to balance the convergence and the feasibility by dynamically adjusting the offspring number of the two populations. In the early stage of evolution, the algorithm mainly focuses on the convergence and more offspring of the first population are generated. In the late stage of evolution, the algorithm mainly focuses on the feasibility and more offspring of the second population are generated. Finally, feasible solutions with good convergence could be obtained. To further enhance the diversity of the offspring and obtain feasible solutions with a wide spread of distribution, the evolution operators of the genetic algorithm and the differential evolution are chosen as the search engines for the first population and the second population, respectively. The performance of DDCMEA is further tested through thirty-one bench-mark test problems and two real-world problems in comparison with other five state-of-the-art algorithms. The results show the proposed algorithm DDCMEA achieves competitive performance when handling constrained multi-objective optimization problems.

A dual-population constrained multi-objective evolutionary algorithm with variable auxiliary population size

Constrained multi-objective optimization problems (CMOPs) exist widely in the real world, which simultaneously contain multiple constraints to be satisfied and multiple conflicting objectives to be optimized. Therefore, the challage in addressing CMOPs is how to better balance constraints and objectives. To remedy this issue, this paper proposes a novel dual-population based constrained multi-objective evolutionary algorithm to solve CMOPs, in which two populations with different functions are employed. Specifically, the main population considers both objectives and constraints for solving the original CMOPs, while the auxiliary population is used only for optimization of objectives without considering constraints. In addition, a dynamic population size reducing mechanism is proposed, which is used to adjust the size of the auxiliary population, so as to reduce the consumption of computing resoruces in the later stage. Moreover, an independent external archive is set to store feasible solutions found by the auxiliary population, so as to provide high-quality feasible solutions for the main population. The experimental results on 55 benchmark functions show that the proposed algorithm exhibits superior or at least competitive performance compared to other state-of-the-art algorithms.

Evolutionary Multiobjective Design Approach for Robust Balancing of the Shaking Force, Shaking Moment, and Torque under Uncertainties: Application to Robotic Manipulators

In this paper, the environmental uncertainties are taken into account when designing a robotic manipulator to balance the shaking force, shaking moment, and torque. The proposed robust balancing design approach does not consider the probability distributions of the uncertainties and is addressed without dependence on specific trajectories. This is expressed as a nonlinear constrained multiobjective optimization problem in which the nominal performance in the time-independent terms of the shaking force balancing, the shaking moment balancing, and the torque delivery, as well as their three sensitivities to uncertainties, are simultaneously optimized to provide a set of link shapes that match link mass distributions in a single stage. The proposal is applied to a three-degree-of-freedom serial-parallel manipulator, and the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is used to solve the associated problem. Comparative results with other design approaches reveal that the selected design achieves a suitable tradeoff in balancing the shaking force balancing, the shaking moment balancing, and the torque delivery and their sensitivities, leading to a reduction in their values and variations under mass changes in the manipulator end-effector with different operating conditions (tasks).

A Survey on Evolutionary Constrained Multiobjective Optimization

Handling constrained multi-objective optimization problems (CMOPs) is extremely challenging, since multiple conflicting objectives subject to various constraints require to be simultaneously optimized. To deal with CMOPs, numerous constrained multi-objective evolutionary algorithms (CMOEAs) have been proposed in recent years, and they have achieved promising performance. However, there has been few literature on the systematic review of the related studies currently. This article provides a comprehensive survey for evolutionary constrained multi-objective optimization. We first review a large number of CMOEAs through categorization, and analyze their advantages and drawbacks in each category. Then, we summarize the benchmark test problems and investigate the performance of different constraint handling techniques and different algorithms, followed by some emerging and representative applications of CMOEAs. Finally, we discuss some new challenges and point out some directions of the future research in the field of evolutionary constrained multi-objective optimization.

Design of PCB Rogowski Coil Current Sensor With Low Droop Distortion

Wide bandgap devices (SiC and GaN) bring the potential for a great increase in power density of power electronics, but also pose a challenge to the bandwidth of the current sensor because of the high operating frequency. Rogowski coil is suitable for high-frequency current measurement due to its characteristic of differential output, but the existing design is difficult to take into account the sensitivity and mutual inductance at the same time, and faces the inherent defect of droop distortion. This article proposes an optimal printed circuit board (PCB) Rogowski coil structure design method for SiC MOSFET power module and a novel sample-filter-hold (SFH) method to minimize droop distortion in integrators. First, the optimal parameters with multiobjective constrained optimization of the Rogowski coil current sensor for the SiC-MOSFET power module are obtained. Then, the influence factors of droop distortion of the sensor in buck converters and inverters are quantitatively analyzed for the first time and an SFH suppression method is proposed, which can eliminate the offset caused by droop distortion reliably and accurately. Finally, the experimental results demonstrate that the proposed Rogowski coil design method and the SFH method for lower droop distortion achieve better low-frequency performance compared to the commercial Rogowski coil.

A Multi-Objective Carnivorous Plant Algorithm for Solving Constrained Multi-Objective Optimization Problems

Satisfying various constraints and multiple objectives simultaneously is a significant challenge in solving constrained multi-objective optimization problems. To address this issue, a new approach is proposed in this paper that combines multi-population and multi-stage methods with a Carnivorous Plant Algorithm. The algorithm employs the ϵ-constraint handling method, with the ϵ value adjusted according to different stages to meet the algorithm's requirements. To improve the search efficiency, a cross-pollination is designed based on the trapping mechanism and pollination behavior of carnivorous plants, thus balancing the exploration and exploitation abilities and accelerating the convergence speed. Moreover, a quasi-reflection learning mechanism is introduced for the growth process of carnivorous plants, enhancing the optimization efficiency and improving its global convergence ability. Furthermore, the quadratic interpolation method is introduced for the reproduction process of carnivorous plants, which enables the algorithm to escape from local optima and enhances the optimization precision and convergence speed. The proposed algorithm's performance is evaluated on several test suites, including DC-DTLZ, FCP, DASCMOP, ZDT, DTLZ, and RWMOPs. The experimental results indicate competitive performance of the proposed algorithm over the state-of-the-art constrained multi-objective optimization algorithms.

Optimization and Decision-Making Framework for Small Unmanned Aircraft Systems Fleet Design

Many unmanned aircraft applications require or benefit from deploying a fleet to perform a mission. Technology innovations such as rapid prototyping and microelectronics enable the use of unmanned aircraft systems (UAS) at lower costs, so the ability to design a bespoke UAS for a specific mission is an additional advantage. The problem of designing the new UAS and allocating the fleet of these aircraft is a challenging optimization problem. Using constrained multi-objective design optimization with an interactive multicriteria decision-making method, which involves the mission customer, the approach in this paper generates UAS designs and fleet allocations and determines preferential solutions for the customer. The paper presents a case study involving a UAS weather profiling mission for the aircraft sizing and fleet allocation and uses inputs from meteorological subject-matter experts acting as customers to demonstrate the functionality of the approach.

A coevolutionary algorithm based on reference line guided archive for constrained multiobjective optimization

The objective space of the constrained multiobjective optimization problem (CMOP) is constantly torn by the applied constraints. This makes evolutionary algorithms, which are driven by objectives, face greater difficulties in feasibility, convergence, and diversity. Most evolutionary algorithms will be trapped in local optimums such as a fake Pareto-optimal front or a mutilated Pareto-optimal front. To address this issue, this paper proposes an archive-assisted evolutionary framework with a novel archive structure and cooperative mechanism. A reference line guided archive (RA) is established to record the evolution of the unconstrained solutions. The updated criteria of RA are based on the distance from the individual to the reference line and the unconstrained dominance relation. A specially designed adaptive mating selection operator will select mating parents from RA and the main population according to the convergence of the main population and RA, respectively. The participation of RA in offspring reproduction is conducive to skipping infeasible regions for extensive searches. The performance of the archive-assisted nondominated sorting genetic algorithm (AA-NSGA), which embeds the proposed archive strategy into the nondominated sorting genetic algorithm II, is compared with four state-of-the-art constrained multiobjective evolutionary algorithms (MOEAs). The experimental results on 38 benchmark CMOPs and a reactor network design problem show that the proposed AA-NSGA has a high performance among the existing MOEAs in terms of feasibility, convergence, and diversity.

Probabilistic optimal power dispatch in a droop controlled islanded microgrid in presence of renewable energy sources and PHEV load demand

This paper proposes a stochastic optimal power allocation strategy for a droop controlled islanded microgrid (DCIMG) with a medium X/R ratio in presence of combined heat and power-based distributed generators (DGs), renewable generators, switched capacitor banks, plug-in hybrid electric vehicle (PHEV) loads, voltage dependent electric loads and heat loads. Optimal values of stochastic power demand due to PHEV charging and discharging load are determined based on the goals of achieving target state-of-charge of PHEV batteries and effective peak shaving during peak load periods, respectively. Active power of dispatchable DGs is optimally allocated depending on the simultaneous fulfillment of an economic objective, an environmental objective along with three network related objectives. Hong's 2m+1 point estimate method (PEM) and Hong's 2m+1 PEM coupled with Nataf transformation (NT) are employed to handle the uncorrelated uncertainties accompanied with PHEV load demand and the correlated uncertainties associated with spatially correlated renewable generation and load demand. Active and reactive power static droop coefficients, and frequency and voltage reference settings of the dispatchable DG units are considered as the control variables for optimal dispatching of active power. A modified hybrid particle swarm and grey wolf optimizer embedded in fuzzy framework is used to solve the constrained multi-objective optimization problem. The efficacy of the proposed framework is validated on a 33-node droop controlled islanded microgrid test system.

DeepMorpher: deep learning-based design space dimensionality reduction for shape optimisation

ABSTRACT In engineering design optimisations, the dimensionality of the design space plays a critical role in determining the number of computationally expensive evaluations required to explore the design space and reach convergence. Dimensionality reduction allows engineers to represent high-dimensional design spaces in a more compact form to reduce the computational cost while retaining design flexibility. In order to reduce the dimensions of design space for shape optimisation problems, we propose a deep learning-based architecture named DeepMorpher. Our proposed architecture is a PointNet-based encoder-decoder network, which can directly be trained on 3D point-cloud geometries, and generate simulation-ready high-quality geometries without any pre-processing or post-processing steps. Our proposed DeepMorpher can work with multiple baseline templates and allows explainability and disentanglement of learned low-dimensional latent space through sampling, interpolation and feature space visualisation. To evaluate our approach, we created an engineering dataset consisting of 3D ship hull designs. The quantitative and qualitative experimentation results of the present study demonstrate that DeepMorpher can outperform previous state-of-the-art autoencoder-based shape-generative models by several orders of magnitudes. To demonstrate the high representation capacity and compactness of learned low-dimensional latent space, we employed DeepMorpher within a simulation-based design optimisation framework to perform a real-world constrained multi-objective optimisation to find optimal ship hull designs. Highlights A deep learning-based autoencoder network for reducing the dimensionality of design space in shape optimisation is proposed. The proposed network learns an explainable and disentangled low-dimensional latent space where each dimension captures different attributes of high-dimensional input shape. The proposed network incorporates free-form deformation within the architecture to generate simulation-ready geometries with high visual quality. The efficacy of the proposed network was demonstrated inside a simulation-based design optimisation framework to perform multi-objective Bayesian optimisation using multiple baseline templates to find optimal designs.

Data Offloading Enabled by Heterogeneous UAVs for IoT Applications Under Uncertain Environments

With the continuous expansion of the Internet of Things (IoT) application scope, there are growing IoT scenarios that lack the coverage of wireless communication networks have the demand for data offloading. Efficient data transmission has been the main concern of these applications. Thus, data offloading within the limited communication environment has become a hotspot in both industry and academia. Since unmanned aerial vehicles (UAVs) can move across regions to make up for the communication gap caused by the loss of wireless communication networks, a lot of in-depth studies on UAV-enabled data offloading have been conducted. Nevertheless, few studies to date consider the uncertain user status and the heterogeneous UAV capabilities, which is however more practical and needs more attention. In this article, we propose an innovative framework to dynamically estimate user status information and determine the UAV scheduling strategy. On this basis, the heterogeneous UAV-enabled data offloading is modeled as a constrained multiobjective optimization problem, whose purpose is to lower the user data queue length while extending the working time of the UAV. Moreover, a differential evolution-based dynamic objective approximation method—RUDDER is proposed to solve the constrained multiobjective optimization problem. Through rigorous mathematical proof, we prove that RUDDER can consistently guide the population to approach the optimization solution with polynomial-level time complexity. To verify the effectiveness of the proposed RUDDER, extensive experiments are conducted to compare it with five comparison algorithms. The experimental results demonstrate the superiority of the RUDDER in terms of energy saving, time efficiency, and adaptability.

Constrained multi-objective optimization problem model to design multi-band terahertz metamaterial absorbers

The multi-band metamaterial absorbers studied today offer optimal sensing performance by maximizing the absorption at resonance frequencies. A constrained multi-objective optimization problem (CMOP) model is proposed to intelligently obtain the optimized geometrical parameters of the designed MA for optimal multi-band absorption. The proposed multi-band terahertz metamaterial absorber is formed by a patterned metallic patches (symmetric snowflake-shaped resonators) layer and a continuous metallic layer separated by a dielectric layer. The simulation results show that there are three discrete narrow resonance peaks with the absorption of 99.1%, 90.0%, and 99.9% in the range of 0.5–2 THz after being optimized by the proposed CMOP model. The reflection loss of all resonance modes is improved significantly compared with the conventional brute-force approach. Specifically, reflection loss at the highest resonance frequency is suppressed from -6.76 dB to -28.17 dB. Consequently, the reported MA design can be used as a refractive index sensor with the highest sensitivity of 495 GHz/RIU and the figure of merit (FoM) of 8.9 RIU−1 through a refractive index ranging from 1.0 to 1.6 at the analyte thickness of 18.5 μm. It is worth noting that most of the liquid samples have a refractive index ranging from 1.0 to 1.6. Therefore, the reported sensor can be used for liquid detection with high sensitivity.

Placement and size-oriented heat dissipation optimization for antenna module in space solar power satellite based on interval dimension-wise method

Space solar power satellite is one of the large space systems for supplying solar energy in the future, and its effective thermal management and heat dissipation can affect safety and efficiency. To balance the mass and temperature distribution in the antenna module, a novel placement and size-oriented heat dissipation optimization is proposed in space solar power satellite based on an interval dimension-wise method considering both design and uncertain variables. To improve the accuracy of the interval propagation, a novel interval dimension-wise method is investigated based on the Legendre polynomial approximation model and the union operator of the boundary points combinations. A novel mapping matrix with sensitivity analysis is developed to determine the final design variables from the candidate ones, which constitutes a relationship between the uncertain parameters and design variables. The placement and size-oriented heat dissipation optimization is solved using the coevolutionary constrained multi-objective optimization framework (CCMO), and thermophysical response intervals of optimal heat dissipation layout can be obtained using the interval dimension-wise method again. The effectiveness of the proposed method is verified by two numerical examples, which can reflect the applicability of the proposed heat dissipation.

The Hypervolume Newton Method for Constrained Multi-Objective Optimization Problems

Recently, the Hypervolume Newton Method (HVN) has been proposed as a fast and precise indicator-based method for solving unconstrained bi-objective optimization problems with objective functions. The HVN is defined on the space of (vectorized) fixed cardinality sets of decision space vectors for a given multi-objective optimization problem (MOP) and seeks to maximize the hypervolume indicator adopting the Newton–Raphson method for deterministic numerical optimization. To extend its scope to non-convex optimization problems, the HVN method was hybridized with a multi-objective evolutionary algorithm (MOEA), which resulted in a competitive solver for continuous unconstrained bi-objective optimization problems. In this paper, we extend the HVN to constrained MOPs with in principle any number of objectives. Similar to the original variant, the first- and second-order derivatives of the involved functions have to be given either analytically or numerically. We demonstrate the applicability of the extended HVN on a set of challenging benchmark problems and show that the new method can be readily applied to solve equality constraints with high precision and to some extent also inequalities. We finally use HVN as a local search engine within an MOEA and show the benefit of this hybrid method on several benchmark problems.

Transient motion of an industrial mixer with elastically suspended work head

This paper investigates the transient behavior of a novel design of an industrial mixer incorporating additional work head movement. In contrast to conventional models, this mixer consists of an elastic torsion bar with its longitudinal axis oriented orthogonally to the main drive shaft axis. Using the elastic torsion bar to hang the workhead adds an additional degree of freedom, greatly improving the mixing of fluids or powders. The motion of the electro-mechanical device was simulated using a system of nonlinear differential equations that consider the dynamic characteristics of the driving electric motor, as derived from the theorem for the rate of change of angular momentum. A numerical analysis was carried out to study the transient processes in the system under different working conditions. Finally, a constrained multiobjective optimization was performed to identify the optimal design variable values that minimize the force and power characteristics of the mixer.

Shift-Based Penalty for Evolutionary Constrained Multiobjective Optimization and its Application.

This article presents a new constraint-handling technique (CHT), called shift-based penalty (ShiP), for solving constrained multiobjective optimization problems. In ShiP, infeasible solutions are first shifted according to the distributions of their neighboring feasible solutions. The degree of shift is adaptively controlled by the proportion of feasible solutions in the current parent and offspring populations. Then, the shifted infeasible solutions are penalized based on their constraint violations. This two-step process can encourage infeasible solutions to approach/enter the feasible region from diverse directions in the early stage of evolution, and guide diverse feasible solutions toward the Pareto optimal solutions in the later stage of evolution. Moreover, ShiP can achieve an adaptive transition from both diversity and feasibility in the early stage of evolution to both diversity and convergence in the later stage of evolution. ShiP is flexible and can be embedded into three well-known multiobjective optimization frameworks. Experiments on benchmark test problems demonstrate that ShiP is highly competitive with other representative CHTs. Further, based on ShiP, we propose an archive-assisted constrained multiobjective evolutionary algorithm (CMOEA), called ShiP+, which outperforms two other state-of-the-art CMOEAs. Finally, ShiP is applied to the vehicle scheduling of the urban bus line successfully.

Balancing Constraints and Objectives by Considering Problem Types in Constrained Multiobjective Optimization.

Constrained multiobjective optimization problems widely exist in real-world applications. To handle them, the balance between constraints and objectives is crucial, but remains challenging due to non-negligible impacts of problem types. In our context, the problem types refer particularly to those determined by the relationship between the constrained Pareto-optimal front (PF) and the unconstrained PF. Unfortunately, there has been little awareness on how to achieve this balance when faced with different types of problems. In this article, we propose a new constraint handling technique (CHT) by taking into account potential problem types. Specifically, inspired by the prior work, problems are classified into three primary types: 1) I; 2) II; and 3) III, with the constrained PF being made up of the entire, part and none of the unconstrained counterpart, respectively. Clearly, any problem must be one of the three types. For each possible type, there exists a tailored mechanism being used to handle the relationships between constraints and objectives (i.e., constraint priority, objective priority, or the switch between them). It is worth mentioning that exact problem types are not required because we just consider their possibilities in the new CHT. Conceptually, we show that the new CHT can make a tradeoff among different types of problems. This argument is confirmed by experimental studies performed on 38 benchmark problems, whose types are known, and a real-world problem (with unknown types) in search-based software engineering. Results demonstrate that within both decomposition-based and nondecomposition-based frameworks, the new CHT can indeed achieve a good tradeoff among different problem types, being better than several state-of-the-art CHTs.

Parameter space exploration for the probabilistic damage stability method for dry cargo ships

The prediction of the statutory attained subdivision index is a challenging issue for the initial design of ships due to the design freedom offered by a probabilistic damage stability assessment. To this end, optimisation techniques integrated with a parametric model of the internal layout may generate a preliminary subdivision design, fulfilling damage stability regulations and cargo volume requirements. The present study explores using a multi-objective constrained optimisation algorithm coupled with a parametric model of a single hold cargo vessel, first investigating two design goal alternatives and secondly performing a global sensitivity analysis on the design variables for the most promising solution. The adoption, in parallel, of state-of-the-art practices shows the validity of the obtained solutions and the time benefits for designers. Nonetheless, the non-linear nature of probabilistic damage stability does not allow for clearly identifying the most impactful parameters on the attained survivability index.

Parameter Optimization of Intercalated Meltblown Nonwovens Based on NSGA-II

The preparation process parameters of intercalated meltblown nonwoven materials are complicated, and the relationship between process parameters, structural variables, and product performance needs to be investigated to establish a good mechanism for product performance regulation. In this study, we first used Wilcoxon test and Pearson correlation analysis to investigate the effect of intercalation rate on structural variables and product performance. Then, regression models were constructed to predict the values of each structural variable under different combinations of process parameters. Finally, we constructed a multi-objective constrained optimization problem based on the stepwise regression model and the product variable conditions. The problem was solved using the NSGA-II algorithm. The optimal was achieved when the acceptance distance was 2.892 cm and the hot air speed was 2000 r/min.

An Adaptive Two-Population Evolutionary Algorithm for Constrained Multi-Objective Optimization Problems

An Adaptive Two-Population Evolutionary Algorithm for Constrained Multi-Objective Optimization Problems