Constrained Multi-objective Optimization Research Articles

Paired Offspring Generation for Constrained Large-Scale Multiobjective Optimization

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization.

Multi-Objective Optimization of Deadline and Budget-Aware Workflow Scheduling in Uncertain Clouds

Cloud technologies are being used nowadays to cope with the increased computing and storage requirements of services and applications. Nevertheless, decisions about resources to be provisioned and the corresponding scheduling plans are far from being easily made especially because of the variability and uncertainty affecting workload demands as well as technological infrastructure performance. In this paper we address these issues by formulating a multi-objective constrained optimization problem aimed at identifying the optimal scheduling plans for scientific workflows to be deployed in uncertain cloud environments. In particular, we focus on minimizing the expected workflow execution time and monetary cost under probabilistic constraints on deadline and budget. According to the proposed approach, this problem is solved offline, that is, prior to workflow execution, with the intention of allowing cloud users to choose the plan of the Pareto optimal set satisfying their requirements and preferences. The analysis of the combined effects of cloud uncertainty and probabilistic constraints has shown that the solutions of the optimization problem are strongly affected by uncertainty. Hence, to properly provision cloud resources, it is compelling to precisely quantify uncertainty and take explicitly into account its effects in the decision process.

A Constrained Multi/Many-Objective Particle Swarm Optimization Algorithm With a Two-Level Balance Scheme

Constrained multi-objective optimization problems are common in practical engineering and are more difficult to handle than unconstrained problems. In general, it is necessary to find a balance between the convergence and diversity of solutions, as well as its feasibility. For the constrained multi/many-objective optimization problem, a particle swarm optimization algorithm based on a two-level balance strategy is proposed. In contrast to existing views, the first level of the proposed algorithmic framework emphasizes convergence, while diversity and feasibility are considered together as the second-level scheme. An ensemble fitness ranking was used to improve the convergence of the proposed algorithm. To balance the diversity and solution feasibility, the solutions are selected by combining the angles between the solutions using the constraint dominance principle. A penalty-based boundary-crossing approach is used as a utility function to calculate the fitness of the populations, which is compared with six state-of-the-art constrained multi/many-objective evolutionary optimization algorithms on multiple constrained test suites, and the experimental results show that the proposed algorithm is highly competitive in most test problems. Furthermore, to illustrate the effect of different utility functions on the performance of the algorithm, the Chebyshev decomposition method is employed and compared with the former, and the results show that different utility functions need to be chosen to cope with problems of different characteristics.

A constraint handling technique using compound distance for solving constrained multi-objective optimization problems

Guiding the working population to evenly explore the valuable areas which are not dominated by feasible solutions is important in the process of dealing with constrained multi-objective optimization problems (CMOPs). To this end, according to the angular distance and $ \ell_p $-norm, this paper introduces a new compound distance to measure individual's search diameter in the objective space. After that, we propose a constraint handling technique using the compound distance and embed it in evolutionary algorithm for solving CMOPs. In the proposed algorithm, the individuals with large search diameters in the valuable areas are given priority to be preserved. This can prevent the working population from getting stuck in the local areas and then find the optimal solutions for CMOPs more effectively. A series of numerical experiments show that the proposed algorithm has better performance and robustness than several existing state-of-the-art constrained multi-objective evolutionary algorithms in dealing with different CMOPs.

Adaptive ε-Constraint Multi-Objective Evolutionary Algorithm Based on Decomposition and Differential Evolution

To improve distribution and convergence of the obtained solution set in constrained multi-objective optimization problems, this paper presents an adaptive $\varepsilon $ -constraint multi-objective evolutionary algorithm based on decomposition and differential evolution ( $\varepsilon $ -MOEA/D-DE). First, an adaptive $\varepsilon $ -constraint strategy based on both evolution generation and constraint violation is designed to make better use of excellent evolution individuals and improve population diversity. Then, an adaptive differential evolution (DE) mutation strategy with full utilization of infeasible individuals is proposed to increase search efficiency and avoid falling into the local optimum. Finally, a replacement mechanism is suggested to take advantage of the infeasible individuals in the population with better objective function values and constraint violation degree, and thus both diversity and convergence are well coordinated. A comparative experiment with four other excellent constrained multi-objective algorithms was implemented on standard constrained multi-objective optimization problems (CF series), and the results showed that the diversity and convergence of our algorithm were both improved.

Quasi Oppositional Population Based Global Particle Swarm Optimizer With Inertial Weights (QPGPSO-W) for Solving Economic Load Dispatch Problem

In recent years, power companies have shown increasing interest in making strategic decisions to maintain profitable energy systems. Economic Load Dispatch (ELD) is a complex decision-making process where the output power of the entire power generating units must be set in a way that results in the overall economic operation of the power system. Moreover, it is a constrained multi-objective optimization problem. Now a days, there is a tendency to use metaheuristic methods to deal with the complexity of the ELD problem. Particle swarm optimization (PSO) is a subclass of metaheuristic methods inspired by fish schooling and bird flocking behaviors. However, the optimization performance of the PSO is highly dependent on fitness landscapes and can lead to local optima stagnation and premature convergence. Therefore, in the proposed study, two new variants of the PSO called global particle swarm optimizer with inertial weights (GPSO-w) and quasi-oppositional population based global particle swarm optimizer with inertial weights (QPGPSO-w) are proposed to address the complexity of the ELD problem. The ELD problem is formulated as an optimization problem and validation of the proposed methods is performed on IEEE standards (3, 6, 13, 15, 40 & 140) unit Korean grid ELD test systems under numerous constraints and the obtained results are compared with the several recent techniques presented in the literature. The results obtained with convex systems showed excellent cost-effectiveness, while for non-convex systems sequential quadratic programming (SQP) optimizer was added to discover global minima even more efficiently. The proposed techniques were successful in solving the ELD problem and yielded better results compared to the reported results in the selected cases. It is further inferred that the proposed techniques with less algorithmic parameters reflected improved exploration and convergence characteristics.

A New Arithmetic Optimization Algorithm for Solving Real-World Multiobjective CEC-2021 Constrained Optimization Problems: Diversity Analysis and Validations

In this paper, a new Multi-Objective Arithmetic Optimization Algorithm (MOAOA) is proposed for solving Real-World constrained Multi-objective Optimization Problems (RWMOPs). Such problems can be found in different fields, including mechanical engineering, chemical engineering, process and synthesis, and power electronics systems. MOAOA is inspired by the distribution behavior of the main arithmetic operators in mathematics. The proposed multi-objective version is formulated and developed from the recently introduced single-objective Arithmetic Optimization Algorithm (AOA) through an elitist non-dominance sorting and crowding distance-based mechanism. For the performance evaluation of MOAOA, a set of 35 constrained RWMOPs and five ZDT unconstrained problems are considered. For the fitness and efficiency evaluation of the proposed MOAOA, the results obtained from the MOAOA are compared with four other state-of-the-art multi-objective algorithms. In addition, five performance indicators, such as Hyper-Volume (HV), Spread (SD), Inverted Generational Distance (IGD), Runtime (RT), and Generational Distance (GD), are calculated for the rigorous evaluation of the performance and feasibility study of the MOAOA. The findings demonstrate the superiority of the MOAOA over other algorithms with high accuracy and coverage across all objectives. This paper also considers the Wilcoxon signed-rank test (WSRT) for the statistical investigation of the experimental study. The coverage, diversity, computational cost, and convergence behavior achieved by MOAOA show its high efficiency in solving ZDT and RWMOPs problems.

The Pareto Tracer for General Inequality Constrained Multi-Objective Optimization Problems

Problems where several incommensurable objectives have to be optimized concurrently arise in many engineering and financial applications. Continuation methods for the treatment of such multi-objective optimization methods (MOPs) are very efficient if all objectives are continuous since in that case one can expect that the solution set forms at least locally a manifold. Recently, the Pareto Tracer (PT) has been proposed, which is such a multi-objective continuation method. While the method works reliably for MOPs with box and equality constraints, no strategy has been proposed yet to adequately treat general inequalities, which we address in this work. We formulate the extension of the PT and present numerical results on some selected benchmark problems. The results indicate that the new method can indeed handle general MOPs, which greatly enhances its applicability.

A multi-objective evolutionary algorithm for steady-state constrained multi-objective optimization problems

Many multi-objective evolutionary algorithms (MOEAs) are developed to solve constrained multi-objective optimization problems (CMOPs). However, they encounter low efficiency for steady-state CMOPs which are to optimize a known feasible solution named the current steady-state operation point. This paper proposes a multi-objective evolutionary algorithm (MOEA), termed FACE, for tackling the steady-state CMOPs. In FACE, the known feasible solution is maintained in the second population and evolves together with the main population. The main population is evolved by global search without handling constraints to accelerate the convergence. And the second population is evolved by local search to hold and achieve more feasible solutions. Two kinds of performance comparison between FACE and five representative MOEAs are made: the first using two cases of steady-state multi-source compressed-air pipeline optimization problems to evaluate the performance of FACE on real life applications, and the second using a set of bench-mark test problems of CMOPs assigned with a known feasible solution to further assess the scalability of FACE. The results demonstrate the efficiency and the scalability of FACE with a potential for steady-state CMOPs.

Purpose-directed two-phase multiobjective differential evolution for constrained multiobjective optimization

Abstract When solving constrained multiobjective optimization problems by evolutionary algorithm, the key challenge is how to achieve the balance among convergence, diversity, and feasibility. To deal with this challenge, a purpose-directed two-phase multiobjective differential evolution (PDTP-MDE) algorithm is developed in this paper. The main idea of PDTP-MDE is that the whole evolution process is divided into two sequential phases according to the expected purpose of each stage. To be specific, the first phase aims at keeping the balance between convergence and diversity, while the feasibility is taken as an auxiliary indicator. In this way, the population is capable of exploring different potential areas and avoiding to be trapped into local ones, thus providing more information about convergence and diversity for the later evolution process. Afterwards, the second phase mainly tends to maintain feasibility and diversity by selecting and using some promising infeasible solutions according to the population evolution status. In addition, an archive is maintained after each phase to preserve the superior feasible Pareto solutions found so far. By the above processes, the feasible Pareto front with well convergence and well diversity is obtained. The comprehensive experiments on 42 benchmark problems from three test suites demonstrate the superiority and competitiveness of the proposed PDTP-MDE, in comparison with other state-of-the-art constrained multiobjective evolutionary algorithms.

MOEA/UE: A novel multi-objective evolutionary algorithm using a uniformly evolving scheme

In multi-objective optimization, the Pareto-optimal solutions must be widely and uniformly distributed on the Pareto-optimal front. When this condition is satisfied, decision makers can choose satisfactory solutions. In this paper, gaps in the evolving nondominated front are filled by a multi-objective evolutionary algorithm using a uniformly evolving scheme (MOEA/UE), thus generating a uniform and wide Pareto-optimal front. The MOEA/UE first calculates the distance between the points on the nondominated front and identifies those with relatively wide gaps. At the widely spaced points, the objective and constraint functions are approximated by linear functions expressed as Taylor series in auxiliary linear programs. Finally, the MOEA/UE solves these auxiliary linear programs and retains the optimal or intermediate solutions as the potentially superior individuals that will be selected in the next generation of the population. Convergence toward potentially superior solutions is accelerated by a mutation operator (a basic differential evolution operator). In constrained multi-objective optimization problems, the MOEA/UE implements a new constraint-handling strategy that pulls infeasible solutions into or nearby the feasible domain. Experimental results illustrate that MOEA/UE obtains wider and more uniform Pareto-optimal fronts and better Pareto-optimal solutions.

Multi-objective constrained Bayesian optimization for structural design

The planning and design of buildings and civil engineering concrete structures constitutes a complex problem subject to constraints, for instance, limit state constraints from design codes, evaluated by expensive computations such as finite element (FE) simulations. Traditionally, the focus has been on minimizing costs exclusively, while the current trend calls for good trade-offs of multiple criteria such as sustainability, buildability, and performance, which can typically be computed cheaply from the design parameters. Multi-objective methods can provide more relevant design strategies to find such trade-offs. However, the potential of multi-objective optimization methods remains unexploited in structural concrete design practice, as the expensiveness of structural design problems severely limits the scope of applicable algorithms. Bayesian optimization has emerged as an efficient approach to optimizing expensive functions, but it has not been, to the best of our knowledge, applied to constrained multi-objective optimization of structural concrete design problems. In this work, we develop a Bayesian optimization framework explicitly exploiting the features inherent to structural design problems, that is, expensive constraints and cheap objectives. The framework is evaluated on a generic case of structural design of a reinforced concrete (RC) beam, taking into account sustainability, buildability, and performance objectives, and is benchmarked against the well-known Non-dominated Sorting Genetic Algorithm II (NSGA-II) and a random search procedure. The results show that the Bayesian algorithm performs considerably better in terms of rate-of-improvement, final solution quality, and variance across repeated runs, which suggests it is well-suited for multi-objective constrained optimization problems in structural design.

Seismic response control of adjacent high-rise buildings linked by the Tuned Liquid Column Damper-Inerter (TLCDI)

A novel passive control device termed Tuned Liquid Column Damper-Inerter (TLCDI) is adopted to control the seismic response of adjacent high-rise buildings. The inerter of the proposed TLCDI scheme exploits the large relative acceleration between the two adjacent buildings, producing a significant resisting force. Firstly, the governing nonlinear equations of motion of the TLCDI-linked adjacent high-rise buildings under earthquake excitation are formulated. For design purposes, the inherent nonlinear behavior of the TLCDI is linearized by adopting an equivalent stochastic linearization method. Subsequently, a benchmark structure consisting of two adjacent high-rise buildings linked by a corridor with a roller support and equipped with a TLCDI is examined. Eight characteristic parameters of the TLCDI are optimized through two distinct constrained multi-objective optimization problems by an evolutionary algorithm aimed at minimizing either the peak acceleration or the interstory-drift ratio of the two adjacent benchmark structures, respectively. Mitigation effects of the optimally designed TLCDIs on the responses of the benchmark structure are assessed considering a set of ten natural ground-motion records. It is found that the optimum TLCDI achieves mean reduction ratios of 15.7 and 26.1% on the peak acceleration of both the buildings, respectively. Conversely, the TLCDI fails to significantly reduce the interstory drift ratios. Robustness analysis of the optimally designed TLCDI is investigated under perturbations to the frequency and damping ratios. Some practical considerations are finally discussed concerning the feasibility of employing the proposed TLCDI in the considered benchmark project.

Constrained Multiobjective Optimization: Test Problem Construction and Performance Evaluations

Constrained multiobjective optimization abounds in practical applications and is gaining growing attention in the evolutionary computation community. Artificial test problems are critical to the progress in this research area. Nevertheless, many of them lack important characteristics, such as scalability and variable dependencies, which may be essential in benchmarking modern evolutionary algorithms. This article first proposes a new framework for constrained test problem construction. This framework splits a decision vector into position and distance variables and forces their optimal values to lie on a nonlinear hypersurface such that the interdependencies can be introduced among the position ones and among the distance ones individually. In this framework, two kinds of constraints are designed to introduce convergence-hardness and diversity-hardness, respectively. The first kind introduces infeasible barriers in approaching the optima, and at the same time, makes the position and distance variables interrelate with each other. The second kind restricts the feasible optimal regions such that different shapes of Pareto fronts can be obtained. Based on this framework, we construct 16 scalable and constrained test problems covering a variety of difficulties. Then, in the second part of this article, we evaluate the performance of some state of the art on the proposed test problems, showing that they are quite challenging and there is room for further enhancement of the existing algorithms. Finally, we discuss in detail the source of difficulties presented in these new problems.

Differential evolution with infeasible-guiding mutation operators for constrained multi-objective optimization

Constrained multi-objective optimization problems (CMOPs) are common in engineering design fields. To solve such problems effectively, this paper proposes a new differential evolution variant named IMDE with infeasible-guiding mutation operators and a multistrategy technique. In IMDE, an infeasible solution with lower objective values is maintained for each individual in the main population, and this infeasible solution is then incorporated into some common differential evolution’s mutation operators to guide the search toward the region with promising objective values. Moreover, multiple mutation strategies and control parameters are adopted during the trial vector generation procedure to enhance both the convergence and the diversity of differential evolution. The superior performance of IMDE is validated via comparisons with some state-of-the-art constrained multi-objective evolutionary algorithms over 3 sets of artificial benchmarks and 4 widely used engineering design problems. The experiments show that IMDE outperforms other algorithms or obtains similar results. It is an effective approach for solving CMOPs, basically due to the use of infeasible-guiding mutation operators and multiple strategies.

A Kriging Model-Based Expensive Multiobjective Optimization Algorithm Using R2 Indicator of Expectation Improvement

Most of the multiobjective optimization problems in engineering involve the evaluation of expensive objectives and constraint functions, for which an approximate model-based multiobjective optimization algorithm is usually employed, but requires a large amount of function evaluation. Aiming at effectively reducing the computation cost, a novel infilling point criterion EIR2 is proposed, whose basic idea is mapping a point in objective space into a set in expectation improvement space and utilizing the R2 indicator of the set to quantify the fitness of the point being selected as an infilling point. This criterion has an analytic form regardless of the number of objectives and demands lower calculation resources. Combining the Kriging model, optimal Latin hypercube sampling, and particle swarm optimization, an algorithm, EIR2-MOEA, is developed for solving expensive multiobjective optimization problems and applied to three sets of standard test functions of varying difficulty and comparing with two other competitive infill point criteria. Results show that EIR2 has higher resource utilization efficiency, and the resulting nondominated solution set possesses good convergence and diversity. By coupling with the average probability of feasibility, the EIR2 criterion is capable of dealing with expensive constrained multiobjective optimization problems and its efficiency is successfully validated in the optimal design of energy storage flywheel.

A Coevolutionary Framework for Constrained Multiobjective Optimization Problems

Constrained multiobjective optimization problems (CMOPs) are challenging because of the difficulty in handling both multiple objectives and constraints. While some evolutionary algorithms have demonstrated high performance on most CMOPs, they exhibit bad convergence or diversity performance on CMOPs with small feasible regions. To remedy this issue, this article proposes a coevolutionary framework for constrained multiobjective optimization, which solves a complex CMOP assisted by a simple helper problem. The proposed framework evolves one population to solve the original CMOP and evolves another population to solve a helper problem derived from the original one. While the two populations are evolved by the same optimizer separately, the assistance in solving the original CMOP is achieved by sharing useful information between the two populations. In the experiments, the proposed framework is compared to several state-of-the-art algorithms tailored for CMOPs. High competitiveness of the proposed framework is demonstrated by applying it to 47 benchmark CMOPs and the vehicle routing problem with time windows.

Sizing of a Battery Pack Based on Series/Parallel Configurations for a High-Power Electric Vehicle as a Constrained Optimization Problem

The design of electric vehicle batteries is a challenge requiring significant computational, human and material resources. This paper proposes a new framework to design a battery pack that allows quick iterations between designs and computes optimized configurations to give more information about the best available solutions. The battery pack is represented as a series/parallel configuration in a Constraint Satisfaction Problem (CSP) that is solved to exploit every possible configuration for specific vehicle requirements. The result is then used in a multi-objective Constrained Optimization Problem (COP) to determine the non-dominated (Pareto optimal) solutions according to selected objectives functions. Non-dominated solutions are computed by optimizing weighted-sum scalar functions as well as minimizing different types of distances to a design reference. A case study is explored to design a battery pack for an electric motorcycle. A 24s/6p battery module designed with this methodology has been validated by experiments and the battery pack has successfully been used at MotoStudent in 2018.

A Cooperative Evolutionary Framework Based on an Improved Version of Directed Weight Vectors for Constrained Multiobjective Optimization With Deceptive Constraints.

When solving constrained multiobjective optimization problems (CMOPs), the most commonly used way of measuring constraint violation is to calculate the sum of all constraint violations of a solution as its distance to feasibility. However, this kind of constraint violation measure may not reflect the distance of an infeasible solution from feasibility for some problems, for example, when an infeasible solution closer to a feasible region does not have a smaller constraint violation than the one farther away from a feasible region. Unfortunately, no set of artificial benchmark problems focusing on this area exists. To remedy this issue, a set of CMOPs with deceptive constraints is introduced in this article. It is the first attempt to consider CMOPs with deceptive constraints (DCMOPs). Based on our previous work, which designed a set of directed weight vectors to solve CMOPs, this article proposes a cooperative framework with an improved version of directed weight vectors to solve DCMOPs. Specifically, the cooperative framework consists of two switchable phases. The first phase uses two subpopulations-one to explore feasible regions and the other to explore the entire space. The two subpopulations provide useful information about the optimal direction of objective improvement to each other. The second phase aims mainly at finding Pareto-optimal solutions. Then an infeasibility utilization strategy is used to improve the objective function values. The two phases are switchable based on the information found to date at any time in the evolutionary process. The experimental results show that this method significantly outperforms the algorithms with which it is compared on most of the DCMOPs, in terms of reliability and stability in finding a set of well-distributed optimal solutions.

Efficient Unbalanced Three-Phase Network Modelling for Optimal PV Inverter Control

High penetration levels of renewable energy generation in the distribution network require voltage regulation to avoid excessive voltage at generating nodes. To effectively control the network and optimize network hosting capacity, the distribution system operator must have an efficient model for power flow analysis. This paper presents the formulas and steps to express the power flow analysis equations of an unbalanced 3-phase network in matrix form suited to programmed solutions. A benchmark MATLAB/Simulink network with unbalanced distribution lines, photovoltaic inverters, and loads is built to verify the matrix model. To demonstrate the application of the model, the control of reverse energy flow from the photovoltaic inverters to keep the voltage in the network below the regulated level is simulated. Two decentralized control algorithms are applied in the network, including an on/off and a multi-objective constrained optimization controller. The detailed construction of the optimization problem for the 3-phase network in matrix form, which is consistent with the power flow calculation, is described. Simulation with the control methods over a day shows that the total active power of the on/off and optimized controllers deliver 41.92% and 99.39% of the available solar power, respectively, while maintaining the network node voltages within limits.