Real-world optimization problems often involve multiple conflicting objective functions to be optimized simultaneously. The optima to such multiobjective optimization problems are trade-off solutions constituting a so-called Pareto front. Practically, a few representative solutions are of interest for decision-making in lieu of the entire Pareto front as the latter is computationally and cognitively demanding to generate and analyze, respectively. However, finding a set of a few uniformly distributed solutions on the Pareto front is challenging for large-scale problems with irregularly-shaped Pareto fronts. We address this gap for a special type of problems inspired by a real-world case study in forest management. The proposed iterative Pareto representer uses an achievement scalarizing function to identify and prune certain search directions that are likely to yield duplicate or densely packed solutions. A subset selection technique is then used to identify the next search direction for achieving a relatively uniform distribution. Numerical experiments on 23 problems demonstrate the improved performance of the proposed approach over three alternate ones.
The state-of-the-art evolutionary algorithms (EAs), developed to solve constrained multi/many-objective optimization problems (M/MaOPs), mostly deal with deterministic design variables causing no uncertainty in their implementation. However, from a practical point of view, it is imperative to consider unavoidable uncertainties in implementing design variables and parameters. In the presence of hard constraints, a slight change in one or more variables may cause a feasible optimal solution to become infeasible upon implementation and result in a failure during operation. The literature suggests reliability-based techniques for solving such M/MaOPs to obtain a Reliable Front (ReF), rather than a Pareto-optimal front (PF). A ReF is usually either a part of the PF or a completely different set of trade-off solutions dominated by the deterministic PF. However, in the presence of decision-making, computing the complete ReF may not be necessary, as the focus would be to locate only the preferred part of the ReF, dictated by the objective preference information provided by a decision-maker (DM). The proposed approach incorporates DM's preferences and variable uncertainty information a priori. The proposed Reliability-based Multi-criteria Decision-making (ReMCDM) approach uses the hybrid mean value (HMV) method for constrained handling under variable uncertainties and R-NSGA-III for preference incorporation by DMs to conduct MCDM. Results obtained by the proposed method, implemented on several benchmark and realworld engineering examples, encourage future EMO research combining constraint handling, uncertainty in decision variables, and preference incorporation.
Coverage tasks have garnered significant interest in recent years due to their applications in fields such as environmental monitoring, search and rescue, and robotic exploration. Swarm robotics has emerged as a promising approach to tackle these challenges, leveraging collective behaviors to enhance efficiency. However, the effectiveness of swarm algorithms often hinges on the careful tuning of parameters, which can be influenced by varying environmental features. This study investigates the impact of these features on the performance of optimized collective motion in swarm robotics for solving multi-objective coverage problems. Through a comprehensive sensitivity analysis, we evaluate how changes in parameters-such as arena size, number of robots, obstacle density, and obstacle structure-affect swarm performance in solving coverage problems. The analysis focuses on key performance metrics across multiple objectives: coverage percentage, coverage time, robot connectivity, and collision rates. Our findings provide critical insights into the adaptability of swarm algorithms, highlighting the importance of environmental context in optimizing swarm efficacy across competing objectives. These results contribute to the development of more robust and adaptive swarm-based solutions for multi-objective coverage tasks.
In this paper, we propose an efficient optimization approach for multi-objective optimization problems (MOPs). The proposed utilizes an autoencoder-based surrogate model and gradient descent to search for Pareto optimal solutions. Evolutionary multi-objective optimization algorithms excel in finding diverse non-dominated solutions but often suffer from excessive execution times, particularly for real-world applications. Surrogate models offer a promising alternative by reducing the computational cost of objective function evaluations, and as such, have become the focus of increasing research interest. However, challenges persist in achieving accurate approximations and reducing surrogate model training times, limiting their practicality. To address this, we propose a method that uses an autoencoder with residual connections as surrogate models, introducing active learning to reduce sampling costs. The problem is then decomposed into multiple single-objective problems using weighted sum decomposition, and the solutions are optimized based on the approximate gradients established by the surrogate model. Evaluation experiments using the ZDT series and a human-powered aircraft design problem demonstrate that the proposed method effectively reduces sampling and training time, while achieving better approximation accuracy compared to existing surrogate methods. Furthermore, it achieves Pareto optimal fronts more quickly, while maintaining solution accuracy comparable to other representative evolutionary multiobjective optimization algorithms.
Most real-world optimization problems have multiple objectives and constraints. To address constrained multi-objective optimization problems (CMOPs), researchers have proposed many constrained evolutionary multi-objective optimization (EMO) algorithms. They usually evaluate the proposed constrained EMO algorithms on artificial CMOPs in the literature. As a result, some newly specialized constrained EMO algorithms are designed for those artificial problems. Our previous work demonstrated that an old EMO algorithm NSGA-II with a simple constraint handling mechanism can outperform some newly proposed algorithms on real-world CMOPs. In this paper, we further examine the performance of constrained EMO algorithms on artificial and real-world CMOPs. We also analyze the relationship between algorithms' performance and problems' properties.
We propose a novel optimisation algorithm designed for multi-condition multi-objective problems, utilising a single-step Twin Delayed Deep Deterministic Policy Gradient (TD3) framework with parallel function evaluations. The proposed algorithm's performance was tested on two mathematical test problems and an airfoil shape optimisation scenario. In all cases, our approach outperformed both a previously proposed deep reinforcement learning algorithm and the wellknown NSGA-II genetic algorithm by discovering a more comprehensive Pareto front at a lower number of function evaluations. Additionally, the proposed algorithm demonstrated more stable training of the deep neural networks across all problems.
A low-fidelity model using surrogate functions helps reduce the overall computational time for optimizing problems involving timeconsuming evaluation of objectives and constraints. Most surrogateassisted evolutionary multi- and many-objective optimization (SAEM(a)O) algorithms evaluate a newly-created population member either using high-fidelity or low-fidelity models for all objectives and constraints. However, recent studies have shown that a mixed-fidelity evaluation of a population member, in which some objectives and constraints are evaluated using a high-fidelity model and others are evaluated using a low-fidelity model, may be more efficient. This is because the time saved by not evaluating constraint functions for largely infeasible or largely feasible solutions, and objective functions for largely infeasible solutions can be utilized to evaluate potential solutions near the constraint boundaries and feasible regions, respectively. In this paper, we propose a metric that determines the potential benefit for evaluating every objective and constraint function for every population member independently. The metric uses the potential for the solution's probability to dominate other neighboring members, modeling error, and extent of constraint violation. We demonstrate the efficacy of the proposed approach by presenting results on two to 10-variable test and engineering design problems having up to eight objectives and up to 10 constraints. We compare the proposed approach with a SA-EMO algorithm which evaluates every constraint and objective function for every new offspring. This preliminary study shows promise as a new direction of research in the area of surrogate-assisted evolutionary multi- and many-objective optimization.
In the evolutionary multi-objective optimization (EMO) field, evolutionary algorithms have a population to store a well-converged and well-distributed solution set. In the literature, the population size is usually specified in the range of 100–300 to approximate the entire Pareto front. Recently, a new framework with an external archive has been actively studied where a solution set selected from the archive is used to approximate the Pareto front. In this framework, a small population size can be better than a standard population size since a smaller population size means a larger number of generations under the termination condition specified by the same number of evaluated solutions. In this paper, we numerically examine whether better results are obtained from a smaller population size specification than a standard specification when the algorithm performance is evaluated using a solution set selected from an external archive. In our experiments, we examine the performance of four representative EMO algorithms (NSGA-II, NSGA-III, MOEA/D, and SMS-EMOA) with four specifications of the population size (36, 66, 120, and 210) on two frequently-used artificial test suites (DTLZ and WFG) and three real-world test suites (RWA, DDMOP, and RE). Our experimental results show that better results are obtained from a smaller population size specification in many cases when we cannot evaluate many solutions (i.e., when solution evaluation is expensive).
So far, many constraint-handling techniques (CHTs) exist that allow specialized multi-objective evolutionary algorithms (MOEAs) to deal with constrained multi-objective optimization problems (CMOPs). In contrast, all existing external archivers that yield certain approximation qualities in the limit still assume the feasibility of each incoming candidate solution. While this is acceptable for unconstrained or lightly constrained MOPs, this assumption is inadequate for the consideration of problems with complex domains. In this study, we make a first effort to investigate how these external archivers can be prepared for CMOPs. To this end, we discuss if and how existing CHTs can be leveraged. In this study, we consider the problem of capturing the entire Pareto front (ideally in the mathematical sense). As the base algorithm we will take ArchiveUpdateP_Q that stores all non-dominated solutions found during the run of the algorithm. Our discussion and results indicate that while existing CHTs work adequately, they come with certain issues, such as introducing additional parameters that need adjustment, lacking theoretical results, or incurring a certain computational overhead. Finally, we performe a numerical analysis, integrating several MOEAs with the investigated CHTs and comparing the performance of all methods using HV and _p as indicators.
Scalarisation-based approaches to multi-objective Bayesian optimization, such as the seminal ParEGO algorithm, may be either single-surrogate or multi-surrogate. In the former case, a single surrogate model is built of the scalarised function; in the latter case, separate surrogates are built for each objective function. A recent study argued that the multi-surrogate approach should be preferred and presented empirical findings supportive of this case. However, these findings were based on an outdated approach to benchmarking algorithm performance and were limited to low-dimensional problems. In this study, we use the modern COCO benchmarking framework to analyse the performance of single-surrogate and multi-surrogate ParEGO algorithms and compare these to random sampling, Sobol space-filling, and the highperforming optimizer known as TPB. Our findings broadly support the original findings for low-dimensional problems, but we find that multisurrogate ParEGO performs comparatively poorly in higher dimensions. TPB tends to outperform both ParEGOs, suggesting that initial budget investment in ideal and nadir point identification is a favourable strategy.
As advanced driver-assistant systems move towards higher automation levels, the industry now frequently employs data-driven techniques like Reinforcement Learning over rule-based systems. While the current literature on this topic contains analyses of the multiobjective and multi-agent settings, the combination of these two aspects is largely unexplored. This paper proposes a Q-learning-based Reinforcement Learning algorithm combining the multi-objective and multi-agent aspects. The algorithm was tested on a highway traffic environment, displaying its ability to produce policies that consider user preferences, exhibit social behavior, and adhere to basic traffic regulations to a certain extent. As such, the proposed approach is deemed to provide a promising foundation to be used as a baseline algorithm for future research advancements.
Coverage path planning (CPP) is the problem of determining a path that covers any given area and is mainly applied in the field of robotics. To ensure efficient coverage, objectives like path length, overlaps, and traversal time are considered. However, in certain scenarios, a simplistic total coverage approach may not be optimal, necessitating a trade-off strategy. This paper addresses a novel challenge in CPP: the multi-objective weighted coverage path planning problem, where total coverage is not strictly required but balanced against other objectives and constraints. We present an approach to solve this problem using evolutionary multi-objective algorithms with free-form path representations. The focus lies on comparing different path representations, ranging from polygonal chains to Bézier curves and B-splines, to Non-Uniform Rational B-splines (NURBs). Additionally, we incorporate an overlaid rectangular grid for comparison with a graph-based approach.
In practice, users are often stuck with an existing solution, despite agreeing on the fact that the solution needs a substantial change. The hesitation might stem from the hefty cost or amount of effort required to adopt a new solution or could just be human apathy to large changes. In this regard, a recent preliminary study proposed a stepconstrained based bi-objective optimization approach which attempts to discover a set of acceptable intermediate solutions starting from the current solution to the desired target solution leading to an innovation path (IP). Intermediate solutions, obtained using a multiobjectivization approach, reduce the amount of change required between two successive steps, thereby facilitating multiple gradual changes more acceptable by the users. In this paper, we propose a directed domination concept to make the IP-seeking algorithm more computationally efficient. Results on a number of test and engineering problems reveal that the proposed new approach reaches closer to the target solution and finds closer to optimally trade-off solutions than the previous IP approach.
Real world optimization problems contain multiple complexities that are often not tractable if one completely relies on mathematical programming or metaheuristic approaches, as each approach has unique strengths and limitations in dealing with the complexities in optimization problems. For instance, metaheuristic methods have an edge in tackling non-regularities like discontinuity, non-convexity, and discreteness, given that problem has few decision variables (low dimensionality). Contrary to that, mathematical programming based classical methods can efficiently solve high dimensional problems if the problem encompasses regular functions as constraints and objective function. Thus, non-regular and highdimensional optimization problems pose challenges to both approaches for getting solved using the method explicitly from any one approach. To excel in this situation, this study proposes a difficulties separation approach that enables to convert a single level optimization problem into bilevel optimization problem and solve the problem synergistically by applying methods from both complementary approaches. We demonstrate the benefits of proposed bilevel based decomposition approach on a wide range of single and multiobjective test problems.
In bilevel optimization, an upper level (UL) decision maker seeks to optimize an objective function while considering the optimal solutions of a lower level (LL) optimization problem. This hierarchical structure poses modeling and solution challenges, especially when the LL problem has multiple solutions. In such a case, the UL needs to make assumptions about the LL reaction. In the optimistic approach, the UL assumes that the LL reaction will be favorable, while in the pessimistic approach the opposite is true. In this study, we consider the case of a multiobjective bilevel optimization problem, where the UL has multiple objectives, while the LL has a single objective, but multiple optimal solutions for any given UL decision. Given that the LL can choose any solution from its optimal set, and in case the UL is not aware of the LL choice function, it leads to the possibility of two Pareto-optimal fronts at the UL, i.e. the optimistic and the pessimistic frontiers. To approximate both Pareto-optimal fronts, a δ -perturbation approximation is proposed in this paper, where the LL objective is perturbed by a small δ by utilizing the UL objectives at the LL. The perturbed reformulated bilevel problem is then solved via an extension of m-BLEAQ, an evolutionary bilevel algorithm that can deal with bilevel problems with multiple objectives at the UL and a single objective at the LL. The application of the m-BLEAQ algorithm to the reformulated bilevel problem leads to the identification of the optimistic and pessimistic frontiers for the multiobjective bilevel problem. In this proof-of-concept study, the proposed reformulation strategy is demonstrated on two test problems, showing the effectiveness of the proposal in accurately finding the optimistic and pessimistic frontiers.
This paper presents a supply chain (SC) optimization model that balances economic, environmental, and social objectives by aiming to maximize profits while minimizing greenhouse gas emissions and service level inequalities. It simulates real-world SC issues using a fourechelon facility model with variable demands from three markets. We utilize multi-objective Markov decision processes (MOMDP) through multiobjective reinforcement learning with decomposition (MORL/D), paired with weighted sum proximal policy optimization (PPO), and compare them using a non-dominated sorting genetic algorithm II (NSGA-II). The decision variables are production and delivery quantities, leading to Pareto front sets that illustrate optimal trade-offs. Key contributions include defining a three-objective SC under a MOMDP framework, introducing a Python-based SC simulation tool called Messiah, pioneering MORL/D in multi-objective SC optimization, and comparing it with PPO and NSGA-II. Our findings reveal that MORL/D achieves more balanced outcomes in optimality, diversity, and density, with enhanced hypervolume and expected utility metrics through knowledge sharing.
Dealing with multiple conflicting objectives in a multi-agent system is challenging, as agents' interactions complicate decision-making, especially when managing multiple Pareto-optimal fronts. In competitive co-evolutionary frameworks, not only do the objectives of each agent conflict with one another, but the agents' goals are also at odds. One such application domain is wargame strategy optimization, where the strategies of one agent must adapt based on the moves of opposing agents. Despite advancements in modern warfare, strategy analysis and decisionmaking are still largely manual, leaving room for great application of computational methods to automate different parts of the system. To address this, we propose a co-evolutionary optimization algorithm that integrates strategy search with interactive decision-making, allowing coevolving populations to collaboratively identify their respective Paretooptimal strategies. Central to this approach is a progressive-shrinking method that aligns feasible moves with those previously taken, ensuring smoother transitions. Our framework introduces a novel decision-making strategy using opposition front hypervolume improvement, particularly suited for competitive co-evolutionary contexts, combined with Penaltybased Boundary Intersection selection, to optimize strategy selections. We also examine the influence of various decision-making approaches, shrinking techniques, and parameter settings on the final results. This coevolutionary framework, combining multi-agent interaction, evolutionary multi-objective optimization, and progressive shrinking, is not only effective for wargame strategy optimization but is also adaptable to other multi-agent conflicting systems.
Recently, neural network-based inverse models have been used for multi-objective optimization. The basic idea is to approximate the mapping from the Pareto front to the Pareto set. In general, inverse modeling from a low-dimensional objective space to a high-dimensional decision space is difficult. However, good results have been reported in the literature. In this paper, we numerically examine the performance of inverse modeling. For visual examination, we use 2- and 3-objective distance minimization problems with 2 to 1,000 decision variables. Our experimental results show that inverse modeling improves the quality of the final population, which is used as the training data. One interesting observation is that improving fitting accuracy by increasing the neural network complexity does not improve the quality of the model outputs. We also demonstrate that inverse modeling works well even for large-scale multi-objective problems with 1,000 decision variables when high-quality training data are available.
This study introduces an interactive evolutionary algorithm (EA) for optimizing path planning in groundfish surveys. The approach employs interactive reoptimization to iteratively refine plans by adjusting constraints and incorporating new information, addressing the limitations of traditional optimization models and including real-world factors not initially captured. Developed with input from experienced surveyors, the method provides a practical way to refine survey plans at sea, where fast optimization is crucial. We compare the EA's performance with a mixed integer programming model applied to a groundfish survey in Iceland involving four vessels, aiming to find the shortest route while considering vessel capacity and port unloading. Results show the EA's potential in complex marine surveys, highlighting its advantages in interactive optimization and sustainable fisheries management.
In evolutionary multi-objective optimization (EMO), the quality of Pareto front (PF) approximations obtained using various algorithms are benchmarked using set-based unary indicators. Hypervolume (HV) is an indicator that has attracted widespread attention as it attempts to capture both convergence and diversity, is scalable in terms of objectives, and is Pareto-compliant. Even so, HV is not without its limitations, ignoring which may lead to erroneous judgments about algorithm performance, especially for problems with more than three objectives, where visualization of the objective space is not straightforward. Various theoretical studies have therefore been conducted to improve understanding of HV behavior and its implications in EMO benchmarking, including a recent one that focuses on computing HV for some common PF shapes analytically. This paper aims to extend these theoretical results and discussions, by analyzing continuous subsets (symmetric patches) on some common shapes of PFs (linear and quadratic). Towards this end, a simple parametrization scheme is proposed to represent the symmetric patches on linear and quadratic PFs, and their HV is analytically derived. The resulting expressions are used to observe some HV trends with respect to the patch size, reference point specification and number of objectives, to provide related insights for benchmarking.
