Constrained Multi-objective Optimization Research Articles

Deep reinforcement learning-guided coevolutionary algorithm for constrained multiobjective optimization

Deep reinforcement learning-guided coevolutionary algorithm for constrained multiobjective optimization

Optimal siting and sizing of battery energy storage systems in unbalanced distribution systems: A multi objective problem under uncertainty

In this paper the siting and sizing problem of battery energy storage systems in unbalanced active distribution systems is formulated as a mixed-integer, non-linear, constrained multi objective (MO) optimization problem under uncertainties. The problem is cumbersome from the computational point of view due to the presence of intertemporal constraints, a great number of state variables and the presence of uncertainties in the problem input data. A new approach based on the trade-off/risk analysis is proposed to obtain with acceptable computational efforts a solution that may not be the optimal solution but represents a reasonable and robust compromise. We use the trade-off/risk analysis, because it was specifically developed for power system planning problems in which we deal with a wide range of options, with possible conflicting objectives, and with uncertainty and risk. The proposed approach includes new procedures to select an adequate set of planning alternatives to be considered in the trade-off/risk analysis framework and to assist the planning engineer when difficulties arise in setting probabilities of the input data. Numerical applications to an IEEE unbalanced test system demonstrate the effectiveness of the proposed procedure and indicate the best alternatives of storage systems in the range from 450 kW to 600 kW globally installed in a reduced set of nodes.

An inexact restoration direct multisearch filter approach to multiobjective constrained derivative-free optimization

Direct Multisearch (DMS) is a well-established class of methods for multiobjective derivative-free optimization, where constraints are addressed by an extreme barrier approach, only evaluating feasible points. In this work, we propose the replacement of this extreme barrier approach by a filter strategy, combined with an inexact feasibility restoration step, to address constraints in the DMS framework. The filter approach treats feasibility as an additional component of the objective function that needs to be minimized. The inexact restoration step attempts to generate new feasible points, contributing to prioritize this feasibility, a requirement for the good performance of any filter approach. Theoretical results are provided, analysing the different types of sequences of points generated by the new algorithm, and numerical experiments on a set of nonlinearly constrained biobjective problems are reported, stating the good algorithmic performance of the proposed approach.

A staged fuzzy evolutionary algorithm for constrained large-scale multiobjective optimization

Constrained multiobjective optimization problems (CMOPs) are prevalent in practical applications, where constraints play a significant role. Building on techniques from constrained single-objective optimization, classical methods such as the constrained dominance principle have been extended to CMOPs. However, these methods struggle with CMOPs characterized by complex infeasible regions. Furthermore, as the number of decision variables increases, the search efficiency of algorithms deteriorates dramatically. To solve those issues, we propose a staged fuzzy evolutionary algorithm (i.e., SFEA) for constrained large-scale problems. To balance exploration and exploitation, a fuzzy stage adjustment strategy based on the sigmoid function is proposed. Furthermore, this article develops an improved fuzzy operator to perform fuzzy operations on various vectors (e.g., solutions or constraint violations). Computational experiments were conducted on CMOP test suites with up to 500 decision variables and a series of real-world applications. The experimental results demonstrate that, compared to existing peer algorithms, our algorithm exhibits superior or competitive performance.

An adaptive co-evolutionary competitive particle swarm optimizer for constrained multi-objective optimization problems

In constrained multi-objective optimization problems, it is challenging to balance the convergence, diversity and feasibility of the population, especially encountering complex infeasible regions. In order to effectively balance the three indicators, from the aspects of the handling of infeasible solution and the quality of individuals, a multi-population co-evolutionary competitive particle swarm optimization algorithm hybridized with infeasible solution transfer and an adaptive technique (ACCPSO) is proposed. Firstly, the information of feasible and infeasible individuals is fully utilized and the individuals are classified by Hamming distance. Then, a novel constraint handling technique based on learning from the promising feasible direction is designed to make individuals cross large infeasible regions and explore more potential feasible regions. Moreover, aiming to provide robust search capability and consequently further generate high-quality solutions, the genetic operators and the particle swarm optimization operator with the competitive mechanism are introduced as operators with an adaptive mechanism. Finally, compared with the state-of-the-art methods, the performance of the proposed algorithm is verified on LIR-CMOP, MW and DTLZ, as well as two real-world problems. The results indicate that ACCPSO exhibits stronger competitiveness in terms of convergence, the solution quality, and distribution diversity on the feasible Pareto front.

Evolutionary Dynamic Constrained Multiobjective Optimization: Test Suite and Algorithm

Evolutionary Dynamic Constrained Multiobjective Optimization: Test Suite and Algorithm

Joint Energy and Completion Time Difference Minimization for UAV-Enabled Intelligent Transportation Systems: A Constrained Multi-Objective Optimization Approach

Joint Energy and Completion Time Difference Minimization for UAV-Enabled Intelligent Transportation Systems: A Constrained Multi-Objective Optimization Approach

Adaptive Constraint Relaxation-Based Evolutionary Algorithm for Constrained Multi-Objective Optimization

The key problem to solving constrained multi-objective optimization problems (CMOPs) is how to achieve a balance between objectives and constraints. Unfortunately, most existing methods for CMOPs still cannot achieve the above balance. To this end, this paper proposes an adaptive constraint relaxation-based evolutionary algorithm (ACREA) for CMOPs. ACREA adaptively relaxes the constraints according to the iteration information of population, whose purpose is to induce infeasible solutions to transform into feasible ones and thus improve the ability to explore the unknown regions. Completely ignoring constraints can cause the population to waste significant resources searching for infeasible solutions, while excessively satisfying constraints can trap the population in local optima. Therefore, balancing constraints and objectives is a crucial approach to improving algorithm performance. By appropriately relaxing the constraints, it induces infeasible solutions to be transformed into feasible ones, thus obtaining more information from infeasible solutions. At the same time, it also establishes an archive for the storage and update of solutions. In the archive update process, a diversity-based ranking is proposed to improve the convergence speed of the algorithm. In the selection process of the mating pool, common density selection metrics are incorporated to enable the algorithm to obtain higher-quality solutions. The experimental results show that the proposed ACREA algorithm not only achieved the best Inverse Generation Distance (IGD) value in 54.6% of the 44 benchmark test problems and the best Hyper Volume (HV) value in 50% of them, but also obtained the best results in seven out of nine real-world problems. Clearly, CP-TSEA outperforms its competitors.

Multi-Objective Optimisation and Deformation Analysis of Double-System Composite Guideway Based on NSGA-II

To study the optimal design of the section of the double-system composite guideway under the economic, steel consumption, and carbon emission characteristics, this paper introduced the multi-objective constrained optimisation model, which was established by the non-dominated sorting genetic algorithm II. In addition, the finite element model was established to further analyse the optimised section’s deformation and summarise the rail girder’s deformation law under different loads. The results showed that compared with the original design scheme, the optimised scheme can effectively reduce carbon emission during the construction of the double-system composite guideway, by 23.67% for Scheme I and 42.03% for Scheme II. On the other hand, steel had the largest share in the economic targets of the three design options, accounting for about 75% to 88.5% of the total cost. Concrete had the highest share of carbon emissions, ranging from 90% to 95% of the total carbon emissions. The distribution patterns of horizontal and vertical deformations in the three design options were independent of the load type as well as the load magnitude, but the vertical deformations were related to the load type, especially the self-weight load. The conclusions of this paper aim to fill the gap in the theoretical study of section optimisation of the double-system composite guideway and lay the theoretical foundation for developing the multi-system monorail transportation system.

An Improved Coevolutionary Algorithm for Constrained Multi-Objective Optimization Problems

Constrained multi-objective optimization problems are ubiquitous in engineering applications. In recent years, constrained multi-objective optimization algorithms based on the dual population coevolutionary framework have been widely studied due to their excellent performance. However, when facing optimization problems with complex constraints, the performance of existing algorithms still needs further improvement. This paper proposes an improved constrained multi-objective coevolutionary algorithm (iCMOCA). The algorithm mainly includes two populations: One population takes into account constraints, while the other population disregards them. Meanwhile, the iCMOCA employs effective collaboration between two populations during the process of offspring generation and environmental selection, and it utilizes an environmental selection strategy based on multi-objective to multi-objective decomposition to improve the performance. Comparative analysis conducted on the DAS-CMOP and MW test suites provides empirical evidence that iCMOCA outperforms five state-of-the-art algorithms.

A surrogate-assisted expensive constrained multi-objective global optimization algorithm and application

Expensive multi-objective optimization problems (MOPs) have seen the successful applications of surrogate-assisted evolutionary algorithms (SAEAs). Nevertheless, the majority of SAEAs are developed for costly unconstrained optimization, and costly constrained MOPs (CMOPs) have received less attention. Therefore, this article proposes a surrogate-assisted global optimization algorithm (named CTEA) for solving CMOPs within a very limited number of fitness evaluations. The proposed algorithm combines two selection frameworks, a bi-level selection framework, and an adaptive sampling framework, to enhance optimization performance. Leveraging on a constraint-improving strategy and a Pareto-based three-indicator criterion (convergence, constraint, and diversity indicators) at the different levels, the proposed bi-level selection framework can select more promising solutions. Moreover, an adaptive sampling framework is developed to prioritize objective and constraint functions and select the candidate solutions for real function evaluations according to the priority. Experimental results demonstrate that the proposed CTEA exhibits superior performance when compared with five state-of-the-art algorithms, achieving the best results in 61.9 % out of the 64 test instances. Finally, CTEA is applied to the multidisciplinary design optimization of blended-wing-body underwater gliders, and an impressive solution set is obtained.

Surrogate-assisted push and pull search for expensive constrained multi-objective optimization problems

In many real-world engineering optimizations, a large number of objective and constraint function values often need to be obtained through simulation software or physical experiments, which incurs significant computational costs and/or time expenses. These problems are known as expensive constraint multi-objective optimization problems (ECMOPs). This paper combines the push and pull search (PPS) framework and proposes a surrogate-assisted evolutionary algorithm to solve ECMOPs through Bayesian active learning, naming it the surrogate-assisted PPS (SA-PPS). Specifically, during the push search stage, candidate solutions are selected based on two indicators: hypervolume improvement and objective uncertainty. These aim to quickly guide the population towards the unconstrained Pareto front while ensuring diversity. During the pull search stage, the population is partitioned into many subregions through reference vectors, and different selection strategies are assigned to each subregion based on its state, aiming to guide the population towards the constrained Pareto front while ensuring diversity. Furthermore, we introduce a batch data selection strategy that utilizes Bayesian active learning to enable the surrogate model to focus on regions of interest in the pull search stage. Extensive experimental results have shown that the proposed SA-PPS algorithm exhibits superior convergence and diversity compared to 9 state-of-the-art algorithms across a variety of benchmark problems and a real-world optimization problem.

Interval Constrained Multi-Objective Optimization Scheduling Method for Island-Integrated Energy Systems Based on Meta-Learning and Enhanced Proximal Policy Optimization

Multiple uncertainties from source–load and energy conversion significantly impact the real-time dispatch of an island integrated energy system (IIES). This paper addresses the day-ahead scheduling problems of IIES under these conditions, aiming to minimize daily economic costs and maximize the output of renewable energies. We introduce an innovative algorithm for Interval Constrained Multi-objective Optimization Problems (ICMOPs), which incorporates meta-learning and an improved Proximal Policy Optimization with Clipped Objective (PPO-CLIP) approach. This algorithm fills a notable gap in the application of DRL to complex ICMOPs within the field. Initially, the multi-objective problem is decomposed into several single-objective problems using a uniform weight decomposition method. A meta-model trained via meta-learning enables fine-tuning to adapt solutions for subsidiary problems once the initial training is complete. Additionally, we enhance the PPO-CLIP framework with a novel strategy that integrates probability shifts and Generalized Advantage Estimation (GAE). In the final stage of scheduling plan selection, a technique for identifying interval turning points is employed to choose the optimal plan from the Pareto solution set. The results demonstrate that the method not only secures excellent scheduling solutions in complex environments through its robust generalization capabilities but also shows significant improvements over interval-constrained multi-objective evolutionary algorithms, such as IP-MOEA, ICMOABC, and IMOMA-II, across multiple multi-objective evaluation metrics including hypervolume (HV), runtime, and uncertainty.

Coevolutionary multitasking for constrained multiobjective optimization

Addressing the challenges of constrained multiobjective optimization problems (CMOPs) with evolutionary algorithms requires balancing constraint satisfaction and optimization objectives. Coevolutionary multitasking (CEMT) offers a promising strategy by leveraging synergies from distinct, complementary tasks. The primary challenge in CEMT frameworks is constructing suitable auxiliary tasks that effectively complement the main CMOP task. In this paper, we propose an adaptive CEMT framework (ACEMT), which customizes two adaptive auxiliary tasks to enhance CMOP-solving efficiency. The first auxiliary task dynamically narrows constraint boundaries, facilitating exploration in regions with smaller feasible spaces. The second task focuses specifically on individual constraints, continuously adapting to expedite convergence and uncover optimal regions. In solving the main CMOP task, this dual-auxiliary-task strategy not only improves search thoroughness but also clarifies the balance between constraints and objectives. Concretely, ACEMT incorporates an adaptive constraint relaxation technique for the first auxiliary task and a specialized constraint selection strategy for the second. These innovations foster effective knowledge transfer and task synergy, addressing the key challenge of auxiliary task construction in CEMT frameworks. Extensive experiments on three benchmark suites and real-world applications demonstrate ACEMT’s superior performance compared to state-of-the-art constrained evolutionary algorithms. ACEMT sets a new standard in CMOP-solving by strategically constructing and adapting auxiliary tasks, representing a significant advancement in this research direction.

Manifold-assisted coevolutionary algorithm for constrained multi-objective optimization

In constrained multi-objective optimization problems (CMOPs), constraints often fragment the Pareto solution space into multiple feasible and infeasible regions. This fragmentation presents a challenge for evolutionary optimization methods as feasible regions can be discrete and isolated by infeasible areas, making exploration difficult and leading to populations getting trapped in local optima. To address these issues, this paper introduces a manifold assisted coevolutionary algorithm for solving CMOPs. Firstly, a guided feasible search strategy is proposed to explore feasible regions, especially those isolated by infeasible barriers. This is achieved by estimating directions to the Constrained Pareto Set (CPS). Secondly, a manifold learning-based exploration strategy is employed to spread the population along the Pareto Set (PS) manifold by estimating the manifold distribution. Moreover, two populations are exploited, where the first population serves as the primary population, considering both constraints and objectives to explore the feasible region and search along the CPS. The second population, on the other hand, does not consider constraints and serves as an auxiliary population to explore the Unconstrained PS. By cooperating, these two populations effectively approach and cover separated CPS segments. The proposed algorithm is evaluated against seven state-of-the-art algorithms on 37 CMOP test functions and 5 CMOPs with fraudulent constraints. The experimental results clearly demonstrate that our algorithm can reliably locate multiple CPSs and is considered state-of-the-art in handling CMOPs.

Second-order Karush/Kuhn–Tucker conditions and duality for constrained multiobjective optimization problems

In this paper, we investigate the second-order strong Karush/Kuhn–Tucker conditions and duality of a constrained multiobjective optimization problem (CMOP). Exploiting a second-order regularization condition, we obtain second-order strong KKT necessary conditions of Borwein-properly efficient solution of CMOP without convexity assumptions. Further, second-order sufficient conditions for the second-order KKT point to be an efficient solution of CMOP are derived under the generalized second-order convexity assumptions. Finally, we establish duality results between CMOP and its second-order Wolfe-type dual problem.

EvolutionViT: Multi-objective evolutionary vision transformer pruning under resource constraints

Vision Transformer (ViT) has emerged as a pivotal model for a variety of visual tasks, surpassing convolutional neural networks by a substantial margin. However, the performance of ViT is seriously impaired by intensive computational and storage costs requirements, posing significant barriers for real-world applications or deployment on resource-constrained edge devices. To address this limitation, compressing the ViT to accelerate its inference at no appreciable degradation of vision performance has attracted widespread attention. Although there are some studies on accelerating ViT, they seldom consider resource constraints and multi-criteria decision making in the process. This article formulates ViT pruning as a large-scale constrained multi-objective optimization problem, and proposes a patch pruning framework for accelerating ViT, called EvolutionViT, based on the developed multi-objective optimization model. EvolutionViT can effectively tradeoff between computational cost and performance under resource constraints, automatically searching for solutions while optimizing two conflicting objectives. In particular, exploiting the knee solution and boundary solutions to directly guide the entire evolutionary process, EvolutionViT can efficiently identify a knee solution that satisfies the resource constraints, which in turn avoids the manual search for a good trade-off. To verify and evaluate our proposed method, we compare EvolutionViT with a number of representative ViT models on the ImageNet dataset. The comprehensive simulation results show that the proposed EvolutionViT demonstrates a competitive advantage compared to peers, with significantly reduced computational expense at the cost of slightly degraded performance.

Trajectory Optimization to Enhance Observability for Bearing-Only Target Localization and Sensor Bias Calibration.

This study addresses the challenge of bearing-only target localization with sensor bias contamination. To enhance the system's observability, inspired by plant phototropism, we propose a control barrier function (CBF)-based method for UAV motion planning. The rank criterion provides only qualitative observability results. We employ the condition number for a quantitative analysis, identifying key influencing factors. After that, a multi-objective, nonlinear optimization problem for UAV trajectory planning is formulated and solved using the proposed Nonlinear Constrained Multi-Objective Gray Wolf Optimization Algorithm (NCMOGWOA). Simulations validate our approach, showing a threefold reduction in the condition number, significantly enhancing observability. The algorithm outperforms others in terms of localization accuracy and convergence, achieving the lowest Generational Distance (GD) (7.3442) and Inverted Generational Distance (IGD) (8.4577) metrics. Additionally, we explore the effects of the CBF attenuation rates and initial flight path angles.

A constrained multi-objective evolutionary algorithm based on fitness landscape indicator

Constrained multi-objective optimization problems (CMOPs) with constraints in both the decision and objective space are shown to be great challenges to be solved. Considering the different requirements of different problems on resource allocation of exploration and exploitation, this paper proposes a new constrained multi-objective evolutionary algorithm based on a novel fitness landscape indicator. The indicator regards the fitness landscape and evolutionary generation among the population to determine the selection of the offspring generation mechanism. The proposed algorithm uses the new indicator to select different differential evolutions during the evolutionary process to balance exploration and exploitation. Numerical experiments on three test suites and three practical examples compared with six existing algorithms show the proposed algorithm can effectively deal with different types of CMOPs, especially in CMOPs with constraints in both the decision and objective spaces.

Dynamic-multi-task-assisted evolutionary algorithm for constrained multi-objective optimization

Compared with common multi-objective optimization problems, constrained multi-objective optimization problems demand additional consideration of the treatment of constraints. Recently, many constrained multi-objective evolutionary algorithms have been presented to reconcile the relationship between constraint satisfaction and objective optimization. Notably, evolutionary multi-task mechanisms have also been exploited in solving constrained multi-objective problems frequently with remarkable outcomes. However, previous methods are not fully applicable to solving problems possessing all types of constraint landscapes and are only superior for a certain type of problem. Thus, in this paper, a novel dynamic-multi-task-assisted constrained multi-objective optimization algorithm, termed DTCMO, is proposed, and three dynamic tasks are involved. The main task approaches the constrained Pareto front by adding new constraints dynamically. Two auxiliary tasks are devoted to exploring the unconstrained Pareto front and the constrained Pareto front with dynamically changing constraint boundaries, respectively. In addition, the first auxiliary task stops the evolution automatically after reaching the unconstrained Pareto front, avoiding the waste of subsequent computational resources. A series of experiments are conducted with eight mainstream algorithms on five benchmark problems, and the results confirm the generality and superiority of DTCMO.