Many-objective Optimization Problems Research Articles

A coevolutionary algorithm using multi-operator ensemble for many-objective optimisation problems

A coevolutionary algorithm using multi-operator ensemble for many-objective optimisation problems

Dynamic Multi-Objective Evolutionary Algorithms: An Overview

Evolutionary algorithms have been widely explored and applied in optimization problems. The introduction of multi-objective evolutionary algorithms (MOEAs) has facilitated the adaptation and creation of new methods to handle more complex and realistic optimizations, such as dynamic multi-objective optimization problems (DMOPs). A dynamic MOEA (DMOEA) can be constructed by changing the MOEA structure and variation operators used to solve DMOPs. Furthermore, DMOEAs can implement change-detection strategies and mechanisms to handle the dynamics of the environment. DMOEAs are often designed to solve unconstrained DMOPs. However, several studies have been conducted to solve dynamic constrained MOPs and more recently, to solve dynamic many-objective optimization problems. There are many real-world DMOPs; however, few DMOEAs have been evaluated with respect to those problems. Evaluation of algorithm performance is an essential aspect of using DMOEAs. Several measures used to validate MOEAs have been adapted to assess DMOEAs and quantitatively compare them. This paper presents a broad review of DMOEAs. The information is summarized and organized according to current research branches. Furthermore, a more general classification scheme for DMOEAs is proposed, based on the method of incorporating diversity to solve DMOPs.

Neuro-evolution-based generic missile guidance law for many-scenarios

Designing efficient aerial interceptors is an important task. Using Pareto-optimality, this paper presents a novel approach for generating guidance laws that are generic for many aerial pursuit-evasion scenarios. In particular, the pure-proportional navigation law is combined with a neural network to create adaptive guidance laws, which adapt according to the current state of the system. First, a many-objective optimization problem is formulated in which each objective aims at the best performance in one of the scenarios. Next, using simulations with a many-objective evolutionary algorithm, a population of guidance laws is evolved towards the Pareto-optimal ones. The obtained guidance laws from multiple runs are statistically analyzed and compared with a set of Pareto-optimal pure-proportional navigation laws. The results suggest that the proposed approach provides a significant improvement as compared with the pure-proportional navigation law over the entire set of scenarios.

Radial projection-based adaptive sampling strategies for surrogate-assisted many-objective optimization

Intelligent manufacturing and industrial control systems frequently encounter expensive many-objective optimization problems (EMaOPs). Surrogate-assisted evolutionary algorithms (SAEAs) build predictive models to substitute the expensive fitness evaluation, enabling them to solve optimization more efficiently. In SAEAs, to enhance the exploration of optimization and the generalization of the surrogate model, the diversity of infill offspring and training database should be well-maintained, which is challenging in high-dimensional spaces or problems with disconnected Pareto front. This paper suggests a radial projection-based surrogate-assisted framework for solving EMaOPs. The radial projection can map the high-dimensional objective space into a 2-dimensional radial space. Based on this, a dynamic quadratic division method is proposed to enhance the diversity of solutions. Furthermore, an adaptive infill sampling criterion is introduced based on the distribution of selected convergent solutions, and a training database updating strategy is designed under the premise of maintaining its diversity and the model training efficiency. The presented framework exhibits a notable level of flexibility and adaptability as it can be effortlessly combined with other multi-objective optimization algorithms. Several experimental results on a set of expensive multi/many-objective test problems have demonstrated the superiority of the framework.

An adaptive parental guidance strategy and its derived indicator-based evolutionary algorithm for multi- and many-objective optimization

The indicator-based multi-objective evolutionary algorithms have demonstrated their superiority in handling diverse types of multi- and many-objective optimization problems. However, these evolutionary algorithms still face significant challenges in balancing convergence and diversity of the evolutionary population, despite numerous auxiliary mechanisms designed to improve their performance. To address this issue, an adaptive parental guidance strategy (APGS) is proposed in this paper. On the one hand, APGS leverages the current population to evaluate the quality of the newly generated offspring. On the other hand, it employs an adaptive threshold to select offspring individuals with better convergence and diversity. This approach enhances the convergence and diversity of the candidate solution set throughout the evolutionary process, thereby ensuring high-quality obtained solutions. By incorporating the APGS, this paper proposes a new indicator-based evolutionary algorithm with parental guidance (IEAPG). Simulation results on several test suites and real-world problem show that compared to PREA, SPEA/R, GrEA, TS-NSGA-II, HEA and MaOEA/IGD, the proposed IEAPG has better performance and robustness in dealing with different types of multi- and many-objective optimization problems. Furthermore, further investigation reveals that the incorporation of the APGS can significantly improve the performance of different categories of multi- and many-objective evolutionary algorithms.

A many-objective evolutionary algorithm assisted by ideal hyperplane

In many-objective optimization problems (MaOPs), balancing convergence and diversity while rapidly converging to the Pareto front is an arduous task for evolutionary algorithms. In addition, with the increase of the number of targets, the number of non-dominant solutions increases exponentially, and the individual selection pressure is insufficient. For this problem, we propose a many-objective evolution algorithm assisted by an ideal hyperplane (MaOEA-IH). To begin, the ideal hyperplane is built from the extremums of each dimension of objective space, guiding the individual to the Pareto front of search. Second, a parallel p-norm mating selection strategy based on the ideal hyperplane is proposed to improve convergence. In addition, two other factors are taken into account: (1) different p-norms are used to measure different spatial scales; and (2) individual selection uncertainty is defined by incorporating a probabilistic perturbation mechanism. Following that, the sum of objectives is applied to shift-based density estimation, which serves as an evaluation criterion in the environmental selection operation. This method increases the chances of solutions with high convergence and diversity entering the next generation, thereby increasing selection pressure. On three benchmark problems of DTLZ, WFG, and MaF, we compare MaOEA-IH with seven excellent algorithms. The results demonstrate that the MaOEA-IH proposed is highly competitive in solving MaOPs.

Multi-and many-objective optimization: present and future in de novo drug design.

de novo Drug Design (dnDD) aims to create new molecules that satisfy multiple conflicting objectives. Since several desired properties can be considered in the optimization process, dnDD is naturally categorized as a many-objective optimization problem (ManyOOP), where more than three objectives must be simultaneously optimized. However, a large number of objectives typically pose several challenges that affect the choice and the design of optimization methodologies. Herein, we cover the application of multi- and many-objective optimization methods, particularly those based on Evolutionary Computation and Machine Learning techniques, to enlighten their potential application in dnDD. Additionally, we comprehensively analyze how molecular properties used in the optimization process are applied as either objectives or constraints to the problem. Finally, we discuss future research in many-objective optimization for dnDD, highlighting two important possible impacts: i) its integration with the development of multi-target approaches to accelerate the discovery of innovative and more efficacious drug therapies and ii) its role as a catalyst for new developments in more fundamental and general methodological frameworks in the field.

A solution potential-based adaptation reference vector evolutionary algorithm for many-objective optimization

Decomposition-based multiobjective evolutionary algorithms are widely utilized for solving multiobjective optimization problems (MOPs). These algorithms have a higher degree of diversity due to a uniform distribution of the reference vectors in the objective space. However, these algorithms may struggle to select a suitable candidate set for solving many-objective optimization problems (MaOPs) with irregular Pareto fronts (PFs). To solve this problem, a solution potential-based adaptation reference vector evolutionary algorithm (SPARVEA) is proposed in this paper. In the algorithm, a concept called solution potential is presented to assess whether the direction of convergence of the solution to the ideal solution has potential. The solution potential is obtained with an evaluation function to calculate the potentials of the corresponding solutions. A solution potential-based adaptation strategy is then designed for adapting the reference vector to better guide the direction of convergence of the solution when solving MaOPs with irregular PFs. A modified angle penalty distance (mAPD) is adjusted to improve the convergence rate of the solution set in many-objective optimization problems. The proposed algorithm SPARVEA is compared with state-of-the-art many-objective evolutionary algorithms on four test sets with irregular PFs and four practical problems. Experimental results demonstrate the superior performance of SPARVEA on many-objective optimization problems. This contribution helps to advance the development of efficient algorithms for solving many-objective optimization problems.

Multipopulation-Based Differential Evolution for Large-Scale Many-Objective Optimization.

In recent years, numerous efficient many-objective optimization evolutionary algorithms have been proposed to find well-converged and well-distributed nondominated optimal solutions. However, their scalability performance may deteriorate drastically to solve large-scale many-objective optimization problems (LSMaOPs). Encountering high-dimensional solution space with more than 100 decision variables, some of them may lose diversity and trap into local optima, while others may achieve poor convergence performance. This article proposes a multipopulation-based differential evolution algorithm, called LSMaODE, which can solve LSMaOPs efficiently and effectively. In order to exploit and explore the exponential decision space, the proposed algorithm divides the population into two groups of subpopulations, which are optimized with different strategies. First, the randomized coordinate descent technique is applied to 10% of individuals to exploit the decision variables independently. This subpopulation maintains diversity in the decision space to avoid premature convergence into local optimum. Second, the remaining 90% of individuals are optimized with the nondominated guided random interpolation strategy, which interpolates individual among three nondominated solutions randomly. The strategy can guide the population convergent toward the nondominated solutions quickly, meanwhile, maintain good distribution in objective space. Finally, the proposed LSMaODE is evaluated on the LSMOP test suites from the scalability in both decision and objective dimensions. The performance is compared against five state-of-the-art large-scale many-objective evolutionary algorithms. The experimental results show that LSMaODE provides highly competitive performance.

MaOEA/D with adaptive external population guided weight vector adjustment

In order to make multiobjective evolutionary algorithm based on decomposition (MOEA/D) have better performance in dealing with many-objective optimization problems (MaOPs) with different Pareto fronts (PFs), this paper proposes a MOEA/D with adaptive external population guided weight vector adjustment (MaOEA/D-AEW). The algorithm modifies the MOEA/D method of constructing mating pools by calculating the utility function value of individuals in the population to produce high-quality offspring. For the maintenance of non-dominated solutions in the external population (EP), solutions exceeding the capacity of the EP are removed by the method of minimum Euclidean distance combined with the aggregation function value to ensure the convergence and diversity of non-dominated solutions in the EP. For the addition of weight vectors, a new sparsity evaluating method (SL+) is proposed to determine the sparsity of weight vectors by multiplying parallel distances between individuals. For the deletion of the weight vector, the minimum Euclidean distance combined with the aggregation function value is used to delete the weight vector corresponding to invalid or crowded individuals in the population. Comparing MaOEA/D-AEW with six state-of-the-art algorithms on regular PF and irregular PF test problems respectively, simulation results show that the proposed algorithm can adapt well to MaOPs with different PFs and significantly outperforms other comparative algorithms in terms of convergence and diversity of solution sets.

A new proximity metric based on optimality conditions for single and multi-objective optimization: Method and validation

With the widespread application of Evolutionary Algorithms (EAs), their performance needs to be evaluated using more than the usual performance metrics. In the EA literature, various metrics assess the convergence ability of these algorithms. However, many of them require prior knowledge of the Pareto-optimal front. Recently, two Karush–Kuhn–Tucker Proximity Metrics (KKTPMs) have been introduced to measure convergence without needing prior knowledge. One relies on the Augmented Achievement Scalarization Function (AASF) method (AASF-KKTPM), and the other on Benson’s method (B-KKTPM). However, both require specific parameters and reference points, making them computationally expensive. In this paper, we introduce a novel version of KKTPM applicable to single-, multi-, and many-objective optimization problems, utilizing the Penalty-based Boundary Intersection (PBI) method (PBI-KKTPM). Additionally, we introduce an approximate approach to reduce the computational burden of solving PBI-KKTPM optimization problems. Through extensive computational experiments across 23 case studies, our proposed metric demonstrates a significant reduction in computational cost, ranging from 20.68% to 60.03% compared to the computational overhead associated with the AASF-KKTPM metric, and from 16.48% to 61.15% compared to the computational overhead associated with the B-KKTPM metric. Noteworthy features of the proposed metric include its independence from knowledge of the true Pareto-optimal front and its applicability as a termination criterion for EAs. Another feature of the proposed metric is its ability to deal with black box problems very efficiently.

Learning Decision Variables in Many-Objective Optimization Problems

Learning Decision Variables in Many-Objective Optimization Problems

An information entropy-driven evolutionary algorithm based on reinforcement learning for many-objective optimization

Many-objective optimization problems (MaOPs) are challenging tasks involving optimizing many conflicting objectives simultaneously. Decomposition-based many-objective evolutionary algorithms have effectively maintained a balance between convergence and diversity in recent years. However, these algorithms face challenges in accurately approximating the complex geometric structure of irregular Pareto fronts (PFs). In this paper, an information entropy-driven evolutionary algorithm based on reinforcement learning (RL-RVEA) for many-objective optimization with irregular Pareto fronts is proposed. The proposed algorithm leverages reinforcement learning to guide the evolution process by interacting with the environment to learn the shape and features of PF, which adaptively adjusts the distribution of reference vectors to cover the PFs structure effectively. Moreover, an information entropy-driven adaptive scalarization approach is designed in this paper to reflect the diversity of nondominated solutions, which facilitates the algorithm to balance multiple competing objectives adaptively and select solutions efficiently while maintaining individual diversity. To verify the effectiveness of the proposed algorithm, the RL-RVEA compared with seven state-of-the-art algorithms on the DTLZ, MaF, and WFG test suites and four real-world MaOPs. The results of the experiments demonstrate that the suggested algorithm provides a novel and practical method for addressing MaOPs with irregular PFs.

On the adaptation of reference sets using niching and pair-potential energy functions for multi-objective optimization

Multi-objective evolutionary algorithms (MOEAs) have shown a good capability to approximate the Pareto front (PF) of complex multi-objective optimization problems (MOPs). Recently, several decomposition-based, indicator-based, and reference set-based MOEAs have been proposed to tackle MOPs with four or more objective functions, known as many-objective optimization problems (MaOPs). Some well-known MOEAs from these design strategies use a predefined reference set generated with uniformly distributed weight vectors on a unit simplex. These MOEAs have demonstrated good convergence, coverage, and uniformity of solutions on MOPs with regular PF shapes, i.e., simplex-like shapes. However, it has been shown that their performance degrades on MOPs with irregular PF shapes. In this paper, we proposed a pluggable reference set adaptation method based on niching and pair-potential energy functions (PPFs), called AdaK, to enhance the performance of these MOEAs on MOPs with irregular PF shapes while maintaining their good behavior on MOPs with regular PF shapes. Our adaptation method was validated by plugging it into three well-known MOEAs that use a predefined weight vector-based reference set. We perform an empirical study using different PPFs for our adaptation method on the DTLZ, WFG, Minus-DTLZ, Minus-WFG, IMOP, and VNT test suites. Our experimental results show the capability of our adaptation method to promote an invariant performance regardless of the PF shape. Additionally, we show that MOEAs with our adaptation method have a competitive performance against state-of-the-art MOEAs.

A many-objective optimization evolutionary algorithm based on hyper-dominance degree

Compared with multi-objective optimization, solving many-objective optimization problems usually require more strong selection pressure. However, too strong selection pressure usually leads to the loss of diversity, while insufficient selection pressure often results in the failure of convergence. How to control the selection pressure to balance convergence and diversity remains a challenge in many-objective optimization. To tackle this challenge, a many-objective optimization evolutionary algorithm based on the hyper-dominance degree is proposed in this paper. In the proposed algorithm, the convergence of each solution is quantified by hyper-dominance degree so that the convergence of the population can be controlled by setting a tolerance to screen solutions. To better balance the convergence and the diversity, a tolerance adjusting strategy is designed to control selection pressure during optimization, an improved reference vectors-based diversity preservation strategy is proposed to make the solutions well-distributed in the objective space, and a population reselection strategy based on hyper-dominance degree is proposed to further improve the convergence. The experimental results on various benchmark problems with up to 20 objectives verify that the proposed algorithm outperforms the state-of-the-art peer many-objective optimization algorithms.

Solving optimal power flow problems via a constrained many-objective co-evolutionary algorithm

The optimal power flow problem in power systems is characterized by a number of complex objectives and constraints, which aim to optimize the total fuel cost, emissions, active power loss, voltage magnitude deviation, and other metrics simultaneously. These conflicting objectives and strict constraints challenge existing optimizers in balancing between active power and reactive power, along with good trade-offs among many metrics. To address these difficulties, this paper develops a co-evolutionary algorithm to solve the constrained many-objective optimization problem of optimal power flow, which evolves three populations with different selection strategies. These populations are evolved towards different parts of the huge objective space divided by large infeasible regions, and the cooperation between them renders assistance to the search for feasible and Pareto-optimal solutions. According to the experimental results on benchmark problems and the IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems, the proposed algorithm is superior over peer algorithms in solving constrained many-objective optimization problems, especially the optimal power flow problems.

Multiple Populations for Multiple Objectives Framework With Bias Sorting for Many-Objective Optimization

The convergence and diversity enhancement of multiobjective evolutionary algorithms (MOEAs) to efficiently solve many-objective optimization problems (MaOPs) is an active topic in evolutionary computation. By considering the advantages of the multiple populations for multiple objectives (MPMO) framework in solving multi-objective optimization problems and even MaOPs, this paper proposes an MPMO-based algorithm with a bias sorting (BS) method (termed MPMO-BS) for solving MaOPs to achieve both good convergence and diversity perfor-mance. For convergence, the BS method is applied to each popu-lation of the MPMO framework to enhance the role of nondomi-nated sorting by biasedly paying more attention to the objective optimized by the corresponding population. This way, all the populations in the MPMO framework evolve together to promote the convergence performance on all objectives of the MaOP. For diversity, an elite learning strategy is adopted to generate locally mutated solutions, and a reference vector-based maintenance method is adopted to preserve diverse solutions. The performance of the proposed MPMO-BS algorithm is assessed on 29 widely used MaOP test problems and two real-world application prob-lems. The experimental results show its high effectiveness and competitiveness when compared with seven state-of-the-art MOEAs for many-objective optimization.

A composite surrogate-assisted evolutionary algorithm for expensive many-objective optimization

In many practical many-objective optimization problems, the computational cost of real evaluation is often high. To solve such problems, surrogate-assisted evolutionary algorithms (SAEAs) are effective methods that use low-cost surrogate models to approximate expensive real evaluation and select better-performing individuals for further evaluation. However, existing surrogate models accumulate prediction errors with increasing numbers of objectives, and the computational cost of constructing surrogate models also increases sharply. This paper proposes a composite surrogate-assisted evolutionary algorithm for expensive many-objective optimization. The crux of this method involves categorizing the surrogate models into two distinct types: a global model, constructed based on all costly real evaluations, and local models tailored to subsets derived from clustering all evaluations. Additionally, a new filling sampling criteria is proposed to guide the selection of solutions, considering the requirements of convergence, diversity, and uncertainty. Experimental results on multiple benchmark problems demonstrate that the proposed method is competitive compared to other popular surrogate-assisted evolutionary algorithms in solving expensive many-objective optimization problems within limited computational budgets.

An improved two-archive artificial bee colony algorithm for many-objective optimization

Artificial bee colony (ABC) algorithm has shown good performance on many optimization problems. However, these problems mainly focus on single-objective and ordinary multi-objective optimization problems (MOPs). For many-objective optimization problems (MaOPs), ABC encounters some difficulties. The selection pressure based on Pareto-dominance degrades severely. It is hard to balance convergence and population diversity. To help ABC solve MaOPs, this paper proposes an improved two-archive many-objective ABC (called MaOABC-TA) algorithm. Inspired by the improved two-archive (Two_Arch2) method, MaOABC-TA uses two archives namely convergence archive (CA) and diversity archive (DA) to promote convergence and diversity. Based on CA and DA, three different search strategies are designed to strengthen convergence or diversity in different search stages. In addition, a new probability selection strategy is proposed to choose solutions with good diversity. To verify the performance of MaOABC-TA, it is compared with 10 many-objective evolutionary algorithms (MaOEAs) and 3 many-objective ABCs on DTLZ and MaF benchmark sets with 3, 5, 8, and 15 objectives. Two performance indicators including inverted generational distance (IGD) and hypervolume (HV) and utilized. Experimental results show that MaOABC-TA is more competitive than the compared algorithms in term of the IGD and HV values.

Cooperative-competitive two-stage game mechanism assisted many-objective evolutionary algorithm

It is critical to maintain significant convergence and diversity in many-objective optimization problems (MaOPs) for the performance of many-objective evolutionary algorithms (MaOEAs). However, some issues pose serious challenges to MaOEAs, such as the intensification of the conflict between convergence and diversity, and the lack of Pareto selection pressure. To address the above issues, we propose a cooperative-competitive two-stage game mechanism assisted many-objective evolutionary algorithms (MaOEA-GM). The algorithm is divided into two stages, such as competition and cooperative. In competitive game stage, a strategy pool is constructed, including angle penalty distance strategy and favorable convergence strategy. In addition, a new game utility function is designed to balance convergence and diversity. This method promotes the selection of genetically superior parents for inheritance, so that the population can quickly approach the true Pareto front. In cooperative game stage, individuals choose their preferred environmental selection mechanism by voting. The final scheme is determined using the criterion of minority obedience to the majority, thereby increasing the algorithm Pareto selection pressure. Experimental results demonstrate that compared with five advanced MaOEAs, The MaOEA-GM algorithm has not only preponderance in convergence and diversity indicators, but also higher competitiveness in solving MaOPs.