Many-objective Problems Research Articles

Integrating $$\varepsilon $$-dominance and RBF surrogate optimization for solving computationally expensive many-objective optimization problems

Multi-objective optimization of computationally expensive, multimodal problems is very challenging, and is even more difficult for problems with many objectives (more than three). Optimization methods that incorporate surrogates within iterative frameworks, can be effective for solving such problems by reducing the number of expensive objective function evaluations that need to be done to find a good solution. However, only a few surrogate algorithms have been developed that are suitable for solving expensive many-objective problems. We propose a novel and effective optimization algorithm, $$\varepsilon $$ -MaSO, that integrates $$\varepsilon $$ -dominance with iterative Radial Basis Function surrogate-assisted framework to solve problems with many expensive objectives. $$\varepsilon $$ -MaSO also incorporates a new strategy for selecting points for expensive evaluations, that is specially designed for many-objective problems. Moreover, a bi-level restart mechanism is introduced to prevent the algorithm from remaining in a local optimum and hence, increase the probability of finding the global optimum. Effectiveness of $$\varepsilon $$ -MaSO is illustrated via application to DTLZ test suite with 2 to 8 objectives and to a simulation model of an environmental application. Results on both test problems and the environmental application indicate that $$\varepsilon $$ -MaSO outperforms the other two surrogate-assisted many-objective methods, CSEA and K-RVEA, and an evolutionary many-objective method Borg within limited budget.

Tensor decomposition-based alternate sub-population evolution for large-scale many-objective optimization

• Introduce tensor decomposition to divide higher-dimensional heterogeneous decision variables. • The alternate sub-population evolution asymptotically optimizes the whole population. • A cross-population matching scheme is designed to reconstruct the whole population accurately. • Tensor decomposition-based model can be applied to other MOEAs with large-scale variables. A novel alternate evolution of sub-populations based on tensor decomposition called, TASE is proposed, for solving multi- and many-objective optimization problems with large-scale decision variables in this work. Tensor canonical polyadic (CP) decomposition is introduced for the first time to divide the heterogeneous variables of higher-dimensional decision space into several lower-dimensional sub-components. Furthermore, these sub-populations are optimized alternatively to search for improved solutions in their respective lower-dimensional decision subspace . Finally, a cross-population matching scheme is designed to reconstruct the whole population accurately. The experiments use some largescale multi- and many-objective problems with 2–6 objectives and 1000–5000 variables. The proposed algorithm is compared with other state-of-the-art algorithms, and the experimental results indicate that it can solve some problems that other well-known large-scale optimization algorithms cannot, as well as outperforming other algorithms in terms of solution quality and convergence rate.

Decomposition in decision and objective space for multi-modal multi-objective optimization

Multi-modal multi-objective optimization problems (MMMOPs) have multiple subsets within the Pareto-optimal Set, each independently mapping to the same Pareto-Front. Prevalent multi-objective evolutionary algorithms are not purely designed to search for multiple solution subsets, whereas, algorithms designed for MMMOPs demonstrate degraded performance in the objective space. This motivates the design of better algorithms for addressing MMMOPs. The present work identifies the crowding illusion problem originating from using crowding distance globally over the entire decision space. Subsequently, an evolutionary framework, called graph Laplacian based Optimization using Reference vector assisted Decomposition (LORD), is proposed, which uses decomposition in both objective and decision space for dealing with MMMOPs. Its filtering step is further extended to present LORD-II algorithm, which demonstrates its dynamics on multi-modal many-objective problems. The efficacies of the frameworks are established by comparing their performance on test instances from the CEC 2019 multi-modal multi-objective test suite and polygon problems with the state-of-the-art algorithms for MMMOPs and other multi- and many-objective evolutionary algorithms. The manuscript is concluded by mentioning the limitations of the proposed frameworks and future directions to design still better algorithms for MMMOPs. The source code is available at https://worksupplements.droppages.com/lord.

Modified non-dominated sorting genetic algorithm III with fine final level selection

Dominance resistance is a challenge for Pareto-based multi-objective evolutionary algorithms to solve the high-dimensional optimization problems. The Non-dominated Sorting Genetic Algorithm III (NSGA-III) still has such disadvantage even though it is recognized as an algorithm with good performance for many-objective problems. Thus, a variation of NSGA-III algorithm based on fine final level selection is proposed to improve convergence. The fine final level selection is designed in this way. The θ-dominance relation is used to sort the solutions in the critical layer firstly. Then ISDE index and favor convergence are employed to evaluate convergence of individuals for different situations. And some better solutions are selected finally. The effectiveness of our proposed algorithm is validated by comparing with nine state-of-the-art algorithms on the Deb-Thiele-Laumanns-Zitzler and Walking-Fish-Group test suits. And the optimization objectives are varying from 3 to 15. The performance is evaluated by the inverted generational distance (IGD), hypervolume (HV), generational distance (GD). The simulation results show that the proposed algorithm has an average improvement of 55.4%, 60.0%, 63.1% of 65 test instances for IGD, HV, GD indexes over the original NSGA-III algorithm. Besides, the proposed algorithm obtains the best performance by comparing 9 state-of-art algorithms in HV, GD indexes and ranks third for IGD indicator. Therefore, the proposed algorithm can achieve the advantages over the benchmarks.

A New Decomposition-Based Many-Objective Algorithm Based on Adaptive Reference Vectors and Fractional Dominance Relation

Decomposition-based evolutionary multi-objective algorithms (MOEAs) and many-objective algorithms (MaOEAs) divide a multi-objective problem (MOP) or a many-objective problem (MaOP) into several subproblems by using a set of predefined uniformly distributed reference vectors and can achieve good overall performance especially in maintaining population diversity. However, they encounter huge difficulties in addressing problems with irregular Pareto Fronts (PFs) since many reference vectors do not work during the searching process. To cope with this problem, this paper aims to improve an existing decomposition-based algorithm called reference vector guided evolutionary algorithm (RVEA) by designing an adaptive reference vectors adjustment strategy and strengthening the poor selection pressure. By adding the adaptive strategy, the predefined reference vectors will be dynamically adjusted according to the distribution of promising solutions with good overall performance and the subspaces where the PF lies may be further divided so as to contribute more to the searching process. Besides, the selection pressure with respect to convergence performance posed by RVEA is mainly from the length of normalized objective vectors and the metric is poor in evaluating the convergence performance of a solution with the increasing of objective size. Motivated by that, an improved angle-penalized distance (APD) method based on a newly proposed fractional dominance relation is developed to better distinguish solutions with sound convergence performance in each subspace. To investigate the performance of the proposed algorithm, extensive experiments are conducted to compare it with 5 state-of-the-art decomposition-based algorithms on 3-, 5-, 8-, 10- objective MaF1-MaF9. The results demonstrate that the proposed algorithm obtains the best overall performance.

Preference-inspired coevolutionary algorithm based on differentiated space for many-objective problems

Preference-inspired coevolutionary algorithm based on differentiated space for many-objective problems

A decomposition-based multiobjective evolutionary algorithm with weight vector adaptation

Multi-objective Evolutionary Algorithms (MOEAs) have been concerned and studied with great achievements in the last two decades. As a typical decomposition-based MOEA, MOEA/D aims to decompose a multi-objective optimization problem (MOP) into several subproblems through a set of predefined weight vectors and then optimizes these problems simultaneously. However, performance degradation occurs when complex optimization problems with complicated Pareto Front shape (i.e., irregular and discontinuous PF) are handled. This paper proposes a decomposition-based multi-objective evolutionary algorithm with weight vector adaptation (WVA-MOEA/D) to adjust the weight vectors uniformly distribute in the solution space. The algorithm decomposes a MOP into several subproblems, the new environment selection mechanism defines several neighborhoods with weight vectors as the center of the circle, and elite solutions are selected based on the density of each neighborhood. Weight vector adaptation is employed to guide solution selection and obtain a set of uniformly distributed solutions. The proposed WVA-MOEA/D can improve the performance of MOEA/D on MOPs and many-objective problems with irregular PFs. Besides, the neighborhood adaptation strategy used in the algorithm aims to maintain the diversity solutions and decrease the selection pressure in many-objective optimization problems. Experimental results indicate that WVA-MOEA/D could further effectively solve MOPs with various types of Pareto Fronts for multi-objective and many-objective optimization compared with several state-of-the-art evolutionary algorithms.

Large-scale many-objective particle swarm optimizer with fast convergence based on Alpha-stable mutation and Logistic function

The challenges of the most multi-objective particle swarm optimization (MOPSO) algorithms are to improve the selection pressure, equilibrate the convergence and diversity when tackling large-scale many-objective problems. To overcome these challenges, this paper proposes a novel PSO-based large-scale many-objective algorithm, named as LMPSO. In LMPSO, the Alpha-stable mutation is performed to enhance the diversity of swarm for avoiding premature convergence. And the parameters of PSO and Alpha-stable mutation are dynamically set following the Logistic function, which emphasize different convergence and diversity at different optimization stages. Moreover, LMPSO adopts a fitness to maintain the external archive, and the calculation of fitness is based on binary additive epsilon indicator. The binary indicator is also used to update the personal best of particles to avoid wrongly selecting dominance resistance solutions (DRSs). Aims for improving the selection pressure, the proposed algorithm employs a concept of dominance resistance error to identify the DRSs. To verify this idea, the DTLZ, ZDT, and LSMOP test suites with up to 1000 decision variables and 10-objective are used to qualify the performance of LMPSO. The simulations reveal the fact that the LMPSO significantly outruns the several chosen state-of-the-art algorithms when solving large-scale many-objective test instances.

Non-dominated sorting on performance indicators for evolutionary many-objective optimization

Much attention has been paid to evolutionary multi-objective optimization approaches to efficiently solve real-world engineering problems with multiple conflicting objectives. However, the loss of selection pressure and the non-uniformity in the distribution of the Pareto optimal solutions in the objective space can impede both dominance-based and decomposition-based multi-objective optimizers when solving many-objective problems. In this work, we circumvent this issue by exploiting two performance indicators, and use these in an optimizer’s environmental selection via non-dominated sorting. This effectively converts the original many-objective problem into a bi-objective one. Our convergence performance criterion tries to balance the performance of individuals in different parts of the objective space. The angle between solutions on objective space is adopted to measure the diversity of each individual. Using these solutions can be separated into different layers easily, which is often not possible for the original many-objective optimization representation. The performance of the proposed method is evaluated on the DTLZ benchmark problems with up to 30 objectives, and MaF test suite with 10, 15, 20 and 30 objectives. The experimental results show that our proposed method is competitive compared to six recently proposed algorithms, especially for solving problems with a large number of objectives.

An indicator and adaptive region division based evolutionary algorithm for many-objective optimization

Problems involving four or more objectives are termed Many-objective problems (MaOPs), which pose serious challenges to existing evolutionary algorithms (EAs). Although EAs via decomposition exhibit encouraging performance in handling MaOPs, they need a set of predefined weight vectors, which is not well adaptable to problems possessing various Pareto front (PF) shapes. Besides, the nature of subproblem formulations renders the overwhelming convergence property. In this paper, we introduce an indicator and adaptive region division based EA, referred to as IREA, which is free from the presetting of weight vectors. To be specific, an angular distance based space division module and a proximity-oriented indicator are incorporated into IREA, where the former highlights diversity adaptively via maximum angular distance while the latter emphasizes convergence in a local manner. Furthermore, the quality of mating pool is leveraged by a coordinate transformation assisted niche technique. The proposed IREA is compared with several prevalent many-objective EAs on scalable MaOPs with varying characteristics, as well as on the many-objective cloud manufacturing service composition problems. The experimental results demonstrate that IREA is highly competitive and can be used as an alternative for handling MaOPs.

Achievement scalarizing function sorting for strength Pareto evolutionary algorithm in many-objective optimization

Multi-objective evolutionary algorithms (MOEAs) have proven their effectiveness in solving two or three objective problems. However, recent research shows that Pareto-based MOEAs encounter selection difficulties facing many similar non-dominated solutions in dealing with many-objective problems. In order to reduce the selection pressure and improve the diversity, we propose achievement scalarizing function sorting strategy to make strength Pareto evolutionary algorithm suitable for many-objective optimization. In the proposed algorithm, we adopt density estimation strategy to redefine a new fitness value of a solution, which can select solution with good convergence and distribution. In addition, a clustering method is used to classify the non-dominated solutions, and then, an achievement scalarizing function ranking method is designed to layer different frontiers and eliminate redundant solutions in the environment selection stage, thus ensuring the convergence and diversity of non-dominant solutions. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems with 3, 5, 8, 10 objectives. Experimental studies demonstrate that the proposed algorithm shows very competitive performance.

An adaptive decomposition evolutionary algorithm based on environmental information for many-objective optimization

The performance of traditional penalty boundary intersection (PBI) decomposition-based evolutionary algorithm is totally determined by the penalty factor. The fixed penalty factor causes the imbalance between the convergence and the diversity when solving many-objective problems. So, an adaptive decomposition evolutionary algorithm based on environmental information (MaOEA/ADEI) is proposed to solve the imbalance. The penalty factor of PBI decomposition is determined by the environmental information (include distribution information of weight vectors and population). Furthermore, the parent individual selection strategy is introduced to select promising individuals for variation and the weight vectors adaption strategy is used to handle problems with scaled objectives. Comparisons with 4 algorithms on 24 benchmark instances are used to test the property of MaOEA/ADEI. The experimental results show MaOEA/ADEI performs best on 14 test instances.

A surrogate-ensemble assisted expensive many-objective optimization

The surrogate ensemble has been shown to have a good performance for assisting the evolutionary algorithms to solve computationally expensive single-objective problems. However, it has not been paid much attention to assist in multi-/many-objective optimization. In this paper, we propose to train multiple surrogate models to assist many-objective optimization algorithm for solving expensive many-objective problems. In the proposed method, some non-dominated solutions, which have been evaluated using the expensive objectives, will be utilized together with the current population to generate the offspring in order to prevent the population from moving forward to a bad region. In the model management, the way to determine the method used for selecting individuals to be evaluated using the exact expensive objectives depends on the convergence degree of the current population. A new method to measure the uncertainty is also proposed in our method by considering the distance to the samples in the decision space and the approximation variance in the objective space. The experimental results on a number of benchmark problems show that our proposed method is competitive to two state-of-the-art surrogate-assisted evolutionary algorithms for solving expensive many-objective problems in a limited computational budget.

A many-objective evolutionary algorithm based on rotation and decomposition

Abstract Evolutionary algorithms have shown their promise in addressing multiobjective problems (MOPs). However, the Pareto dominance used in multiobjective optimization loses its effectiveness when addressing many-objective problems (MaOPs), which are defined as having more than three objectives. This is because the Pareto dominance loses its ability to distinguish between individuals. In this paper, a many-objective evolutionary algorithm based on rotation and decomposition is proposed (MaOEA-RD) to overcome the shortcoming of insufficient selection pressure caused by the Pareto dominance. First, the coordinates system is rotated and a hyperplane is established to distinguish between the nondominated individuals. Then, a novel individual selection mechanism incorporating decomposition is adopted to maintain the diversity of the population. In order to compensate for the deficiency of the predefined reference vectors, a reference vector adjustment mechanism is proposed. Experimental studies on several well-known benchmark problems show that the proposed algorithm is competitive compared with nine state-of-the-art many-objective algorithms.

A novel encoding for separable large-scale multi-objective problems and its application to the optimisation of housing stock improvements

Large-scale optimisation problems, having thousands of decision variables, are difficult as they have vast search spaces and the objectives lack sensitivity to each decision variable. Metaheuristics work well for large-scale single-objective optimisation, but there has been little work for large-scale, multi-objective optimisation. We show that, for the special case problem where the objectives are each additively-separable in isolation and share the same separability, the problem is not separable when considering the objectives together. We define a problem with this property: optimisation of housing stock improvements, which seeks to distribute limited public investment to achieve the optimal reduction in the housing stock’s energy demand. We then present a two-stage approach to encoding solutions for additively-separable, large-scale, multi-objective problems called Sequential Pareto Optimisation (SPO), which reformulates the global problem into a search over Pareto-optimal solutions for each sub-problem. SPO encoding is demonstrated for two popular MOEAs (NSGA-II and MOEA/D), and their relative performance is systematically analysed and explained using synthetic benchmark problems. We also show that reallocating seed solutions to the most appropriate sub-problems substantially improves the performance of MOEA/D, but overall NSGA-II still performs best. SPO outperforms a naive single-stage approach, in terms of the optimality of the solutions and the computational load, using both algorithms. SPO is then applied to a real-world housing stock optimisation problem with 4424 binary variables. SPO finds solutions that save 20% of the cost of seed solutions yet obtain the same reduction in energy consumption. We also show how application of different intervention types vary along the Pareto front as cost increases but energy use decreases; e.g., solid wall insulation replacing cavity wall insulation, and condensing boilers giving way to heat pumps. We conclude with proposals for how this approach may be extended to non-separable and many-objective problems.

A many-objective particle swarm optimization with grid dominance ranking and clustering

The existing MOPSOs face a great challenge in dealing with many-objective problems, due to the low discriminability of particles in many-objective spaces, which will affect the selection of leaders, thereby deteriorating the effectiveness of the algorithm. By using the property of domination, with the mapping of grid coordinates, this paper presents a framework of grid-based ranking scheme to sort the particles in grid space, which can select the swarm leaders (gbest, pbest) efficiently in MOPSO and enhance the convergence. Moreover, the clustering operation is conducted to update and maintain the external archive in the grid space to output a more diverse Pareto front. The performance of the proposed algorithm is verified by benchmark comparisons with several state-of-the-art MOPSOs and MOEAs. The DTLZ and WFG test suites with 4 to 10 objectives are used to assess the performance of our proposal. Experimental results demonstrate the promising performance of the proposed algorithm in terms of both optimization quality and convergence speed. Additionally, we discuss the influence of grid partition on efficiency, which verifies the effectiveness of the proposal from the other aspect.

The hybrid bacterial foraging algorithm based on many-objective optimizer.

A new multi-objective optimized bacterial foraging algorithm - Hybrid Multi-Objective Optimized Bacterial Foraging Algorithm (HMOBFA) is presented in this article. The proposed algorithm combines the crossover-archives strategy and the life-cycle optimization strategy, look for the best method through research area. The crossover-archive strategy with an external archive and internal archive is assigned to different selection principles to focus on diversity and convergence separately. Additionally, according to the local landscape to satisfy population diversity and variability as well as avoiding redundant local searches, individuals can switch their states periodically throughout the colony lifecycle with the life-cycle optimization strategy. all of which may perform significantly well. The performance of the algorithm was examined with several standard criterion functions and compared with other classical multi-objective majorization methods. The examiner results show that the HMOBFA algorithm can achieve a significant enhancement in performance compare with other method and handles many-objective issues with solid complexity, convergence as well as diversity. The HMOBFA algorithm has been proven to be an excellent alternative to past methods for solving the improvement of many-objective problems.

A decomposition-based many-objective ant colony optimization algorithm with adaptive reference points

The discrete many-objective problem (MaOP) is challenging in practice. Improving the convergence speed and making the nondominated solutions close to the Pareto front (PF) are vital issues in the optimization of the discrete MaOP. This pper proposed a modified decomposition-based many objective ant colony optimization (ACO) algorithm and employs an adaptive reference point mechanism that chooses the ideal point or nadir point as the reference point according to the distribution of the candidate solutions. This mechanism is utilized to improve the selection operator, accelerate the convergence speed and enhance the optimization ability. A comparative experiment is conducted on traveling salesman problems (TSPs) constrained by two, five, and ten objectives, which are built using test cases from the TSPLIB. The experimental results indicate that the inverted generational distance (IGD) indicator of the nondominated solution of the proposed algorithm has a rapid convergence speed and that the proposed algorithm achieves competitive performance regarding its optimization quality.

Robust optimization of multi-scenario many-objective problems with auto-tuned utility function

ABSTRACT The main focus in optimization studies, including multi-objective and multi-scenario optimization, is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. Manual preferences and selection are required to obtain a single implementable solution from the obtainable Pareto set. For achieving such a single optimum, the multi-scenario property is applied to adjust the required preference weights automatically; as a result, robust performance for various scenarios is provided. The robust quality definition is believed to decrease the performance uncertainty owing to any possible scenario in real-world settings. These challenges have motivated the creation of a method that searches a robust solution using evolutionary algorithms and well-applied IT tools. This selection process applies two robustness aspects simultaneously: the necessary weights for the aggregation of multi-scenario and multi-objective dimensions are adjusted to obtain similar performances for scenarios as much as possible; the robustness against system parameters is evaluated by the calculation of the robustness indices. The method can be used universally where the robust optimum is required for a model having multi-scenario and multi-objective properties. The proposed method has been tested on test functions, and in conclusion it has been applied to two engineering problems.

RODE: Ranking-Dominance-Based Algorithm for Many-Objective Optimization with Opposition-Based Differential Evolution

During the period of 1990s and early 2000s, the Pareto-dominance (PD) relation was successfully applied for solving multiobjective optimization problems (MOPs) with small number of objectives (typically not exceeding four objectives). However, the performance of these PD-based multiobjective evolutionary algorithms (MOEAs) becomes hopeless when it comes to solving problems with larger number of objectives. Many alternative dominance relations have been proposed in the last few years to improve the search ability of EMO algorithms. In this paper, we present the RODE algorithm as a novel scalable approach for many-objective problems which adopts the ranking-dominance relation for evaluating the fitness of solutions that provides improved convergence. The evolutionary search mechanism employed in our algorithm is the conventional differential evolution (DE) approach. However, for attaining the improved diversity of solutions, we have incorporated the weight vectors and opposition-based differential evolution (ODE) in a unique way. In order to validate our RODE approach, we have compared it with other state-of-the-art MOEAs, namely GDE3, NSGAII and two versions of MOEA/D, namely MOEA/D-TCH and MOEA/D-PBI. All the MOEAs have been executed on the standard benchmark problems DTLZ1-DTLZ4 with 5-, 8-, 10-, 15-, and 20-D objective spaces and MaF03, MaF05 and MaF06 with 8-, 10- and 15-D objective spaces. In almost all the simulation experiments (especially with higher than 5-objectives), our approach has achieved promising results in terms of convergence and diversity of solutions.