Evolutionary Algorithm For Many-objective Optimization Research Articles

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.

Enhanced decomposition-based hybrid evolutionary and gradient-based algorithm for many-objective optimization

Enhanced decomposition-based hybrid evolutionary and gradient-based algorithm for many-objective optimization

A Two-Stage Hypervolume-Based Evolutionary Algorithm for Many-Objective Optimization

Many-objective optimization is a critical research topic in the evolutionary computing community. Many algorithms have been proposed to tackle this problem, with evolutionary algorithms based on the hypervolume being among the most effective ones. However, calculating the hypervolume indicator in high-dimensional objective spaces remains time-consuming. To address this issue, we propose a two-stage hypervolume-based evolutionary algorithm (ToSHV) that separates global search and local search to ensure both convergence and diversity. ToSHV performs a global search in the first stage by generating multiple offspring per generation. We modified the R2HCA method to estimate the overall hypervolume contribution, avoiding the time-consuming nature of updating the hypervolume contribution with the greedy method. In the second stage, only one offspring is produced per generation to emphasize local exploration and enhance population diversity. Furthermore, a stage-switching mechanism is designed to dynamically select the appropriate search mode based on the prevailing population distribution. We evaluate our algorithm on WFG and DTLZ test suites, comparing it with three hypervolume-based algorithms and four state-of-the-art algorithms. Experimental results show that our approach is competitive in most cases.

A point crowding-degree based evolutionary algorithm for many-objective optimization

A point crowding-degree based evolutionary algorithm 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.

A Projection-Based Evolutionary Algorithm for Multi-Objective and Many-Objective Optimization

Many-objective optimization problems (MaOPs) are challenging optimization problems in scientific research. Research has tended to focus on algorithms rather than algorithm frameworks. In this paper, we introduce a projection-based evolutionary algorithm, MOEA/PII. Applying the idea of dimension reduction and decomposition, it divides the objective space into projection plane and free dimension(s). The balance between convergence and diversity is maintained using a Bi-Elite queue. The MOEA/PII is not only an algorithm, but also an algorithm framework. We can choose a decomposition-based or dominance-based algorithm to be the free dimension algorithm. When it is an algorithm framework, it exhibits a better performance. We compare the performance of the algorithm and the algorithm with the MOEA/PII framework. The performance is evaluated by benchmark test instances DTLZ1-7 and WFG1-9 on 3, 5, 8, 10, and 15 objectives using IGD-metric and HV-metric. In addition, we investigated its superior performance on the wireless sensor networks deployment problem using C-metric. Moreover, determining objective domain for the objects of the wireless sensor networks deployment problem reduces the time and makes the solution set more responsive to user needs.

An incremental learning evolutionary algorithm for many-objective optimization with irregular Pareto fronts

Multi objective optimization problems (MOPs) with irregular Pareto Fronts (PFs) appear frequently in a large of practical cases. Proximity and diversity are the key indicators for decomposition-based multi-objective evolutionary algorithms (MOEA/D) to solve MOPs, which are decided by the neighborhood structure and a set of predefined direction vectors. In certain MOPs with irregular PFs, the effectiveness of MOEA/D is restricted by the shape of the PF. This limitation arises because MOEA/D generates direction vectors in a uniform manner, which can result in invalid direction vectors due to the irregularity of the PF. When dealing with many-objective optimization problems (MaOP) that have irregular PFs, such as degenerate or disconnected PFs, this decline in effectiveness becomes particularly problematic. To address this limitation, a copula incremental learning (CIL) scheme has been developed to progressively extract implicit knowledge on the appropriate distribution of direction vectors for generating non-uniform direction vectors. Additionally, a niche hierarchical selection (NHS) methodology is employed to construct the neighborhood structure and prevent the generation of duplicate solutions. These enhancements are designed to improve the overall effectiveness of MOEA/D. Efficiency is further ensured through the use of convergence-guided direction (CGD) to approximate irregular PFs. Statistical analysis indicates that the proposed method outperforms other competitive algorithms across most test benchmarks, particularly in its ability to effectively address MaOPs with irregular PFs.

A Line Complex-Based Evolutionary Algorithm for Many-Objective Optimization

A Line Complex-Based Evolutionary Algorithm for Many-Objective Optimization

Dynamical decomposition and selection based evolutionary algorithm for many-objective optimization

Decomposition-based many-objective evolutionary algorithms decompose the objective space into multiple subregions, with the help of a set of predefined reference vectors. These vectors serve to guide coevolution between subproblems, and while they show potential for maintaining the diversity of solutions, they have limited exploration capabilities in complex problems and high-dimensional objective space. The main issue is that the predefined reference vectors cannot maintain uniformity of the intersection points between search directions and the irregular Pareto front (PF). To address this problem, this paper proposes a dynamical decomposition and selection strategy (DDS). In DDS, the predefined reference vectors are replaced by solutions themselves and normal-boundary directions (NBI), which guide the population to automatically adapt to the shape of PF. The subregions adapt to divide the objective space to increase diversity. Then, to adjust the relationship between diversity and convergence, a dynamical selection strategy based on the course of evolution is proposed. The process of dynamical decomposition and selection strategy is repeated until the termination condition is met. The proposed algorithm is compared with the state-of-the-art many-objective optimization algorithms on several benchmark problems with 5 to 15 objectives in evolutionary computation. Experimental results show that it outperforms other algorithms in most test instances and is less sensitive to irregular PFs.

Objective Space-Based Population Generation to Accelerate Evolutionary Algorithms for Large-Scale Many-Objective Optimization

The generation and updating of solutions, e.g., crossover and mutation, of many existing evolutionary algorithms directly operate on decision variables. The operators are very time consuming for large-scale and many-objective optimization problems. Different from them, this work proposes an objective space-based population generation method to obtain new individuals in the objective space and then map them to decision variable space and synthesize new solutions. It introduces three new objective vector generation methods and uses a linear mapping method to tightly connect objective space and decision one to jointly determine new-generation solutions. A loop can be formed directly between two spaces, which can generate new solutions faster and use more feedback information in the objective space. In order to demonstrate the performance of the proposed algorithm, this work performs a series of empirical experiments involving both large-scale decision variables and many objectives. Compared with the state-of-the-art traditional and large-scale algorithms, the proposed method exceeds or at least reaches its peers’ best level in overall performance while achieving great saving in execution time.

Growing neural gas assisted evolutionary many-objective optimization for handling irregular Pareto fronts

Reference vector adjustment approaches pose a considerable potential to strengthen the capability of decomposition-based evolutionary many-objective algorithms in handling many-objective optimization problems (MaOPs) with irregular Pareto-optimal fronts (PFs). However, most existing approaches often rely on the current population’s distribution, which may not well match MaOPs’ PFs. This can easily mislead the reference vector adjustment and impair algorithms’ performance. To address the abovementioned issue, this paper designs a growing neural gas-assisted evolutionary many-objective optimization algorithm, called GNG-EMO. Specifically, the proposal contains an environmental selection without reference vectors to maintain a well-diversified and converged archive over the PFs. Also, the GNG-EMO leverages a growing neural gas (GNG) network to learn Pareto-optimal fronts’ topologies using the solutions in the archive. In addition, to alleviate the issue that the nodes of the GNG network cannot reach the boundaries of Pareto-optimal fronts, we design an expansion mechanism to further diversify the nodes of the GNG network for constructing a set of better reference vectors. To evaluate the performance of the GNG-EMO, we compare it with five state-of-the-art competitors in the context of 48 test cases with irregular PFs. The numerical results demonstrate the GNG-EMO’s superior competitiveness by significantly outperforming all the five competitors on 27 test cases with respect to the indicator Hypervolume.

A double association-based evolutionary algorithm for many-objective optimization.

In this paper, a double association-based evolutionary algorithm (denoted as DAEA) is proposed to solve many-objective optimization problems. In the proposed DAEA, a double association strategy is designed to associate solutions with each subspace. Different from the existing association methods, the double association strategy takes the empty subspace into account and associates it with a promising solution, which can facilitate the exploration of unknown areas. Besides, a new quality evaluation scheme is developed to evaluate the quality of each solution in subspace, where the convergence and diversity of each solution is first measured, and in order to evaluate the diversity of solutions more finely, the global diversity and local diversity is designed to measure the diversity of each solution. Then, a dynamic penalty coefficient is designed to balance the convergence and diversity by penalizing the global diversity distribution of solutions. The performance of DAEA is validated by comparing with five state-of-the-art many-objective evolutionary algorithms on a number of well-known benchmark problems with up to 20 objectives. Experimental results show that our DAEA has high competitiveness in solving many-objective optimizatiopn problems compared with the other compared algorithms.

ACDB-EA: Adaptive convergence-diversity balanced evolutionary algorithm for many-objective optimization

Recently, evolutionary algorithms (EAs) have shown their strong competitiveness in handling many-objective optimization problems (MaOPs) with different Pareto fronts (PFs). However, maintaining convergence and diversity simultaneously in high-dimensional problems can be further explored. This paper suggests an adaptive convergence-diversity balanced evolutionary algorithm (ACDB-EA) to handle the above issue, which maintains balance between convergence and diversity adaptively during the evolutionary process. In the proposed algorithm, a novel diversity maintenance mechanism based on the global and local diversities is developed to promote the diversity by considering them collaboratively. To be specific, the average similarity and the maximal similarity represent respectively the global and local diversities of the solution, where the similarity between two solutions is defined as the cosine similarity between their objective vectors. In the environmental selection, the proposed adaptive convergence-diversity balanced strategy is used to adjust weights of convergence (defined as the L 2 norm in the objective space), global diversity and local diversity according to the population adaptively. Under this strategy, each solution produces a score and the solution with the highest score will enter the next generation, which means it acquires the optimal performance in terms of convergence and diversity. In each iteration, scores of candidate solutions will be recalculated to continuously search for the most suitable one, which strengthens the selection pressure toward the true PFs. We conduct experimental study on 111 benchmark testing instances with 2-20 objectives. The proposed method is shown to be superior to seven state-of-the-art algorithms in maintaining balance between convergence and diversity.

An effective and efficient evolutionary algorithm for many-objective optimization

In evolutionary multiobjective optimization, effectiveness refers to how an evolutionary algorithm performs in terms of converging its solutions into the Pareto front and also diversifying them over the front. This is not an easy job, particularly for optimization problems with more than three objectives, dubbed many-objective optimization problems. In such problems, classic Pareto-based algorithms fail to provide sufficient selection pressure towards the Pareto front, whilst recently developed algorithms, such as decomposition-based ones, may struggle to maintain a set of well-distributed solutions on certain problems (e.g., those with irregular Pareto fronts). Another issue in some many-objective optimizers is rapidly increasing computational requirement with the number of objectives, such as hypervolume-based algorithms and shift-based density estimation (SDE) methods. In this paper, we aim to address this problem and develop an effective and efficient evolutionary algorithm (E3A) that can handle various many-objective problems. In E3A, inspired by SDE, a novel population maintenance method is proposed to select high-quality solutions in the environmental selection procedure. We conduct extensive experiments and show that E3A performs better than 11 state-of-the-art many-objective evolutionary algorithms in quickly finding a set of well-converged and well-diversified solutions.

An adaptive switching-based evolutionary algorithm for many-objective optimization

Pareto-based evolutionary algorithms are challenging in dealing with many-objective problems encountering many incomparable nondominated solutions. To reduce selection pressure and improve diversity, this paper proposes an adaptive switching strategy-based evolutionary algorithm for many-objective optimization. This strategy contains two deletion criteria, which are switched adaptively between generations, aiming to delete poor solutions one by one in environmental selection. The first criterion is devised to delete the solution with poor convergence among the two most similar solutions. The second criterion is developed to delete the worse solution according to a designed indicator that takes into account both convergence and diversity. Finally, comparisons with five state-of-the-art many-objective evolutionary algorithms on some widely used benchmark problems and the water resource planning problem are given to illustrate the effectiveness and advantages of the proposed algorithm.

A C[formula omitted]-dominance-based solution estimation evolutionary algorithm for many-objective optimization

Balancing convergence and diversity is a key issue for many-objective optimization problems (MaOPs), which is a great challenge to the classical Pareto-based multi-objective algorithms due to its severe lack of selection pressure. To relieve the above challenge, a Cα-dominance-based solution estimation evolutionary algorithm is proposed for MaOPs. In the proposed algorithm, a new dominance method, called Cα-dominance, is proposed to provide reasonable selection pressure for MaOPs. By designing a nonlinear function to transform the original objectives, Cα-dominance expands the dominated area where dominance resistant solutions located, while remains the solutions to be non-dominated in area close to Pareto optimal solutions. Furthermore, an adaptive parameter adjustment mechanism on the unique parameter α of Cα-dominance is designed to control the expansion degree of the dominance area based on the number of objectives and the stages of evolution. Finally, a new solution estimation scheme based on Cα-dominance is designed to evaluate the quality of each solution, which incorporates convergence information and diversity information of each solution. The experimental results on widely used benchmark problems having 5–20 objectives have shown the proposed algorithm is more effective in terms of both convergence enhancement and diversity maintenance.

A surrogate-assisted evolutionary algorithm for expensive many-objective optimization in the refining process

Computationally expensive optimization problems are difficult for an evolutionary algorithm within limited fitness evaluations, especially for many-objective optimization. To remedy this issue, a surrogate-assisted decomposition-based evolutionary algorithm is proposed in this work where the Kriging model is used to approximate each objective function. In order to balance exploration and exploitation, the model management strategy integrates the infill criterion with the distribution information of both weight vectors and population. Additionally, a one-by-one replacement strategy is developed to select a fixed number of training data to limit computational time for the model training as well as maintain the approximation accuracy. The empirical results show that the performance of the proposed algorithm is competitive to the state-of-the-art algorithms on a number of benchmarks. Finally, the proposed algorithm is applied for a real-world optimization problem in the refining process and exhibits superior performance.

Performance limits of the single screw expander in organic Rankine cycle with ensemble learning and hyperdimensional evolutionary many-objective optimization algorithm intervention

The operating performance of the expander, as the core component of the ORC system, directly determines the comprehensive performance of the system. Single screw expander (SSE) has received widespread attention due to its advantages such as large single-stage expansion pressure ratio, insensitivity to liquid, wide working range, and balanced forces. This study fully considers the strong coupling, high nonlinearity, large time variability and uncertainty between SSE performance evaluation indexes and operating parameters. On this basis, a scatter–bilinear interpolation hybrid feature selection approach is proposed, and the feature selection and analysis of the correlation degree among SSE performance evaluation indexes are conducted. In the high-dimensional space, the influence of operating parameters on different performance evaluation indexes is analyzed. Then, this study proposes the ensemble learning-driven non-dominated sorting genetic algorithm-III (NSGA-III) approach. An ensemble learning-driven NSGA-III model for SSE performance limit optimization is constructed on the basis of the scatter–bilinear interpolation hybrid feature selection method and the ensemble learning-driven NSGA-III approach. In the hyperdimensional space, the performance limits of SSE are obtained. When the comprehensive performance of SSE reaches its limit, the shaft efficiency, expansion pressure ratio, and power output are 53.71%, 7.76, and 10.83 kW, respectively. This research provides a reliable reference for analyzing, designing, and optimizing the SSE performance limits, and it also provides a direct guidance for obtaining the SSE performance limits in the ORC system.

A multistage evolutionary algorithm for many-objective optimization

Convergence and diversity are of high significance to many-objective optimization, which are considered by most state-of-the-art many-objective evolutionary algorithms (MaOEAs) simultaneously. However, it is not easy to balance them during the optimization process due to their conflicting nature. This study proposes a multistage MaOEA to address this issue, where convergence and diversity are processed respectively at different optimization stages. At the first stage, the population approaches Pareto front rapidly and the diversity is ignored. After the population is converged, the diversity will be emphasized by applying the decision variable clustering method at the second stage. When the population achieves high convergence and diversity, the algorithm will enter the last stage, where the quality of the solution set is fine-tuned by substituting those solutions with worse convergence and diversity degrees. As demonstrated by the experimental results with peer competitors on common benchmark problems, that the proposed algorithm is promising.

Adaptive Decomposition-Based Evolutionary Algorithm for Many-Objective Optimization with Two-Stage Dual-Density Judgment

Adaptive Decomposition-Based Evolutionary Algorithm for Many-Objective Optimization with Two-Stage Dual-Density Judgment