Algorithm For Many-objective Optimization Problems Research Articles

An improved indicator-based two-archive algorithm for many-objective optimization problems

An improved indicator-based two-archive algorithm for many-objective optimization problems

A Coevolutionary Algorithm for Many-Objective Optimization Problems with Independent and Harmonious Objectives

Evolutionary algorithm is an effective strategy for solving many-objective optimization problems. At present, most evolutionary many-objective algorithms are designed for solving many-objective optimization problems where the objectives conflict with each other. In some cases, however, the objectives are not always in conflict. It consists of multiple independent objective subsets and the relationship between objectives is unknown in advance. The classical evolutionary many-objective algorithms may not be able to effectively solve such problems. Accordingly, we propose an objective set decomposition strategy based on the partial set covering model. It decomposes the objectives into a collection of objective subsets to preserve the nondominance relationship as much as possible. An optimization subproblem is defined on each objective subset. A coevolutionary algorithm is presented to optimize all subproblems simultaneously, in which a nondominance ranking is presented to interact information among these sub-populations. The proposed algorithm is compared with five popular many-objective evolutionary algorithms and four objective set decomposition based evolutionary algorithms on a series of test problems. Numerical experiments demonstrate that the proposed algorithm can achieve promising results for the many-objective optimization problems with independent and harmonious objectives.

An adaptive convergence enhanced evolutionary algorithm for many-objective optimization problems

For evolutionary computation, how to balance the convergence and diversity of populations is a challenging problem for solving many-objective optimization problems. In particular, with the increasing of the number of objectives, the non-dominated solutions in the population increase sharply, and there is no sufficient convergence pressure for the population to converge to the true Pareto front. How to solve this problem, we propose a new preferred solution selection strategy by finding non-dominated solutions with better convergence to enhance the convergence pressure of evolution, where a number of special solutions are selected with the best convergence index value from the surviving solutions of the previous generation. The number of special solutions determines the convergence pressure, so an adaptive updating method of the number of special solutions is further proposed to dynamically adjust the convergence pressure of the population, the aim is to guide the convergence direction to the Pareto front. Based on these strategies, we propose a convergence enhanced evolutionary algorithm (CEEA) to balance the convergence and diversity of the search algorithm. A large number of experiments have been carried out on some benchmark many-objective problems with 3–15 objectives, experimental results demonstrate that CEEA has better results compared with some state-of-the-art many-objective evolutionary algorithms.

A knowledge guided bacterial foraging optimization algorithm for many-objective optimization problems

A knowledge guided bacterial foraging optimization algorithm for many-objective optimization problems

A bi-layer decomposition algorithm for many-objective optimization problems

A bi-layer decomposition algorithm for many-objective optimization problems

An improved farmland fertility algorithm for many-objective optimization problems

Recent studies on many-objective optimization problems (MaOPs) have tended to employ some promising evolutionary algorithms with excellent convergence accuracy and speed. However, difficulties in scalability upon MaOPs including the selection of leaders, etc., are encountered because the most evolutionary algorithms are proposed for single-objective optimization. To further improve the performance of many-objective evolutionary algorithms in solving MaOPs when the number of the objectives increases, this paper proposes a many-objective optimization algorithm based on the improved Farmland Fertility algorithm (MOIFF). In MOIFF, a novel bio-inspired meta heuristic method proposed in 2018, called Farmland Fertility algorithm (FF), is employed to serve as the optimization strategy. In order to handle MaOPs effectively, FF has been tailored from the following aspects. An individual fitness assessment approach based on cumulative ranking value has been proposed to distinguish the quality of each individual; a novel method based on individual cumulative ranking value to constitute and update the global memory and local memory of each individual is proposed, and a hybrid subspace search and full space search method has been designed to update individuals in the stages of soil optimization and soil fusion. In addition, adaptive environmental selection has been proposed. Finally, MOIFF is compared with four state-of-the art many-objective evolutionary algorithms on many test problems with various characteristics, including the DTLZ and WFG test suites. Experimental results demonstrate that the proposed algorithm has competitive convergence and diversity on MaOPs.

A Survey of Decomposition Based Evolutionary Algorithms for Many-Objective Optimization Problems

The framework of decomposition-based multi-objective evolutionary algorithms(MOEA/D) has evolved for more than ten years, and it has become irreplaceable tool for solving multi-objective optimization problems. In recent years, many scholars have investigated improved strategies from different directions. This paper gives a systematic comparison of six different components for decomposition-based algorithms, including framework analysis, weight vector generation scheme, aggregation evaluation function construction, reproduction operator, individual selection and update strategy, and the characteristics and application scope of various algorithms are also analyzed in detail in the survey. Different from previous survey on decomposition-based multi-objective evolutionary algorithms, a more detailed classification and experimental comparison are elaborated in the proposed paper.

A Non-dominated Sorting based Evolutionary Algorithm for Many-objective Optimization Problems

The optimization problems with more than three objectives are many-objective optimization problems exist in various scientific and engineering domains. The existing multi-objective evolutionary algorithmic approaches primarily developed to address problems up to three objectives. Such multi-objective evolutionary algorithms do not found effective to address the many-objective optimization problems. The limitations of existing multi-objective evolutionary algorithms initiated the need to develop a specific algorithm which efficiently solves the many-objective optimization problems. The proposed work presents the design of the MaOHDE to address MaOPs. Initially, NS-MODE & NS-MOPSO algorithms developed by incorporating the non-dominated sorting approach from NSGA-II, the ranking approach, weight vector, and reference points. The widely used Tchebycheff – a decomposition-based approach applied to decompose the MaOPs. The MaOHDE algorithm developed by hybridizing the NS-MODE with NS-MOPSO. The presented approach’s strength is revealed using 20 instances of DTLZ functions. The effectiveness and efficiency are verified by comparing with MaOJaya, RD-EMO, NSGA-III, MOEA/D, MOEA/DD, RVEA, and MOEA/D-M2M algorithms. From the results, it is observed that the hybridization of NS-MODE and NS-MOPSO as MaOHDE responds better than its competitors for most of the test instances or it is competitive. The convergence rate is also good as compared with other state-of-art algorithms.

A Novel Nonlinear Expanded Dominance Relation Based Evolutionary Algorithm for Many-Objective Optimization Problems

Multi-objective optimization problems exist widely in scientific research and engineering applications. With the number of objectives increasing, the proportion of non-dominated individuals in the population of many-objective optimization problems increases sharply, resulting in a reduction of convergence pressure of the traditional multi-objective optimization algorithms. In some cases, the optimal solutions may be located in the special regions, such as many discrete regions and the regions with very few feasible solutions. In this case, the existing nonlinear expanded evolutionary algorithm can not find the true Pareto fronts. To address the limitation, a novel nonlinear expanded dominance relation based many-objective evolutionary algorithm is proposed to handle many-objective optimization problems. Experimental results show that compared with the state of art algorithms, the proposed algorithm is effective for DTLZs, in terms of IGD, PD and GD metrics.

A chaotic-based improved many-objective Jaya algorithm for many-objective optimization problems

The Jaya algorithm is a recently developed novel population-based algorithm. The proposed work presents the modifications in the existing many-objective Jaya (MaOJaya) algorithm by integrating the chaotic sequence to improve the performance to optimize many-objective benchmark optimization problems. The MaOJaya algorithm has exploitation more dominating, due to which it traps in local optima. The proposed work aims to reduce these limitations by modifying the solution update equation of the MaOJaya algorithm. The purpose of the modification is to balance the exploration and exploitation, improve the divergence and avoid premature convergence. The well-known chaotic sequence - a logistic map integrated into the solution update equation. This modification keeps the MaOJaya algorithm simple as well as, preserves its parameterless feature. The other component of the existing MaOJaya algorithm, such as non-dominated sorting, reference vector and tournament selection scheme of NSGA-II is preserved. The decomposition approach used in the proposed approach simplifies the complex many-objective optimization problems. The performance of the proposed chaotic based many-objective Jaya (C-MaOJaya) algorithm is tested on DTLZ benchmark functions for three to ten objectives. The IGD and Hypervolume performance metrics evaluate the performance of the proposed C-MaOJaya algorithm. The statistical tests are used to compare the performance of the proposed C-MaOJaya algorithm with the MaOJaya algorithm and other algorithms from the literature. The C-MaOJaya algorithm improved the balance between exploration and exploitation and avoids premature convergence significantly. The comparison shows that the proposed C-MaOJaya algorithm is a promising approach to solve many-objective optimization problems.

Many-objective optimization for scheduling of crude oil operations based on NSGA-Ⅲ with consideration of energy efficiency

Abstract Crude oil operations in refineries are characterized as a hybrid system since it contains both discrete-event and continuous processes, and it is extremely difficult to schedule such a system. For scheduling such a system, initially the discrete tasks to be performed during the scheduling horizon is unknown such that heuristics and meta-heuristics are not directly applicable, which further complicates its scheduling problem. Moreover, there are large number of objectives to be optimized, including the minimization of energy consumption due to that crude oil operations consume large amount of energy and therefore lead to large amount of emissions. Furthermore, the energy optimization problem is characterized as highly non-linearity. Hence, the scheduling problem of crude oil operations in refineries belongs to the many-objective optimization problems and it is extremely challenging. This paper addresses this challenging scheduling problem of crude oil operations. This scheduling problem is first converted to a discrete dynamic resource allocation problem such that meta-heuristics can be applicable. Then, with the results of large number of experiments, this work innovatively proposes an NSGA-Ⅲ-based optimization algorithm to efficiently solve the problem for Pareto-optimal solutions. By the proposed method, the genes in a chromosome are generated one by one and, when generating a gene, safeness check is done according to the derived safeness conditions such that each gene is feasible. In this way, the schedule feasibility can be ensured. An industrial case study is given to test its performance and comparison is made with the existing state-of-the-art algorithms for many-objective optimization problems. The results show its good performance in terms of convergence, solution diversity, time efficiency, and its applicability to real-life refinery scheduling problems.

An adaptive clustering-based evolutionary algorithm for many-objective optimization problems

This paper proposes an adaptive clustering-based evolutionary algorithm for many-objective optimization problems (MaOPs), called MaOEA/AC. In this algorithm, an adaptive clustering strategy (ACS) is first introduced to divide the population into multiple clusters, which can properly fit various Pareto fronts (PFs) of the target MaOPs. Then, the environmental selection of MaOEA/AC is designed based on these clusters to collect the solutions with balanceable convergence and diversity. To be more detail, the similarity between solutions in ACS is appropriately measured by computing the Euclidean distance between their projections on an adaptive unit hyper-surface, whose curving rate is controlled by a parameter p. A simple yet effective estimation method is proposed to get a suitable value of p based on the distribution of the current non-dominated solution set, so that the estimated unit hyper-surface can roughly reflect the characteristics of PFs in the target MaOPs. The effectiveness of MaOEA/AC is validated by numerous experimental studies on solving test MaOPs with various PFs, which have the characteristics with convex, concave, inverted, disconnected, degenerated, and other mixed or irregular PFs. The experiments also show that MaOEA/AC has the superior performance over several recent many-objective evolutionary algorithms, when solving most of these test MaOPs.

Multi-dimensional tree guided efficient global association for decomposition-based evolutionary many-objective optimization

The suitable association between solutions and subproblems or reference vectors (RVs) is very critical to decomposition-based evolutionary algorithms for many-objective optimization problems (MaOPs). However, the original local association approach leads to the mismatch often and the currently existing global ones have to exhaust all subproblems expensively. In this paper, a multi-dimensional tree guided global association (TGA) mechanism is proposed to associate a solution with the nearest RV more efficiently. The TGA mechanism first constructs a nonlinear multi-dimensional tree (MDTree) to organize all RVs of subproblems. It further introduces a direction dissimilarity metric to measure the mismatches of associations between solutions and RVs. More significantly, owing to the compatibility between this metric and the RV MDTree, the TGA mechanism is capable to prune the RV MDTree to find the nearest RV to a solution in a logarithmic time complexity. In addition, an instantiation of a decomposition-based evolutionary algorithm using the TGA mechanism together with an adaptive aggregation approach is further designed to facilitate the empirical validation of the mechanism. The performance of the mechanism is extensively assessed on the normalized and scaled DTLZ benchmark MaOPs, WFG test suite, as well as two engineering problems. A statistical comparison with several existing local and global association approaches demonstrates the superior effectiveness and computational efficiency of the mechanism.

A [Math Processing Error]-indicator-based shuffled frog leaping algorithm for many-objective optimization problems

Many-objective optimization problems take challenges to multi-objective evolutionary algorithms. A number of nondominated solutions in population cause a difficult selection towards the Pareto front. To tackle this issue, a series of indicatorbased multi-objective evolutionary algorithms (MOEAs) have been proposed to guide the evolution progress and shown promising performance. This paper proposes an indicator-based manyobjective evolutionary algorithm called e-indicator-based shuffled frog leaping algorithm (e-MaOSFLA), which adopts the shuffled frog leaping algorithm as an evolutionary strategy and a simple and effective e-indicator as a fitness assignment scheme to press the population towards the Pareto front. Compared with four stateof-the-art MOEAs on several standard test problems with up to 50 objectives, the experimental results show that e-MaOSFLA outperforms the competitors.

A Dimension Convergence-Based Evolutionary Algorithm for Many-Objective Optimization Problems

Recently, multi-objective evolutionary algorithms have become the most popular and efficient approach for multi-objective optimization problems involving two and three objectives. However, as the number of objectives increases, the performance of multi-objective evolutionary algorithms tends to deteriorate. This can be mainly attributed to the loss of sufficient selection pressure towards the Pareto front. To address this issue, this paper proposes a dimension convergence-based many-objective evolutionary algorithm to solve many-objective optimization problems (MaOPs). To be specific, a convergence indicator, named dimension convergence, is presented to enhance the selection pressure toward the Pareto front. When the Pareto dominance-based indicator loses the discrimination, the new indicator can further measure the convergence performance of the candidates. Moreover, a new selection strategy is designed to balance the convergence and diversity of the evolutionary process. The mating selection is applied to strengthen the selection pressure, while the environmental selection is developed to comprehensive evaluate the candidates. The proposed algorithm is tested on 36 instances of 12 many-objective benchmarks and compared with five state-of-the-art algorithms. 3 real-world problems are also used to evaluate the performance of the comparison algorithms. Experimental results show that DC-MaOEA is competitive concerning the peer algorithms.

A Novel Objective Grouping Evolutionary Algorithm for Many-Objective Optimization Problems

The thorniest difficulties for multi-objective evolutionary algorithms (MOEAs) handling many-objective optimization problems (MaOPs) are the inefficiency of selection operators and high computational cost. To alleviate such difficulties and simplify the MaOPs, objective reduction algorithms have been proposed to remove the redundant objectives during the search process. However, those algorithms can only be applicable to specific problems with redundant objectives. Worse still, the Pareto solutions obtained by reduced objective set may not be the Pareto solutions of the original MaOPs. In this paper, we present a novel objective grouping evolutionary algorithm (OGEA) for general MaOPs. First, by dividing original objective set into several overlapping lower-dimensional subsets in terms of interdependence correlation information, we aim to separate the MaOPs into a number of sub-problems so that each of them can be able to preserve as much dominance structure in the original objective set as possible. Subsequently, we employ the nondominated sorting genetic algorithm II (NSGA-II) to generate Pareto solutions. Besides, instead of nondominated sorting on the whole population, a novel dual selection mechanism is proposed to choose individuals either having high ranks in subspaces or locating sparse region in the objective space for better proximity and diversity. Finally, we compare the proposed strategy with the other two classical space partition methods on benchmark DTLZ5 (I, M), DTLZ2 and a practical engineering problem. Numerical results show the proposed objective grouping algorithm can preserve more dominance structure in original objective set and achieve better quality of Pareto solutions.

A Clustering-Based Evolutionary Algorithm for Many-Objective Optimization Problems

This paper suggests a novel clustering-based evolutionary algorithm for many-objective optimization problems. Its main idea is to classify the population into a number of clusters, which is expected to solve the difficulty of balancing convergence and diversity in high-dimensional objective space. The individuals showing high similarities on the vector angles are gathered into the same cluster, such that the population’s distribution can be well portrayed by the clusters. To efficiently find these clusters, partitional clustering is first used to classify the union population into ${m}$ main clusters based on the ${m}$ axis vectors ( ${m}$ is the number of objectives), and then hierarchical clustering is further run on these ${m}$ main clusters to get ${N}$ final clusters ( ${N}$ is the population size and ${N>m}$ ). At last, in environmental selection, one individual from each of ${N}$ clusters closest to the axis vectors is selected to maintain diversity, while one individual from each of the other clusters is preferred by a simple convergence indicator to ensure convergence. When tackling some well-known test problems with 5–15 objectives, extensive experiments validate the superiority of our algorithm over six competitive many-objective EAs, especially on problems with incomplete and irregular Pareto-optimal fronts.

IGD Indicator-Based Evolutionary Algorithm for Many-Objective Optimization Problems

Inverted Generational Distance (IGD) has been widely considered as a reliable performance indicator to concurrently quantify the convergence and diversity of multi- and many-objective evolutionary algorithms. In this paper, an IGD indicator-based evolutionary algorithm for solving many-objective optimization problems (MaOPs) has been proposed. Specifically, the IGD indicator is employed in each generation to select the solutions with favorable convergence and diversity. In addition, a computationally efficient dominance comparison method is designed to assign the rank values of solutions along with three newly proposed proximity distance assignments. Based on these two designs, the solutions are selected from a global view by linear assignment mechanism to concern the convergence and diversity simultaneously. In order to facilitate the accuracy of the sampled reference points for the calculation of IGD indicator, we also propose an efficient decomposition-based nadir point estimation method for constructing the Utopian Pareto front which is regarded as the best approximate Pareto front for real-world MaOPs at the early stage of the evolution. To evaluate the performance, a series of experiments is performed on the proposed algorithm against a group of selected state-of-the-art many-objective optimization algorithms over optimization problems with $8$-, $15$-, and $20$-objective. Experimental results measured by the chosen performance metrics indicate that the proposed algorithm is very competitive in addressing MaOPs.

Hyperplane Assisted Evolutionary Algorithm for Many-Objective Optimization Problems

In many-objective optimization problems (MaOPs), forming sound tradeoffs between convergence and diversity for the environmental selection of evolutionary algorithms is a laborious task. In particular, strengthening the selection pressure of population toward the Pareto-optimal front becomes more challenging, since the proportion of nondominated solutions in the population scales up sharply with the increase of the number of objectives. To address these issues, this paper first defines the nondominated solutions exhibiting evident tendencies toward the Pareto-optimal front as prominent solutions, using the hyperplane formed by their neighboring solutions, to further distinguish among nondominated solutions. Then, a novel environmental selection strategy is proposed with two criteria in mind: 1) if the number of nondominated solutions is larger than the population size, all the prominent solutions are first identified to strengthen the selection pressure. Subsequently, a part of the other nondominated solutions are selected to balance convergence and diversity and 2) otherwise, all the nondominated solutions are selected; then a part of the dominated solutions are selected according to the predefined reference vectors. Moreover, based on the definition of prominent solutions and the new selection strategy, we propose a hyperplane assisted evolutionary algorithm, referred here as hpaEA, for solving MaOPs. To demonstrate the performance of hpaEA, extensive experiments are conducted to compare it with five state-of-the-art many-objective evolutionary algorithms on 36 many-objective benchmark instances. The experimental results show the superiority of hpaEA which significantly outperforms the compared algorithms on 20 out of 36 benchmark instances.

A pigeon-inspired optimization algorithm for many-objective optimization problems

A pigeon-inspired optimization algorithm for many-objective optimization problems