High-dimensional Objective Space Research Articles

Multi-objective optimization and inverse design of complementary field-effect transistor using combined approach of machine learning and non-dominated sorting genetic algorithms for next-generation semiconductor devices

Complementary field-effect transistors (CFETs), which are structures in which different types of transistors are vertically stacked with a shared control gate, are being focused on for continuing to satisfy Moore's law by overcoming the limitations in pitch scaling. The structures of semiconductor devices become more complex as technology node shrinks, and interrelated multivariate parameters increase. In addition, predicting problems and proposing solutions by identifying complex patterns within extensive data collected for emerging semiconductor designs pose significant computational challenges and are inherently difficult. As a breakthrough in design technology co-optimization for advanced devices, this study developed a novel optimization framework integrating technology computer-aided design simulations, machine learning, and non-dominated sorting genetic algorithms. The developed framework provides unbiased optimal solutions, even in a high-dimensional objective space, while considering the tradeoff relationships between multiple variables. In addition, it enables inverse design to identify the design parameters of devices that satisfy specific electrical performance criteria using only a forward model, while achieving an error rate of less than 2%. Using this framework, we analyzed the operational mechanism of CFETs by comparing the inverse designs of various devices. This novel approach is particularly important when the design space is complex and extensive and is well suited for developing devices that emerge with technological advancements in the semiconductor industry.

Multi-objective optimization with Q-learning for cruise and power allocation control parameters of connected fuel cell hybrid vehicles

Fuel cell hybrid vehicles (FCHVs) are significant for achieving zero carbon emissions. Connected FCHVs can leverage traffic information to collaboratively optimize cruise and power allocation control, enhancing various performance aspects. For urban driving scenarios, this paper introduces a multi-strategy series control architecture for longitudinal cruise and power allocation control in connected FCHVs. However, particle swarm optimization (PSO) algorithms face challenges in high-dimensional decision and objective spaces when optimizing multiple strategies. Additionally, manually preset PSO parameters hinder particle evolution from dynamically adapting to unknown multi-objective spaces, thereby limiting the development of multiple performance metrics. To address this issue, this paper proposes a Q-learning multi-objective PSO (QMOPSO) algorithm. This algorithm tackles high-dimensional optimization challenges by improving population initialization distribution and subpopulation division, and enables particles to dynamically adjust exploration strategies, thereby maximizing multiple objective performances. The results indicate that compared to a control scheme optimized with PSO under predefined driving conditions, the multi-strategy series control framework optimized with the QMOPSO algorithm improves tracking stability by 50.20%, driving comfort by 1.77%, fuel economy by 6.10%, and reduces power source degradation by 2.04% in urban driving scenarios. Compared to PSO and multi-objective PSO algorithms, the QMOPSO algorithm demonstrates superior trade-offs. This research provides a collaborative optimization solution for FCHVs in connected environments.

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.

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 Many-Objective Evolutionary Algorithm Based on Dual Selection Strategy.

In high-dimensional space, most multi-objective optimization algorithms encounter difficulties in solving many-objective optimization problems because they cannot balance convergence and diversity. As the number of objectives increases, the non-dominated solutions become difficult to distinguish while challenging the assessment of diversity in high-dimensional objective space. To reduce selection pressure and improve diversity, this article proposes a many-objective evolutionary algorithm based on dual selection strategy (MaOEA/DS). First, a new distance function is designed as an effective distance metric. Then, based distance function, a point crowding-degree (PC) strategy, is proposed to further enhance the algorithm's ability to distinguish superior solutions in population. Finally, a dual selection strategy is proposed. In the first selection, the individuals with the best convergence are selected from the top few individuals with good diversity in the population, focusing on population convergence. In the second selection, the PC strategy is used to further select individuals with larger crowding distance values, emphasizing population diversity. To extensively evaluate the performance of the algorithm, this paper compares the proposed algorithm with several state-of-the-art algorithms. The experimental results show that MaOEA/DS outperforms other comparison algorithms in overall performance, indicating the effectiveness of the proposed algorithm.

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.

Integrated multiple criteria decision-making framework for ranking Pareto optimal solutions of the multiobjective optimization problem of tuned mass dampers

Tuned mass dampers (TMD) have demonstrated effectiveness and robustness in controlling vibrations in marine structures. Solutions to the TMD's Multi-objectives optimization problem (MOOP) always yield a set of Pareto compromise solutions for different conflicting objectives. However, because all the Pareto solutions are non-dominated, the solution of the MOOP still needs to be completed since a single Pareto solution must be selected from a large set of Pareto solutions. Researchers in the literature concentrated on obtaining the Pareto front by solving the MOOP of the TMD for different objective functions. Nonetheless, no effort has been put into selecting the best solution from the Pareto set. Therefore, this paper presents the novel application of the multiple-criteria decision-making (MCDM) techniques in the solution of the MOOP of the TMD applied to a marine structure. The presented framework ensures that the decision-makers (DMs) obtain the most optimal solution from the Pareto set of solutions of the TMD. In this paper, the problem of the TMD applied to a jacket platform is formulated as a MOOP, which comprises conflicting objectives related to TMD cost and performance. NSGA-II is utilized to find a large set of non-dominated TMD tuning parameters from which the DMs must select only one solution. Then, the DMs' preferences are prioritized using the Analytical Hierarchy Process (AHP). Meanwhile, the reasonableness of the DMs' preferences is assessed. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is utilized for ranking the non-dominated set of solutions by assessing the relative closeness of each solution to the ideal solution and farness away from the negative ideal solution. A Finite Element (FE) model of a fixed offshore platform under the dynamic disturbance of irregular waves is numerically solved using COMSOL multiphysics to assess the goals mentioned earlier. A TMD is applied to the marine structure to mitigate wave-induced vibrations. The hydrodynamic forces affecting the structure are computed using Morison's equation, and the irregular waves are characterized using the Joint North Sea Wave Project (JONSWAP) wave spectrum. The presented framework will be used for ranking the Pareto solutions of the TMD applied to the deck of the FE model. The TMD tuning parameters considered in this regard are the TMD mass ratio μ, damping ratio ξopt, and frequency ratio fopt. The findings indicate that the framework presented provides an efficient decision-making methodology for selecting the best solution in various design scenarios. These scenarios can be presented in 2D, 3D, or high-dimensional objective space. Meanwhile, the DMs will be able to select the best compromise among the objectives without the need for numerical judgments. As a result, an efficient TMD that provides valuable response mitigation in a low-cost and effective space was presented.

Many-objective many-task optimization using reference-points-based nondominated sorting approach

Multi-task optimization utilizes knowledge transfer to optimize multiple tasks simultaneously. When the number of tasks is increased to many-task optimization, the computational burden of the algorithm increases and the positive knowledge transfer rate decreases, which leads to the algorithm performance degradation. In the face of high-dimensional objective space, the existing many-task optimization algorithms have some problems, such as decreasing population diversity and slowing down the search speed of the optimal solution. For many-objective and many-task optimization problems, we propose a many-objective many-task optimization using reference-points-based nondominated sorting approach (MOMaTO-RP). MOMaTO-RP can use multiple tasks with high similarity for knowledge transfer, and the population diversity can also be maintained in the high-dimensional objective space, which greatly improves the population convergence speed. The proposed algorithm is compared with other related algorithms on the classical benchmark set. The results show that the algorithm has a faster convergence speed and better distribution performance.

A Multiobjective Framework for Many-Objective Optimization.

It is known that many-objective optimization problems (MaOPs) often face the difficulty of maintaining good diversity and convergence in the search process due to the high-dimensional objective space. To address this issue, this article proposes a novel multiobjective framework for many-objective optimization (Mo4Ma), which transforms the many-objective space into multiobjective space. First, the many objectives are transformed into two indicative objectives of convergence and diversity. Second, a clustering-based sequential selection strategy is put forward in the transformed multiobjective space to guide the evolutionary search process. Specifically, the selection is circularly performed on the clustered subpopulations to maintain population diversity. In each round of selection, solutions with good performance in the transformed multiobjective space will be chosen to improve the overall convergence. The Mo4Ma is a generic framework that any type of evolutionary computation algorithm can incorporate compatibly. In this article, the differential evolution (DE) is adopted as the optimizer in the Mo4Ma framework, thus resulting in an Mo4Ma-DE algorithm. Experimental results show that the Mo4Ma-DE algorithm can obtain well-converged and widely distributed Pareto solutions along with the many-objective Pareto sets of the original MaOPs. Compared with seven state-of-the-art MaOP algorithms, the proposed Mo4Ma-DE algorithm shows strong competitiveness and general better performance.

A Two-Archive Many-Objective Optimization Algorithm Based on D-Domination and Decomposition

Decomposition-based evolutionary algorithms are popular with solving multi-objective optimization problems. It uses weight vectors and aggregate functions to keep the convergence and diversity. However, it is hard to balance diversity and convergence in high-dimensional objective space. In order to discriminate solutions and equilibrate the convergence and diversity in high-dimensional objective space, a two-archive many-objective optimization algorithm based on D-dominance and decomposition (Two Arch-D) is proposed. In Two Arch-D, the method of D-dominance and adaptive strategy adjusting parameter are used to apply selection pressure on the population to identify better solutions. Then, it uses the two archives’ strategy to equilibrate convergence and diversity, and after classifying solutions in convergence archive, the improved Tchebycheff function is used to evaluate the solution set and retain the better solutions. For the diversity archive, the diversity is maintained by making any two solutions as far apart and different as possible. Finally, the Two Arch-D is compared with other four multi-objective evolutionary algorithms on 45 many-objective test problems (including 5, 10 and 15 objectives). Good performance of the algorithm is verified by the description and analysis of the experimental results.

A Two-Stage Evolutionary Algorithm With Balanced Convergence and Diversity for Many-Objective Optimization

Multiobjective optimization evolutionary algorithms (MOEAs) have received significant achievements in recent years. However, they encounter many difficulties in dealing with many-objective optimization problems (MaOPs) due to the weak selection pressure. One possible way to improve the ability of MOEAs for these MaOPs is to balance the convergence and diversity in the high-dimensional objective space. Based on this consideration, this article proposes a novel generic two-stage (TS) framework for MaOPs. The entire evolutionary search process is divided into two stages: in the first stage, a new subregion dominance and a modified subregion density-based mating selection mainly purse the convergence and in the second stage, a novel level-based Pareto dominance cooperates with the traditional Pareto dominance that mainly promotes diversity. Integrated into NSGA-II, the TS NSGA-II, referred to as TS-NSGA-II, is proposed. To extensively evaluate the performance of our approach, 29 benchmark problems were used as the test suite. The experimental results demonstrate our approach obtained superior or competitive performance compared with eight state-of-the-art many-objective optimization evolutionary algorithms. To study its generality, the proposed TS strategy was also combined with four other advanced methods for MaOPs. The results show that it can also improve the performance of these four methods in terms of convergence and diversity.

A convergence and diversity guided leader selection strategy for many-objective particle swarm optimization

Recently, particle swarm optimizer (PSO) is extended to solve many-objective optimization problems (MaOPs) and becomes a hot research topic in the field of evolutionary computation. Particularly, the leader particle selection (LPS) and the search direction used in a velocity update strategy are two crucial factors in PSOs. However, the LPS strategies for most existing PSOs are not so efficient in high-dimensional objective space, mainly due to the lack of convergence pressure or loss of diversity. In order to address these two issues and improve the performance of PSO in high-dimensional objective space, this paper proposes a convergence and diversity guided leader selection strategy for PSO, denoted as CDLS, in which different leader particles are adaptively selected for each particle based on its corresponding situation of convergence and diversity. In this way, a good tradeoff between the convergence and diversity can be achieved by CDLS. To verify the effectiveness of CDLS, it is embedded into the PSO search process of three well-known PSOs. Furthermore, a new variant of PSO combining with the CDLS strategy, namely PSO/CDLS, is also presented. The experimental results validate the superiority of our proposed CDLS strategy and the effectiveness of PSO/CDLS, when solving numerous MaOPs with regular and irregular Pareto fronts (PFs).

An improved NSGA-III algorithm based on distance dominance relation for many-objective optimization

There are two main aspects of research in many-objective optimization algorithm, namely, convergence and diversity. However, it is difficult for original algorithms to maintain the diversity of solutions in the high-dimensional objective space. The NSGA-III algorithm is an advanced algorithm based on Pareto dominance. In the high-dimensional objective space, the diversity maintenance of this algorithm is obviously lacking. In order to enhance the diversity of algorithms in many-objective optimization problems, a new distance dominance relation is proposed in this paper. First, in order to ensure the convergence of the algorithm, the distance dominance relation calculates the distance from the candidate solution to the ideal point as the fitness value, and selects the candidate solution with good fitness value as the non-dominant solution. Then, in order to enhance the diversity of the algorithm, the distance dominance relation sets each candidate solution to have the same niche and ensures that only one optimal solution is retained in the same territory. Finally, the NSGA- III algorithm is improved based on the proposed distance dominance relation. On the DTLZ and MaF test problems with 3, 5, 8, 10, and 15 objectives, the improved algorithm is compared with seven commonly used algorithms. The experimental results show that the improved algorithm is highly competitive and can significantly enhance the diversity of the algorithm.

MMOEA-SP: A multistage many-objective evolutionary algorithm based on sampling points

For the optimization algorithm based on Pareto dominance, how to obtain excellent performance in the high-dimensional objective space is a crucial issue. If the extreme solutions are correctly identified, the algorithm process can be accelerated. However, the existence of the dominance resistant solutions makes it suffer a devastating blow, especially when solving many-objective optimization problems. To address this issue, a multistage many-objective evolutionary algorithm based on sampling points (MMOEA-SP) is proposed. First of all, a state judgment strategy based on sampling points is developed to guide the population to explore or exploit, which divides the population into two stages and assigns different matching pools. The purpose of this strategy is to balance the convergence and diversity of the population. In the environment selection mechanism, two strategies are considered. On the one hand, a cluster-based dominance resistant solutions elimination strategy is introduced to accelerate population convergence and narrow the search area. On the other hand, solutions with larger differences are given priority as candidate solutions in the environmental selection mechanism to increase the diversity. The proposed algorithm is compared with plentiful state-of-the-art algorithms on a number of well-known benchmarks with 5–20 objectives. Finally, the experimental results show that the proposed algorithm can perform well on most of the test functions and generally outperforms its competitors.

A decomposition-based many-objective ant colony optimization algorithm with adaptive solution construction and selection approaches

The ant colony optimization algorithm (ACO) had an exceptional performance in solving discrete optimization problems because of its design in solution construction and search strategy. However, the study of ACO in discrete many-objective optimization remains insufficient. This paper proposes a decomposition-based ACO for discrete many-objective optimization. The proposed algorithm utilizes a reinforcement learning-based adaptive pheromone updating strategy that enhances the solution construction phase’s searching ability in the high-dimensional objective space. Furthermore, an adaptive selection strategy is adopted to improve its convergence performance using different reference points. And A comparative experimental study is conducted on many-objective benchmark test cases. The experimental results indicate that the proposed algorithm achieves competitive performance on optimization quality.

Design optimization of broadband extreme ultraviolet polarizer in high-dimensional objective space

With the purpose of designing the extreme ultraviolet polarizer with many objectives, a combined application of multi-objective genetic algorithms is theoretically proposed. Owing to the multi-objective genetic algorithm, the relationships between different designing objectives of extreme ultraviolet polarizer have been obtained by analyzing the distribution of nondominated solutions in the four-dimensional objective space, and the optimized multilayer design can be obtained by guiding the searching in the desired region based on the multi-objective genetic algorithm with reference direction. Compared with the conventional method of multilayer design, our method has a higher probability of achieving the optimal multilayer design. Our work should be the first research in optimizing the optical multilayer designs in the high-dimensional objective space, and our results demonstrate a potential application of our method in the designs of optical thin films.

An improved competitive particle swarm optimization for many-objective optimization problems

Multi-objective particle swarm optimization (MOPSO) has been widely applied to solve multi-objective optimization problems (MOPs), due to its efficient implementation and fast convergence. However, most MOPSOs are ineffective in achieving the balance between convergence and diversity in the high-dimensional objective space. In this paper, an improved competitive particle swarm optimization is proposed for solving many-objective optimization problems. To improve the quality of the first generation population, a decision variable dividing-based multi-step initialization mechanism is presented, decision variables are divided into two groups and optimized individually. Moreover, an improved competitive learning strategy is suggested as the main part to further optimization, where particles are updated via leader information from winner particles with well convergence and diversity. The performance of the proposed algorithm is verified by benchmark comparisons with several state-of-the-art evolutionary algorithms. Experimental results demonstrate the promising performance of the algorithm in terms of balance convergence and diversity.

An Elite Gene Guided Reproduction Operator for Many-Objective Optimization.

Traditional reproduction operators in many-objective evolutionary algorithms (MaOEAs) seem to not be so effective to tackle many-objective optimization problems (MaOPs). This is mainly because the population size cannot be set to an arbitrarily large value if the computational efficiency is of concern. In such a case, the distance between the parents becomes remarkably large and, consequently, it is not easy to reproduce a superior offspring in high-dimensional objective space. To alleviate this problem, an elite gene-guided (EGG) reproduction operator is proposed to tackle MaOPs in this article. In this operator, an elite gene pool is built by collecting the knee points from the current population. Then, the offspring is produced by exchanging the genes with this elite gene pool under an exchange rate, aiming to reserve more promising genes into the next generation. In order to provide new genes for the population, other genes will be disturbed under a disturbance rate. The settings and functional analysis of the exchange rate and disturbance rate are studied using several experiments. The proposed EGG operator is easy to implement and can be embedded to any MaOEA. As examples, we show the embedding of the proposed EGG operator into four competitive MaOEAs, that is, MOEA/D, NSGA-III, θ -DEA, and SPEA2-SDE provide some advantages over simulated binary crossover, differential evolution, and an evolutionary path-based reproduction operator on solving a number of benchmark problems with 3 to 15 objectives.

An Evolutionary Many-Objective Algorithm Based on a Novel Tournament Selection Strategy

Recent studies in scientific research and engineering practice have the tendency to employ evolutionary algorithms to solve multi-objective optimization problems (MOPs), which has a certain effect. In the evolutionary process, the mating selection that aims to make a good preparation for exchanging the information of individuals plays an important role in multi-objective evolutionary algorithms (MOEAs). However, existing MOEAs usually use random selection strategy to form the mating pools. This strategy of generating offspring has a certain randomness, which will affect the quality of offspring, thereby deteriorating the effectiveness of the algorithm. To address this issue, we propose a novel tournament selection strategy, in which a type of binary tournament selection strategy based on the grid dominance relation and density information is adopted to select individuals for variation. The experimental results indicate that the proposed method performed well in terms of convergence and diversity, especially due to the significant benefits of high-dimensional objective space handling.

Handling Constrained Many-Objective Optimization Problems via Problem Transformation.

Objectives optimization and constraints satisfaction are two equally important goals to solve constrained many-objective optimization problems (CMaOPs). However, most existing studies for CMaOPs can be classified as feasibility-driven-constrained many-objective evolutionary algorithms (CMaOEAs), and they always give priority to satisfy constraints, while ignoring the maintenance of the population diversity for dealing with conflicting objectives. Consequently, the population may be pushed toward some locally feasible optimal or locally infeasible areas in the high-dimensional objective space. To alleviate this issue, this article presents a problem transformation technique, which transforms a CMaOP into a dynamic CMaOP (DCMaOP) for handling constraints and optimizing objectives simultaneously, to help the population cross the large and discrete infeasible regions. The well-known reference-point-based NSGA-III is tailored under the problem transformation model to solve CMaOPs, namely, DCNSGA-III. In this article, ε -feasible solutions play an important role in the proposed algorithm. To this end, in DCNSGA-III, a mating selection mechanism and an environmental selection operator are designed to generate and choose high-quality ε -feasible offspring solutions, respectively. The proposed algorithm is evaluated on a series of benchmark CMaOPs with three, five, eight, ten, and 15 objectives and compared against six state-of-the-art CMaOEAs. The experimental results indicate that the proposed algorithm is highly competitive for solving CMaOPs.