The Nearest-Better Network (NBN) is a powerful method to visualize sampled data for continuous optimization problems while preserving multiple landscape features. However, the calculation of NBN is very time-consuming, and the extension of the method to combinatorial optimization problems is challenging but very important for analyzing the algorithm’s behavior. This paper provides a straightforward theoretical derivation showing that the NBN network essentially functions as the maximum probability transition network for algorithms. This paper also presents an efficient NBN computation method with logarithmic linear time complexity to address the time-consuming issue. By applying this efficient NBN algorithm to the OneMax problem and the Traveling Salesman Problem (TSP), we have made several remarkable discoveries for the first time: The fitness landscape of OneMax exhibits neutrality, ruggedness, and modality features. The primary challenges of TSP problems are ruggedness, modality, and deception. Three state-of-the-art TSP algorithms (EAX, LKH, and NLKH) have limitations when addressing challenges related to modality and deception, respectively. LKH, based on local search operators, fails when there are deceptive solutions near global optima. EAX, which is based on a single population, can efficiently maintain diversity. However, when multiple attraction basins exist, EAX retains individuals within multiple basins simultaneously, reducing inter-basin interaction efficiency and leading to algorithm’s stagnation. NLKH improves over LKH by leveraging learned edge weights to increase the chance of reaching the global basin, but it remains vulnerable to deceptive funnels due to biased learning from underrepresented complex instances.

Dynamic optimization focuses on solving problems where the search space changes over time. The multi-population method is the most widely used approach for addressing such problems. Traditional multi-population methods often lack a deep understanding of the problem’s structural characteristics, such as the boundaries of basins of attraction (BoAs), which leads to redundant searches in less promising regions. Without guidance from these structural features, most populations are regenerated randomly, resulting in inefficient exploration. Furthermore, the search range for each population remains fixed and does not adapt to the BoAs, leading to the loss of tracking for certain peaks. To address these challenges, this paper proposes a solution space partitioning based multi-population method. The algorithm partitions the solution space into subspaces and leverages historical population data to assign an uncertainty property to each subspace. It further learns the problem’s BoAs to guide populations in exploiting within the BoAs while exploring outside them. A dual-layer exclusion mechanism dynamically adjusts the search and exclusion ranges based on the BoAs, ensuring precise control, preventing overlaps, and preserving diversity. Experimental results demonstrate that the proposed algorithm significantly outperforms state-of-the-art algorithms on moving peaks benchmark, generalized moving peaks benchmark, and a real-world problem: marine magnetic compensation problem.

Fitness landscape analysis (FLA) is quite important in evolutionary computation. In this article, we propose a novel FLA method, the nearest-better network (NBN), which uses the nearest-better relationship to simplify the original fitness landscape of continuous optimization problems. We introduce an efficient algorithm to calculate NBN for continuous problems. We also propose four numerical measurements and a 3-D visualization method based on NBN. Experiments show that compared to the other main FLA methods, the four numerical measurements proposed here can effectively measure the four intended features: 1) neutrality; 2) ruggedness; 3) modality; and 4) Basin of Attraction, respectively, and common features of the fitness landscape can be maintained in 3-D NBN visualization, regardless of the scale of the problem. NBN also provides a view of how algorithms search in high-dimensional problems with the help of the 3-D NBN visualization.

Environmental selection is an important process in multi-objective evolutionary algorithms (MOEAs). As the evolution progresses, the number of non-dominated solutions increases. This paper is focused on selecting a subset from excess non-dominated solutions for evolutionary or the final output. However, traditional selection methods in classical MOEAs encounter difficulties when dealing with candidate solutions that possess irregular topologies. Although the distance-based subset selection methods are not sensitive to the topologies of the candidate points, they have significant room for reducing computational complexity. In order to address the above issues, a subspace selection method is proposed in this paper. It partitions the objective space into multiple subspaces that have comparable volumes and shapes. The maximal minimum distance of each solution is considered to ensure that the sparsest solution is always chosen first. To save computational costs, only the solutions in the neighboring subspaces are taken into account. The experimental results demonstrate that the proposed subspace selection method outperforms classical selection methods in solving problems with various shapes of the Pareto front.

Model management is an essential component in data-driven surrogate-assisted evolutionary optimization. In model management, the solutions with a large degree of uncertainty in approximation play an important role. They can strengthen the exploration ability of algorithms and improve the accuracy of surrogates. However, there is no theoretical method to measure the uncertainty of prediction of Non-Gaussian process surrogates. To address this issue, this article proposes a method to measure the uncertainty. In this method, a stationary random field with a known zero mean is used to measure the uncertainty of prediction of Non-Gaussian process surrogates. Based on experimental analyses, this method is able to measure the uncertainty of prediction of Non-Gaussian process surrogates. The method's effectiveness is demonstrated on a set of benchmark problems in single surrogate and ensemble surrogates cases.

In data-driven evolutionary optimization, most existing Gaussian processes (GPs)-assisted evolutionary algorithms (EAs) adopt stationary GPs (SGPs) as surrogate models, which might be insufficient for solving most optimization problems. This article finds that GPs in the optimization problems are nonstationary with great probability. We propose to employ a nonstationary GP (NSGP) surrogate model for data-driven evolutionary optimization, where the mean of the NSGP is allowed to vary with the decision variables, while its residue variance follows an SGP. In this article, the nonstationarity of GPs in the tested functions is theoretically analyzed. In addition, this article constructs an NSGP where the SGP is a degenerate case. Performance comparisons of the NSGP with the SGP and the NSGP-assisted EA (NSGP-MAEA) with the SGP-assisted EA (SGP-MAEA) are carried out on a set of benchmark problems and an antenna design problem. These comparison results demonstrate the competitiveness of the NSGP model.

Although evolutionary computing (EC) methods are stochastic optimization methods, it is usually difficult to find the global optimum by restarting the methods when the population converges to a local optimum. A major reason is that many optimization problems have basins of attraction (BoAs) that differ widely in shape and size, and the population always prefers to converge toward BoAs that are easy to search. Although heuristic restart based on tabu search is a theoretically feasible idea to solve this problem, existing EC methods with heuristic restart are difficult to avoid repetitive search results while maintaining search efficiency. This article tries to overcome the dilemma by online learning the BoAs and proposes a search mode called history-guided hill exploration (HGHE). In the search mode, evaluated solutions are used to help separate the search space into hill regions which correspond to the BoAs, and a classical EC method is used to locate the optimum in each hill region. An instance algorithm for continuous optimization named HGHE differential evolution (HGHE-DE) is proposed to verify the effectiveness of HGHE. Experimental results prove that HGHE-DE can continuously discover unidentified BoAs and locate optima in identified BoAs.

Gaussian processes (GPs) are a kind of non-parametric Bayesian approach. They are widely used as surrogate models in data-driven optimization to approximate the exact functions. However, the cubic computation complexity is involved in building GPs. This paper proposes hyperparameters adaptive sharing based on transfer learning for scalable GPs to address the limitation. In this method, the hyperparameters across source tasks are adaptively shared to the target task by the linear predictor. This method can reduce the computation cost of building GPs without losing capability based on experimental analyses. The method's effectiveness is demonstrated on a set of benchmark problems.

In complex optimization problems, learning where to search is a difficult but critical decision for all search algorithms. Evolutionary computation methods also encounter a dilemma about where to explore or exploit. In this paper, a Monte-Carlo tree is constructed to guide evolutionary algorithms to search multiple promising regions simultaneously. In the Monte-Carlo tree model, a root node that contains all historical solutions represents the whole solution space. In each node of the tree, with k-means clustering method to partition solutions into different groups, group labels of the solutions are used to train support vector regression, which can learn a boundary to partition a region into different sub-regions. According to state values of nodes, reproduction operators of evolutionary algorithms are strengthened by selecting solutions in the most promising regions. From experimental results on multimodal problems, the proposed algorithm shows a competitive performance, which also indicates a great potential for applications to other kinds of optimization problems.

Modeling dynamic multi-objective optimization problems (DMOPs) has been one of the most challenging tasks in the field of dynamic evolutionary optimization. Based on the analysis of the existing DMOPs, several features widely existed in real-world applications are not taken into account: different objectives may have different function models and variables to be optimized; and the number of conflicting variables should be independent from the number of objectives; the time-linkage property is not considered. In order to overcome the above issues, a novel framework for constructing DMOPs is proposed, where all objectives can be designed independently, and the number of the conflicting variables can be tuned by users. Moreover, it is easy to add new dynamic features to this framework. Several classical dynamic multi-objective optimization algorithms are tested on four scenarios, results show that these characteristics are challenging for the existing algorithms.

In evolutionary algorithms, how to effectively select interactive solutions for generating offspring is a challenging problem. Though many operators are proposed, most of them select interactive solutions (parents) randomly, having no specificity for the features of landscapes in various problems. To address this issue, this paper proposes a reinforcement-learning-based evolutionary algorithm to select solutions within the approximated basin of attraction. In the algorithm, the solution space is partitioned by the k-dimensional tree, and features of subspaces are approximated with respect to two aspects: objective values and uncertainties. Accordingly, two reinforcement learning (RL) systems are constructed to determine where to search: the objective-based RL exploits basins of attraction (clustered subspaces) and the uncertainty-based RL explores subspaces that have been searched comparatively less. Experiments are conducted on widely used benchmark functions, demonstrating that the algorithm outperforms three other popular multimodal optimization algorithms.

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.

This paper proposes a general algorithm framework for solving dynamic sequence optimization problems (DSOPs). The framework adapts a novel genetic learning (GL) algorithm to dynamic environments via a clustering-based multi-population strategy with a memory scheme, namely, multi-population GL (MPGL). The framework is instantiated for a 3D dynamic shortest path problem, which is developed in this paper. Experimental comparison studies show that MPGL is able to quickly adapt to new environments and it outperforms several ant colony optimization variants.

The following topics are dealt with: evolutionary computation; genetic algorithms; search problems; optimisation; learning (artificial intelligence); particle swarm optimisation; Pareto optimisation; pattern classification; pattern clustering; computational complexity.

Many artificial benchmark problems have been proposed for different kinds of continuous optimization, e.g., global optimization, multimodal optimization, multiobjective optimization, dynamic optimization, and constrained optimization. However, there is no unified framework for constructing these types of problems and possible properties of many problems are not fully tunable. This will cause difficulties for researchers to analyze strengths and weaknesses of an algorithm. To address these issues, this paper proposes a simple and intuitive framework, which is able to construct different kinds of problems for continuous optimization. The framework utilizes the k -d tree to partition the search space and sets a certain number of simple functions in each subspace. The framework is implemented into global/multimodal optimization, dynamic single objective optimization, multiobjective optimization, and dynamic multiobjective optimization, respectively. Properties of the proposed framework are discussed and verified with traditional evolutionary algorithms.

For computationally expensive constrained optimization problems, one crucial issue is that the existing expected improvement (EI) criteria are no longer applicable when a feasible point is not initially provided. To address this challenge, this paper uses the expected improvement of constraint violation to reach feasible region. A new constrained expected improvement criterion is proposed to select sample solutions for the update of Gaussian process (GP) surrogate models. The validity of the proposed constrained expected improvement criterion is proved theoretically. It is also verified by experimental studies and results show that it performs better than or competitive to compared criteria.

A novel multiobjective technique is proposed for solving constrained optimization problems (COPs) in this paper. The method highlights three different perspectives: 1) a COP is converted into an equivalent dynamic constrained multiobjective optimization problem (DCMOP) with three objectives: a) the original objective; b) a constraint-violation objective; and c) a niche-count objective; 2) a method of gradually reducing the constraint boundary aims to handle the constraint difficulty; and 3) a method of gradually reducing the niche size aims to handle the multimodal difficulty. A general framework of the design of dynamic constrained multiobjective evolutionary algorithms is proposed for solving DCMOPs. Three popular types of multiobjective evolutionary algorithms, i.e., Pareto ranking-based, decomposition-based, and hype-volume indicator-based, are employed to instantiate the framework. The three instantiations are tested on two benchmark suites. Experimental results show that they perform better than or competitive to a set of state-of-the-art constraint optimizers, especially on problems with a large number of dimensions.

This paper proposes dynamic constrained version of NSGA-III to handle constraints for constrained optimization problems (COPs). The methodology first constructs a dynamic constrained multi-objective optimization problem (DCMOP) equivalent to the COP by converting the constraints into some violation objective functions and gradually shrinking the initially broadened boundary to the original one. Then a dynamic constrained version of the state-of-the-art NSGA-III is implemented to solve the DCMOP. Differential evolution (DE) is used as the evolutionary algorithm to generate offspring. Experimental results show that it is competitive to peer algorithm referred in this paper, and has better performance on global search.

A difficulty in solving nonlinear equation systems (NESs) stays in finding all the solutions for NES. This paper uses multi-objective evolutionary techniques to overcome it. We converted the NES into a multi-objective optimization problem (MOP) with a parameter C. The Pareto-optimal set of the MOP becomes the solutions of the NES when the parameter C gets to infinity. Next, a multi-objective evolutionary algorithm (MOEA) is used to solve the transformed MOP, during which C is gradually approaching infinity. A significant feature of this algorithm is that there is one-to-one relationship between the Pareto optimal set and the Pareto front, which suggests that different solutions have different objective values in the MOP. Thus the MOEA can find multi-solutions of the NES in a single run. Since the MOP is a multi-objective problem in many cases, this paper applies an advanced multi-objective evolutionary algorithm (i.e., Hype algorithm) to solve NES. Our experiment shows better results than or competitive to the four mentioned single-objective optimization in a set of test cases.

This paper evolves a 2.4GHz microstrip antenna for WiFi application by using the differential evolution (DE). The antenna structure is square substrate with a size of 50mm × 50mm×1.6mm. The bottom side of the substrate is printed with a microstrip feed line. The top is metallic painted where a ellipse cut off. The structure is parameterized as a solution vector, and the requirements of the antenna are modeled as objective and constraints. Then a constrained optimization problem (COP) is formulated. The DE algorithm finds an antenna which satisfy the requirements in the simulating. The measured performance of the fabricated antenna matches the simulated one on the whole. This evolved antenna can be applied as a antenna for router and personal computer in indoor use. Notably, an exceptional achievement is the bandwidth of the antenna reaches over 1 GHz.

The design of linear sparse array is a constrained multi-objective optimization problem(CMOP). There are three objectives: minimization of peak sidelobe level(PSLL), half-power beam width(HPBW) and spatial aperture. The amplitude coefficients of elements and sensor positions of the array are decision variables. Dynamic constrained multi-objective evolutionary algorithm(DCMOEA) is used to design linear sparse arrays in this paper. It makes a difference that the output is a set of Pareto solutions (antenna arrays), not just only one solution. The users can choose an array from the set to meet their preferences for low PSLL, small HPBW, small spatial aperture or a trade-off among them. Experimental results showed that the DCMOEA performs better than peer state-of-art algorithms referred in this paper, especially on the arrays' spatial aperture optimization.

Naive Bayes (NB) is a popularly used classification method. One potential weakness of NB is the strong conditional independence assumption between attributes, which may deteriorate the classification accuracy. In this paper, we propose a new Artificial Immune System based Weighted Naive Bayes (AISWNB) classifier. AISWNB uses immunity theory in artificial immune systems to find optimal weight values for each attribute. The adjusted weight values will alleviate the conditional independence assumption and help calculate the conditional probability in an accurate way. Because AISWNB uses artificial immune system search mechanism to find optimal weights, it does not need to know the importance of individual attributes nor the relevance among attributes. As a result, it can obtain optimal weight value for each attribute during the learning process. Experiments and comparisons on 36 benchmark data sets demonstrate that AISWNB outperforms other state-of-the-art attribute weighted NB algorithms.

Dynamic constrained multi-objective differential evolution (DCMODE) is designed for solving constrained optimization problem (COP). Main feature presented in this paper is to construct dynamic multi-objective optimization problem (DMOP) from COP. The two evolved objectives are original function objective and violation objective. Constraints are controlled by dynamic environments, where the relaxed constraints boundaries are gradually tightened to original boundaries. After this dynamic process, DMOP solutions are close to COP solution. This new algorithm is tested on benchmark problems of special session at CEC2006 with 100% success rates of all problems. Compared with several state-of-the-art DE variants referred in this paper, our algorithm outperforms or performs similarly to them. The satisfactory results suggest that it is efficient and generic when handling inequality/equality constraints.

Constrained optimization problem (COP) is skillfully converted into dynamic constrained multi-objective optimization problem (DCMOP) in this paper. Then dynamic constrained multi-objective evolutionary algorithms (DCMOEAs) can be used to solve the COP problem by solving the DCMOP problem. Seemingly, a complex DCMOEA algorithm is used to solve a relatively simple COP problem. However, the DCMOEA algorithm can adopt Pareto domination to achieve a good trade off between fast converging and global searching, and therefore a DCMOEA algorithm can effectively solve a COP problem by solving the DCMOP problem. An instance of DCMOEA was used to to solve 13 widely used constraint benchmark problems, The experimental results suggest it outperforms or performs similarly to other state-of-the-art algorithms referred to in this paper. The efficient performance of the DCMOEA algorithm shows, to some extend, the DCMOP model works well.

This paper proposes an multi-objective evolutionary algorithm. The algorithm is based on OMOEA-II. A new linear breeding operator with lower-dimensional crossover and copy operation is used. By using the lower-dimensional crossover, the complexity of searching is decreased so the algorithm converges faster. The orthogonal crossover increase probability of producing potential superior solutions, which helps the algorithm get better results. Ten unconstrained problems are used to test the algorithm. For three problems, the obtained solutions are very close to the true Pareto front, and for one problem, the obtained solutions distribute on part of the true Pareto front.

Particle swarm optimization (PSO) has shown its fast search speed in many complicated optimization and search problems. However, PSO could often easily fall into local optima. This paper presents an improved PSO with adaptive jump. The proposed method combines a novel jump strategy and an adaptive Cauchy mutation operator to help escape from local optima. The new algorithm was tested on a suite of well-known benchmark functions with many local optima. Experimental results were compared with some similar PSO algorithms based on Gaussian distribution and Cauchy distribution, and showed better performance on those test functions.

The quality of an approximation set usually includes two aspects-- approaching distance and spreading diversity. This paper introduces a new technique for assessing the diversity of an approximation to an exact Pareto-optimal front. This diversity is assessed by using an ldquoexposure degreerdquo of the exact Pareto-optimal front against the approximation set. This new technique has three advantages: Firstly, The ldquoexposure degreerdquo combines the uniformity and the width of the spread into a direct physical sense. Secondly, it makes the approaching distance independent from the spreading diversity at the most. Thirdly, the new technique works well for problems with any number of objectives, while the widely used diversity metric proposed by Deb would work poor in problems with 3 objectives or over. Experimental computational results show that the new technique assesses the diversity well.

This paper proposes a new evolutionary algorithm with lower dimensional crossover and gradient-based mutation for real-valued optimization problems with constraints. The crossover operator of the new algorithm searches a lower dimensional neighbor of the parent points where the neighbor center is the barycenter of the parents, and therefore the new algorithm converges fast. The gradient-based mutation is used to converge fast for the problems with equality constraints and active inequality constraints. And the new algorithm is simple and easy to be implemented. We have used 24 constrained benchmark problems to test the new algorithm. The experimental results show it works better than or competitive to a known effective algorithm.

Particle swarm optimization (PSO) has shown its fast search speed in many complicated optimization and search problems. However, PSO could often easily fall into local optima. This paper presents an Opposition-based PSO (OPSO) to accelerate the convergence of PSO and avoid premature convergence. The proposed method employs opposition-based learning for each particle and applies a dynamic Cauchy mutation on the best particle. Experimental results on many well- known benchmark optimization problems have shown that OPSO could successfully deal with those difficult multimodal functions while maintaining fast search speed on those simple unimodal functions in the function optimization.

This paper proposes a new evolutionary algorithm, called lower-dimensional-search evolutionary algorithm (LDSEA). The crossover operator of the new algorithm searches a lower-dimensional neighbor of the parent points where the neighbor center is the barycenter of the parents therefore the new algorithm converges fast, especially for high-dimensional constrained optimization problems. The niche-impaction operator and the mutation operator preserve the diversity of the population to make the LDSEA algorithm not to be trapped in local optima as much as possible. What's more is that the LDSEA algorithm is simple and easy to be implemented. We have used the 24 constrained benchmark problems [18] to test the LDSEA algorithm. The experimental results show it works better than or competitive to a known effective algorithm [7] for higher-dimensional constrained optimization problems.

Particle swarm optimization (PSO) has shown its fast search speed in many complicated optimization and search problems. However, PSO could often easily fall into local optima because the particles could quickly get closer to the best particle. At such situations, the best particle could hardly be improved. This paper proposes a new hybrid PSO (HPSO) to solve this problem by adding a Cauchy mutation on the best particle so that the mutated best particle could lead all the rest of particles to the better positions. Experimental results on many well-known benchmark optimization problems have shown that HPSO could successfully deal with those difficult multimodal functions while maintaining fast search speed on those simple unimodal functions in the function optimization

An orthogonal hill-climbing algorithm for dynamic optimization problems with continuous variables (labeled ODHC ) is proposed in present paper. The local peak climber is not a solution x, but rather a "niche", a small hyperrectangle. An orthogonal design method is employed on the niches for the niche to climb a potentially peak fast. An archive is used to store the latest found higher peaks for the ODHC algorithm learning from the past search. The randomly creating niches implement the global search. Numerical experiments show that the ODHC algorithm performs a lot better than the SOS (self organizing scouts) algorithm [J. Branke, T. Kaufler, C. Schmidt, and H. Schmeck. A multipopulation approach to dynamic optimization problems. Adaptive Computing in Design and Manufacturing. Springer, 2000.].

A robust dynamic evolutionary algorithm (labeled RODEA), where both the robust calculation and mutation operator are based on an orthogonal design, is proposed in this paper. Previous techniques calculate the mean effective objective (for robust) by using samples without much evenly distributing over the neighborhood. The samples by using orthogonal array distribute evenly. Therefore the calculation of mean effective objective more robust. The new technique is generalized from the ODEA algorithm [1]. An orthogonal design method is employed on the niches for the mutation operator to find a potentially good solution that may become the representative in the niche. The fitness of the offspring is therefore likely to be higher than that of its parent. We propose a complex benchmark, consisting of moving function peaks, to test our new approach. Numerical experiments show that the moving solutions of the algorithm are a little worse in objective value but robust.

This paper introduces an orthogonal evolutionary algorithm for dynamic optimization problems with continuous variables (called ODEA). Its population does not consist of individuals, but rather of "niches". Each niche selects the best solution found so far as its representative. An orthogonal design method is employed to find a potentially good solution that may become the representative of the niche in the mutation operator. We employ a complex benchmark to test the new approach. Numerical experiments show that the ODEA algorithm performs a lot better than the SOS (self organizing scouts) algorithm in Branke, Kaufler, Schmidt and Schmeck, (2000).

In this paper, an orthogonal multi-objective evolutionary algorithm (OMOEA) is proposed for multi-objective optimization problems (MOPs) with constraints. Firstly, these constraints are taken into account when determining Pareto dominance. As a result, a strict partial-ordered relation is obtained, and feasibility is not considered later in the selection process. Then, the orthogonal design and the statistical optimal method are generalized to MOPs, and a new type of multi-objective evolutionary algorithm (MOEA) is constructed. In this framework, an original niche evolves first, and splits into a group of sub-niches. Then every sub-niche repeats the above process. Due to the uniformity of the search, the optimality of the statistics, and the exponential increase of the splitting frequency of the niches, OMOEA uses a deterministic search without blindness or stochasticity. It can soon yield a large set of solutions which converges to the Pareto-optimal set with high precision and uniform distribution. We take six test problems designed by Deb, Zitzler et al., and an engineering problem (W) with constraints provided by Ray et al. to test the new technique. The numerical experiments show that our algorithm is superior to other MOGAS and MOEAs, such as FFGA, NSGAII, SPEA2, and so on, in terms of the precision, quantity and distribution of solutions. Notably, for the engineering problem W, it finds the Pareto-optimal set, which was previously unknown.

A fast algorithm is proposed to find the nondominated set for multiobjective optimization problems in this paper. Two accelerated techniques are adopted in the algorithm. One is that the algorithm can yield an integer rank set after it indexes the search space. Based on this, the goal is changed into determination of the nondominated set of the integer rank set. The other is that the nondominated check sequence follows that the likely nondominated members are first checked, and that the check process is stopped when the remaining members in the nondominated check sequence are dominated. The computational complexity of the new algorithm is analyzed theoretically. Experimental results show that the new method performs much better than KLP (a famous effective algorithm) when the search space contains a large nondominated set. Moreover, the two new techniques introduced in this paper are very useful for multiobjective evolutionary algorithms (MOEAs) to improve the computational speed.

A new algorithm is proposed to solve constrained multiobjective problems. The constraints of the MOPs are taken account of in determining Pareto dominance. As a result, the feasibility of solutions is not an issue. At the same time, it takes advantage of both the orthogonal design method to search evenly, and the statistical optimal method to speed up the computation. The output of the technique is a large set of solutions with high precision and even distribution. Notably, for an engineering problem WATER, it finds the Pareto-optimal set, which was previously unknown.