Differentiable Neural Architecture Search (NAS) and traditional evolutionary approaches frequently struggle with premature convergence to local optima. To overcome this limitation, we propose LLMENAS, a hierarchical framework that introduces trajectory-aware fitness design as the upper-level optimizer. By analyzing the convergence state of historical optimization trajectories, the Large Language Model (LLM) acts as a fitness designer to dynamically design fitness functions. This mechanism enables the search to navigate complex landscapes and escape local optima. Furthermore, we introduce a closed-loop self-improving mechanism, enabling the LLM to iteratively enhance its design strategies through self-reflection and self-refinement based on feedback. Extensive experiments show that LLMENAS achieves competitive results, with top-1 accuracies of 97.58% on CIFAR-10, 83.52% on CIFAR-100, and 75.6% on ImageNet-1k. Furthermore, it is achieved with remarkable efficiency, costing only 0.15 GPU days on the CIFAR 10 datasets and 2 GPU days on ImageNet. The source code is publicly available at: https://github.com/LLMENAS/LLMENAS.

In bilevel optimization, the upper-level optimization problem (ULOP) requires to be solved under the constraint of the inner lower-level optimization problem (LLOP). However, it is computationally expensive to always consider the constraint caused by the LLOP in a higher priority because heavy evaluation budgets are required for validating the constraint satisfaction. From this aspect, this paper investigates bilevel evolutionary multi-objective optimization with multiple lower-level search modes (BLEMO-MLS). Assisted by self-learning and reinforcement, BLEMO-MLS could adaptively adjust the priority of considering more on the upper-level objective optimization or the constraint caused by the LLOP. In BLEMO-MLS, three lower-level search modes are designed to handle the constraint caused by the LLOP. The former two search modes have higher priorities on the satisfaction of the constraint caused by the LLOP, where lower-level decisions of solutions are optimized by a hybrid lower-level search method, while the last search mode has a higher priority on the upper-level objective optimization with the constraint caused by the LLOP temporarily ignored. Through self-learning and reinforcement, BLEMO-MLS dynamically selects proper lower-level search modes in the bilevel optimization process, aiming to obtain approximate bilevel Pareto-optimal solutions fulfilling the constraint caused by the LLOP as much as possible. Compared with five existing bilevel evolutionary algorithms, BLEMO-MLS could effectively solve bilevel multi-objective optimization problems with function evaluations saved in both levels.

ABSTRACT Addressing multi‐objective optimization problems (MOPs) with complex constraints presents a significant challenge due to their diverse nature. While existing algorithms can effectively handle specific types of complex constraints, they often struggle with a variety of such constraints. To address this issue, we propose an innovative evolutionary algorithm for constrained multi‐objective optimization. A key feature is the integration of a novel differential operator that generates offspring based on the presence of feasible solutions within the main population. This strategy is particularly effective for handling complex constraints characterised by small feasible spaces and deceptive infeasible regions. Additionally, the algorithm employs a dual‐indicator ranking mechanism to evaluate and select individuals from the auxiliary population based on the quality and quantity of feasible solutions generated by the main population. Promising individuals are then migrated back to the main population, thereby enhancing the exploration of the solution space. This approach demonstrates significant superiority in solving MOPs with discontinuous feasible regions or extensive infeasible areas. Empirical comparisons across a range of benchmark problems show that the proposed algorithm outperforms current state‐of‐the‐art methods in evolutionary constrained multi‐objective optimization, underscoring its potential as a robust tool for handling MOPs with complex constraints.

Evolutionary algorithms (EAs) for optimization have received wide attention due to their robustness and practicality. However, the traditional way of asynchronously handling bilevel optimization problems (BLOPs) ignores the benefits brought by effective upper- and lower-level collaboration. To address this issue, this article proposes a collaborative multiobjective transformation (MOT)-based EA (MOTEA-II). In MOTEA-II, the BLOP is handled within a decomposition-based multiobjective optimization paradigm using a two-stage collaborative MOT strategy. The stage-1 MOT focuses on multiple lower-level optimizations and collaboration, while stage-2 collaborates the upper-level optimization with lower-level optimization, which makes simultaneously horizontal and vertical optimization information sharing in bilevel optimization possible. In addition, a dynamic decomposition strategy is further proposed to reconstruct the hierarchy relationship in collaborative multiobjective optimization, facilitating the adaptive and flexible importance control of the upper-level objective optimization and lower-level optimality satisfaction for better-bilevel search efficiency. Empirical studies are conducted on two groups of commonly used BLOP benchmark suites and four practical applications. Experimental results show that the proposed collaborative MOTEA-II can achieve performance comparable to that of the previous MOTEA and three other representative EA-based bilevel optimization approaches, but using much fewer computational resources.

Bilevel multiobjective optimization problem (BLMOP) can be seen as a special constrained multiobjective optimization problem (CMOP), where the optimality constraint of the lower-level (LL) problem cannot be easily and quickly checked, but must be verified by solving the LL problem. If there was a relatively simple alternative formulation for effectively identifying LL optimal solutions, the abovementioned special constraint could become as simple as the usual constraints and the nested optimization structure of BLMOP would be altered, which can greatly improve efficiency and effectively find LL optimal solutions with excellent upper-level (UL) performance. In this article, we improve the training mechanism for conditional generative adversarial network (cGAN) by introducing multiple generators to construct a reasonable reference distribution, so as to prevent the discriminator from performance degradation. With a discriminator identifying LL optimal solutions effectively, the nested optimization structure of BLMOP is altered by adding the objective that maximizing the discriminator output score into the UL objectives and removing the LL optimality constraint, realizing synchronous optimization of UL and LL vectors. The cooperation of generators and discriminator greatly reduces the computational overhead for solving BLMOPs. The proposed algorithm has achieved the best or competitive results in comparison with 6 state-of-the-art algorithms and a nested method on benchmark problems and a real-world problem, whose effectiveness has been demonstrated.

Optimizing antenna placement in in-building distributed antenna systems is critical for achieving comprehensive 5G coverage. Due to the utilization of high-frequency signal bands, the propagation of 5G signals is significantly influenced by distance and obstacles in indoor environments. Consequently, devising effective placement schemes faces challenges in complex indoor scenarios. This paper presents a multiobjective genetic algorithm for antenna placement optimization. It tracks the signal propagation by introducing the ray-tracing propagation model. Although the ray-tracing propagation model provides high simulation accuracy, it also incurs substantial computation costs, rendering the antenna placement problem expensive. To tackle this challenge, the algorithm introduces the idea of Bayesian optimization, where a surrogate model replaces certain calculations in the ray tracing propagation model, thereby reducing the computational burden. Experimental results demonstrate the superior performance of the proposed algorithm compared to other algorithms in two real-world scenarios, as evidenced by evaluations based on hypervolume and inverted generational distance metrics.

The bilevel optimization problem is a kind of commonly existing optimization problem, which includes a nested lower-level optimization problem as a constraint condition. The nested lower-level optimization problem should be solved with every upper-level decision fixed as a parameter. Consequently, it is usually computationally very expensive to solve bilevel optimization problems. In this paper, we propose a bilevel evolutionary algorithm based on upper-level-driven lower-level search (BLEA-UDLS). Driven by the upper-level optimization, the lower-level search in BLEA-UDLS is carried out on some upper-level superior solutions rather than equally and indiscriminately on all solutions, which makes sure that the front solutions of the population have more accurate lower-level decisions and saves lots of evaluation budgets on less important solutions. In the lower-level search, the lower-level decisions of different solutions are optimized cooperatively with the computation resources dynamically adjusted, where more computation resources are assigned for less explored solutions. Compared with some other bilevel evolutionary algorithms, the experimental results have confirmed the effectiveness of the proposed BLEA-UDLS for solving BLOPs and meanwhile saving evaluation budgets.

Face recognition (FR) has encountered great difficulties due to the label noise and low-quality samples in the face recognition database. Previous studies addressed these issues through hard sample mining, focusing on preventing the overfitting of label noise and low-quality samples. However, existing loss functions struggle to handle both label noise and low-quality samples simultaneously. This paper proposes a novel loss function, NLMFace, to address these challenges. By considering a sample’s ground truth class center and its nearest negative class center, NLMFace incorporates a new mining framework with the margin-based loss function. This method adaptively corrects label noise and identifies high-quality samples through data mining. Extensive experiments on CASIA WebFace datasets, along with evaluations on benchmarks like LFW, CPLFW, and CALFW, demonstrate the superior performance of NLMFace.

Nested evolutionary algorithms (EAs) have been regarded as very promising tools for bi-level optimization. Due to the nested structure, the upper level population evaluation requires a set of complete lower level optimizations, thereby reducing the efficiency and practicability of EA methods. In this paper, a multi-objective transformation-based evolutionary algorithm (MOTEA) is proposed to perform multiple lower level optimizations in a parallel and collaborative manner. Specifically, the corresponding multiple lower level optimizations for each generation of the upper level population evaluation are transformed into locating a set of Pareto optimal solutions of a constructed multi-objective optimization problem. By utilizing the built-in implicit parallelism of evolutionary multi-objective optimization, multiple lower level problems can thus be optimized in parallel. Within one multi-objective search population, the collaboration among the parallel lower level optimization can be realized by exploiting and utilizing the implicit similarities among them for better efficiency. The effectiveness and efficiency of the proposed MOTEA are verified by comparing it with four state-of-the-art evolutionary bi-level optimization algorithms on two sets of popular bi-level optimization benchmark test problems and three application problems.

In bilevel multiobjective optimization problems (BLMOPs), the mapping from an upper-level vector to the corresponding lower-level optimal vectors is a complex set valued mapping. Existing methods require numerous surrogate models to fit such a set valued mapping by grouping the lower-level optimal vectors, and the effects are not satisfactory because the correlation among lower-level optimal vectors corresponding to the same upper-level vector is disregarded. In this paper, introducing conditional generative adversarial network (cGAN), we use only one surrogate model to effectively fit such a set valued mapping, which extracts knowledge from lower-level optimal vectors corresponding to the same upper-level vector. Then, a BLMOP is transformed into a single-level constraint multiobjective optimization problem (CMOP). By adaptively allocating computational resources to optimize the CMOP, promising upper-level vectors are obtained. Furthermore, a lower-level search is executed for these promising upper-level vectors, thus obtaining high-quality solutions. Because of the excellent performance of cGAN and the lower-level search conducted only for promising upper-level vectors, the computational overhead is greatly reduced. The proposed algorithm has achieved the best results in comparison with 5 state-of-the-art algorithms on benchmark problems and a real-world problem, whose effectiveness has been demonstrated.

In this paper, we propose a stochastic bilevel optimization approach based on a hypergradient estimator, called BHE- DARTS, as a remedy for this issue that it is easy to search for locally optimal structures rather than globally optimal ones in Differentiable Architecture Search (DARTS) bilevel optimization model. To be specific, we apply a stochastic gradient for updating the lower level variable ω and design a hypergradient estimator, which is built by the Jacobian- and Hessianvector product, to assist in updating the upper level variable α. This operation can more fully apply the gradient information to escape the trap of local optimal in the NAS bilevel model. Compared to state-of-the-art DARTS methods, experimental studies have shown the competitive performance of the proposed BHE-DARTS in the DARTS search space (CIFAR-100: a test accuracy rate of 82.69 % ) and NAS-Bench-201 search space (ImageNet16-120: a test accuracy rate of 42.44%).

The radial basis function (RBF) model and the Kriging model have been widely used in the surrogate-assisted evolutionary algorithms (SAEAs). Based on their characteristics, a global and local surrogate-assisted differential evolution algorithm (GL-SADE) for high-dimensional expensive problems is proposed in this article, in which a global RBF model is trained with all samples to estimate a global trend, and then its optima is used to significantly accelerate the convergence process. A local Kriging model prefers to select points with good predicted fitness and great uncertainty, which can effectively prevent the search from getting trapped into local optima. When the local Kriging model finds the best solution so far, a reward search strategy is executed to further exploit the local Kriging model. The experiments on a set of benchmark functions with dimensions varying from 30 to 200 are conducted to evaluate the performance of the proposed algorithm. The experimental results of the proposed algorithm are compared to four state-of-the-art algorithms to show its effectiveness and efficiency in solving high-dimensional expensive problems. Besides, GL-SADE is applied to an airfoil optimization problem to show its effectiveness.

Multiobjective multitask optimization (MMO) attempts to solve several problems simultaneously. This is commonly done by identifying useful knowledge to transfer between tasks, thereby producing optimal solutions more quickly. In this study, an MMO algorithm using transfer rank and a KNN model is proposed to achieve this goal. The definition of transfer rank is first introduced for quantifying the priority of transfer solutions, to improve the probability of a positive result. The solution with the higher rank was assumed to be the most suitable for transfer, as solutions were sorted in descending order based on transfer rank. Priority was given to previous and positive-transfer solutions and those with the same transfer rank were distinguished using a KNN model classifier. The effectiveness of the proposed algorithm was verified by studying benchmark MMO problems. The experimental results showed the proposed algorithm was more effective than other conventional MMO techniques.

In multi-objective bilevel optimisation problems, the upper-level performance of different lower-level optimal solutions may be very different, even though they belong to the same lower-level problem. It may lead to poor optimisation results. Therefore, the lower-level search should search lower-level non-dominated solutions that are also non-dominated in the upper-level objective space. In this paper, we use two populations in the lower-level search. The first population maintains non-dominance and diversity in the lower-level objective space and provides the second population with convergence pressure from the lower level. The second population selects the upper-level non-dominated solutions that are not dominated by the first population in the lower-level objective space, which make the second population maintain the non-dominance at both upper and lower levels. Besides, to improve the search efficiency, we set up the upper-level mating pool to generate the upper-level vectors of offsprings near the upper-level vectors of the better individuals in the current population. To balance convergence and diversity, the selection operator of a decomposition based multi-objective evolutionary algorithm is adopted. The proposed algorithm has been evaluated on a set of benchmark problems and a real-world optimisation problem. Experimental results demonstrate that the proposed algorithm is efficient and effective.

This article presents a rough-to-fine evolutionary multiobjective optimization algorithm based on the decomposition for solving problems in which the solutions are initially far from the Pareto-optimal set. Subsequently, a tree is constructed by a modified k -means algorithm on N uniform weight vectors, and each node of the tree contains a weight vector. Each node is associated with a subproblem with the help of its weight vector. Consequently, a subproblem tree can be established. It is easy to find that the descendant subproblems are refinements of their ancestor subproblems. The proposed algorithm approaches the Pareto front (PF) by solving a few subproblems in the first few levels to obtain a rough PF and gradually refining the PF by involving the subproblems level-by-level. This strategy is highly favorable for solving problems in which the solutions are initially far from the Pareto set. Moreover, the proposed algorithm has lower time complexity. Theoretical analysis shows the complexity of dealing with a new candidate solution is O(M logN) , where M is the number of objectives. Empirical studies demonstrate the efficacy of the proposed algorithm.

Evolutionary algorithms (EAs) have been recognized as a promising approach for bilevel optimization. However, the population-based characteristic of EAs largely influences their efficiency and effectiveness due to the nested structure of the two levels of optimization problems. In this article, we propose a transfer learning-based parallel EA (TLEA) framework for bilevel optimization. In this framework, the task of optimizing a set of lower level problems parameterized by upper level variables is conducted in a parallel manner. In the meanwhile, a transfer learning strategy is developed to improve the effectiveness of each lower level search (LLS) process. In practice, we implement two versions of the TLEA: the first version uses the covariance matrix adaptation evolutionary strategy and the second version uses the differential evolution as the evolutionary operator in lower level optimization. The experimental studies on two sets of widely used bilevel optimization benchmark problems are conducted, and the performance of the two TLEA implementations is compared to that of four well-established evolutionary bilevel optimization algorithms to verify the effectiveness and efficiency of the proposed algorithm framework.

To prevent the population from getting stuck in local areas and then missing the constrained Pareto front fragments in dealing with constrained multiobjective optimization problems (CMOPs), it is important to guide the population to evenly explore the promising areas that are not dominated by all examined feasible solutions. To this end, we first introduce a cost value-based distance into the objective space, and then use this distance and the constraints to define an indicator to evaluate the contribution of each individual to exploring the promising areas. Theoretical studies show that the proposed indicator can effectively guide population to focus on exploring the promising areas without crowding in local areas. Accordingly, we propose a new constraint handling technique (CHT) based on this indicator. To further improve the diversity of population in the promising areas, the proposed indicator-based CHT divides the promising areas into multiple subregions, and then gives priority to removing the individuals with the worst fitness values in the densest subregions. We embed the indicator-based CHT in evolutionary algorithm and propose an indicator-based constrained multiobjective algorithm for solving CMOPs. Numerical experiments on several benchmark suites show the effectiveness of the proposed algorithm. Compared with six state-of-the-art constrained evolutionary multiobjective optimization algorithms, the proposed algorithm performs better in dealing with different types of CMOPs, especially in those problems that the individuals are easy to appear in the local infeasible areas that dominate the constrained Pareto front fragments.

Evolutionary algorithms for bi-level optimization suffer from the low efficiency in dealing with the multiple lower level optimization tasks required by the population based upper level search. In this paper, we design a transfer learning based covariance matrix adaptation evolution strategy (TL-CMA-ES) for bi-level optimization problems, where the tasks of searching for multiple lower level optimal solutions are conducted by a set of CMA-ES optimizers in a parallel manner. Furthermore, a transfer learning strategy is introduced in the parallel lower level CMA-ES search such that each CMA-ES optimizer can learn and utilize useful features gained by its neighbours. Experimental comparison and analysis are carried out to verify the effectiveness and efficiency of the proposed method.

An objective normalization strategy is essential in any evolutionary multiobjective or many-objective optimization (EMO or EMaO) algorithm, due to the distance calculations between objective vectors required to compute diversity and convergence of population members. For the decomposition-based EMO/EMaO algorithms involving the Penalty Boundary Intersection (PBI) metric, normalization is an important matter due to the computation of two distance metrics. In this article, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBI-based MOEA/D and a proposed PBI-based NSGA-III procedure. Although the effect is well recognized in the literature, few theoretical studies have been done so far to understand its true nature and the choice of a suitable penalty parameter value for an arbitrary problem. The developed theoretical results have been corroborated with extensive experimental results on three to 15-objective convex and non-convex instances of DTLZ and WFG problems. The article, makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study.

For solving constrained multi-objective optimization problems (CMOPs), an effective constraint-handling technique (CHT) is of great importance. Recently, many CHTs have been proposed for solving CMOPs. However, no single CHT can outperform all kinds of CMOPs. This paper proposes an algorithm, namely, ACHT-M2M, which adaptively allocates the existing CHTs in an M2M framework for solving CMOPs. To be more specific, a CMOP is first decomposed into several constrained multi-objective optimization subproblems by ACHT-M2M. Each subproblem has a subpopulation in a subregion. CHT for each subregion is adaptively allocated according to a proposed composite performance measure. Population for the next generation is selected from subregions by selection operators with different CHTs and the obtained nondominated feasible solutions in each generation are used to update a predefined archive. ACHT-M2M assembles the advantages of different CHTs and makes them cooperate with each other. The proposed ACHT-M2M is finally compared with the other 12 representative algorithms on benchmark CMOPs and the experimental results further confirm the effectiveness of ACHT-M2M for solving CMOPs.

Multiobjective multitasking optimization (MTO), which is an emerging research topic in the field of evolutionary computation, was recently proposed. MTO aims to solve related multiobjective optimization problems at the same time via evolutionary algorithms. The key to MTO is the knowledge transfer based on sharing solutions across tasks. Notably, positive knowledge transfer has been shown to facilitate superior performance characteristics. However, how to find more valuable transferred solutions for the positive transfer has been scarcely explored. Keeping this in mind, we propose a new algorithm to solve MTO problems. In this article, if a transferred solution is nondominated in its target task, the transfer is positive transfer. Furthermore, neighbors of this positive-transfer solution will be selected as the transferred solutions in the next generation, since they are more likely to achieve the positive transfer. Numerical studies have been conducted on benchmark problems of MTO to verify the effectiveness of the proposed approach. Experimental results indicate that our proposed framework achieves competitive results compared with the state-of-the-art MTO frameworks.

When solving constrained multiobjective optimization problems (CMOPs), the most commonly used way of measuring constraint violation is to calculate the sum of all constraint violations of a solution as its distance to feasibility. However, this kind of constraint violation measure may not reflect the distance of an infeasible solution from feasibility for some problems, for example, when an infeasible solution closer to a feasible region does not have a smaller constraint violation than the one farther away from a feasible region. Unfortunately, no set of artificial benchmark problems focusing on this area exists. To remedy this issue, a set of CMOPs with deceptive constraints is introduced in this article. It is the first attempt to consider CMOPs with deceptive constraints (DCMOPs). Based on our previous work, which designed a set of directed weight vectors to solve CMOPs, this article proposes a cooperative framework with an improved version of directed weight vectors to solve DCMOPs. Specifically, the cooperative framework consists of two switchable phases. The first phase uses two subpopulations-one to explore feasible regions and the other to explore the entire space. The two subpopulations provide useful information about the optimal direction of objective improvement to each other. The second phase aims mainly at finding Pareto-optimal solutions. Then an infeasibility utilization strategy is used to improve the objective function values. The two phases are switchable based on the information found to date at any time in the evolutionary process. The experimental results show that this method significantly outperforms the algorithms with which it is compared on most of the DCMOPs, in terms of reliability and stability in finding a set of well-distributed optimal solutions.

This article investigates the properties of ratio and difference-based indicators under the Minkovsky distance and demonstrates that a ratio-based indicator with infinite norm is the best for solution evaluation among these indicators. Accordingly, a promising-region-based evolutionary many-objective algorithm with the ratio-based indicator is proposed. In our proposed algorithm, a promising region is identified in the objective space using the ratio-based indicator with infinite norm. Since the individuals outside the promising region are of poor quality, we can discard these solutions from the current population. To ensure the diversity of population, a strategy based on the parallel distance is introduced to select individuals in the promising region. In this strategy, all individuals in the promising region are projected vertically onto the normal plane so that crowded distances between them can be calculated. Afterward, two solutions with a smaller distance are selected from the candidate solutions each time, and the solution with the smaller indicator fitness value is removed from the current population. Empirical studies on various benchmark problems with 3-20 objectives show that the proposed algorithm performs competitively on all test problems. Compared with a number of other state-of-the-art evolutionary algorithms, the proposed algorithm is more robust on these problems with various Pareto fronts.

This paper addresses a class of optimization problems in which either part of the objective function is differentiable while the rest is nondifferentiable or the objective function is differentiable in only part of the domain. Accordingly, we propose a dual-decomposition-based approach that includes both objective decomposition and domain decomposition. In the former, the original objective function is decomposed into several relatively simple subobjectives to isolate the nondifferentiable part of the objective function, and the problem is consequently formulated as a multiobjective optimization problem (MOP). In the latter decomposition, we decompose the domain into two subdomains, that is, the differentiable and nondifferentiable domains, to isolate the nondifferentiable domain of the nondifferentiable subobjective. Subsequently, the problem can be optimized with different schemes in the different subdomains. We propose a population-based optimization algorithm, called the simulated water-stream algorithm (SWA), for solving this MOP. The SWA is inspired by the natural phenomenon of water streams moving toward a basin, which is analogous to the process of searching for the minimal solutions of an optimization problem. The proposed SWA combines the deterministic search and heuristic search in a single framework. Experiments show that the SWA yields promising results compared with its existing counterparts.

Multiobjective multitasking optimization (MTO) is an emerging research topic in the field of evolutionary computation. In contrast to multiobjective optimization, MTO solves multiple optimization tasks simultaneously. MTO aims to improve the overall performance of multiple tasks through knowledge transfer among tasks. Recently, MTO has attracted the attention of many researchers, and several algorithms have been proposed in the literature. However, one of the crucial issues, finding useful knowledge, has been rarely studied. Keeping this in mind, this article proposes an MTO algorithm based on incremental learning (EMTIL). Specifically, the transferred solutions (the form of knowledge) will be selected by incremental classifiers, which are capable of finding valuable solutions for knowledge transfer. The training data are generated by the knowledge transfer at each generation. Furthermore, the search space of the tasks will be explored by the proposed mapping (among tasks) approach, which helps these tasks to escape from their local Pareto Fronts. Empirical studies have been conducted on 15 MTO problems to assess the effectiveness of EMTIL. The experimental results demonstrate that EMTIL works more effectively for MTO compared to the existing algorithms.

In this paper, we propose a computationally efficient variant of elitist covariance matrix evolution strategy for continuous local search in high dimensional space. It focuses on searching in a low-dimensional subspace expanded by a small number of promising search directions. This leads to the linear internal computational complexity of each iteration, which enables the algorithm to scale to high dimensional problems. We conduct comprehensive experiments to evaluate the parameter sensitivity and the algorithm’s performance. The experimental results validate that the proposed algorithm reduces the running time by a factor of ten, and it can be easily scaled up to n>1000 on a set of commonly used test functions.

For a many-objective optimization problem with redundant objectives, we propose two novel objective reduction algorithms for linearly and, nonlinearly degenerate Pareto fronts. They are called LHA and NLHA respectively. The main idea of the proposed algorithms is to use a hyperplane with non-negative sparse coefficients to roughly approximate the structure of the PF. This approach is quite different from the previous objective reduction algorithms that are based on correlation or dominance structure. Especially in NLHA, in order to reduce the approximation error, we transform a nonlinearly degenerate Pareto front into a nearly linearly degenerate Pareto front via a power transformation. In addition, an objective reduction framework integrating a magnitude adjustment mechanism and a performance metric [Formula: see text] are also proposed here. Finally, to demonstrate the performance of the proposed algorithms, comparative experiments are done with two correlation-based algorithms, LPCA and NLMVUPCA, and with two dominance-structure-based algorithms, PCSEA and greedy [Formula: see text]MOSS, on three benchmark problems: DTLZ5(I,M), MAOP(I,M), and WFG3(I,M). Experimental results show that the proposed algorithms are more effective.

Decomposition-based evolutionary algorithms have shown great potential in many-objective optimization. However, the lack of theoretical studies on decomposition methods has hindered their further development and application. In this paper, we first theoretically prove that weight sum, Tchebycheff, and penalty boundary intersection decomposition methods are essentially interconnected. Inspired by this, we further show that highly customized dominance relationship can be derived from decomposition for any given decomposition vector. A new evolutionary algorithm is then proposed by applying the customized dominance relationship with adaptive strategy to each subpopulation of multiobjective to multiobjective framework. Experiments are conducted to compare the proposed algorithm with five state-of-the-art decomposition-based evolutionary algorithms on a set of well-known scaled many-objective test problems with 5 to 15 objectives. Simulation results have shown that the proposed algorithm can make better use of the decomposition vectors to achieve better performance. Further investigations on unscaled many-objective test problems verify the robust and generality of the proposed algorithm.

Based on the ideas of minimizing the loss of convergence and diversity of the candidate solution set, this paper proposes a cost value based evolutionary many-objective algorithm with neighbor selection strategy. In this work, the cost value of each solution is the mutual evaluation from other ones in current population. By this way, the proposed algorithm, named MEMO, can easily recognize the dominated and the nondominated solutions and assess the contribution of convergence and diversity of each solution among the candidate solution set. To further enhance the performance of proposed algorithm, a neighbor selection strategy is also suggested in this paper. Simulation experiments on MaF series indicate that the proposed MEMO is superior to IBEA, MOEA/D, NSGA-III and RVEA in terms of effectiveness and robustness.

An effective allocation of search effort is important in multiobjective optimization, particularly in many-objective optimization problems (MaOPs). This paper presents a new adaptive search effort allocation strategy for multiobjective evolutionary algorithm based on decomposition MOEA/D-M2M, a recent MOEA/D algorithm for challenging MaOPs. This proposed method adaptively adjusts the subregions of its subproblems by detecting the importance of different objectives in an adaptive manner. More specifically, it periodically resets the subregion setting based on the distribution of the current solutions in the objective space such that the search effort is not wasted on unpromising regions. The basic idea is that the current population can be regarded as an approximation to the Pareto front (PF) and thus one can implicitly estimate the shape of the PF and such estimation can be used for adjusting the search focus. The performance of proposed algorithm has been verified by comparing it with eight representative and competitive algorithms on a set of degenerated MaOPs with disconnected and connected PFs. Performances of the proposed algorithm on a number of nondegenerated test instances with connected and disconnected PFs are also studied.

Despite the effectiveness of the decomposition-based multi-objective evolutional algorithm (MOEA/D-M2M) in solving continuous multi-objective optimization problems (MOPs), its performance in addressing 0/1 multi-objective knapsack problems (MOKPs) has not been fully explored. In this paper, we use MOEA/D-M2M with an improved greedy repair strategy to solve MOKPs. It first decomposes an MOKP into a number of simple optimization subproblems and solves them in a collaborative way. Each subproblem has its own subpopulation, and then an improved greedy strategy is introduced to improve the performance of the proposed algorithm on MOKPs. Therein, a weight vector chosen randomly from a corresponding subpopulation is utilized to repair infeasible individuals or improve feasible individuals to have a better fitness, which improves the convergence of the population. Experimental studies on a set of test instances indicate that the MOEA/D-M2M with the improved greedy strategy is superior to MOGLS and MOEA/D in terms of finding better approximations to the Pareto front.

There are two main tasks involved in addressing a multiobjective optimization problem (MOP) by evolutionary multiobjective (EMO) algorithms: 1) make the population converge close to the Pareto-optimal front and 2) maintain adequate population diversity. However, most state-of-the-art EMO algorithms are designed based on the “convergence first and diversity second” principle. It has been observed that although these EMO algorithms have been successful in optimizing many real-world MOPs, they fail to solve certain problems that feature a severe imbalance between diversity preservation and achieving convergence. This paper characterizes an imbalanced MOP by clearly defining properties and indicating the reasons for the existing EMO algorithms' difficulties in solving them. We then present 14 imbalanced problems, with and without constraints. Computational results using four existing EMO algorithms-elitist non-dominated sorting genetic algorithm (NSGA-II), multiobjective evolutionary algorithm based on decomposition (MOEA/D), strength Pareto evolutionary algorithm 2 (SPEA2), and S metric selection EMO algorithm (SMS-EMOA) and a proposed generalized vector-evaluated genetic algorithm are then presented. It is seen that these EMO algorithms cannot solve these imbalanced problems, but they are able to solve the problems when augmented by multiobjective to multiobjective (M2M), an approach that decomposes the population into several interacting subpopulations. These results and the successful application of the EMO methods with the M2M approach even on standard so-called balanced problems indicate the usefulness of using the M2M approach.

In this paper, we propose a simple but effective objective reduction algorithm (ORA) for many-objective optimization problems (MaOPs). It uses a hyperplane involving sparse non-negative coefficients to roughly approximate the conflicting structure of the Pareto front in the objective space. Then the objectives with non-zero coefficients are considered as essential objectives. In order to verify the performance of proposed algorithm, we compare the proposed algorithm with two correlation-based ORA, i.e., L-PCA and NL-MVU-PCA and a dominance structure-based ORA, i.e., PCSEA on the benchmark problem DTLZ5(I, M). The experimental results show the effectiveness of the proposed algorithm.

When optimizing an multiobjective optimization problem, the evolution of population can be regarded as a approximation to the Pareto Front (PF). Motivated by this idea, we propose an adaptive region decomposition framework: MOEA/D-AM2M for the degenerated Many-Objective optimization problem (MaOP), where degenerated MaOP refers to the optimization problem with a degenerated PF in a subspace of the objective space. In this framework, a complex MaOP can be adaptively decomposed into a number of many-objective optimization subproblems, which is realized by the adaptively direction vectors design according to the present population's distribution. A new adaptive weight vectors design method based on this adaptive region decomposition is also proposed for selection in MOEA/D-AM2M. This strategy can timely adjust the regions and weights according to the population's tendency in the evolutionary process, which serves as a remedy for the inefficiency of fixed and evenly distributed weights when solving MaOP with a degenerated PF. Five degenerated MaOPs with disconnected PFs are generated to identify the effectiveness of proposed MOEA/D-AM2M. Contrast experiments are conducted by optimizing those MaOPs using MOEA/D-AM2M, MOEA/D-DE and MOEA/D-M2M. Simulation results have shown that the proposed MOEA/D-AM2M outperforms MOEA/D-DE and MOEA/D-M2M.

This paper proposes two constraint-handling techniques based on multiobjective optimization with biased dynamic weights for constrained optimization problems (COPs). Transforming a COP into an unconstrained biobjective optimization, two popular strategies based on decomposition, i.e. Tchebycheff approach (TEA) and weighted sum approach (WSA) are used in this paper respectively. In order to keep a good balance between convergence and diversity of the population, this paper uses the weights, which are designed with bias and change dynamically as the generation increases, to select different individuals with smaller objective values and lower degree of constraint violations. Furthermore, 13 benchmark test functions are used to investigate the effectiveness of TEA and WSA. Experimental results demonstrate that TEA not only works better than WSA, but also is superior to the compared algorithms, i.e. MDPE, GDE and SR in terms of reliability and stabilization of converging to a global solution.

In this paper, we propose a decomposition-based evolutionary algorithm with boundary search and archive for constrained multi-objective optimization problems (CMOPs), named CM2M. It decomposes a CMOP into a number of optimization subproblems and optimizes them simultaneously. Moreover, a novel constraint handling scheme based on the boundary search and archive is proposed. Each subproblem has one archive, including a subpopulation and a temporary register. Those individuals with better objective values and lower constraint violations are recorded in the subpopulation, while the temporary register consists of those individuals ever found before. To improve the efficiency of the algorithm, the boundary search method is designed. This method makes the feasible individuals with a higher probability to perform genetic operator with the infeasible individuals. Especially, when the constraints are active at the Pareto solutions, it can play its leading role. Compared with two algorithms, i.e. CMOEA/D-DE-CDP and Gary’s algorithm, on 18 CMOPs, the results show the effectiveness of the proposed constraint handling scheme.

Research on multiobjective optimization problems becomes one of the hottest topics of intelligent computation. In order to improve the search efficiency of an evolutionary algorithm and maintain the diversity of solutions, in this paper, the learning automata (LA) is first used for quantization orthogonal crossover (QOX), and a new fitness function based on decomposition is proposed to achieve these two purposes. Based on these, an orthogonal evolutionary algorithm with LA for complex multiobjective optimization problems with continuous variables is proposed. The experimental results show that in continuous states, the proposed algorithm is able to achieve accurate Pareto-optimal sets and wide Pareto-optimal fronts efficiently. Moreover, the comparison with the several existing well-known algorithms: nondominated sorting genetic algorithm II, decomposition-based multiobjective evolutionary algorithm, decomposition-based multiobjective evolutionary algorithm with an ensemble of neighborhood sizes, multiobjective optimization by LA, and multiobjective immune algorithm with nondominated neighbor-based selection, on 15 multiobjective benchmark problems, shows that the proposed algorithm is able to find more accurate and evenly distributed Pareto-optimal fronts than the compared ones.

For many-objective optimization problems (MaOPs), in which the number of objectives is greater than three, the performance of most existing evolutionary multi-objective optimization algorithms generally deteriorates over the number of objectives. As some MaOPs may have redundant or correlated objectives, it is desirable to reduce the number of the objectives in such circumstances. However, the Pareto solution of the reduced MaOP obtained by most of the existing objective reduction methods, based on objective selection, may not be the Pareto solution of the original MaOP. In this paper, we propose an objective extraction method (OEM) for MaOPs. It formulates the reduced objective as a linear combination of the original objectives to maximize the conflict between the reduced objectives. Subsequently, the Pareto solution of the reduced MaOP obtained by the proposed algorithm is that of the original MaOP, and the proposed algorithm can thus preserve the dominance structure as much as possible. Moreover, we propose a novel framework that features both simple and complicated Pareto set shapes for many-objective test problems with an arbitrary number of essential objectives. Within this framework, we can control the importance of essential objectives. As there is no direct performance metric for the objective reduction algorithms on the benchmarks, we present a new metric that features simplicity and usability for the objective reduction algorithms. We compare the proposed OEM with three objective reduction methods, i.e., REDGA, L-PCA, and NL-MVU-PCA, on the proposed test problems and benchmark DTLZ5 with different numbers of objectives and essential objectives. Our numerical studies show the effectiveness and robustness of the proposed approach.

This paper proposes an improved version of the single objective optimization evolutionary algorithm based on Covariance Matrix Learning and Searching Preference (CMLSP), named ICMLSP. ICMLSP uses the same way with CMLSP to generate high quality solutions by sampling a multivariate Gauss distribution, which uses the best solutions found so far as its mean value. However, unlike the previous one, ICMLSP uses different covariance matrix learning philosophy, that is, the principal component analysis (PCA) method is used to estimate the covariance matrix. Furthermore, ICMLSP tends to use smaller population size than CMLSP to achieve a faster search. In order to get enough information for a reliable estimation, a new cumulation strategy is designed in ICMLSP. Solutions are selected from an archive set which stores the best λ individuals in present population and last t(t >= 1) populations to estimate the covariance matrix. A new adaptive rule, which makes use of the history successful information to generate different searching step from two different Cauchy distributions, is designed for ICMLSP to balance global exploration and local exploitation. Finally, the performance of the ICMLSP has been tested on 15 noiseless optimization problems designed for the CEC 2015 Competition on Learning-based Real-Parameter Single Objective Optimization. The results are reported at the end of this paper.

Recently, energy efficiency (EE) has drawn more and more attentions due to the need for green communication. It is well-known that the base station (BS) sleeping strategy and radio resource allocation can effectively improve the EE for the wireless communication networks. Meanwhile, renewable energy is important to decrease the carbon emissions which is also subject to the need for green communication. In this paper, we jointly optimize the BS sleeping strategy, resource allocation and renewable energy procurement scheme to maximize the profit of the network operators and minimize the carbon emissions. We formulate the joint optimization problem as a mixed integer programming problem which is difficult to tackle in general. In the first, we adopt the bi-velocity discrete particle swarm optimization (BVDPSO) algorithm to optimize the BS sleeping strategy. When the BS sleeping strategy is fixed, the original optimization problem only consists of the power allocation, subcarrier assignment and energy procurement which is also difficult to solve due to the combinatorial nature of the subcarrier assignment. Then, we propose an optimal algorithm based on Lagrange dual domain method to optimize the power allocation, subcarrier assignment and energy procurement. Numerical results illustrate the effectiveness of our proposed scheme and algorithm.

In this paper, we propose a single objective optimization evolutionary algorithm (EA) based on Covariance Matrix Learning and Searching Preference (CMLSP) and design a switching method which is used to combine CMLSP and Covariance Matrix Adaptation Evolution Strategy (CMAES). Then we investigate the performance of the switch method on a set of 30 noiseless optimization problems designed for the special session on real-parameter optimization of CEC 2014. The basic idea of the proposed CMLSP is that it is more likely to find a better individual around a good individual. That is to say, the better an individual is, the more resources should be invested to search the region around the individual. To achieve it, we discard the traditional crossover and mutation and design a novel method based on the covariance matrix leaning to generate high quality solutions. The best individual found so far is used as the mean of a Gaussian distribution and the covariance of the best λ individuals in the population are used as the evaluation of its covariance matrix and we sample the next generation individual from the Gaussian distribution other than using crossover and mutation. In the process of generating new individuals, the best individual is changed if ever a better one is found. This search strategy emphasizes the region around the best individual so that a faster convergence can be achieved. The use of switch method is to make best use of the proposed CMLSP and existing CMAES. At last, we report the results.

In this paper, a novel multi-objective particle swarm optimization algorithm based on MOEA/D-M2M decomposition strategy (MOPSO-M2M) is proposed. MOPSO-M2M can decompose the objective space into a number of subregions and then search all the subregions using respective sub-swarms simultaneously. The M2M decomposition strategy has two very desirable properties with regard to MOPSO. First, it facilitates the determination of the global best (gbest) for each sub-swarm. A new global attraction strategy based on M2M decomposition framework is proposed to guide the flight of particles by setting an archive set which is used to store the historical best solutions found by the swarm. When we determine the gbest for each particle, the archive set is decomposed and associated with each sub-swarm. Therefore, every sub-swarm has its own archive subset and the gbest of the particle in a sub-swarm is selected randomly in its archive subset. The new global attraction strategy yields a more reasonable gbest selection mechanism, which can be more effective to guide the particles to the Pareto Front (PF). This strategy can ensure that each sub-swarm searches its own subregion so as to improve the search efficiency. Second, it has a good ability to maintain the diversity of the population which is desirable in multi-objective optimization. Additionally, MOPSO-M2M applies the Tchebycheff approach to determine the personal best position (pbest) and no additional clustering or niching technique is needed in this algorithm. In order to demonstrate the performance of the proposed algorithm, we compare it with two other algorithms: MOPSO and DMS-MO-PSO. The experimental results indicate the validity of this method.

This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA/D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA/D-M2M with classic MOEA/D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA/D-M2M performs better.

We propose a novel constrained multiobjective evolutionary algorithm based on decomposition and temporary register in this paper. It decomposes the constrained multiobjective optimization problem into a number of subproblems and then optimizes each subproblem in a collaborative way. We also propose a novel constraint handling technique based on temporary register. Each subproblem has its own subpopulation and one temporary register. The subpopulation is composed of those individuals which has better objective values and lower constraint violations of this subproblem, while the temporary register is composed of those individuals that are found before. We perform the crossover operator between each individual in the subpopulations and an individual which is randomly chosen from the corresponding temporary register. Therefore, the temporary register strategy makes the individuals which have better objective values and lower constraint violations have an opportunity to participate in the crossover and mutation, but don't been eliminated at once. Moreover, this constraint handling technique does not need any parameter setting. The numerical simulations show the proposed algorithm outperforms existing ones.

A novel multiobjective DE algorithm using the subregion and external set strategy (MOEA/S-DE) is proposed in this paper, in which the objective space is divided into some subregions and then independently optimize each subregion. An external set is introduced for each subregion to save some individuals ever found in this subregion. An alternative of mutation operators based the idea of direct simplex method of mathematical programming are proposed: local and global mutation operator. The local mutation operator is applied to improve the local search performance of the algorithm and the global mutation operator to explore a wider area. Additionally, a reusing strategy of difference vector also is proposed. It reuses the difference vector of the better individuals according to a given probability. Compared with traditional DE, the crossover operator also is improved. In order to demonstrate the performance of the proposed algorithm, it is compared with the MOEA/D-DE and the hybrid-NSGA-II-DE. The result indicates that the proposed algorithm is efficient.

In order to overcome the co-channel interference between neighbor cells in Long Term Evolution (LTE) system, a novel power allocation algorithm based on non-cooperative game theory is proposed in this paper. In the downlink of LTE, the pricing mechanism in the non-cooperative game theory is used to model the power allocation scheme in Orthogonal Frequency Division Multiple Access (OFDMA) cellular systems. We set the data rate as the pricing factor, and the objective function is the variance of power. Then we use Genetic algorithm to obtain the optimal solution. Both the QoS of user and the system throughput are considered in our proposed algorithm, simulation results show that the co-channel interference is reduced maximally. Comparing with the traditional water filling algorithm, the user can obtain more system throughput as the number of user increasing. It is suitable for the crowded user areas.

By dividing the objective space into several small regions, this paper proposes an improved NSGA-II algorithm, which updates the population in each sub-region by using non-dominated sorting and crowded distance selection operator (NSGA-II). Since performing the evolutionary operator is independent in each sub-region and the number of the individuals in a sub-region is far less than the size of the population, the computational complexity at each generation is lower than NSGA-II. The computational complexity of each generation in the proposed algorithm is O(mN3/2), where m is the number of the objective and N is its population size. For enhancing the capability of proposed algorithm, The algorithm exchanges the information between sub-regions through re-dividing their offsprings and the evolutionary operators between individuals are operated in the same sub-region. Such evolutionary operators can largely play a role of exploring the good individuals in this area and improve the local search capabilities of the algorithm. A specific selection in this paper surmounts the intrinsic shortcoming of the sub region decomposition technique, which there may be no Pareto optimal solutions in some sub-region. Numerical results show that the proposed algorithm has a good performance.

Lip contour extraction is crucial to the success of a lipreading system. This paper presents a lip contour extraction algorithm using localized active contour model with the automatic selection of proper parameters. The proposed approach utilizes a minimum-bounding ellipse as the initial evolving curve to split the local neighborhoods into the local interior region and the local exterior region, respectively, and then compute the localized energy for evolving and extracting. This method is robust against the uneven illumination, rotation, deformation, and the effects of teeth and tongue. Experiments show its promising result in comparison with the existing methods.

For the purpose of estimating the mixing matrix under the nonstrictly sparse condition, this paper presents the algorithms to approximate the mixing matrix in two different situations in which the source vectors are 1-sparse and (m - 1)-sparse. When the source signals are 1-sparse, we use the generalized spherical coordinate transformation to convert the matrix of observation signals into the new one, which makes the process of estimating column A become the process of finding the center point of these new data. For the situation that source signals are (m - 1)-sparse, we propose a new algorithm for the underdetermined mixtures blind source separation based on hyperplane clustering. The algorithm firstly finds out the linearly independent vectors from the observations, and secondly determines all the normal vectors of hyperplanes by analyzing the number of observations that are in the same hyperplane. Finally, we identify the column vectors of the mixing matrix A by calculating the vectors which are orthogonal to the clustered normal vectors. These two new algorithms for estimating the mixing matrix are more suitable for the general cases as they have lower requirement for the sparsity of the observations.