Image processing is a fundamental field in computer science with applications across various real-life areas. Image enhancement in the preprocessing stage is crucial for tasks in computer vision. Contrast enhancement in images aims to improve visual quality by increasing contrast and highlighting significant details. Although classical contrast enhancement techniques are widely used, they often suffer from issues such as over-enhancement due to the lack of mechanisms to control this improvement. To improve the contrast, transformation functions assign new intensities to each pixel in the image. One of the main drawbacks of the transformation functions is tuning their parameters. On the other hand, most contrast-enhancement techniques are typically used to improve the contrast in color images. In this regard, this study examines the effectiveness of three color spaces-HSV, HSI, and CIELAB-in enhancing contrast, with the goal of identifying the most effective space for this purpose. Additionally, the performance of a widely used metaheuristic is evaluated in tuning the parameters of the transformation function. The model is evaluated using standard quality indicators on a public image dataset. Preliminary findings suggest that the HSI color space is better suited for optimization using metaheuristics and effectively improves image contrast.

The analysis of chest radiographs holds significant importance in healthcare and medical diagnostics. Also known as chest X-rays, these images offer valuable insights into the condition of a patient’s chest area, including vital organs like the lungs, heart, and surrounding structures. Amid the ongoing COVID-19 pandemic, the analysis of chest radiographs has become even more crucial. These images are extensively utilized in detecting and monitoring COVID-19 pneumonia, aiding in identifying characteristic lung patterns associated with the disease. This paper presents a method for automatically detecting and normalizing the Region of Interest (ROI) in chest radiographs. It evaluates three contrast enhancement techniques: Contrast Enhancement by Particle Swarm Optimization (CE-PSO), Contrast Limited Adaptive Histogram Equalization (CLAHE), and CE-PSO+CLAHE applied to the ROI, aiming to determine the most effective method for classifying radiographs in COVID-19 detection. For this purpose, six Convolutional Neural Networks (CNNs) are evaluated, including MobileNetV2, ResNet-50, ResNet-18, Enhanced, Compact, and AlexNet. The best-performing methods were the ResNet-50 using CE-PSO, with an accuracy of 95.6%, and MobileNetV2 using CLAHE, with an accuracy of 96%. The experimental results reveal that CE-PSO+CLAHE is not recommended for image preprocessing. Furthermore, CE-PSO and CLAHE show similar results, depending on the classifier used. These results are competitive with state-of-the-art algorithms and highlight the importance of image preprocessing to classify radiographs accurately.

Optical Coherence Tomography (OCT) is an imaging technique that enables medical experts to obtain transverse scans of various small tissues, including relevant eye elements, such as the retina. The analysis of retinal OCT scans holds significant clinical relevance as it provides physicians with anatomical and pathological insights. However, speckle noise, inherent to OCT’s operational nature, negatively impacts image quality and hinders precise interpretation and diagnosis. In this regard, this paper presents the Metaheuristic-Driven Bayesian Speckle Denoising (MDBSD) framework: a fusion of a metaheuristic-based optimization using the Optimized Bayesian Non-local means with block selection (OBNLM) as the transformation function while introducing a specific fitness function for the evaluation of the potential solutions. Different classical metaheuristic algorithms are utilized and applied to a public retinal OCT dataset, followed by a numerical evaluation using contrast and denoising performance metrics to validate this framework. Regardless of the metaheuristic technique used for optimization, enhanced images present a notorious scan denoising while keeping retinal layers’ edges quite delimited, proving this framework’s efficiency. Additionally, future research directions are outlined for further advancement in this area.

In recent years, significant advancements have been witnessed in the field of mobile robot path-planning research. However, navigating through complex scenarios poses a formidable challenge for mobile robots, given the multitude of objectives they must satisfy. Traditional path-planning algorithms often struggle in such environments, highlighting the need for more sophisticated approaches. Among these, the multi-objective evolutionary optimization paradigm stands out for its ability to tackle the complexities inherent in mobile robot navigation. This study delves into the efficacy of three prominent multi-objective evolutionary approaches based on different principles, specifically tailored to address the challenges of minimizing path time and enhancing trajectory smoothness for ground robots. Through a comprehensive analysis, we explore their performance across four distinct environments, each presenting unique navigational hurdles. The comparative evaluation reveals that NSGA-II emerges as the frontrunner among the trio of algorithms, consistently delivering superior results across varied scenarios. Its adeptness in balancing conflicting objectives and generating optimized paths underscores its efficacy in real-world applications. By synthesizing empirical findings, this study sheds light on the evolving landscape of mobile robot path planning and underscores the pivotal role of multi-objective evolutionary optimization in overcoming navigational complexities.

This paper introduces Dynamic Social Particle Swarm Optimization (DS-PSO), a novel adaptation of the traditional Particle Swarm Optimization (PSO) technique specifically engineered for complex optimization challenges. DS-PSO innovatively incorporates dynamic social interactions within the swarm, enhancing adaptability and addressing the typical limitations of premature convergence and limited exploration in conventional PSO. A key feature of DS-PSO is its ability to balance exploration and exploitation efficiently, making it particularly suitable for dynamic environments. The primary application highlighted in this study is automatic clustering, a crucial task in data analysis involving unsupervised data grouping without prior knowledge of cluster numbers. DS-PSO’s flexibility and improved search capability demonstrate its potential as an effective tool for automatic clustering, promising significant advancements in data-driven optimization and analysis.

In the context of evolutionary algorithms, having the ability to adapt to any search space within an optimization problem is an essential task. Appropriately adapting the population can lead to better solutions and more efficient use of function call resources. This article presents a renewed approach to population management inspired by modifying the well-known Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm. The proposed strategy aims to improve the algorithm’s population adaptability to the search space and optimize function evaluations. Statistically evaluated experimental test outcomes demonstrate significantly better performance on high-dimensional problems in comparison to the original CMA-ES and seven other known evolutionary algorithms in the literature.

This paper introduces an archive-based multi- objective algorithm based on simulated annealing to deal with the time/weight-balanced cluster problem. In the presented paper, we adapted the necessary components into the Archive Multi- Objective Simulated Annealing (AMOSA) framework to deal appropriately with the clustering problem. Due to the computational cost of the objective functions, we introduced a parallelization of such objectives to reduce the algorithm's running time, achieving an improvement of 13%. The introduced algorithm was evaluated in a real-world scenario, and its parameters were tuned using the Iterated Local Search in Parameter Configuration Space (ParamILS) method. AMOSA, using its best configuration, was compared concerning a popular Pareto-based multi-objective evolutionary algorithm. The preliminary results indicate the viability of using the proposed approach to deal with the type of problem tackled in this study. The proposed method outperformed the Nondominated Sorting Genetic Algorithm II (NSGA II) concerning the quality of the solutions and the execution time, as will be seen later on.

For more than a decade, the efficiency and effectiveness of MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition) when solving complicated problems has been shown. Due to this, several researchers have focused their investigations on MOEA/D's extensions that can deal with CMOPs (Constrained Multi-objective Optimization Problems). In this paper, we adhere to the MOEA/D framework, a simple penalty function to deal with CMOPs. The penalty function is dynamically adapted during the search. In this way, the interaction between feasible and infeasible solutions is promoted. As a result, the proposed approach (namely MOEA/D-DPF) extends MOEA/D to handle constraints. The proposed approach performance is evaluated on the well-known CF test problems taken from the CEC'2009 suite. Using convergence and feasibility indicators, we compare the solutions produced by our algorithm against those produced by state-of-the-art MOEAs. Results show that MOEA/D-DPF is highly competitive and, in some cases, it performs better than the MOEAs adopted in our comparative study.

Pareto compliance is a critical property of quality indicators (QIs) focused on measuring convergence to the Pareto front. This property allows a QI not to contradict the order imposed by the Pareto dominance relation. Hence, Pareto-compliant QIs are remarkable when comparing approximation sets of multi-objective evolutionary algorithms (MOEAs) since they do not produce misleading conclusions. However, the practical effect of Pareto-compliant QIs as the backbone of MOEAs' survival mechanisms is not entirely understood. In this paper, we study the practical effect of IGD++ (which is a combination of IGD+ and the hypervolume indicator), IGD+, and IGD, which are Pareto-compliant, weakly Pareto-compliant, and not Pareto-compliant QIs, respectively. To this aim, we analyze the convergence and diversity properties of steady-state MOEAs based on the previously mentioned QIs throughout the whole evolutionary process. Our experimental results showed that, in general, the practical effect of a Pareto-compliant QI in a selection mechanism is not very significant concerning weaker QIs, taking into account the whole evolutionary process.

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has attracted the attention of several investigators working on multi-objective optimization. At each iteration, MOEA/D generates an offspring solution from a parent's neighborhood. The new solution is evaluated, and, according to the decomposition approach, it can replace one or more solutions from the neighborhood, maintaining the population updated. In this sense, MOEA/D can be considered as a steady-state algorithm that maintains updated its population once a new solution is generated. In this work, we investigate the performance of MOEA/D in the multi-objective 0/1 knapsack problem considering a steady-state version and a proposed generational version. We explore the benefits of the generational version proposed in this paper. According to results, we show that the proposed approach can obtain a suitable performance in the multi-objective 0/1 knapsack problem employing between two and eight objective functions. Additionally, we propose a two-stage hybrid algorithm that employs the two different approaches of MOEA/D (i.e., the steady-state and generational versions). Our results reveal that the proposed hybrid approach can outperform the original MOEA/D in the many-objective settings of the 0/1 knapsack problem.

The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) has shown high-performance levels when solving complicated multi-objective optimization problems. However, its adaptation for dealing with constrained multi-objective optimization problems (cMOPs) keeps being under the scope of recent investigations. This paper introduces a novel selection mechanism inspired by the ε-constraint method, which builds a bi-objective problem considering the scalarizing function (used into the decomposition approach of MOEA/D) and the constraint violation degree as an objective function. During the selection step of MOEA/D, the scalarizing function is considered to choose the best solutions to the cMOP. Preliminary results obtained over a set of complicated test problems drawn from the CF test suite indicate that the proposed algorithm is highly competitive regarding state-of-the-art MOEAs adopted in our comparative study.

In multiobjective optimization, a scalable test problem is one that can be formulated for an arbitrary number of objectives. Scalable test problems evaluate the conceptual foundations of the so-called many-objective evolutionary algorithms. As an important class of problems, scalable test problems should contemplate a wide variety of features allowing us to evaluate and judge specific components of many-objective evolutionary algorithms. This, in fact, should promote the development of new strategies and/or methods in the design of many-objective optimization approaches. For this reason, the study of features and difficulties of this class of problems, plays a salient role in the development of many-objective approaches. As a result, a number of multiobjective scalable test problems have been proposed in recent years. In this paper, we present a review of features and limitations of existing multiobjective test problems formulated in continuous and unconstrained search spaces. We examine some features observed in some test problems which have not been properly discussed before. Additionally, we summarize a list of features and recommendations that should be considered in the design of scalable multiobjective test instances. Then, we preset a review of the state-of-the-art scalable test suites, including their features and limitations according to the recommended guidelines discussed herein. Finally, some possible paths for future research in this area are briefly discussed.

A Wireless Sensor Network (WSN) is composed of a set of energy- and processing-constrained devices that gather data about a set of phenomena (e.g., temperature, humidity, pressure, etc.). An efficient way to extend the lifetime of a WSN is using a clustering organization, which hierarchically structures the sensors in groups and one of them is chosen as a cluster head (leader). Such cluster head performs specific tasks such as gathering data from other cluster sensors (cluster members) and resending this data through the network to the base station. By using a cluster-head organization, data gathering process is improved and, by extension, the network lifetime is extended. In this paper, we formulate a multi-objective approach for the optimal cluster-head selection in a WSN aiming to minimize: i) the distance from the cluster head to the base station, ii) the distance from cluster members to their leader, and iii) the residual energy of the leaders. The goal of our study is two-fold. First, as currently it is not known the conflict relation among those objectives, we carry out an analysis to discover which objectives are essential for solving the cluster-head selection problem. Second, we investigate the performance of multi-objective evolutionary approaches based on decomposition and the impact of their main parameters when solving the multi-objective cluster-head selection problem.

This paper presents an analysis of parameters defined within some of the most representative decomposition-based multi-objective evolutionary algorithms (MOEAs) namely: MOEA/D, MOEA/D-DE, and MOEA/D-DRA. Our main interest is focused in the many-objective context, where decomposition-based MOEAs have been successfully applied, but a lack of analysis on the relevance of their parameters is evidently observable. We review the literature related to those parameter values that have been commonly adopted and we perform some experiments oriented to validate these decisions. Our results show that some alternative parameter configurations can allow these methods to obtain better solutions than the standard values. Moreover, some of our recommendations can conduct to inspect in detail the design of these algorithms.

In this paper, we analyze the behavior and compare the performance of three state-of-the-art Multi-objective Evolutionary Algorithms (MOEAs) based on three different approaches when solving the Multi-Objective Unit Commitment Problem (MO-UCP). Particularly, we study the performance of representative Pareto-, indicator- and decomposition-based MOEAs (namely NSGA-II, SMS-EMOA and MOEA/D) when solving standard MO-UCP test instances. The MOEAs employed in our comparative study, handle binary representation while lambda-iteration method is probabilistically used for assigning the economic/environmental power real dispatch. In our experiments, each evolutionary approach adopts the window crossover and the window mutation. A detailed study of the impact of these operators is carried out when different crossover and mutation ratios are employed. The comparative study presented here, shows that for low-dimensional instances, the performance of the three evolutionary approaches became very similar. However, when the dimension of the problem (large bit strings) increases, the performance of NSGA-II and SMS-EMOA became better than MOEA/D.

Due to its simplicity and effectiveness in solving many optimization problems, Particle Swarm Optimization (PSO) has attracted the attention of many researchers in the last few years. Nonetheless, in more complicated problems (involving multi-modality, non-separable, etc.), the use of PSO becomes limited and sometimes impractical. In this paper, we proposed an algorithm which is able to deal with optimization problems having several features. More specific, we introduce a refine mechanism into the evolutionary process of PSO for deep exploration of the local search space in which a particle is located. The proposed mechanism is inspired by the animal foraging behaviour, where searching is a mixture of systematic and random movements. In contrast to other existing PSO variants which aimed to improve the exploration ability by using random walk, the proposed approach exploits the locality of the particles by performing local variations in the flight of the individuals according to a Gaussian distribution. In our study, we analyze the effects of the proposed refinement mechanism when it is coupled into different PSO variants which are adopted in our experimental analysis. We show that our proposed approach not only was able to outperform the adopted PSO variants, but also was significantly better in most of the test functions employed in our comparative study.

Multi-objective evolutionary algorithms (MOEAs) based on decomposition are aggregation-based algorithms which transform a multi-objective optimization problem (MOP) into several single-objective subproblems. Being effective, efficient, and easy to implement, Particle Swarm Optimization (PSO) has become one of the most popular single-objective optimizers for continuous problems, and recently it has been successfully extended to the multi-objective domain. However, no investigation on the application of PSO within a multi-objective decomposition framework exists in the context of combinatorial optimization. This is precisely the focus of the paper. More specifically, we study the incorporation of Geometric Particle Swarm Optimization (GPSO), a discrete generalization of PSO that has proven successful on a number of single-objective combinatorial problems, into a decomposition approach. We conduct experiments on many-objective 1/0 knapsack problems i.e. problems with more than three objectives functions, substantially harder than multi-objective problems with fewer objectives. The results indicate that the proposed multi-objective GPSO based on decomposition is able to outperform two version of the well-know MOEA based on decomposition (MOEA/D) and the most recent version of the non-dominated sorting genetic algorithm (NSGA-III), which are state-of-the-art multi-objec\-tive evolutionary approaches based on decomposition.

In spite of the success of the multi-objective evolutionary algorithm based on decomposition (MOEA/D), the generation of weights for problems having many objectives, continues to be an open research problem. In this paper, we introduce a new methodology based on low-discrepancy sequences to generate the weights vectors employed by MOEA/D. We analyze and compare the proposed methodology using different low-discrepancy sequences and its impact in the search process of MOEA/D. The proposed approach is evaluated in problems having many objective functions (up to 15 objectives). We show the flexibility and ease of use of this type of sequences when adopting them to generate the weights of MOEA/D.

MOEA/D is an aggregation-based evolutionary algorithm which has been proved extremely efficient and effective for solving multi-objective optimization problems. It is based on the idea of decomposing the original multi-objective problem into several single-objective subproblems by means of well-defined scalarizing functions. Those single-objective subproblems are solved in a cooperative manner by defining a neighborhood relation between them. This makes MOEA/D particularly interesting when attempting to plug and to leverage single-objective optimizers in a multi-objective setting. In this context, we investigate the benefits that MOEA/D can achieve when coupled with CMA-ES, which is believed to be a powerful single-objective optimizer. We rely on the ability of CMA-ES to deal with injected solutions in order to update different covariance matrices with respect to each subproblem defined in MOEA/D. We show that by cooperatively evolving neighboring CMA-ES components, we are able to obtain competitive results for different multi-objective benchmark functions.

In spite of the popularity of the Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D), its use in Constrained Multi-objective Optimization Problems (CMOPs) has not been fully explored. In the last few years, there have been a few proposals to extend MOEA/D to the solution of CMOPs. However, most of these proposals have adopted selection mechanisms based on penalty functions. In this paper, we present a novel selection mechanism based on the well-known ε-constraint method. The proposed approach uses information related to the neighborhood adopted in MOEA/D in order to obtain solutions which minimize the objective functions within the allowed feasible region. Our preliminary results indicate that our approach is highly competitive with respect to a state-of-the-art MOEA which solves in an efficient way the constrained test problems adopted in our comparative study.

In this paper, we present a Multi-objective Particle Swarm Optimizer (MOPSO) based on a decomposition approach, which is proposed to solve Constrained Multi-Objective Aerodynamic Shape Optimization Problems (CMO-ASOPs). The constraint-handling technique adopted in this approach is based on the well-known epsilon-constraint method. Since the ε-constraint method was initially proposed to deal with constrained single-objective optimization Problems, we adapted it so that it could be incorporated into a MOPSO. Our main focus is to solve CMO-ASOPs in an efficient and effective manner. The proposed constrained MOPSO guides the search by updating the position of each particle using a set of solutions considered as the global best according to both the decomposition approach and the epsilon-constraint method. Our preliminary results indicate that our proposed approach is able to outperform a state-of-the-art MOEA in several CMO-ASOPs.

The development of multi-objective evolutionary algorithms (MOEAs) assisted by surrogate models has significantly increased in the last few years. However, in realworld applications, the high modality and dimensionality that functions normally have, often causes problems to such models. Therefore, if the Pareto optimal set of a multi-objective optimization problem is located in a search space in which the surrogate model is not able to shape the corresponding region, the search could be misinformed and thus converge to wrong regions. This has motivated the idea of incorporating refinement mechanisms to such approaches, in order to improve the search. In this paper, we present a local search mechanism which improves the search of a MOEA assisted by surrogate models. Our preliminary results indicate that our proposed approach can produce good quality results when it is restricted to performing only between 1,000 and 5,000 fitness function evaluations. Our proposed approach is validated using a set of standard test problems and an airfoil design problem.

The development of multi-objective evolutionary algorithms assisted by surrogate models has increased in the last few years. However, in real-world applications, the high modality and dimensionality that functions have, often causes problems to such models. In fact, if the Pareto optimal set of a multi-objective optimization problem is located in a search space in which the surrogate model is not able to shape the corresponding region, the search could be misinformed and thus converge to wrong regions. Because of this, a considerable amount of research has focused on improving the prediction of the surrogate models by adding the new solutions to the training set and retraining the model. However, when the size of the training set increases, the training complexity can significantly increase. In this paper, we present a surrogate model which maintains the size of the training set, and in which the prediction of the function is improved by using radial basis function networks in a cooperative way. Preliminary results indicate that our proposed approach can produce good quality results when it is restricted to performing only 200, 1,000 and 5,000 fitness function evaluations. Our proposed approach is validated using a set of standard test problems and an airfoil design problem.

In recent years, the development of multi-objective evolutionary algorithms (MOEAs) hybridized with mathematical programming techniques has significantly increased. However, most of these hybrid approaches are gradient-based, and tend to require a high number of extra objective function evaluations to estimate the gradient information required. The use of direct search methods—i.e., methods that do not require gradient information—has been, however, less popular in the specialized literature (although such approaches have been used with single-objective evolutionary algorithms). This paper precisely focuses on the design of a hybrid between the wellknownMOEA/ D and Nelder and Mead's algorithm. Clearly, the mathematical programming technique adopted here, acts as a local search mechanism, whose goal is to improve the search performed by MOEA/D. Because of its nature, the proposed local search mechanism can be easily coupled to any other decomposition-based MOEA. Our preliminary results indicate that this sort of hybridization is quite promising for dealing with multi-objective optimization problems (MOPs) having high dimensionality (in decision variable space).

The linear antenna array design problem is one of the most important in electromagnetism. While designing a linear antenna array, the goal of the designer is to achieve the “minimum average side lobe level” and a “null control” in specific directions. In contrast to the existing methods that attempt to minimize a weighted sum of these two objectives considered here, in this paper our contribution is twofold. First, we have considered these as two distinct objectives which are optimized simultaneously in a multi-objective framework. Second, for directivity purposes, we have introduced another objective called the “maximum side lobe level” in the design formulation. The resulting multi-objective optimization problem is solved by using the recently-proposed decomposition-based Multi-Objective Particle Swarm Optimizer (dMOPSO). Our experimental results indicate that the proposed approach is able to obtain results which are better than those obtained by two other state-of-the-art Multi-Objective Evolutionary Algorithms (MOEAs). Additionally, the individual minima reached by dMOPSO outperform those achieved by two single-objective evolutionary algorithms.

This paper proposes an algorithm for dealing with nonlinear and unconstrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple approximations of the Pareto optimal set. The search is directed by a well-distributed set of weighted vectors. Each weighted vector defines a scalarization problem which is solved by deforming a simplex according to the movements described by Nelder and Mead's method. The simplex is constructed with a set of solutions which minimize different scalarization problems defined by a set of neighbor weighted vectors. The solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a well distributed set of solutions along the Pareto front. The main aim of this work is to show that a well-designed strategy using just mathematical programming techniques can be competitive with respect to a state-of-the-art multi-objective evolutionary algorithm.

The simplicity and success of particle swarm optimization (PSO) algorithms, has motivated researchers to extend the use of these techniques to the multi-objective optimization field. This paper presents a multi-objective particle swarm optimization (MOPSO) algorithm based on a decomposition approach, which is intended for solving continuous and unconstrained multi-objective optimization problems (MOPs). The proposed decomposition-based multi-objective particle swarm optimizer (dMOPSO), updates the position of each particle using a set of solutions considered as the global best according to the decomposition approach. dMOPSO is mainly characterized by the use of a memory reinitialization process which aims to provide diversity to the swarm. Our proposed approach is compared with respect to two decomposition-based multi-objective evolutionary algorithms (MOEAs) which are representative of the state-of-the-art in the area. Our results indicate that our proposed approach is competitive and it outperforms the two MOEAs with respect to which it was compared in most of the test problems adopted.

Diversity plays an important role in evolutionary multi-objective optimization. Because of this, a number of density estimators (i.e., mechanisms that help to maintain diversity) have been proposed since the early days of multi-objective evolutionary algorithms (MOEAs). Fitness sharing and niching were among the most popular density estimator used with non-elitist MOEAs, but their main drawback was their high dependence on the niche radius, which was normally difficult to set. In recent years, the use of external archives to store the nondominated solutions found by an elitist MOEA has become popular. This has motivated an important amount of research related to archiving techniques for MOEAs. In this paper, we contribute to such literature by introducing a new archiving strategy based on the Convex Hull of Individual Minima (CHIM). Our proposed approach is compared with respect to two competitive MOEAs (NSGA-II and SPEA2) using standard test problems and performance measures taken from the specialized literature.

In this paper, we present an approach in which a local search mechanism is coupled to a multi-objective evolutionary algorithm. The local search mechanism is assisted by a meta-model based on support vector machines. Such a mechanism consists of two phases: the first one involves the use of an aggregating function which is defined by different weighted vectors. For the (scalar) optimization task involved, we adopt a non-gradient mathematical programming technique: the Hooke-Jeeves method. The second phase computes new solutions departing from those obtained in the first phase. The local search engine generates a set of solutions which are used in the evolutionary process of our algorithm. The preliminary results indicate that our proposed approach is quite promising.

In recent years, the development of multi-objective evolutionary algorithms (MOEAs) hybridized with mathematical programming techniques has significantly increased. However, most of these hybrid approaches are gradient-based, and tend to require a high number of extra objective function evaluations to estimate the gradient information required. The use of nonlinear optimization approaches taken from the mathematical programming literature has been, however, less popular (although such approaches have been used with single-objective evolutionary algorithms). This paper precisely focuses on the design of a hybrid between a well-known MOEA (the NSGA-II) and two direct search methods taken from the mathematical programming literature (Nelder and Mead.s method and the golden section algorithm). The idea is to combine the explorative power of the evolutionary algorithm with the exploitative power of the direct search methods previously indicated (one is used for unidimensional functions and the other for multidimensional functions). Clearly, these mathematical programming techniques act as local search engines, whose goal is to refine the search performed by the MOEA. Our preliminary results indicate that this sort of hybridization is quite promising.