Interactive evolutionary multiobjective methods enable a decision maker to solve optimization problems involving multiple conflicting objective functions by iteratively incorporating preference information. When applying interactive methods, two phases can often be identified: a learning phase, where a decision maker gains insights on trade-offs and identifies a region of interest based on their preferences, and a decision phase focused on fine-tuning and selecting the most preferred solution. The configuration of evolutionary operators, like selection, crossover, and mutation, heavily influences the performance of evolutionary methods. However, despite extensive research on parameter tuning, identifying optimal configurations for these operators within interactive methods while accounting for the specific goals of each phase has not been studied. This study introduces a framework for the automatic configuration of interactive methods, taking the first step toward addressing this research gap. The framework systematically identifies phase-specific optimal configurations by combining the PHI indicator with the irace automatic configuration tool. Experiments with interactive RVEA and interactive RNSGA-II on problems involving three, five, and seven objective functions reveal notable differences in optimal configurations between the learning and decision phases. These findings lay a foundation for enhancing the performance of interactive evolutionary multiobjective methods and highlight the importance of phase-specific configurations.

Various interactive evolutionary multiobjective optimization methods have been proposed in the literature for problems with multiple, conflicting objective functions. In these methods, a decision maker, who is a domain expert, iteratively provides preference information to guide the solution process while gaining insight into the problem. To compare interactive evolutionary multiobjective optimization methods, a preference-based hypervolume indicator (PHI) has been proposed to quantify the performance of the methods. PHI was the first indicator designed based on some desirable properties of indicators for interactive evolutionary multiobjective optimization methods. However, it has some shortcomings, such as excluding some potentially interesting solutions and being limited to consider a reference point as a type of preference information. In this paper, a modified indicator called PHI+ is proposed to address the mentioned drawbacks. PHI+ modifies the region of interest in PHI. While PHI is directed at methods where a decision maker provides preference information in the form of a reference point, PHI+ is applicable for methods that utilize desirable ranges of objective function values as preference information. Therefore, PHI+ is the first indicator that can handle preference information provided as desirable ranges when evaluating interactive methods. Experimental results show that PHI+ can also better distinguish differences in the performance of interactive evolutionary multiobjective optimization methods.

Solving real-world optimization problems in engineering and design involves various practical challenges. They include simultaneously optimizing multiple conflicting objective functions that may involve computationally expensive simulations. Failed simulations introduce another practical challenge, as it is not always possible to set constraints a priori to avoid failed simulations. Failed simulations are typically ignored during optimization, which leads to wasting computation resources. When the optimization problem has multiple objective functions, failed simulations can also be misleading for the decision maker while choosing the most preferred solution. Utilizing data collected from previous simulations and enabling the optimization algorithm to avoid failed simulations can reduce the computational requirements. We consider data-driven multiobjective optimization of the diffusor of an axial pump and propose an approach to reduce the number of solutions that fail in expensive computational fluid dynamics simulations. The proposed approach utilizes Kriging surrogate models to approximate the objective functions and is inexpensive to evaluate. We utilize a probabilistic selection approach with constraints in a multiobjective evolutionary algorithm to find solutions with better objective function values, lower uncertainty, and lower probability of failing. Finally, a domain expert chooses the most preferred solution using one’s preferences. Numerical tests show significant improvement in the ratio of feasible solutions to all the available solutions without special treatment of failed simulations. The solutions also have a higher quality (hypervolume) and accuracy than the other tested approaches. The proposed approach provides an efficient way of reducing the number of failed simulations and utilizing offline data in multiobjective design optimization.

Abstract We consider multiobjective optimization problems with at least one computationally expensive constraint function and propose a novel surrogate-assisted evolutionary algorithm that can incorporate preference information given a priori. We employ Kriging models to approximate expensive objective and constraint functions, enabling us to introduce a new selection strategy that emphasizes the generation of feasible solutions throughout the optimization process. In our innovative model management, we perform expensive function evaluations to identify feasible solutions that best reflect the decision maker’s preferences provided before the process. To assess the performance of our proposed algorithm, we utilize two distinct parameterless performance indicators and compare them against existing algorithms from the literature using various real-world engineering and benchmark problems. Furthermore, we assemble new algorithms to analyze the effects of the selection strategy and the model management on the performance of the proposed algorithm. The results show that in most cases, our algorithm has a better performance than the assembled algorithms, especially when there is a restricted budget for expensive function evaluations.

In recent years, interactive evolutionary multiobjective optimization methods have been getting more and more attention. In these methods, a decision maker, who is a domain expert, is iteratively involved in the solution process and guides the solution process toward her/his desired region with preference information. However, there have not been many studies regarding the performance evaluation of interactive evolutionary methods. On the other hand, indicators have been developed for a priori methods, where the DM provides preference information before optimization. In the literature, some studies treat interactive evolutionary methods as a series of a priori steps when assessing and comparing them. In such settings, indicators designed for a priori methods can be utilized. In this paper, we propose a novel performance indicator for interactive evolutionary multiobjective optimization methods and show how it can assess the performance of these interactive methods as a whole process and not as a series of separate steps. In addition, we demonstrate the shortcomings of using indicators designed for a priori methods for comparing interactive evolutionary methods.

Solving real-life data-driven multiobjective optimization problems involves many complicated challenges. These challenges include preprocessing the data, modelling the objective functions, getting a meaningful formulation of the problem, and supporting decision makers to find preferred solutions in the existence of conflicting objective functions. In this paper, we tackle the problem of optimizing the composition of microalloyed steels to get good mechanical properties such as yield strength, percentage elongation, and Charpy energy. We formulate a problem with six objective functions based on data available and support two decision makers in finding a solution that satisfies them both. To enable two decision makers to make meaningful decisions for a problem with many objectives, we create the so-called MultiDM/IOPIS algorithm, which combines multiobjective evolutionary algorithms and scalarization functions from interactive multiobjective optimization methods in novel ways. We use the software framework called DESDEO, an open-source Python framework for interactively solving multiobjective optimization problems, to create the MultiDM/IOPIS algorithm. We provide a detailed account of all the challenges faced while formulating and solving the problem. We discuss and use many strategies to overcome those challenges. Overall, we propose a methodology to solve real-life data-driven problems with multiple objective functions and decision makers. With this methodology, we successfully obtained microalloyed steel compositions with mechanical properties that satisfied both decision makers.

For offline data-driven multiobjective optimization problems (MOPs), no new data is available during the optimization process. Approximation models (or surrogates) are first built using the provided offline data, and an optimizer, for example, a multiobjective evolutionary algorithm, can then be utilized to find Pareto optimal solutions to the problem with surrogates as objective functions. In contrast to online data-driven MOPs, these surrogates cannot be updated with new data and, hence, the approximation accuracy cannot be improved by considering new data during the optimization process. Gaussian process regression (GPR) models are widely used as surrogates because of their ability to provide uncertainty information. However, building GPRs becomes computationally expensive when the size of the dataset is large. Using sparse GPRs reduces the computational cost of building the surrogates. However, sparse GPRs are not tailored to solve offline data-driven MOPs, where good accuracy of the surrogates is needed near Pareto optimal solutions. Treed GPR (TGPR-MO) surrogates for offline data-driven MOPs with continuous decision variables are proposed in this paper. The proposed surrogates first split the decision space into subregions using regression trees and build GPRs sequentially in regions close to Pareto optimal solutions in the decision space to accurately approximate tradeoffs between the objective functions. TGPR-MO surrogates are computationally inexpensive because GPRs are built only in a smaller region of the decision space utilizing a subset of the data. The TGPR-MO surrogates were tested on distance-based visualizable problems with various data sizes, sampling strategies, numbers of objective functions, and decision variables. Experimental results showed that the TGPR-MO surrogates are computationally cheaper and can handle datasets of large size. Furthermore, TGPR-MO surrogates produced solutions closer to Pareto optimal solutions compared to full GPRs and sparse GPRs.

Abstract Interactive multiobjective optimization methods have proven promising in solving optimization problems with conflicting objectives since they iteratively incorporate preference information of a decision maker in the search for the most preferred solution. To find the appropriate interactive method for various needs involves analysis of the strengths and weaknesses. However, extensive analysis with human decision makers may be too costly and for that reason, we propose an artificial decision maker to compare a class of popular interactive multiobjective optimization methods, i.e., reference point based methods. Without involving any human decision makers, the artificial decision maker works automatically to interact with different methods to be compared and evaluate the final results. It makes a difference between a learning phase and a decision phase, that is, learns about the problem based on information acquired to identify a region of interest and refines solutions in that region to find a final solution, respectively. We adopt different types of utility functions to evaluation solutions, present corresponding performance indicators and propose two examples of artificial decision makers. A series of experiments on benchmark test problems and a water resources planning problem is conducted to demonstrate how the proposed artificial decision makers can be used to compare reference point based methods.

Abstract In interactive multiobjective optimization methods, the preferences of a decision maker are incorporated in a solution process to find solutions of interest for problems with multiple conflicting objectives. Since multiple solutions exist for these problems with various trade-offs, preferences are crucial to identify the best solution(s). However, it is not necessarily clear to the decision maker how the preferences lead to particular solutions and, by introducing explanations to interactive multiobjective optimization methods, we promote a novel paradigm of explainable interactive multiobjective optimization. As a proof of concept, we introduce a new method, R-XIMO, which provides explanations to a decision maker for reference point based interactive methods. We utilize concepts of explainable artificial intelligence and SHAP (Shapley Additive exPlanations) values. R-XIMO allows the decision maker to learn about the trade-offs in the underlying problem and promotes confidence in the solutions found. In particular, R-XIMO supports the decision maker in expressing new preferences that help them improve a desired objective by suggesting another objective to be impaired. This kind of support has been lacking. We validate R-XIMO numerically, with an illustrative example, and with a case study demonstrating how R-XIMO can support a real decision maker. Our results show that R-XIMO successfully generates sound explanations. Thus, incorporating explainability in interactive methods appears to be a very promising and exciting new research area.

Abstract We introduce novel concepts to solve multiobjective optimization problems involving (computationally) expensive function evaluations and propose a new interactive method called O-NAUTILUS. It combines ideas of trade-off free search and navigation (where a decision maker sees changes in objective function values in real time) and extends the NAUTILUS Navigator method to surrogate-assisted optimization. Importantly, it utilizes uncertainty quantification from surrogate models like Kriging or properties like Lipschitz continuity to approximate a so-called optimistic Pareto optimal set. This enables the decision maker to search in unexplored parts of the Pareto optimal set and requires a small amount of expensive function evaluations. We share the implementation of O-NAUTILUS as open source code. Thanks to its graphical user interface, a decision maker can see in real time how the preferences provided affect the direction of the search. We demonstrate the potential and benefits of O-NAUTILUS with a problem related to the design of vehicles.

Visualizing the search behavior of a series of points or populations in their native domain is critical in understanding biases and attractors in an optimization process. Distance-based many-objective optimization test problems have been developed to facilitate visualization of search behavior in a 2-D design space with arbitrarily many objective functions. Previous works have proposed a few commonly seen problem characteristics into this problem framework, such as the definition of disconnected Pareto sets and dominance resistant regions of the design space. The authors’ previous work has advanced this research further by providing a problem generator to automatically create user-defined problem instances featuring any combination of these problem features as well as newly introduced ones, such as landscape discontinuities, varying objective ranges, and neutrality. This work makes a number of additional contributions including the proposal of an enhanced, open-source feature-rich problem generator that can create user-defined problem instances exhibiting a range of problem features—some of which are newly introduced here or form extensions of existing features. A comprehensive validation of the problem generator is also provided using popular multiobjective optimization algorithms, and some problem generator settings to create instances exhibiting different challenges for an optimizer are identified.

In offline data-driven multiobjective optimization, no new data are available during the optimization process. Approximation models, also known as surrogates, are built using the provided offline data. A multiobjective evolutionary algorithm can be utilized to find solutions by using these surrogates. The accuracy of the approximated solutions depends on the surrogates and approximations typically involve uncertainties. In this article, we propose probabilistic selection approaches that utilize the uncertainty information of the Kriging models (as surrogates) to improve the solution process in offline data-driven multiobjective optimization. These approaches are designed for decomposition-based multiobjective evolutionary algorithms and can, thus, handle a large number of objectives. The proposed approaches were tested on distance-based visualizable test problems and the DTLZ suite. The proposed approaches produced solutions with a greater hypervolume, and a lower root mean squared error compared to generic approaches and a transfer learning approach that do not use uncertainty information.

We address challenges of decision problems when managers need to optimize several conflicting objectives simultaneously under uncertainty. We propose visualization tools to support the solution of such scenario-based multiobjective optimization problems. Suitable graphical visualizations are necessary to support managers in understanding, evaluating, and comparing the performances of management decisions according to all objectives in all plausible scenarios. To date, no appropriate visualization has been suggested. This paper fills this gap by proposing two visualization methods: a novel extension of empirical attainment functions for scenarios and an adapted version of heatmaps. They help a decision-maker in gaining insight into realizations of trade-offs and comparisons between objective functions in different scenarios. Some fundamental questions that a decision-maker may wish to answer with the help of visualizations are also identified. Several examples are utilized to illustrate how the proposed visualizations support a decision-maker in evaluating and comparing solutions to be able to make a robust decision by answering the questions. Finally, we validate the usefulness of the proposed visualizations in a real-world problem with a real decision-maker. We conclude with guidelines regarding which of the proposed visualizations are best suited for different problem classes.

Abstract Deterministic global optimization algorithms like Piyavskii–Shubert, direct, ego and many more, have a recognized standing, for problems with many local optima. Although many single objective optimization algorithms have been extended to multiple objectives, completely deterministic algorithms for nonlinear problems with guarantees of convergence to global Pareto optimality are still missing. For instance, deterministic algorithms usually make use of some form of scalarization, which may lead to incomplete representations of the Pareto optimal set. Thus, all global Pareto optima may not be obtained, especially in nonconvex cases. On the other hand, algorithms attempting to produce representations of the globally Pareto optimal set are usually based on heuristics. We analyze the concept of global convergence for multiobjective optimization algorithms and propose a convergence criterion based on the Hausdorff distance in the decision space. Under this light, we consider the well-known global optimization algorithm direct, analyze the available algorithms in the literature that extend direct to multiple objectives and discuss possible alternatives. In particular, we propose a novel definition for the notion of potential Pareto optimality extending the notion of potential optimality defined in direct. We also discuss its advantages and disadvantages when compared with algorithms existing in the literature.

In many decision making problems, a decision maker needs computer support in finding a good compromise between multiple conflicting objectives that need to be optimized simultaneously. Interactive multiobjective optimization methods have a lot of potential for solving such problems. However, the growth of complexity in problem formulations and the abundance of data bring new challenges to be addressed by decision makers and method developers. On the other hand, advances in the field of artificial intelligence provide opportunities in this respect. We identify challenges and propose directions of addressing them in interactive multiobjective optimization methods with the help of multiple intelligent agents. We describe a generic architecture of enhancing interactive methods with specialized agents to enable more efficient and reliable solution processes and better support for decision makers.

We formulate and solve a real-world shape design optimization problem of an air intake ventilation system in a tractor cabin by using a preference-based surrogate-assisted evolutionary multi-objective optimization algorithm. We are motivated by practical applicability and focus on two main challenges faced by practitioners in industry: 1) meaningful formulation of the optimization problem reflecting the needs of a decision maker and 2) finding a desirable solution based on a decision maker's preferences when solving a problem with computationally expensive function evaluations. For the first challenge, we describe the procedure of modelling a component in the air intake ventilation system with commercial simulation tools. The problem to be solved involves time consuming computational fluid dynamics simulations. Therefore, for the second challenge, we extend a recently proposed Kriging-assisted evolutionary algorithm K-RVEA to incorporate a decision maker's preferences. Our numerical results indicate efficiency in using the computing resources available and the solutions obtained reflect the decision maker's preferences well. Actually, two of the solutions dominate the baseline design (the design provided by the decision maker before the optimization process). The decision maker was satisfied with the results and eventually selected one as the final solution.

In optimiser analysis and design it is informative to visualise how a search point/population moves through the design space over time. Visualisable distance-based many-objective optimisation problems have been developed whose design space is in two-dimensions with arbitrarily many objective dimensions. Previous work has shown how disconnected Pareto sets may be formed, how problems can be projected to and from arbitrarily many design dimensions, and how dominance resistant regions of design space may be defined. Most recently, a test suite has been proposed using distances to lines rather than points. However, active use of visualisable problems has been limited. This may be because the type of problem characteristics available has been relatively limited compared to many practical problems (and non-visualisable problem suites). Here we introduce the mechanisms required to embed several widely seen problem characteristics in the existing problem framework. These include variable density of solutions in objective space, landscape discontinuities, varying objective ranges, neutrality, and non-identical disconnected Pareto set regions. Furthermore, we provide an automatic problem generator (as opposed to hand-tuned problem definitions). The flexibility of the problem generator is demonstrated by analysing the performance of popular optimisers on a range of sampled instances.

Many-objective optimization problems (MaOPs) contain four or more conflicting objectives to be optimized. A number of efficient decomposition-based evolutionary algorithms have been developed in the recent years to solve them. However, computationally expensive MaOPs have been scarcely investigated. Typically, surrogate-assisted methods have been used in the literature to tackle computationally expensive problems, but such studies have largely focused on problems with 1-3 objectives. In this paper, we present an approach called hybrid surrogate-assisted many-objective evolutionary algorithm to solve computationally expensive MaOPs. The key features of the approach include: 1) the use of multiple surrogates to effectively approximate a wide range of objective functions; 2) use of two sets of reference vectors for improved performance on irregular Pareto fronts (PFs); 3) effective use of archive solutions during offspring generation; and 4) a local improvement scheme for generating high quality infill solutions. Furthermore, the approach includes constraint handling which is often overlooked in contemporary algorithms. The performance of the approach is benchmarked extensively on a set of unconstrained and constrained problems with regular and irregular PFs. A statistical comparison with the existing techniques highlights the efficacy and potential of the approach.

Most evolutionary optimization algorithms assume that the evaluation of the objective and constraint functions is straightforward. In solving many real-world optimization problems, however, such objective functions may not exist. Instead, computationally expensive numerical simulations or costly physical experiments must be performed for fitness evaluations. In more extreme cases, only historical data are available for performing optimization and no new data can be generated during optimization. Solving evolutionary optimization problems driven by data collected in simulations, physical experiments, production processes, or daily life are termed data-driven evolutionary optimization. In this paper, we provide a taxonomy of different data driven evolutionary optimization problems, discuss main challenges in data-driven evolutionary optimization with respect to the nature and amount of data, and the availability of new data during optimization. Real-world application examples are given to illustrate different model management strategies for different categories of data-driven optimization problems.

We consider multiobjective optimization problems where objective functions have different (or heterogeneous) evaluation times or latencies. This is of great relevance for (computationally) expensive multiobjective optimization as there is no reason to assume that all objective functions should take an equal amount of time to be evaluated (particularly when objectives are evaluated separately). To cope with such problems, we propose a variation of the Kriging-assisted reference vector guided evolutionary algorithm (K-RVEA) called heterogeneous K-RVEA (short HK-RVEA). This algorithm is a merger of two main concepts designed to account for different latencies: A single-objective evolutionary algorithm for selecting training data to train surrogates and K-RVEA's approach for updating the surrogates. HK-RVEA is validated on a set of biobjective benchmark problems varying in terms of latencies and correlations between the objectives. The results are also compared to those obtained by previously proposed strategies for such problems, which were embedded in a non-surrogate-assisted evolutionary algorithm. Our experimental study shows that, under certain conditions, such as short latencies between the two objectives, HK-RVEA can outperform the existing strategies as well as an optimizer operating in an environment without latencies.

We propose a surrogate-assisted reference vector guided evolutionary algorithm (EA) for computationally expensive optimization problems with more than three objectives. The proposed algorithm is based on a recently developed EA for many-objective optimization that relies on a set of adaptive reference vectors for selection. The proposed surrogate-assisted EA (SAEA) uses Kriging to approximate each objective function to reduce the computational cost. In managing the Kriging models, the algorithm focuses on the balance of diversity and convergence by making use of the uncertainty information in the approximated objective values given by the Kriging models, the distribution of the reference vectors as well as the location of the individuals. In addition, we design a strategy for choosing data for training the Kriging model to limit the computation time without impairing the approximation accuracy. Empirical results on comparing the new algorithm with the state-of-the-art SAEAs on a number of benchmark problems demonstrate the competitiveness of the proposed algorithm.

Over the past two decades, evolutionary algorithms have successfully been applied to single and multiobjective optimization problems having up to three objectives. Compared to traditional mathematical programming techniques, evolutionary multiobjective algorithms (MOEAs) are particularly powerful in achieving multiple nondominated solutions in a single run. However, the performance of most existing algorithms seriously degrades when the number of objectives is larger than three. Such optimization problems, often referred to as many-objective optimization problems (MaOPs) in the evolutionary computation community, are widely seen in the real-world and therefore it is of great practical importance to efficiently solve them. Challenges to evolutionary algorithms and other meta-heuristics in solving MaOPs include the inability of dominance-based MOEAs to converge to the Pareto frontier while maintaining good diversity, the prohibitively high computational complexity for MOEAs based on performance indicators, and the difficulty for human users or decision makers to clearly understand the relationship between objectives and articulate preferences. In addition, existing performance indicators for multiobjective optimization may become incapable of accurately assessing and comparing the quality of solution sets. Finally, visualization of the solutions of MaOPs also becomes a grand challenge.

We tackle three different challenges in solving a real-world industrial problem: formulating the optimization problem, connecting different simulation tools and dealing with computationally expensive objective functions. The problem to be optimized is an air intake ventilation system of a tractor and consists of three computationally expensive objective functions. We describe the modeling of the system and its numerical evaluation with a commercial software. To obtain solutions in few function evaluations, a recently proposed surrogate-assisted evolutionary algorithm K-RVEA is applied. The diameters of four different outlets of the ventilation system are considered as decision variables. From the set of nondominated solutions generated by K-RVEA, a decision maker having substance knowledge selected the final one based on his preferences. The final selected solution has better objective function values compared to the baseline solution of the initial design. A comparison of solutions with K-RVEA and RVEA (which does not use surrogates) is also performed to show the potential of using surrogates.

Evolutionary multi-objective optimization algorithms are widely used for solving optimization problems with multiple conflicting objectives. However, basic evolutionary multi-objective optimization algorithms have shortcomings, such as slow convergence to the Pareto optimal front, no efficient termination criterion, and a lack of a theoretical convergence proof. A hybrid evolutionary multi-objective optimization algorithm involving a local search module is often used to overcome these shortcomings. But there are many issues that affect the performance of hybrid evolutionary multi-objective optimization algorithms, such as the type of scalarization function used in a local search and frequency of a local search. In this paper, we address some of these issues and propose a hybrid evolutionary multi-objective optimization framework. The proposed hybrid evolutionary multi-objective optimization framework has a modular structure, which can be used for implementing a hybrid evolutionary multi-objective optimization algorithm. A sample implementation of this framework considering NSGA-II, MOEA/D, and MOEA/D-DRA as evolutionary multi-objective optimization algorithms is presented. A gradient-based sequential quadratic programming method as a single objective optimization method for solving a scalarizing function used in a local search is implemented. Hence, only continuously differentiable functions were considered for numerical experiments. The numerical experiments demonstrate the usefulness of our proposed framework.

We propose a procedure to construct evolutionary bilevel optimization algorithms based on recent theoretical advances that have established connections between bilevel optimization and multiobjective optimization. In the proposed procedure, a new algorithm is defined by integrating an evolutionary multiobjective optimization algorithm with a partial order that is compatible with bilevel optimization. The advantages of the procedure include the ability to harness the methodology of evolutionary multiobjective optimization for bilevel optimization and to systematically develop new algorithms for single-objective and multiobjective bilevel optimization. No regularity assumptions are used, which ensures maximal applicability of the optimization algorithms constructed by the procedure. The necessary theoretical foundation is developed and the steps of the procedure are illustrated with an example.

A nadir objective vector is constructed from the worst Pareto-optimal objective values in a multiobjective optimization problem and is an important entity to compute because of its significance in estimating the range of objective values in the Pareto-optimal front and also in executing a number of interactive multiobjective optimization techniques. Along with the ideal objective vector, it is also needed for the purpose of normalizing different objectives, so as to facilitate a comparison and agglomeration of the objectives. However, the task of estimating the nadir objective vector necessitates information about the complete Pareto-optimal front and has been reported to be a difficult task, and importantly an unsolved and open research issue. In this paper, we propose certain modifications to an existing evolutionary multiobjective optimization procedure to focus its search toward the extreme objective values and combine it with a reference-point based local search approach to constitute a couple of hybrid procedures for a reliable estimation of the nadir objective vector. With up to 20-objective optimization test problems and on a three-objective engineering design optimization problem, one of the proposed procedures is found to be capable of finding the nadir objective vector reliably. The study clearly shows the significance of an evolutionary computing based search procedure in assisting to solve an age-old important task in the field of multiobjective optimization.

Evolutionary multi-objective optimization algorithms are commonly used to obtain a set of non-dominated solutions for over a decade. Recently, a lot of emphasis have been laid on hybridizing evolutionary algorithms with MCDM and mathematical programming algorithms to yield a computationally efficient and convergent procedure. In this paper, we test an augmented local search based EMO procedure rigorously on a test suite of constrained and unconstrained multi-objective optimization problems. The success of our approach on most of the test problems not only provides confidence but also stresses the importance of hybrid evolutionary algorithms in solving multi-objective optimization problems.

In this paper, we discuss the idea of incorporating preference information into evolutionary multi-objective optimization and propose a preference-based evolutionary approach that can be used as an integral part of an interactive algorithm. One algorithm is proposed in the paper. At each iteration, the decision maker is asked to give preference information in terms of his or her reference point consisting of desirable aspiration levels for objective functions. The information is used in an evolutionary algorithm to generate a new population by combining the fitness function and an achievement scalarizing function. In multi-objective optimization, achievement scalarizing functions are widely used to project a given reference point into the Pareto optimal set. In our approach, the next population is thus more concentrated in the area where more preferred alternatives are assumed to lie and the whole Pareto optimal set does not have to be generated with equal accuracy. The approach is demonstrated by numerical examples.

Nadir point plays an important role in multi-objective optimization because of its importance in estimating the range of objective values corresponding to desired Pareto-optimal solutions and also in using many classical interactive optimization techniques. Since this point corresponds to the worst Pareto-optimal solution of each objective, the task of estimating the nadir point necessitates information about the whole Pareto optimal frontier and is reported to be a difficult task using classical means. In this paper, for the first time, we have proposed a couple of modifications to an existing evolutionary multi-objective optimization procedure to focus its search towards the extreme objective values front-wise. On up to 20-objective optimization problems, both proposed procedures are found to be capable of finding a near nadir point quickly and reliably. Simulation results are interesting and should encourage further studies and applications in estimating the nadir point, a process which should lead to a better interactive procedure of finding and arriving at a desired Pareto-optimal solution.