Preference-based Evolutionary Multi-objective Optimization Research Articles

Investigating normalization in preference-based evolutionary multi-objective optimization using a reference point

Normalization of objectives plays a crucial role in evolutionary multi-objective optimization (EMO) to handle objective functions with different scales, which can be found in real-world problems. Although the effect of normalization methods on the performance of EMO algorithms has been investigated in the literature, that of preference-based EMO (PBEMO) algorithms is poorly understood. Since PBEMO aims to approximate a region of interest, its population generally does not cover the Pareto front in the objective space. This property may make normalization of objectives in PBEMO difficult. This paper investigates the effectiveness of three normalization methods in three representative PBEMO algorithms. We present a bounded archive-based method for approximating the nadir point. First, we demonstrate that the normalization methods in PBEMO perform significantly worse than that in conventional EMO in terms of approximating the ideal point, nadir point, and range of the PF. Then, we show that PBEMO requires normalization of objectives on problems with differently scaled objectives. Our results show that there is no clear “best normalization method” in PBEMO, but an external archive-based method performs relatively well.

Quality Indicators for Preference-Based Evolutionary Multi-Objective Optimization Using a Reference Point: A Review and Analysis

Quality Indicators for Preference-Based Evolutionary Multi-Objective Optimization Using a Reference Point: A Review and Analysis

Interactive Evolutionary Multiobjective Optimization via Learning to Rank

In practical multi-criterion decision-making, it is cumbersome if a decision maker (DM) is asked to choose among a set of trade-off alternatives covering the whole Pareto-optimal front. This is a paradox in conventional evolutionary multi-objective optimization (EMO) that always aim to achieve a well balance between convergence and diversity. In essence, the ultimate goal of multi-objective optimization is to help a decision maker (DM) identify solution(s) of interest (SOI) achieving satisfactory trade-offs among multiple conflicting criteria. Bearing this in mind, this paper develops a framework for designing preference-based EMO algorithms to find SOI in an interactive manner. Its core idea is to involve human in the loop of EMO. After every several iterations, the DM is invited to elicit her feedback with regard to a couple of incumbent candidates. By collecting such information, her preference is progressively learned by a learning-to-rank neural network and then applied to guide the baseline EMO algorithm. Note that this framework is so general that any existing EMO algorithm can be applied in a plug-in manner. Experiments on 48 benchmark test problems with up to 10 objectives and a real-world multi-objective robot control problem fully demonstrate the effectiveness of our proposed algorithms for finding SOI.

A Reference Point-Based Evolutionary Algorithm Solves Multi and Many-Objective Optimization Problems: Method and Validation.

The integration of a decision maker's preferences in evolutionary multi-objective optimization (EMO) has been a common research scope over the last decade. In the published literature, several preference-based evolutionary approaches have been proposed. The reference point-based non-dominated sorting genetic (R-NSGA-II) algorithm represents one of the well-known preference-based evolutionary approaches. This method mainly aims to find a set of the Pareto-optimal solutions in the region of interest (ROI) rather than obtaining the entire Pareto-optimal set. This approach uses Euclidean distance as a metric to calculate the distance between each candidate solution and the reference point. However, this metric may not produce desired solutions because the final minimal Euclidean distance value is unknown. Thus, determining whether the true Pareto-optimal solution is achieved at the end of optimization run becomes difficult. In this study, R-NSGA-II method is modified using the recently proposed simplified Karush-Kuhn-Tucker proximity measure (S-KKTPM) metric instead of the Euclidean distance metric, where S-KKTPM-based distance measure can predict the convergence behavior of a point from the Pareto-optimal front without prior knowledge of the optimum solution. Experimental results show that the algorithm proposed herein is highly competitive compared with several state-of-the-art preference-based EMO methods. Extensive experiments were conducted with 2 to 10 objectives on various standard problems. Results show the effectiveness of our algorithm in obtaining the preferred solutions in the ROI and its ability to control the size of each preferred region separately at the same time.

A novel dynamic reference point model for preference-based evolutionary multiobjective optimization

AbstractIn the field of preference-based evolutionary multiobjective optimization, optimization algorithms are required to search for the Pareto optimal solutions preferred by the decision-maker (DM). The reference point is a type of techniques that effectively describe the preferences of DM. So far, the reference point is either static or interactive with the evolutionary process. However, the existing reference point techniques do not cover all application scenarios. A novel case, i.e., the reference point changes over time due to the environment change, has not been considered. This paper focuses on the multiobjective optimization problems with dynamic preferences of the DM. First, we propose a change model of the reference point to simulate the change of the preference by the DM over time. Then, a dynamic preference-based multiobjective evolutionary algorithm framework with a clonal selection algorithm (ĝa-NSCSA) and a genetic algorithm (ĝa-NSGA-II) is designed to solve such kind of optimization problems. In addition, in terms of practical applications, the experiments on the portfolio optimization problems with the dynamic reference point model are tested. Experimental results on the benchmark problems and the practical applications show that ĝa-NSCSA exhibits better performance among the compared optimization algorithms.

Finding top-K solutions for the decision-maker in multiobjective optimization

Multiobjective optimization problems (MOPs) are the optimization problem with multiple conflicting objectives. Generally, an optimization algorithm can find a large number of optimal solutions for MOPs, which easily overwhelm decision makers (DMs) and make it difficult for decision-making. Preference-based evolutionary multiobjective optimization (EMO) aims to find the partial optima in the regions preferred by the DM. Although it narrows the scope of the optimal solutions, it usually still returns a population of optimal solutions (typically 100 or larger in EMO) with a small distance between adjacent optima. Top-K, which is a well-established research subject in many fields to find the best K solutions, may be a direction to reduce the number of optimal solutions. In this paper, first, we introduce the top-K notion into preference-based EMO and propose the top-K model to obtain the best K individuals of multiobjective optimization problems (MOPs). Then, with the top-K model, we propose NSGA-II-TopK and SPEA2-TopK to search for the top-K preferred solutions for preference-based continuous and combinatorial MOPs, respectively. Finally, the proposed algorithms with several representative preference-based EMO algorithms are compared in different preference situations for MOPs. Experimental results show the proposed algorithms have strong performances against the compared algorithms.

A novel dynamic reference point model for preference-based evolutionary multiobjective optimization

In the field of preference-based evolutionary multiobjective optimization, optimization algorithms are required to search for the Pareto optimal solutions preferred by the decision maker (DM). The reference point is a type of techniques that effectively describe the preferences of DM. So far, the reference point is either static or interactive with the evolutionary process. However, the existing reference point techniques do not cover all application scenarios. A novel case, i.e., the reference point changes over time due to the environment change, has not been considered. This paper focuses on the multiobjective optimization problems with dynamic preferences of the DM. First, we propose a change model of the reference point to simulate the change of the preference by the DM over time. Then, a dynamic preference-based multiobjective evolutionary algorithm framework with a clonal selection algorithm (ĝa-NSCSA) and a genetic algorithm (ĝa-NSGA-II) is designed to solve such kind of optimization problems. In addition, in terms of practical applications, the experiments on the portfolio optimization problems with the dynamic reference point model are tested. Experimental results on the benchmark problems and the practical applications show that ĝa-NSCSA exhibits better performance among the compared optimization algorithms.

W-dominance: Tradeoff-inspired dominance relation for preference-based evolutionary multi-objective optimization

Abstract The paper presents a method of incorporating decision maker preferences into multi-objective meta-heuristics. It is based on tradeoff coefficients and extends their applicability from bi-objective to multi-objective. The method assumes that a decision maker specifies a priori each objective's importance as a weight interval. Based on this, w-dominance relation is introduced, which extends Pareto dominance. By replacing reference points with weight intervals the method eliminates the need for any knowledge concerning expected solutions. Instead, decision maker reflects his context-independent policy regarding objectives. The proposed w-dominance was incorporated into selected multi-objective metaheuristics. Following this, three new metrics were designed. The metrics include prescreening true Pareto Front and final population according to w-dominance relation. Based on preliminary tests, Vector Angle Evolutionary Algorithm (VaEA) was selected as the best match for w-dominance. W-dominance-extended VAEA (wVAEA) was compared in a series of simulations with four state-of-the-art reference point-based multi-objective algorithms. The results show that wVaEA outperforms the four representative algorithms for selected benchmark problems.

Preference-Based Evolutionary Multiobjective Optimization Through the Use of Reservation and Aspiration Points

Preference-based Evolutionary Multiobjective Optimization (EMO) algorithms approximate the region of interest (ROI) of the Pareto optimal front defined by the preferences of a decision maker (DM). Here, we propose a preference-based EMO algorithm, in which the preferences are given by means of aspiration and reservation points. The aspiration point is formed by objective values which the DM wants to achieve, while the reservation point is constituted by values for the objectives not to be worsened. Internally, the first generations are performed in order to generate an initial approximation set according to the reservation point. Next, in the remaining generations, the algorithm adapts the search for new non-dominated solutions depending on the dominance relation between the solutions obtained so far and both the reservation and aspiration points. This allows knowing if the given points are achievable or not; this type of information cannot be known before the solution process starts. On this basis, the algorithm proceeds according to three different scenarios with the aim of re-orienting the search directions towards the ROI formed by the Pareto optimal solutions with objective values within the given aspiration and reservation values. Computational results show the potential of our proposal in 2, 3 and 5-objective test problems, in comparison to other state-of-the-art algorithms.

Does Preference Always Help? A Holistic Study on Preference-Based Evolutionary Multiobjective Optimization Using Reference Points

The ultimate goal of multiobjective optimization is to help a decision maker (DM) identify solution(s) of interest (SOI) achieving satisfactory tradeoffs among multiple conflicting criteria. This can be realized by leveraging DM's preference information in evolutionary multiobjective optimization (EMO). No consensus has been reached on the effectiveness brought by incorporating preference in EMO (either a priori or interactively) versus a posteriori decision making after a complete run of an EMO algorithm. Bearing this consideration in mind, this article: 1) provides a pragmatic overview of the existing developments of preference-based EMO (PBEMO) and 2) conducts a series of experiments to investigate the effectiveness brought by preference incorporation in EMO for approximating various SOI. In particular, the DM's preference information is elicited as a reference point, which represents her/his aspirations for different objectives. The experimental results demonstrate that preference incorporation in EMO does not always lead to a desirable approximation of SOI if the DM's preference information is not well utilized, nor does the DM elicit invalid preference information, which is not uncommon when encountering a black-box system. To a certain extent, this issue can be remedied through an interactive preference elicitation. Last but not the least, we find that a PBEMO algorithm is able to be generalized to approximate the whole PF given an appropriate setup of preference information.

Ra-dominance: A new dominance relationship for preference-based evolutionary multiobjective optimization

While traditional Pareto-based evolutionary multi-objective optimization (EMO) algorithms have shown an excellent balance between convergence and diversity on a wide range of practical problems with two or three objectives in real applications, the decision maker (DM) is interested in a unique set of solutions rather than the whole population on Pareto optimal front (POF). In addition, Pareto-based EMO algorithms have some shortcomings in dealing with many-objective problems because of insufficient selection pressure toward trade-off solutions. Due to the above, it is crucial to incorporate DM preference information into EMO and seek a representative subset of Pareto optimal solutions with an increase in the number of objectives. This paper proposes a new dominance relationship, called Ra-dominance, which can improve diversity among the Pareto-equivalent solutions increase the selection pressure in evolutionary process. It has the ability to guide the population toward areas more responsive to the needs of the DM according to a reference point and preference angle. We use the new dominance relationship in the NSGA-II algorithm, and the efficacy and usefulness of the modified procedure are assessed through two- to ten-objective problems. Experimental results show that the algorithm applying this new dominance relationship is highly competitive when compared with four state-of-the-art preference-based EMO methods.

Preference-based evolutionary multi-objective optimization in ship weather routing

In evolutionary multi-objective optimization (EMO) the aim is to find a set of Pareto-optimal solutions. Such approach may be applied to multiple real-life problems, including weather routing (WR) of ships. The route should be optimal in terms of passage time, fuel consumption and safety of crew and cargo while taking into account dynamically changing weather conditions. Additionally it must not violate any navigational constraints (neither static nor dynamic). Since the resulting non-dominated solutions might be numerous, some user support must be provided to enable the decision maker (DM) selecting a single “best” solution. Commonly, multi-criteria decision making methods (MCDM) are utilized to achieve this goal with DM’s preferences defined a posteriori. Another approach is to apply DM’s preferences into the very process of finding Pareto-optimal solutions, which is referred to as preference-based EMO. Here the Pareto-set is limited to those solutions, which are compliant with the pre-configured user preferences. The paper presents a new tradeoff-based EMO approach utilizing configurable weight intervals assigned to all objectives. The proposed method has been applied to ship WR problem and compared with a popular reference point method: r-dominance. Presented results prove applicability and competitiveness of the proposed method to solving multi-objective WR problem.

Preference-based evolutionary multi-objective optimization for portfolio selection: a new credibilistic model under investor preferences

We propose a new credibility portfolio selection model, in which a measure of loss aversion is introduced as an objective function, joint to the expected value of the returns and the below-mean absolute semi-deviation as a risk measure. The uncertainty of the future returns is directly approximated using the historical returns on the portfolios, so the uncertain return on a given portfolio is modeled as an LR-power fuzzy variable. Quantifying the uncertainty by means of a credibility distribution allows us to measure the investors’ loss aversion as the credibility of achieving a non-positive return, which is better perceived by investors than other measures of risk. Furthermore, we analyze the relationships between the three objective functions, showing that the risk measure and the loss aversion function are practically uncorrelated. Thus, the information provided by these criteria do not overlap each other. In order to generate several non-dominated portfolios taking into account the investor’s preferences and that the problem is non-linear and non-convex, we apply up to three preference-based EMO algorithms. These algorithms allow to approximate a part of the Pareto optimal front called region of interest. We analyze three investor profiles taking into account their loss-adverse attitudes: conservative, cautious and aggressive. A computational study is performed with data of the Spanish stock market, showing the important role played by the loss aversion function to generate a diversified set of non-dominated portfolios fitting the expectations of each investor.

R-Metric: Evaluating the Performance of Preference-Based Evolutionary Multiobjective Optimization Using Reference Points

Measuring the performance of an algorithm for solving multiobjective optimization problem has always been challenging simply due to two conflicting goals, i.e., convergence and diversity of obtained tradeoff solutions. There are a number of metrics for evaluating the performance of a multiobjective optimizer that approximates the whole Pareto-optimal front. However, for evaluating the quality of a preferred subset of the whole front, the existing metrics are inadequate. In this paper, we suggest a systematic way to adapt the existing metrics to quantitatively evaluate the performance of a preference-based evolutionary multiobjective optimization algorithm using reference points. The basic idea is to preprocess the preferred solution set according to a multicriterion decision making approach before using a regular metric for performance assessment. Extensive experiments on several artificial scenarios, and benchmark problems fully demonstrate its effectiveness in evaluating the quality of different preferred solution sets with regard to various reference points supplied by a decision maker.

A new angle-based preference selection mechanism for solving many-objective optimization problems

Many evolutionary multi-objective optimization (EMO) methodologies have been proposed and performed very well on finding a representative set of Pareto-optimal solutions, but this advantage will be weakened with the increasing number of objectives. And in real applications, what decision makers (DMs) want is a unique solution or a set of solutions rather than the overall Pareto-optimal front. It is a difficult task to solve many-objective problems by using preference information provided by decision maker (DM) during optimization process. In this paper, a new angle-based preference selection mechanism is proposed, which replaces the traditional crowding distance with the aid of preference information provided by the DMs. Particularly, we combine it with a multi-objective immune algorithm with non-dominated neighbor-based selection. The proposed method has been extensively compared with other recently proposed preference-based EMO approaches over DTLZ1, DTLZ2, and DTLZ3 test problems with 4–100 objectives. The results of the experiment indicate that the proposed algorithm can achieve competitive and better results.

Preference-based Evolutionary Multi-objective Optimization for Solving Fuzzy Portfolio Selection Problems

espanolEn este trabajo, abordamos el problema de seleccion de carteras de inversion desde una perspectiva multiobjetivo y consideramos algoritmos evolutivos de optimizacion multiobjetivo basados en preferencias para generar carteras eficientes teniendo en cuenta preferencias del inversor. Por un lado, proponemos un modelo de optimizacion para la seleccion de carteras con tres objetivos, en el que se han considerado numeros fuzzy de tipo LR para modelizar la incertidumbre de los futuros beneficios. Las funciones objetivo a optimizar son el beneficio esperado (a maximizar) y dos medidas del riesgo de la inversion (ambas a minimizar): la semi-desviacion media absoluta por debajo del beneficio esperado y el valor de riesgo (medida de la peor perdida esperada en un horizonte dado). Ademas de la restriccion presupuestaria, se ha introducido una restriccion que limita la cardinalidad de las carteras y cotas superiores e inferiores para la inversion en cada activo. El problema de optimization multiobjetivo resultante, denominado por sus siglas en ingles como modelo MASdVaR (mean-absolute semi-deviation value-at-risk model), es no lineal y no convexo y, por ello, se ha aplicado el algoritmo evolutivo basado en preferencias WASF-GA para generar carteras de inversion eficientes. En WASF-GA, se consideran valores de aspiracion que el decisor desea alcanzar en cada objetivo para expresar las preferencias. Para el modelo MASdVaR, los valores de aspiracion considerados corresponden a un perfil de inversor conservador. Los resultados obtenidos para un caso basado en el mercado de inversion espanol demuestran que las carteras eficientes obtenidas que alcanzan los mayores beneficios, mejoran los valores de referencia considerados para las dos medidas del riesgo. Por otro lado, aquellas carteras con beneficios mas moderados presentan valores de riesgo menores, pero sin dejar de satisfacer las aspiraciones del inversor, por lo que representan estrategias de inversion menos arriesgadas. EnglishIn this work, we address the multi-criteria paradigm of the portfolio selection problem and consider a preference-based evolutionary multi-objective optimization algorithm to find Pareto optimal portfolio solutions based on the investor preferences. Firstly, we propose a three-objective optimization model for portfolio selection, in which the uncertainty of the portfolio returns is modelled by means of an LR-power fuzzy variable. In the model, three criteria are considered, which are the credibility expected value of the returns (to be maximized) and two measures of the risk (both to be minimized): the belowmean absolute semi-deviation and the fuzzy value-at-risk. Besides the budget constraint, a cardinality constraint and lower and upper bound constraints for the assets are also considered. The resulting model, called a credibility mean-absolute semi-deviation value-at-risk (MASdVaR) model, is a non-linear and nonconvex multi-objective optimization problem which is solved by means of the preference-based evolutionary algorithm WASF-GA. In WASF-GA, the preferences are expressed by means of aspiration values that the decision maker would like to achieve for the objectives. In the MASdVaR model, the investor aspiration values considered for the objectives are calculated assuming a conservative profile. The results obtained for a case study based on the Spanish stock market show that the portfolios generated with the highest expected returns improve the aspiration values considered for the two risk measures. Besides, portfolios with intermediate values for the expected return achieve lower values for the risk measures, but they still improve their aspiration values and thus represent less risky investment options for the investor.

An r-dominance-based preference multi-objective optimization for many-objective optimization

Evolutionary multi-objective optimization (EMO) algorithms have been used in finding a representative set of Pareto-optimal solutions in the past decade and beyond. However, most of Pareto domination-based multi-objective optimization evolutionary algorithms (MOEAs) are not suitable for many-objective optimization, in which, a good trade-off among many objectives becomes very difficult. In real-world applications, the fact is that the decision-maker is not interested in the overall Pareto-optimal front since the final decision is a unique or several solutions. So the decision-maker can incorporate his/her preferences into the search process of MOEAs to guide the search toward the preferred parts of the Pareto region rather than the whole Pareto-optimal region. In this paper, we hybridize the classical Pareto dominance principle with reference-based dominance and propose a reference-dominance-based preference multi-objective optimization algorithm (r-PMOA). The proposed method has been extensively compared with other recently proposed preference-based EMO approaches over several benchmark problems of multi-objective optimization having 2–10 objectives. The results of the experiment indicate that r-PMOA achieves competitive results.

A combined interactive procedure using preference-based evolutionary multiobjective optimization. Application to the efficiency improvement of the auxiliary services of power plants

While the auxiliary services required for the operation of power plants are not the main components of the plant, their energy consumption is often significant, and it can be reduced by implementing a series of improvement strategies. However, the cost of implementing these changes can be very high, and has to be evaluated. Indeed, a further economic analysis should be considered in order to maximize the profitability of the investment. In this paper, we propose a multiobjective optimization problem to determine the most suitable strategies to maximize the energy saving, to minimize the economic investment and to maximize the Internal Rate of Return of the investment. Solving this real-life multiobjective optimization problem with a decision maker presents several challenges and difficulties and we have developed a novel interactive procedure which combines three different approaches in order to make use of the main advantages of each method. The idea is to start with the approximation of the Pareto optimal set, in order to gain a global understanding of the trade-offs among the objectives, using evolutionary multiobjective optimization; next step is aiding the decision maker to explore the efficient set and to identify the subset of solutions which fits her/his preferences, for which interactive multiple criteria decision making methodologies are used; and finally we concentrate the search for new solutions into the most interesting part of the efficient set with the help of a preference-based evolutionary algorithm. This allows us to build a flexible scheme that is progressively adapted to the decision maker’s reactions until (s)he finds the most preferred solution. The interactive combined procedure proposed is applied in practice for solving the problem of the auxiliary services with a real decision maker, extracting interesting insights about the efficiency improvement of the auxiliary services. With this practical application, we show the usefulness of the interactive procedure proposed, and we highlight the importance of an understandable feedback and an adaptive process.

Preference representation using Gaussian functions on a hyperplane in evolutionary multi-objective optimization

Many-objective optimization has attracted much attention in evolutionary multi-objective optimization (EMO). This is because EMO algorithms developed so far often degrade their search ability for optimization problems with four or more objectives, which are frequently referred to as many-objective problems. One of promising approaches to handle many objectives is to incorporate the preference of a decision maker (DM) into EMO algorithms. With the preference, EMO algorithms can focus the search on regions preferred by the DM, resulting in solutions close to the Pareto front around the preferred regions. Although a number of preference-based EMO algorithms have been proposed, it is not trivial for the DM to reflect his/her actual preference in the search. We previously proposed to represent the preference of the DM using Gaussian functions on a hyperplane. The DM specifies the center and spread vectors of the Gaussian functions so as to represent his/her preference. The preference handling is integrated into the framework of NSGA-II. This paper extends our previous work so that obtained solutions follow the distribution of Gaussian functions specified. The performance of our proposed method is demonstrated mainly for benchmark problems and real-world applications with a few objectives in this paper. We also show the applicability of our method to many-objective problems.

A preference-based evolutionary algorithm for multiobjective optimization: the weighting achievement scalarizing function genetic algorithm

When solving multiobjective optimization problems, preference-based evolutionary multiobjective optimization (EMO) algorithms introduce preference information into an evolutionary algorithm in order to focus the search for objective vectors towards the region of interest of the Pareto optimal front. In this paper, we suggest a preference-based EMO algorithm called weighting achievement scalarizing function genetic algorithm (WASF-GA), which considers the preferences of the decision maker (DM) expressed by means of a reference point. The main purpose of WASF-GA is to approximate the region of interest of the Pareto optimal front determined by the reference point, which contains the Pareto optimal objective vectors that obey the preferences expressed by the DM in the best possible way. The proposed approach is based on the use of an achievement scalarizing function (ASF) and on the classification of the individuals into several fronts. At each generation of WASF-GA, this classification is done according to the values that each solution takes on the ASF for the reference point and using different weight vectors. These vectors of weights are selected so that the vectors formed by their inverse components constitute a well-distributed representation of the weight vectors space. The efficiency and usefulness of WASF-GA is shown in several test problems in comparison to other preference-based EMO algorithms. Regarding a metric based on the hypervolume, we can say that WASF-GA has outperformed the other algorithms considered in most of the problems.