A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) STUDIES BASED ON USER PREFERENCES <s> Evolutionary algorithms that rely on dominance ranking often suffer from a low selection pressure problem when dealing with many-objective problems. Decomposition and user-preference based methods can help to alleviate this problem to a great extent. In this paper, a user-preference based evolutionary multi-objective algorithm is proposed that uses decomposition methods for solving many-objective problems. Decomposition techniques that are widely used in multi-objective evolutionary optimization require a set of evenly distributed weight vectors to generate a diverse set of solutions on the Pareto-optimal front. The newly proposed algorithm, R-MEAD2, improves the scalability of its previous version, R-MEAD, which uses a simplexlattice design method for generating weight vectors. This makes the population size is dependent on the dimension size of the objective space. R-MEAD2 uses a uniform random number generator to remove the coupling between dimension and the population size. This paper shows that a uniform random number generator is simple and able to generate evenly distributed points in a high dimensional space. Our comparative study shows that R-MEAD2 outperforms the dominance-based method R-NSGA-II on many-objective problems. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) STUDIES BASED ON USER PREFERENCES <s> In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) STUDIES BASED ON USER PREFERENCES <s> Most existing studies on evolutionary multi-objective optimization focus on approximating the whole Pareto-optimal front. Nevertheless, rather than the whole front, which demands for too many points (especially in a high-dimensional space), the decision maker might only interest in a partial region, called the region of interest. In this case, solutions outside this region can be noisy to the decision making procedure. Even worse, there is no guarantee that we can find the preferred solutions when tackling problems with complicated properties or a large number of objectives. In this paper, we develop a systematic way to incorporate the decision maker's preference information into the decomposition-based evolutionary multi-objective optimization methods. Generally speaking, our basic idea is a non-uniform mapping scheme by which the originally uniformly distributed reference points on a canonical simplex can be mapped to the new positions close to the aspiration level vector specified by the decision maker. By these means, we are able to steer the search process towards the region of interest either directly or in an interactive manner and also handle a large number of objectives. In the meanwhile, the boundary solutions can be approximated given the decision maker's requirements. Furthermore, the extent of the region of the interest is intuitively understandable and controllable in a closed form. Extensive experiments, both proof-of-principle and on a variety of problems with 3 to 10 objectives, fully demonstrate the effectiveness of our proposed method for approximating the preferred solutions in the region of interest. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) STUDIES BASED ON USER PREFERENCES <s> User preference is of great importance when dealing with many objective optimization. Using the preference information to obtain preferred parts of the Pareto set has become prevalent in the research domain of Evolutionary Multiobjective Optimization (EMO). In this paper, a target region provided by the decision maker (DM), defined by the preferred range of every objective, is utilized to articulate the preference in-formation. This information is integrated with two well-known multiobjective evolutionary algorithms based on decomposition: MOEA/D and NSGA-III. The newly proposed preference-based algorithms, called T-MOEA/D and T-NSGA-III, can be used both a-priori and interactively. Experiments have demonstrated the benefit of applying them interactively. The DM can easily and quickly adjust the preferences according to the current results, and the proposed algorithms can successfully find non-dominated solutions complying with the preferences. Comparative experiments show that the proposed algorithms outperform the dominance-based algorithm T-NSGA-II on many-objective benchmark problems. <s> BIB004 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 3) STUDIES BASED ON USER PREFERENCES <s> In many-objective optimization, maintaining a good balance between convergence and diversity have turned out to be a considerable challenge for classical evolutionary algorithms. The large scale solution set required to describe the entire Pareto optimal front hinder the decision makers from finding the most satisfactory solution, whereas he/she is only interested in a limited part of the objective space. The dilemma can be handled by incorporating the preference information in the search process. In this paper, an evolutionary algorithm based on region preference is proposed for many-objective optimization. The preference model is constructed in combination of the target region and reference points. To focus the search on the preference region while maintaining well convergence and diversity within the region, a tri-level ranking criterion is introduced into the proposed algorithm, and different rank works at different phase of the search process. A fuzzy theory-based interactive approach is proposed to guide more individuals to further search into the objective space with higher preference degree and help the decision maker choose the most interested solution. The proposed algorithm has been extensively compared with other state-of-the-art preference-based algorithms on DTLZ1~ DTLZ4 test problems having 3–10 objectives. The experimental results indicate that the proposed algorithm can achieve competitive and better performance. Moreover, we extend the algorithm to handle multiple target regions. <s> BIB005
In the high-dimensional objective space, it is impossible to find all Pareto optimal vectors that approximate the real PF. Using preference information on reference points and reference vectors, the search range can be limited to the ROI rather than the whole objective space or more solution vectors can be searched in the ROI, which effectively overcomes the fifth-and-sixth challenges as stated in the beginning of subsection D. User preferences can be incorporated a priori, posteriorly, or interactively into the search process. EAs belong to a posterior algorithm, i.e., DMs are not allowed to express their preferences until a PS has already been obtained, which is not conducive for DMs to making a comprehensive decision. Based on their previous work, the R-MEAD (a reference point based multiobjective evolutionary algorithm with decomposition), Mohammadi et al. BIB001 developed a novel algorithm termed R-MEAD2. This algorithm utilizes weight vectors of decomposition methods to help handle user preferences by guiding solutions toward the preference region, potentially saving quite a huge amount of computational resources. Therefore, the algorithm is less susceptible to the selection pressure and converges to the PF more rapidly. For MOPs with highly nonlinear and complex PFs, a set of uniform weight vectors does not necessarily map to a set of uniform solution vectors in the objective space. Thus, the authors designed a feedback mechanism to adaptively update weight vectors based on the solutions distribution. Besides, a uniform random number generator was employed to decouple the population size from the number of objectives, making the population design easier. Cheng et al. BIB002 defined the preference region by specifying a central vector and a radius, and designed a scalarization approach that is on the basis of the angle penalized distance and has a better capability in handling large-scale problems compared to the PBI approach. The resulting algorithm, called reference vector guided EA (RVEA), not only transforms the original MaOP into a set of single-objective subproblems, but also makes user preferences target the preference region of the PF. The distribution of reference vectors is also adjusted adaptively according to the range of objective function values. The experimental study demonstrated that the RVEA can make a solution set be distributed more uniformly when the PF has a regular geometric structure. However, it does not work well when the PF is irregular. Therefore, the authors proposed a reference vector regeneration strategy, which uses an additional reference vector set to explore the objective space further. Li et al. BIB003 presented a nonuniform mapping scheme (NUMS), in which the original reference points distributed uniformly on the canonical simplex can be mapped to a new position close to the aspiration-level vector that is used to model the DM's preference information. Given the DM's requirements, the NUMS is able to obtain both a set of biased reference points toward the ROI and the ones located on the boundary, which enables MOEA/D not only to find preferred solutions but also to provide DMs with global information about the PF. The above studies use reference points or reference vectors to represent user preferences. Li et al. BIB004 defined the target region that consists of the preferred range of each objective and used it to express preference information from DM. Then a preference-based algorithm of MOEA/D with a target region (T-MOEA/D) was proposed, which aims at well spreading nondominated solutions within the target region and can be used a priori or interactively. Besides, to determine whether a solution is replaced or not, T-MOEA/D compares Pareto dominance and fitness values in turn, while the traditional MOEA/D only compares fitness values of solutions. Comprehensive experiments on a series of benchmark problems with 2 to 15 objectives fully demonstrated the effectiveness of the proposed algorithm, and more experimental problems should further be performed on different object regions. Xiong et al. BIB005 constructed a preference model that combines the target region and reference point together, and introduced an interactive approach using the fuzzy theory to adjust the preference information during the search process for obtaining more reasonable solutions. This interactive method is beneficial to allocating more computational resources to the ROI of users. A tri-level ranking criterion was then designed to focus the search process on the preference region along with balancing convergence and diversity of a PS. Compared with the T-MOEA/D in BIB004 , the proposed algorithm worked well in most of the test cases and showed superior performance in the convergence speed and computational time.