Integrating User Preferences and Decomposition methods for Many-objective Optimization Asad Mohammadi, Mohammad Nabi Omidvar, Xiaodong Li, and Kalyanmoy Deb COIN Report Number 2014004 Abstract— 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 prob- lems. 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 simplex-lattice 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. I. I NTRODUCTION Evolutionary multi-objective optimization (EMO) algo- rithms have been successfully applied to many real-world problems in the past decade [1]. It has been shown that evolu- tionary multi-objective approaches can find evenly distributed and well-converged solutions on two or three objective problems [2]. However, when dealing with many-objective optimization problems which involve four or more objectives, the performance of EMO algorithms degrade rapidly [3], [4]. Hence, there is a need for developing EMO methods that can efficiently solve many-objective problems. A number of challenges exist in solving many-objective optimization problems [3], [5]. Firstly, in a high dimensional objective space, even in the initial random population most of the solutions are non-dominated to each other. Thus, there would not be an adequate selection pressure making the search process very slow or even completely stagnant when a dominance-based algorithm is used. Secondly, generating solutions to approximate the entire Pareto front becomes computationally expensive [3], [4], [6]. Thirdly, it is difficult to visualize the Pareto-optimal front in large dimensions, which makes it difficult for the decision makers to select Asad Mohammadi, Mohammad Nabi Omidvar and Xiaodong Li are with the school of computer science and IT, RMIT University, Melbourne, Australia (email:{asad.mohammadi, mohammad.omidvar, xi- aodong.li}@rmit.edu.au) Kalyanmoy Deb is with Computational Opti- mization and Innovation Laboratory (COIN), Michigan State University, Michigan, USA (email: kdeb@egr.msu.edu) their preferred solutions. Adopting a user-preference based approach can mostly alleviate the second problem. In a user- preference based approach the aim is to find a set of solutions on a smaller region of the Pareto-optimal front which is preferred by the user. This technique requires less compu- tational resources by performing a more focused and guided search rather than approximating the entire Pareto-optimal front. This is of practical value when dealing with many- objective problems. One popular type of user-preference based EMO algorithm is the a priori method, which allows a decision maker to provide the preference information (e.g. a reference point) at the beginning of the search. Reference point based [7], [8], reference direction based [9], and light beam based [10] EMO approaches are a few attempts in this area. Another approach for better handling many-objective op- timization problems is using decomposition-based methods. They convert a multi-objective problem into a set of single- objective problems. This feature makes them less sensitive to the selection pressure issue which is the prime source of the degrading performance of dominance-based approaches in dealing with many-objective problems. In other words, in dominance-based algorithms individuals with many ob- jectives become non-dominated to each other, hence the selection pressure diminishes rapidly, making the population difficult move towards the Pareto-optimal front [11]. Various techniques for decomposing a multi-objective problem have been developed, including boundary intersection [12], [13], Tchebycheff and weighted-sum [14]. Some popular evolu- tionary EMO algorithms that employ such decomposition methods are MOGLS [15], and MOEA/D [16]. As explained before, utilizing user preference informa- tion and a decomposition-based approach can improve the efficiency of an EMO algorithm in solving many-objective problems. Also motivated by the fact that in most real- world situations users are not usually interested in find- ing the entire Pareto-optimal front and the availability of some form of preference information makes preference-based EMO approaches more applicable in real-world settings. This makes a decomposition-based algorithm that takes into account the preference information a promising approach for tackling many-objective problems. R-MEAD [17] is the first attempt in incorporating the user-preference information into a decomposition-based EMO algorithm. R-MEAD uses weighted-sum and Tchebycheff as decomposition methods and a priori approach to search for preferred regions. How- ever, it has only been applied to two and three-objective