Interactive methods support decision-makers in finding the most preferred solution for multiobjective optimization problems, where multiple conflicting objective functions must be optimized simultaneously. These methods let a decision-maker provide preference information iteratively during the solution process to find solutions of interest, allowing them to learn about the trade-offs in the problem and the feasibility of the preferences. Several interactive evolutionary multiobjective optimization methods have been proposed in the literature. In the evolutionary computation community, the so-called decomposition-basedmethods have been increasingly popular because of their good performance in problems with many objective functions. They decompose the multiobjective optimization problem into multiple sub-problems to be solved collaboratively. Various interactive versions of decomposition-based methods have been proposed. However, most of them do not consider the desirable properties of real interactive solution processes, such as avoiding imposing a high cognitive burden on the decision-maker, allowing them to decide when to interact with the method, and supporting them in selecting a final solution. This paper reviews interactive evolutionary decomposition-based multiobjective optimization methods and different methodologies utilized to incorporate interactivity in them. Additionally, desirable properties of interactive decomposition-based multiobjective evolutionary optimization methods are identified, aiming to make them easier to be applied in real-world problems.

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.

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.