Guo Yu, Yaochu Jin, Markus Olhofer, "Benchmark Problems and Performance Indicators for Search of Knee Points in Multi-objective Optimization", IEEE Transactions on Cybernetics, pp. 3531-3544, 2020.

During the preference-based optimization, the decision makers (DMs) are hard to understand the problem without priori knowledge and give their preference information. Also, solutions have many features, and the searching space is very large and usually not homogenious. Depending on the features, the solutions are more less important, while important might be problem dependent. This can be eg. knee points, robust areas, etc. Therefore, the preference-based algorithms can be used to concentrate on the important points, and DM could point the important areas on the basis of the priori knowledge the algorithms provide. Therefore, benchmark functions are defined with dedicated features to evaluate the ability of algorithms to identify these areas. There are two aspects considered in this paper. On the one hand, the problems are embedded with many features (linkage functions, bias relation, knee functions, shape functions) to test the comprehensive ability of the algorithms to cope with the difficulties. On the other hand, the problems can be simplified (by removing the linkage and bias relationship) to test single ability like detecting the knee regions. Notably, the problems (denoted as PMOP) are scalable in decision and objective space with different preference features. On the basis of PMOP, its extension embedding robustness into the knee functions is proposed to test another ability searching the robust knee points. After that, a comprehensive criterion is proposed to evaluate algorithms identifying the preferred regions, and the way how to obtain the PoF and interesting points is also illustrated. All in all, this paper gives a new way for optimization, and appeals that it is practical to learn the existing preference or interesting features the problems have.