Benchmark Problems and Performance Indicators for Search of Knee Points in Multiobjective Optimization

In multi-objective optimization, it is non-trivial for decision makers to articulate preferences without a priori knowledge, which is particular true when the number of objectives becomes large. Depending on the shape of the Pareto front, optimal solutions such as knee points may be of interest. Although several multi- and many-objective optimization test suites have been proposed, little work has been reported focusing on designing multi-objective problems whose Pareto front contains complex knee regions. Likewise, few performance indicators dedicated to evaluating an algorithm’s ability of accurately locating all knee points in high-dimensional objective space have been suggested. This paper proposes a set of multi-objective optimization test problems whose Pareto front consists of complex knee regions, aiming to assess the capability of evolutionary algorithms to accurately identify all knee points. Various features related to knee points have been taken into account in designing the test problems, including symmetry, differentiability, degeneration. These features are also combined with other challenges in solving optimization problems, such as multimodality, linkage between decision variables, non-uniformity and scalability of the Pareto front. The proposed test problems are scalable to both decision and objective spaces. Accordingly, new performance indicators are suggested for evaluating the capability of optimization algorithms in locating the knee points. The proposed test problems together with the performance indicators offer a new means to develop and assess preference-based evolutionary algorithms for solving multi- and many-objective optimization problems.