Searching for knee regions of the Pareto front using mobile reference points

Abstract

Evolutionary Algorithms (EAs) have been recognized to be well suited to approximate the Pareto front of Multi-objective Optimization Problems (MOPs). In reality, the Decision Maker (DM) is not interested in discovering the whole Pareto front rather than finding only the portion(s) of the front that matches at most his/her preferences. Recently, several studies have addressed the decision-making task to assist the DM in choosing the final alternative. Knee regions are potential parts of the Pareto front presenting the maximal trade-offs between objectives. Solutions residing in knee regions are characterized by the fact that a small improvement in either objective will cause a large deterioration in at least another one which makes moving in either direction not attractive. Thus, in the absence of explicit DM's preferences, we suppose that knee regions represent the DM's preferences themselves. Recently, few works were proposed to find knee regions. This paper represents a further study in this direction. Hence, we propose a new evolutionary method, denoted TKR-NSGA-II, to discover knee regions of the Pareto front. In this method, the population is guided gradually by means of a set of mobile reference points. Since the reference points are updated based on trade-off information, the population converges towards knee region centers which allows the construction of a neighborhood of solutions in each knee. The performance assessment of the proposed algorithm is done on two-and three-objective knee-based test problems. The obtained results show the ability of the algorithm to: (1) find the Pareto optimal knee regions, (2) control the extent (We mean by extent the breadth/spread of the obtained knee region.) of the obtained regions independently of the geometry of the front and (3) provide competitive and better results when compared to other recently proposed methods. Moreover, we propose an interactive version of TKR-NSGA-II which is useful when the DM has no a priori information about the number of existing knees in the Pareto optimal front.

Objective reduction based on nonlinear correlation information entropy

“…(3) It will be interesting to evaluate our techniques on hypervolume-based IBEAs. (4) It will be useful to apply our approach to search knee areas of MaOPs (Bechikh et al 2011).…”

Adaptive space transformation: An invariant based method for predicting aerodynamic coefficients of hypersonic vehicles

“…In this paper, the complexity is used as the second objective function, and the trade-off between the overlap degree of transformed curve and the complexity of kernel function is analyzed by multiobjective optimization method. Only the knee of Pareto front (Bechikh et al, 2011) is selected as the best scaling parameter.…”

Robust design in geotechnical engineering – an update

“…This approach does not require a Pareto front to begin with, and it is applicable to optimisation with more than two objectives, although Figure 8 is illustrated with only two objectives. Although many outstanding studies on knee points have been published (e.g., Bechikh et al 2011;Deb et al 2006), the MD approach is easy to use and very efficient computationally. When adopted within the RGD framework, it offers a practical geotechnical design tool.…”