Finding evenly spaced fronts for multiobjective control via averaging Hausdorff-measure

Recently, it was proposed to use the averaged Hausdorff distance Δ_p as a performance indicator for evolutionary multiobjective algorithms (EMOAs). To be more precise, it was proposed to measure the respective distance of the image of the outcome set O of an EMOA to the image of the solution set of a given multiobjective optimization problem (MOP), the so-called Pareto front F. The main advantage of this approach is that the Δ_p value gives a clear idea of the approximation quality of O to F in terms of both convergence and spread. Furthermore, Δ_p is not as sensitive to outliers as the classical Hausdorff distance d_H which is a potential drawback of d_H when using stochastic search algorithms such as evolutionary algorithms. Here we make a first attempt to design an EMOA which aims for low Δ_p values. The challenge is that the Pareto front is of course a priori not known. Instead, we compute approximations of F by linearizing the nondominated front of the current population which we utilize for the selection mechanism. This leads to an EMOA where the images of O are evenly spread along F which is in particular advantageous for multiobjective control problems. We test the novel algorithm on some benchmark problems including comparisons to NSG AII.