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Multi-objective optimization problems require simultaneously optimizing twoor more objective functions. Many studies have reported that the solution setof an M-objective optimization problem often forms an (M-1)-dimensionaltopological simplex (a curved line for M=2, a curved triangle for M=3, a curvedtetrahedron for M=4, etc.). Since the dimensionality of the solution setincreases as the number of objectives grows, an exponentially large sample sizeis needed to cover the solution set. To reduce the required sample size, thispaper proposes a Bezier simplex model and its fitting algorithm. Thesetechniques can exploit the simplex structure of the solution set and decomposea high-dimensional surface fitting task into a sequence of low-dimensionalones. An approximation theorem of Bezier simplices is proven. Numericalexperiments with synthetic and real-world optimization problems demonstratethat the proposed method achieves an accurate approximation of high-dimensionalsolution sets with small samples. In practice, such an approximation will beconducted in the post-optimization process and enable a better trade-offanalysis.

Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization