  Multi-objective optimization problems require simultaneously optimizing two
or more objective functions. Many studies have reported that the solution set
of an M-objective optimization problem often forms an (M-1)-dimensional
topological simplex (a curved line for M=2, a curved triangle for M=3, a curved
tetrahedron for M=4, etc.). Since the dimensionality of the solution set
increases as the number of objectives grows, an exponentially large sample size
is needed to cover the solution set. To reduce the required sample size, this
paper proposes a Bezier simplex model and its fitting algorithm. These
techniques can exploit the simplex structure of the solution set and decompose
a high-dimensional surface fitting task into a sequence of low-dimensional
ones. An approximation theorem of Bezier simplices is proven. Numerical
experiments with synthetic and real-world optimization problems demonstrate
that the proposed method achieves an accurate approximation of high-dimensional
solution sets with small samples. In practice, such an approximation will be
conducted in the post-optimization process and enable a better trade-off
analysis.
