TY - JOUR
AU - Kobayashi, Ken
AU - Hamada, Naoki
AU - Sannai, Akiyoshi
AU - Tanaka, Akinori
AU - Bannai, Kenichi
AU - Sugiyama, Masashi
PY - 2019/07/17
Y2 - 2024/07/15
TI - Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization
JF - Proceedings of the AAAI Conference on Artificial Intelligence
JA - AAAI
VL - 33
IS - 01
SE - AAAI Technical Track: Heuristic Search and Optimization
DO - 10.1609/aaai.v33i01.33012304
UR - https://ojs.aaai.org/index.php/AAAI/article/view/4069
SP - 2304-2313
AB - <p>Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an <em>M</em>-objective optimization problem often forms an (<em>M</em> − 1)-dimensional topological simplex (a curved line for <em>M</em> = 2, a curved triangle for <em>M</em> = 3, a curved tetrahedron for <em>M</em> = 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 Bézier 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 Bézier 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 postoptimization process and enable a better trade-off analysis.</p>
ER -