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

## TL;DR

This paper introduces a Bezier simplex model for efficiently approximating Pareto fronts in multi-objective optimization, reducing sample size requirements by exploiting the geometric structure of solution sets.

## Contribution

The paper proposes a novel Bezier simplex fitting method that decomposes high-dimensional surface fitting into low-dimensional tasks, with proven approximation capabilities.

## Key findings

- Accurately approximates high-dimensional Pareto fronts with small samples
- Demonstrates effectiveness on synthetic and real-world problems
- Enables better trade-off analysis in post-optimization

## Abstract

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.