MPFA: A Pareto Front Approximation Method for Riemannian Bi-objective Optimization

We propose a Pareto front approximation (MPFA) method for smooth bi-objective optimization problems on Riemannian manifolds based on a Hermite interpolation technique. Compared with the existing multiobjective optimization numerical algorithms, the proposed method can generate a continuous approximate Pareto front without multiple initial points. We establish convergence of the proposed method and analyze the approximation error of the resulting Pareto front. Numerical experiments on several test problems demonstrate that the proposed approach can effectively approximate the Pareto front with high accuracy and reasonable computational cost. Furthermore, the method is applied to a bi-objective formulation of sparse principal component analysis, illustrating its practical applicability in data analysis problems.