Faster convergence of conditional gradient method for multiobjective optimization under different step sizes

This paper establishes enhanced convergence rates for the conditional gradient method in multiobjective optimization under assumptions weaker than uniform convexity for the objective functions. Structural conditions are imposed on the constraint set, including uniformly convex sets and polytopes. We analyze both adaptive step sizes and diminishing step sizes of the form $ \frac{l}{k+l} $ with $ l \geq 1 $, improving upon the previously used $ \frac{4}{k+4} $. The results show faster convergence rates surpassing the standard $ \mathcal{O}(\frac{1}{k}) $. Numerical experiments verify the theoretical results.