Generalized Conditional Gradient Methods for Multiobjective Composite Optimization Problems with Hölder Condition

In this paper, we deal with multiobjective composite optimization problems, where each objective function is a combination of smooth and possibly non-smooth functions. We first propose a parameter-dependent generalized conditional gradient method to solve this problem. The step size in this method requires prior knowledge of the parameters related to the Hölder continuity of the gradient of the smooth function. The convergence properties of this method are then established. Given that these parameters may be unknown or, if known, may not be unique, the first method may encounter implementation challenges or slow convergence. To address this, we further propose a parameter-free version of the first method that determines the step size using a local quadratic upper approximation and an adaptive line search strategy, eliminating the need for any problem-specific parameters. The performance of the proposed methods is demonstrated on several test problems involving the indicator function and an uncertainty function.