An implementable descent method for nonsmooth multiobjective optimization on Riemannian manifolds
Chunming Tang, Hao He, Jinbao Jian, Miantao Chao · Optimization methods & software · 2024
In this paper, an implementable descent method for nonsmooth multiobjective optimization problems on complete Riemannian manifolds is proposed. The objective functions are only assumed to be locally Lipschitz continuous instead of convexity used in the existing subgradient method for Riemannian multiobjective optimization. And the constraint manifold is a general manifold rather than some specific manifolds used in the proximal point method. The retraction mapping is introduced to avoid the use of computationally difficult geodesic. A Riemannian version of the necessary condition for Pareto optimality is proposed, which generalized the classical one in Euclidean space to the manifold setting. At every iteration, an acceptable descent direction is obtained by constructing a convex hull of some Riemannian ε-subgradients. And then a Riemannian Armijo-type line search is executed to produce the next iterate. The convergence result is established in the sense that a point satisfying the necessary condition for Pareto optimality can be generated by the algorithm in a finite number of iterations. Finally, some preliminary numerical results are reported, which show that the proposed method is efficient.