A descent method for nonsmooth multiobjective optimization problems on Riemannian manifolds
Chunming Tang, Hao He, Jinbao Jian, Miantao Chao
TL;DR
The paper addresses nonsmooth, multiobjective optimization on general Riemannian manifolds with locally Lipschitz objectives. It introduces an implementable descent method that constructs a common descent direction from a convex hull of Riemannian $\varepsilon$-subgradients and uses a Riemannian Armijo line search to generate iterates, avoiding explicit geodesic computations. The authors establish finite convergence to an $(\varepsilon,\delta)$-critical Pareto point under mild assumptions and provide preliminary numerical results on manifolds like spheres and Stiefel, demonstrating practical efficiency. This work broadens Riemannian multiobjective optimization beyond convexity and offers a scalable framework for nonsmooth problems on general manifolds, with prospects for stronger results by dynamically tuning $(\varepsilon,\delta)$ toward Pareto-critical points.
Abstract
In this paper, a 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 existing methods. A necessary condition for Pareto optimality in Euclidean space is generalized to the Riemannian setting. At every iteration, an acceptable descent direction is obtained by constructing a convex hull of some Riemannian $\varepsilon$-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.