Unconstrained steepest descent method for multicriteria optimization on Riemmanian manifolds
This work provides a theoretical foundation for multicriteria optimization on Riemannian manifolds, which is relevant for applications in geometry and machine learning, but the results are incremental as they extend existing Euclidean methods.
The paper extends the steepest descent method with Armijo's rule to multicriteria optimization on Riemannian manifolds, proving convergence to Pareto critical points under mild assumptions, including quasi-convexity and non-negative curvature.
In this paper we present a steepest descent method with Armijo's rule for multicriteria optimization in the Riemannian context. The well definedness of the sequence generated by the method is guaranteed. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if they exist) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasi-convexity of the multicriteria function and non-negative curvature of the Riemannian manifold, we prove full convergence of the sequence to a Pareto critical.