An inexact proximal point method with quasi-distance for quasiconvex multiobjective optimization problems on Riemannian manifolds

Abstract

In this paper, we investigate a class of unconstrained multiobjective optimization problems in the framework of Riemannian manifolds (abbreviated as, MOP-RM), where the components of the objective function are assumed to be locally Lipschitz and quasiconvex. By employing the powerful tool of Mordukhovich limiting subdifferential, we introduce an inexact proximal point algorithm with quasi-distance (abbreviated as, IPPMQ-RM), to solve MOP-RM. Moreover, we establish the well-definedness of the sequence generated by the IPPMQ-RM algorithm. Based on two different versions of error criteria, we introduce two variants of IPPMQ-RM, namely, IPPMQ-RM1 and IPPMQ-RM2. We deduce that the cluster points of the sequences generated by the IPPMQ-RM1 and IPPMQ-RM2 algorithms are Pareto-Mordukhovich critical points of MOP-RM. Further, we derive that if the components of the objective function of MOP-RM are geodesic convex, then these cluster points become Pareto efficient solutions of MOP-RM. We establish the finite termination of the IPPMQ-RM1 and IPPMQ-RM2 algorithms. By employing MATLAB R2023b, a non-trivial numerical example has been furnished to illustrate the effectiveness of the proposed algorithms, namely, IPPMQ-RM1 and IPPMQ-RM2. Moreover, to demonstrate that the sequence generated by the IPPMQ-RM algorithm converges faster than the algorithms existing in the literature, we furnish several non-trivial numerical examples in the framework of well-known Riemannian manifolds.