Proximal Point Methods for Functions Involving Lojasiewicz, Quasiconvex and Convex Properties on Hadamard Manifolds

This paper extends the full convergence of the proximal point method with Riemannian, Semi-Bregman and Bregman distances to solve minimization problems on Hadamard manifolds. For the unconstrained problem, under the assumptions that the optimal set is nonempty and the objective function is continuous and either quasiconvex or satisfies a generalized Lojasiewicz property, we prove the full convergence of the sequence generated by the proximal point method with Riemannian distances to certain generalized critical point of the problem. For the constrained case, under the same assumption on the optimal set and the quasiconvexity or convexity of the objective function, we study two methods. One of them is the proximal method with semi-Bregman distance, obtaining that any cluster point of the sequence is an optimal solution. The other one, is the proximal method with Bregman distance where we obtain the global convergence of the method to an optimal solution of the problem. In particular, our work recovers some interesting optimization problems, for example, convex and quasiconvex minimization problems in IR, semidefinite problems (SDP), second order cone problems (SOCP), in the same way that, extends the applications of the proximal point methods for solving constrained minimization problems with nonconvex objective functions in Euclidian spaces when the objective function is convex or quasiconvex on the manifold.