ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Full convergence of the proximal point method for quasiconvex functions on Hadamard manifolds

Erik A. Papa Quiroza1 and P. Roberto Oliveiraa2

a1 Universidad Nacional del Callao, Universidad Nacional Mayor de San Marcos, Lima, Peru. [email protected]

a2 PESC-COPPE Federal University of Rio de Janeiro, Rio de Janeiro, Brazil; [email protected]


In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on Hadamard manifolds. To reach this goal, we initially extend the concepts of regular and generalized subgradient from Euclidean spaces to Hadamard manifolds and prove that, in the convex case, these concepts coincide with the classical one. For the minimization problem, assuming that the function is bounded from below, in the quasiconvex and lower semicontinuous case, we prove the convergence of the iterations given by the method. Furthermore, under the assumptions that the sequence of proximal parameters is bounded and the function is continuous, we obtain the convergence to a generalized critical point. In particular, our work extends the applications of the proximal point methods for solving constrained minimization problems with nonconvex objective functions in Euclidean spaces when the objective function is convex or quasiconvex on the manifold.

(Received May 7 2009)

(Revised January 29 2010)

(Online publication June 22 2011)

Key Words:

  • Proximal point method;
  • quasiconvex function;
  • Hadamard manifolds;
  • full convergence.

Mathematics Subject Classification:

  • 90C26