joint work with Gladyston de Carvalho Bento, Lucas Meireles
In this talk, is introduced a definition of approximate Pareto efficient solution as well as a necessary condition for such solutions in the multiobjective setting on Riemannian manifolds. It is also propose an inexact proximal point method for nonsmooth multiobjective optimization by using the notion of approximate solution. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Furthermore, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto optimal is obtained.
joint work with Yldenilson Almeida, Paulo Oliveira, João Xavier da Cruz Neto
In this work, we consider a proximal-type method for solving minimization problems involving smooth DC (difference of convex) functions in Hadamard manifolds. The method accelerates the convergence of the classical proximal point method for DC functions and, in particular, convex functions. We prove that the point which solves the proximal subproblem can be used to define a descent direction of the objective function. Our method uses such a direction together with a line search for finding critical points of the objective function. We illustrate our results with some numerical experiments.
joint work with Orizon Pereira Ferreira, Leandro Prudente
The gradient method for minimize a differentiable convex function on Riemannian manifolds with lower bounded sectional curvature is analyzed in this talk. The analysis of the method is presented with three different finite procedures for determining the step size, namely, Lipschitz stepsize, adaptive stepsize and Armijo's stepsize. Convergence of the whole sequence to a minimizer, without any level set boundedness assumption, is proved. Iteration-complexity bound is also presented. As well as, some examples of functions that satisfy the assumptions of our results.