Convexity of sets and quadratic functions on the hyperbolic space

In this paper some concepts of convex analysis on hyperbolic space are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first and second order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented

September 24-28, 2012 Local Convergence Analysis of Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds

Resumo Neste artigo apresentamos o método de ponto proximal para uma classe especial de funções não-convexas em variedades de Hadamard.É garantida a boa definição das sequência gerada pelo método de ponto proximal. Além disso,é provado que cada ponto de acumulação da sequência satisfaz as condições necessárias de otimalidade e, sob hipóteses adicionais, a convergência para um minimizadoré obtida. Palavras Chave: Método de Ponto Proximal, funções não convexas, variedades de Hadamard. Area principal: Programação matemática. Abstract In this paper we present the proximal point method for a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained

Inexact Multi-Objective Local Search Proximal Algorithms: Application to Group Dynamic and Distributive Justice Problems

International audienceWe introduce and examine an inexact multi-objective proximal method with a proximal distance as the perturbation term. Our algorithm utilizes a local search descent process that eventually reaches a weak Pareto optimum of a multi-objective function, whose components are the maxima of continuously differentiable functions. Our algorithm gives a new formulation and resolution of the following important distributive justice problem in the context of group dynamics: In each period, if a group creates a cake, the problem is, for each member, to get a high enough share of this cake; if this is not possible, then it is better to quit, breaking the stability of the group