Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds

Bento, G. C.; Ferreira, O. P.; Oliveira, Paulo Roberto

doi:10.1016/j.na.2010.03.057

Cited by 57 publications

(21 citation statements)

References 25 publications

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“…Then c ∈ A, and x n = P A (x n−1 ), which together, along with Proposition 2.1, yield α (x n−1 , x n , c) ≥ π/2. Hence we get (6).…”

Section: Proofs Of the Theoremsmentioning

confidence: 91%

“…Gradually, many of the algorithms for solving optimization problems have been generalized from linear spaces (Euclidean, Hilbert, Banach) into differentiable manifolds. In particular, the proximal point algorithm in the context of Riemannian manifold (of nonpositive sectional curvature) was studied in [6,12,20,24]. We continue along these lines and introduce the PPA into geodesic metric spaces of nonpositive curvature, so-called CAT(0) spaces.…”

Section: Introductionmentioning

confidence: 99%

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The proximal point algorithm in metric spaces

2012

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The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of nonpositive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.

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“…Then c ∈ A, and x n = P A (x n−1 ), which together, along with Proposition 2.1, yield α (x n−1 , x n , c) ≥ π/2. Hence we get (6).…”

Section: Proofs Of the Theoremsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

The proximal point algorithm in metric spaces

2012

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“…The proximal method was first proposed and analyzed in the Riemannian setting in [21]. Since then, it has been the subject of intense research; see, for example, [8,22,23,35] and reference therein. As far as we know, all the papers studying convergence of the proximal point method above analyze only its asymptotic convergence property.…”

mentioning

confidence: 99%

Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds

Bento

Ferreira

Melo

2017

J Optim Theory Appl

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This paper considers optimization problems on Riemannian manifolds and analyzes iteration-complexity for gradient and subgradient methods on manifolds with non-negative curvature. By using tools from the Riemannian convex analysis and exploring directly the tangent space of the manifold, we obtain different iterationcomplexity bounds for the aforementioned methods, complementing and improving related results. Moreover, we also establish iteration-complexity bound for the proximal point method on Hadamard manifolds. keywords: Complexity, gradient method, subgradient method, proximal point method, Riemannian manifold.

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“…Tomando límite cuando j → +∞, y considerando elítem (ii) y la hipótesis lim k→+∞ x k j =x, en la desigualdad anterior se tiene que lim j→+∞ x k j +1 −x ≤ 0, de donde se sigue el resultado. A continuación, considerando la acotación de los parámetros regulares {α k }, probaremos que las sucesiones iterantes convergen hacia un punto crítico Pareto-Clarke del problema (6).…”

Section: Algoritmo Mppei1 Y Resultados De Convergenciaunclassified

Un método de punto proximal escalarizado inexacto para minimización multiobjetivo cuasi-convexa en espacios Euclidianos

Quiroz

Acuña

2019

Pesquimat

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Resumen: En este artículo presentamos un método de punto proximal escalarizado inexacto para resolver problemas irrestrictos de minimización multiobjetivo cuasiconvexas definidos en espacios Euclidianos, donde las funciones vectoriales son localmente Lipschitz. Bajo algunas hipótesis naturales, probamos que la sucesión generada por el método está bien definida, y converge globalmente. Seguidamente proporcionando al método propuesto dos criterios de error, se obtienen dos variantes del mismo, y se prueba que las sucesiones generadas por cada una de estas variantes, convergen hacia un punto crítico Pareto-Clarke del problema. Palabras clave: Método de puntoAbstract: In this paper, we present an inexact scalarized proximal point method to solve unconstrained quasiconvex multiobjective minimization problems defined in Euclidean spaces, where the vector functions are locally Lipschitz. Under some natural assumptions, we prove that the sequence generated by the method is well defined and converges globally. Next, introduzing two error criteria on the method, two variants are obtained, and it is proved that the sequences generated by each one of these variants, converge to a Pareto-Clarke critical point of the problem.c Los autores. Este artículo es publicado por la Revista PESQUIMAT de la Facultad de Ciencias Matemáticas, Universidad Nacional Mayor de San Marcos. Este es un artículo de acceso abierto, distribuido bajo los términos de la licencia Creative Commons Atribucion-No Comercia-Compartir Igual 4.0 Internacional.(http://creativecommons.org/licenses/by-nc-sa/4.0/) que permite el uso no comercial, distribución y reproducción en cualquier medio, siempre que la obra original sea debidamente citada. Para información, por favor póngase en contacto con

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Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds

Cited by 57 publications

References 25 publications

The proximal point algorithm in metric spaces

The proximal point algorithm in metric spaces

Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds

Un método de punto proximal escalarizado inexacto para minimización multiobjetivo cuasi-convexa en espacios Euclidianos

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