Subgradient Algorithm on Riemannian Manifolds

Ferreira, O. P.; Oliveira, Paulo Roberto

doi:10.1023/a:1022675100677

Cited by 91 publications

(66 citation statements)

References 14 publications

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“…It is the Hessian of a C ∞ strictly convex selfconcordant function (see Definition 2.1.1 of Nesterov and Nemirovskii [18]), allowing the introduction of new interior point algorithms for convex optimization problems, as proximal and subgradient. We observe that general convergence results could be applied for those methods, through, respectively, the theory developed in [10] and [9].…”

Section: Steepest Descent Algorithms For the Hypercubementioning

confidence: 96%

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

Quiroz

Quispe

Oliveira

2008

Journal of Mathematical Analysis and Applications

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This paper extends the full convergence of the steepest descent method with a generalized Armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete Riemannian manifolds. Previous convergence results are obtained as particular cases and some examples in non-Euclidian spaces are given. In particular, our approach can be used to solve constrained minimization problems with nonconvex objective functions in Euclidian spaces if the set of constraints is a Riemannian manifold and the objective function is quasiconvex in this manifold.

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Section: Steepest Descent Algorithms For the Hypercubementioning

confidence: 96%

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

Quiroz

Quispe

Oliveira

2008

Journal of Mathematical Analysis and Applications

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“…It has been frequently done in recent years, with a theoretical purpose and also to obtain effective algorithms; see [1][2][3][4][5][6][7][8][9]. In particular, we observe that, these extensions allow the solving of some nonconvex constrained problems in Euclidean space.…”

Section: Introductionmentioning

confidence: 97%

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

Bento

Ferreira

Oliveira

2010

Nonlinear Analysis: Theory, Methods & Applications

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a b s t r a c tLocal convergence analysis of the proximal point method for a special class of nonconvex functions on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.

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“…We have the following convergence result, see [31]. For manifolds with curvature bounded from below the subgradient algorithm converges if the iterates stay in bounded sets, see [61] or [62].…”

Section: Subgradient Descentmentioning

confidence: 99%

Recent advances in denoising of manifold-valued images

Bergmann

Laus

Persch

et al. 2019

Handbook of Numerical Analysis

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Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny inaccuracies, which result in noisy data. In this chapter, we review recent advances in denoising of manifoldvalued signals and images, where we restrict our attention to variational models and appropriate minimization algorithms. The algorithms are either classical as the subgradient algorithm or generalizations of the half-quadratic minimization method, the cyclic proximal point algorithm, and the Douglas-Rachford algorithm to manifolds. An important aspect when dealing with real-world data is the practical implementation. Here several groups provide software and toolboxes as the Manifold Optimization (Manopt) package and the manifold-valued image restoration toolbox (MVIRT).

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Subgradient Algorithm on Riemannian Manifolds

Cited by 91 publications

References 14 publications

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

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

Recent advances in denoising of manifold-valued images

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