Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel--Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature

Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm may even converge on manifolds of positive curvature

Métodos de otimização sobre variedades Riemannianas com curvatura limitada inferiormente: gradiente para funções escalares e multi-objetivo e subgradiente para funções escalares

Submitted by Ana Caroline Costa ([email protected]) on 2019-03-12T17:28:11Z No. of bitstreams: 2 Tese - Maurício Silva Louzeiro - 2019.pdf: 1655384 bytes, checksum: a168bf697d30c99e45d2e9d19b65a563 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5)Approved for entry into archive by Liliane Ferreira ([email protected]) on 2019-03-13T10:22:49Z (GMT) No. of bitstreams: 2 Tese - Maurício Silva Louzeiro - 2019.pdf: 1655384 bytes, checksum: a168bf697d30c99e45d2e9d19b65a563 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5)Made available in DSpace on 2019-03-13T10:22:49Z (GMT). No. of bitstreams: 2 Tese - Maurício Silva Louzeiro - 2019.pdf: 1655384 bytes, checksum: a168bf697d30c99e45d2e9d19b65a563 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2019-02-26Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESLet M a Riemannian manifolds with lower bounded curvature. In this thesis, we consider first-order iterative methods to solve optimization problems on M. The gradient method to solve the problem min{f(p) : p M}, where f : M → R is a continuously differentiable convex function is presented with Lipschitz step-size, adaptive step-size and Armijo’s step-size. The first procedure requires that the objective function has Lipschitz continuous gradient, which is not necessary for the other approaches. Convergence of the whole sequence to a minimizer, without any level set boundedness assumption, is proved. Iteration-complexity bound for functions with Lipschitz continuous gradient is also presented. In addition, all these approaches are considered in the multiobjective setting. Here we also consider the subgradient method to solve the problem min{f(p) : p M}, where f : M → R is a convex function. Iteration-complexity bounds of the subgradient method with exogenous step-size and Polyak’s step size are stablished, completing and improving recent results on the subject. Finally, some examples and numerical experiments are presented.Seja M uma variedade Riemanniana com curvatura limitada inferiormente. Nesta tese, consideramos métodos iterativos de primeira ordem para resolver problemas de otimização sobre variedades Riemannianas com curvatura limitada inferiormente. O método do gradiente para resolver o problema min{f(p) : p M}, onde f : M → R é uma função convexa continuamente diferenciável, é apresentado com tamanho de passo Lipshitz, tamanho de passo adaptativo e tamanho de passo de Armijo. O primeiro tipo de passo requer que a função objetivo tenha gradiente continuamente Lipshitz, o que não é necessário para os outros. A convergência total da sequência para um minimizador, sem qualquer hipótese de limitação do conjunto de nível, é provada. Limitantes para a complexidade na iteração para funções com gradiente continuamente Lipschitz também são apresentados. Além disso, todas essas abordagens são consideradas no contexto de otimização multiobjetivo. Aqui também consideramos o método do subgradiente para resolver o problema min{f(p) : p M}, onde f : M → R é uma função convexa. Limitantes para a complexidade na iteração do método do subgradiente com tamanho de passo exógeno e tamanho de passo de Polyak são estabelecidos, completando e melhorando os resultados recentes sobre o assunto. Finalmente, alguns exemplos e experimentos numéricos são apresentados

Fenchel Duality and a Separation Theorem on Hadamard Manifolds

In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate depends on the choice of a certain point on the manifold, as previous definitions required. On the other hand, this new definition still possesses properties known to hold in the Euclidean case. It even yields a broader interpretation of the Fenchel conjugate in the Euclidean case itself. Most prominently, our definition of the Fenchel conjugate provides a Fenchel-Moreau Theorem for geodesically convex, proper, lower semicontinuous functions. In addition, this framework allows us to develop a theory of separation of convex sets on Hadamard manifolds, and a strict separation theorem is obtained