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M. S. Louzeiro

Publications and source records attributed to M. S. Louzeiro.

5 recordsLinked to original sources
arXiv2026

Projected Gradient Method on Hadamard Manifolds

We study constrained smooth optimization problems on Hadamard manifolds with closed geodesically convex feasible sets. We analyze two projected gradient schemes: one with a constant stepsize and another with a backtracking line search. The constant-stepsize scheme is analyzed under the assumption that the objective function has a Lipschitz continuous Riemannian gradient, whereas the backtracking variant does not require this assumption to establish stationarity of accumulation points. For both schemes, we prove that every accumulation point of the generated sequence is first-order stationary under the respective assumptions, without requiring compactness of the feasible set; compactness is needed only to ensure the existence of accumulation points. When the objective function has a Lipschitz continuous Riemannian gradient, we derive iteration-complexity bounds of order \(O(1/\sqrt{N})\) for projection-based stationarity measures for both schemes, together with the corresponding \(\varepsilon\)-complexity estimates. For the backtracking scheme, the complexity analysis additionally requires the trial line-search stepsizes to be uniformly bounded away from zero. Under the same respective assumptions, the generated sequences are also asymptotically regular. Finally, we illustrate the practical performance of the methods by solving constrained Karcher mean problems on the manifold of symmetric positive definite matrices.

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arXiv2026

A subdifferential characterization via Busemann functions and applications to DC optimization on Hadamard manifolds

This paper investigates the properties of Busemann functions on Hadamard manifolds and their use in optimization algorithms in Riemannian settings. We present a new Busemann-based characterization of the subdifferential, which is particularly well suited to Riemannian optimization. In the classical Hadamard manifold framework, a subgradient provides a global lower model of a convex function expressed through the inverse exponential map. However, this model may fail to exhibit a useful convexity or concavity structure. By contrast, our characterization yields a concave bounding function by exploiting key properties of Busemann functions. We use this concavity to design and analyze difference-of-convex (DC) optimization methods on Hadamard manifolds. In particular, we reformulate the classical DC algorithm (DCA) for Riemannian contexts and study its convergence properties. We also report preliminary numerical experiments comparing the proposed Busemann DCA, which leads to geodesically convex subproblems, with the classical Riemannian DCA.

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arXiv2026

A Busemann hybrid projection-proximal point algorithm for optimization problems on Hadamard manifolds

We study optimization problems on Hadamard manifolds, motivated by recent advances in geometric approaches to optimization on curved spaces, particularly those involving the structure of Busemann functions. We introduce a projection based variant of the proximal point algorithm, termed the \emph{Busemann hybrid projection proximal point algorithm}, which replaces Euclidean hyperplanes with horospheres defined via convex Busemann functions. The algorithm performs projections in closed form using the gradients of these functions, resulting in a geometrically intrinsic scheme that requires no tangent space linear solves. We allow for inexact subgradient evaluations and prove global convergence under controlled inexactness, with a relative error level strictly below one. We establish a Fejér type descent and sublinear complexity with a rate proportional to the inverse square root of the iteration count, and show that the exact variant coincides with the classical Riemannian proximal point algorithm. The framework clarifies the role of Busemann based subdifferentials in optimization on spaces of nonpositive curvature.

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arXiv2018

Gradient Method for Optimization on Riemannian Manifolds with Lower Bounded Curvature

The gradient method for minimize a differentiable convex function on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. The analysis of the method is presented with three different finite procedures for determining the stepsize, namely, Lipschitz stepsize, adaptive stepsize and Armijo's stepsize. 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. Numerical experiments are provided to illustrate the effectiveness of the method in this new setting and certify the obtained theoretical results. In particular, we consider the problem of finding the Riemannian center of mass and the so-called Karcher's mean. Our numerical experiences indicate that the adaptive stepsize is a promising scheme that is worth considering.

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