Hybrid Riemannian conjugate gradient methods with global convergence properties

Sakai, Hiroto; Iiduka, Hideaki

doi:10.1007/s10589-020-00224-9

Cited by 14 publications

(24 citation statements)

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“…is not a linear map. Considering the same framework, Sato [34], Sakai and Iiduka [32], and Sakai and Iiduka [33] analyzed the Dai-Yuan-type, some hybrid β k+1 -based, and Hager-Zhang-type of R-CG methods, respectively.…”

Section: Review Of and Discussion On The Existing Riemannian Cg Methodsmentioning

confidence: 99%

“…Sato [34] proposed and analyzed the generalization of β DY k+1 in the same framework as in [36]. Sakai and Iiduka [32] recently discussed a class of β k+1 containing a combination of the generalizations of β DY k+1 and β HS k+1 with the same (scaled) vector transports. Furthermore, Zhu and Sato [39] proposed and analyzed the generalizations of β FR k+1 and β DY k+1 with inverse retraction.…”

Section: Computation Of β K+1 In R-cg Methodsmentioning

confidence: 99%

“…k+1 , where β R-FR k+1 is defined as (32). If, for all k ≥ 0, grad f (x k ) = 0 and step lengths t k satisfy the strong T (k) -Wolfe conditions (11) and (39) with 0 < c 1 < c 2 < 1/2, then we have…”

Section: R-cg Methods With β R-fr and Its Variantmentioning

confidence: 99%

“…It follows from the Cauchy-Schwarz inequality , where β R-FR k+1 is defined as (32). If T (k) satisfies Assumption 3.1 and the step lengths satisfy the strong T (k) -Wolfe conditions (11) and (39)…”

Section: R-cg Methods With β R-fr and Its Variantmentioning

confidence: 99%

“…In this paper, we address the CG methods on Riemannian manifolds, which we refer to as the Riemannian conjugate gradient (R-CG) methods. Several types of R-CG methods have been studied [11,12,24,[31][32][33][34][35][36][37][38][39][40]. In some of these studies, parallel translation along the geodesics are utilized.…”

Section: Introductionmentioning

confidence: 99%

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Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses

Sato¹

2021

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This paper proposes a novel general framework of Riemannian conjugate gradient methods, that is, conjugate gradient methods on Riemannian manifolds. The conjugate gradient methods are important first-order optimization algorithms both in Euclidean spaces and on Riemannian manifolds. While various types of conjugate gradient methods are studied in Euclidean spaces, there have been fewer studies on those on Riemannian manifolds. In each iteration of the Riemannian conjugate gradient methods, the previous search direction must be transported to the current tangent space so that it can be added to the negative gradient of the objective function at the current point. There are several approaches to transport a tangent vector to another tangent space. Therefore, there are more variants of the Riemannian conjugate gradient methods than the Euclidean case. In order to investigate them in more detail, the proposed framework unifies the existing Riemannian conjugate gradient methods such as ones utilizing a vector transport or inverse retraction and also develops other methods that have not been covered in previous studies. Furthermore, sufficient conditions for the convergence of a class of algorithms in the proposed framework are clarified. Moreover, the global convergence properties of several specific types of algorithms are extensively analyzed. The analyses provide the theoretical results for some algorithms in a more general setting than the existing studies and completely new developments for the other algorithms. Numerical experiments are performed to confirm the validity of the theoretical results. The results also compare the performances of several specific algorithms in the proposed framework.

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Section: Review Of and Discussion On The Existing Riemannian Cg Methodsmentioning

confidence: 99%