Gauss–Newton method for convex composite optimizations on Riemannian manifolds

Abstract

A notion of quasi-regularity is extended for the inclusion problem F( p) ∈ C, where F is a differentiable mapping from a Riemannian manifold M to R n . When C is the set of minimum points of a convex real-valued function h on R n and DF satisfies the L-average Lipschitz condition, we use the majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method (with quasiregular initial points) for the convex composite function h • F on Riemannian manifold. Two applications are provided: one is for the case of regularities on Riemannian manifolds and the other is for the case when C is a cone and DF( p 0 )(·) − C is surjective. In particular, the results obtained in this paper extend the corresponding one in Wang et al. (Taiwanese J Math 13:633-656, 2009).

Extending the applicability of Gauss–Newton method for convex composite optimization on Riemannian manifolds

“…Semilocal convergence analysis of (GNAR) using L-average conditions is presented in Section 3. In Section 4, numerical example to illustrate our theoretical results and favorable comparisons to earlier studies (see, e.g., [12,13,23]) are presented.…”

“…The notations and notions about smooth manifolds used in the present paper are standard, see for example [3,7,8,23]. Let M be a complete connected m-dimensional Riemannian manifold with the Levi-Civita connection r on M. Let p 2 M, and let T p M denote the tangent space at p to M. Let < Á; Á > be the scalar product on T p M with the associated norm k Á k p , where the subscript p is sometimes omitted.…”

“…Recently, in the elegant study by Wang, Yao and Li in [23], the notion of quasi-regularity for an initial point x 0 2 R l with respect to inclusion problem was used to present a semilocal convergence analysis of the Gauss-Newton method on Riemannian manifolds. This notion generalizes the case of regularity studied in the seminal paper by Burke and Ferris (see, e.g., [9]) as well as the case when dÀ!DFðx 0 Þd À C is surjective.…”

“…In this paper, motivated by the works in [10,12,23] and optimization considerations, we present a convergence analysis of Gauss-Newton method on Riemannian manifolds (defined by Algorithm (GNAR) in Section 2). In [23], the convergence of (GNAR) is based on the L-average Lipschitz conditions inaugurated by Wang [22] (to be precised in Section 2).…”

“…In [23], the convergence of (GNAR) is based on the L-average Lipschitz conditions inaugurated by Wang [22] (to be precised in Section 2). Using a combination of L-average Lipszhitz condition and a center L 0 -average Lipschitz condition which is a special case of the L-average Lipschitz condition and a more precise condition to use than the L-average Lipschitz for the computation of the upper bounds on the norm of the inverses involved we presented.…”