- Open Access
Convergence criteria of Newton’s method on Lie groups
© He et al.; licensee Springer. 2013
- Received: 20 August 2013
- Accepted: 26 August 2013
- Published: 9 November 2013
In the present paper, we study Newton’s method on Lie groups (independent of affine connections) for finding zeros of a mapping f from a Lie group to its Lie algebra. Under a generalized L-average Lipschitz condition of the differential of f, we establish a unified convergence criterion of Newton’s method. As applications, we get the convergence criteria under the Kantorovich’s condition and the γ-condition, respectively. Moreover, applications to optimization problems are also provided.
- Newton’s method
- Lie group
- L-average Lipschitz condition
Newton’s method is one of the most important methods for finding the approximation solution of the equation , where f is an operator from some domain D in a real or complex Banach space X to another Y. As is well known, one of the most important results on Newton’s method is Kantorovich’s theorem (cf. ). Under the mild condition that the second Fréchet derivative of F is bounded (or more general, the first derivative is Lipschitz continuous) on a proper open metric ball of the initial point , Kantorovich’s theorem provides a simple and clear criterion ensuring the quadratic convergence of Newton’s method. Another important result on Newton’s method is Smale’s point estimate theory (i.e., α-theory and γ-theory) in , where the notions of approximate zeros were introduced and the rules to judge an initial point to be an approximate zero were established, depending on the information of the analytic nonlinear operator at this initial point and at a solution , respectively. There are a lot of works on the weakness and/or the extension of the Lipschitz continuity made on the mappings; see, for example, [3–7] and references therein. In particular, Zabrejko-Nguen parametrized in  the classical Lipschitz continuity. Wang introduced in  the notion of Lipschitz conditions with L-average to unify both Kantorovich’s and Smale’s criteria.
In a Riemannian manifold framework, an analogue of the well-known Kantorovich’s theorem was given in  for Newton’s method for vector fields on Riemannian manifolds while the extensions of the famous Smale’s α-theory and γ-theory in  to analytic vector fields and analytic mappings on Riemannian manifolds were done in . In the recent paper , the convergence criteria in  were improved by using the notion of the γ-condition for the vector fields and mappings on Riemannian manifolds. The radii of uniqueness balls of singular points of vector fields satisfying the γ-conditions were estimated in , while the local behavior of Newton’s method on Riemannian manifolds was studied in [12, 13]. Furthermore, in , Li and Wang extended the generalized L-average Lipschitz condition (introduced in ) to Riemannian manifolds and established a unified convergence criterion of Newton’s method on Riemannian manifolds. Similarly, inspired by previous work of Zabrejko and Nguen in  on Kantorovich’s majorant method, Alvarez et al. introduced in  a Lipschitz-type radial function for the covariant derivative of vector fields and mappings on Riemannian manifolds and established a unified convergence criterion of Newton’s method on Riemannian manifolds.
Note also that Mahony used one-parameter subgroups of a Lie group to develop a version of Newton’s method on an arbitrary Lie group in , where the algorithm presented is independent of affine connections on the Lie group. This means that Newton’s method on Lie groups is different from the one defined on Riemannian manifolds. On the other hand, motivated by looking for approaches to solving ordinary differential equations on Lie groups, Owren and Welfert also studied in  Newton’s method, independent of affine connections on the Lie group, and showed the local quadratical convergence. Recently, Wang and Li  established Kantorovich’s theorem (independent of the connection) for Newton’s method on the Lie group. More precisely, under the assumption that the differential of f satisfies the Lipschitz condition around the initial point (which is in terms of one-parameter semigroups and independent of the metric), the convergence criterion of Newton’s method is presented. Extensions of Smale’s point estimate theory for Newton’s method on Lie groups were given in .
The purpose of the present paper is to establish a unified convergence criterion for Newton’s method (independent of the connection) on Lie groups under a generalized L-average Lipschitz condition. As applications, we get the convergence criteria under the Kantorovich’s condition and the γ-condition, respectively. Hence, our results extend the corresponding results in  and , respectively. Moreover, applications to optimization problems are also provided.
The remainder of the paper is organized as follows. Some preliminary results and notions are given in Section 2, while the main results about a unified convergence criterion are presented in Section 3. In Section 4, applications to optimization problems are explored. Theorems under the Kantorovich’s condition and the γ-condition are provided in the final section.
Most of the notions and notations which are used in the present paper are standard; see, for example, [20, 21]. The Lie group is a Hausdorff topological group with countable bases which also has the structure of an analytic manifold such that the group product and the inversion are analytic operations in the differentiable structure given on the manifold. The dimension of a Lie group is that of the underlying manifold, and we shall always assume that it is m-dimensional. The symbol e designates the identity element of G. Let be the Lie algebra of the Lie group G which is the tangent space of G at e, equipped with Lie bracket .
where we adapt the convention that . It is easy to verify that is a distance on G and the topology induced by this distance is equivalent to the original one on G.
Let denote the set of all linear operators on . Below, we will modify the notion of the Lipschitz condition with L-average for mappings on Banach spaces to suit sections. Let L be a positive nondecreasing integrable function on , where R is a positive number large enough such that . The notion of Lipschitz condition in the inscribed sphere with the L average for operators from Banach spaces to Banach spaces was first introduced in  by Wang for the study of Smale’s point estimate theory.
holds for any and such that and , where .
Some useful properties are described in the following propositions, see .
Proposition 2.1 The function h is monotonic decreasing on and monotonic increasing on . Moreover, if , h has a unique zero respectively in and , which are denoted by and .
Proposition 2.2 Suppose that . Then the sequence generated by (2.18) is monotonic increasing and convergent to .
The following lemma will be useful in the proof of the main theorem.
Thus the conclusion follows from the Banach lemma and the proof is complete. □
Recall that is a -mapping. In the remainder of this section, we always assume that is such that exists and set . Let and b given by (2.16), and be given by Proposition 2.1.
Furthermore, assertions (3.3) and (3.4) hold for each n and the proof of the theorem is completed.
Since , we have . This together with (3.9) gives that (3.5) holds for , which completes the proof of the theorem. □
Newton’s method for solving (4.1) was presented in , where local quadratical convergence result was established for a smooth function ϕ.
Then Newton’s method with initial point considered in  can be written in a coordinate-free form as follows.
Set and repeat.
Then defines a mapping from G to . The following proposition gives the equivalence between and . The following proposition was given in .
Remark 4.1 One can easily see from Proposition 4.1 that, with the same initial point, the sequence generated by Algorithm 4.1 for ϕ coincides with the one generated by Newton’s method (3.1) for f defined by (4.4).
Let be such that exists, and let . Recall that and b are given by (2.16), and is given by Proposition 2.1. Then the main theorem of this section is as follows.
and that satisfies the L-average Lipschitz condition on . Then the sequence generated by Algorithm 4.1 with initial point is well defined and converges to a critical point of ϕ: .
Then is a local solution of (4.1).
Proof Recall that f is defined by (4.4). Then by Proposition 4.1, for each . Hence, by assumptions, satisfies the L-average Lipschitz condition on and condition (3.2) is satisfied because . Thus, Theorem 3.1 is applicable; hence the sequence generated by Newton’s method for f with initial point is well defined and converges to a zero of f. Consequently, by Remark 4.1, one sees that the first assertion holds.
thanks to (4.11). This implies that and completes the proof. □
If is a constant, then the L-average Lipschitz condition is reduced to the classical Lipschitz condition.
holds for any and such that and , where .
Recall that is a -mapping. As in the previous section, we always assume that is such that exists and set . Then, by Theorem 3.1, we obtain the following results, which were given in .
Let be such that exists, and let . Recall that is defined by (5.1). Then, by Theorem 4.1, we get the following results, which were given in .
Theorem 5.2 Suppose that , and that satisfies the L-Lipschitz condition on . Then the sequence generated by Algorithm 4.1 with initial point is well defined and converges to a critical point of ϕ: .
Then is a local solution of (4.1).
The γ-conditions for nonlinear operators in Banach spaces were first introduced and explored by Wang [25, 26] to study Smale’s point estimate theory, which was extended in  for a map f from a Lie group to its Lie algebra in view of the map as given in Definition 5.1 below. Let and be such that .
As shown in Proposition 5.3, if f is analytic at , then f satisfies the γ-condition at .
The following proposition shows that the γ-condition implies the L-average Lipschitz condition.
Proposition 5.1 Suppose that f satisfies the γ-condition at on . Then satisfies the L-average Lipschitz condition on with L defined by (5.3).
Hence, satisfies the L-average Lipschitz condition on with L defined by (5.3). □
Recall that is such that exists, and let . Then, by Theorem 3.1 and Proposition 5.2, we get the following results, which were given in .
where ν is given by (5.4).
Also, we adopt the convention that if is not invertible. Note that this definition is justified and, in the case when is invertible, is finite by analyticity.
The following proposition is taken from .
Proposition 5.3 Let and let . Then f satisfies the -condition at on .
Thus, by Theorem 5.3 and Proposition 5.3, we get the following corollary, which was given in .
where ν is given by (5.4).
The research of the second author was partially supported by the National Natural Science Foundation of China (grant 11001241; 11371325) and by Zhejiang Provincial Natural Science Foundation of China (grant LY13A010011). The research of the third author was partially supported by a grant from NSC of Taiwan (NSC 102-2115-M-037-002-MY3).
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