Abstract
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.
MSC:65H10, 65D99.
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1 Introduction
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. [1]). 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 [2], 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 [7] the classical Lipschitz continuity. Wang introduced in [6] 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 [8] for Newton’s method for vector fields on Riemannian manifolds while the extensions of the famous Smale’s α-theory and γ-theory in [2] to analytic vector fields and analytic mappings on Riemannian manifolds were done in [9]. In the recent paper [10], the convergence criteria in [9] 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 [11], while the local behavior of Newton’s method on Riemannian manifolds was studied in [12, 13]. Furthermore, in [14], Li and Wang extended the generalized L-average Lipschitz condition (introduced in [6]) 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 [7] on Kantorovich’s majorant method, Alvarez et al. introduced in [15] 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 [16], 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 [17] Newton’s method, independent of affine connections on the Lie group, and showed the local quadratical convergence. Recently, Wang and Li [18] 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 [19].
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 [18] and [19], 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.
2 Notions and preliminaries
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 .
In the sequel we make use of the left translation of the Lie group G. We define, for each , the left translation by
The differential of at z is denoted by , which clearly determines a linear isomorphism from to the tangent space . In particular, the differential of at e determines a linear isomorphism from to the tangent space . The exponential map is certainly the most important construction associated to G and , and is defined as follows. Given , let be a one-parameter subgroup of G determined by the left invariant vector field ; i.e., satisfies that
The value of the exponential map exp at u is then defined by
Moreover, we have that
and
Note that the exponential map is not surjective in general. However, the exponential map is a diffeomorphism on an open neighborhood of . In the case when G is Abelian, exp is also a homomorphism from to G, i.e.,
In the non-abelian case, exp is not a homomorphism and, by the Baker-Campbell-Hausdorff (BCH) formula (cf. [[21], p.114]), (2.5) must be replaced by
for all in a neighborhood of , where w is defined by
Let be a -map and let . We use to denote the differential of f at x. Then, by [[22], p.9] (the proof given there for a smooth mapping still works for a -map), for each and any nontrivial smooth curve with and , one has that
In particular,
Define the linear map by
Then, by (2.9),
Also, in view of the definition, we have that for all ,
and
For the remainder of the present paper, we always assume that is an inner product on and is the associated norm on . We now introduce the following distance on G which plays a key role in the study. Let and define
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 and . We denote the corresponding ball of radius r around x of G by , that is,
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 [23] by Wang for the study of Smale’s point estimate theory.
Definition 2.1 Let , , and let T be a mapping from G to . Then T is said to satisfy the L-average Lipschitz condition on if
holds for any and such that and , where .
The majorizing function h defined in the following, which was first introduced and studied by Wang (cf. [23]), is a powerful tool in our study. Let and be such that
For , define the majorizing function h by
Some useful properties are described in the following propositions, see [23].
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 .
Let denote the sequence generated by Newton’s method with initial value for h, that is,
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.
Lemma 2.1 Let and let be such that exists. Suppose that satisfies the L-average Lipschitz condition on . Let be such that there exist and satisfying and . Then exists and
Proof Write and for each . Since (2.15) holds with , one has that
Noting that , we have that
Thus the conclusion follows from the Banach lemma and the proof is complete. □
3 Convergence criteria
Following [17], we define Newton’s method with initial point for f on a Lie group as follows:
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.
Theorem 3.1 Suppose that satisfies the L-average Lipschitz condition on and that
Then the sequence generated by Newton’s method (3.1) with initial point is well defined and converges to a zero of f. Moreover, the following assertions hold for each :
Proof Write for each . Below we shall show that each is well defined and
holds for each . Granting this, one sees that the sequence generated by Newton’s method (3.1) with initial point is well defined and converges to a zero of f, because, by (3.1),
Furthermore, assertions (3.3) and (3.4) hold for each n and the proof of the theorem is completed.
Note that is well defined by assumption and . Hence, . Since , it follows that (3.5) is true for . We now proceed by mathematical induction on n. For this purpose, assume that is well defined and (3.5) holds for each . Then
Thus, we use Lemma 2.1 to conclude that exists and
Therefore, is well defined. Observe that
where the second equality is valid because of (2.13). Therefore, applying (2.15), one has that
where the first equality holds because . Combining this with (3.7) yields that
Since , we have . This together with (3.9) gives that (3.5) holds for , which completes the proof of the theorem. □
4 Applications to optimization problems
Let be a -map. Consider the following optimization problem:
Newton’s method for solving (4.1) was presented in [16], where local quadratical convergence result was established for a smooth function ϕ.
Let . Following [16], we use to denote the left invariant vector field associated with X defined by
and the Lie derivative of ϕ with respect to the left invariant vector field , that is, for each ,
Let be an orthonormal basis of . According to [[24], p.356] (see also [16]), gradϕ is a vector field on G defined by
Then Newton’s method with initial point considered in [16] can be written in a coordinate-free form as follows.
Algorithm 4.1 Find such that and
Set ;
Set and repeat.
Let be a mapping defined by
Define the linear operator for each by
Then defines a mapping from G to . The following proposition gives the equivalence between and . The following proposition was given in [18].
Proposition 4.1 Let and be defined respectively by (4.4) and (4.5). Then
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.
Theorem 4.1 Suppose that
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: .
Furthermore, if is additionally positive definite and the following Lipschitz condition is satisfied:
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.
To prove the second assertion, we assume that is additionally positive definite and the Lipschitz condition (4.8) is satisfied. It is sufficient to prove that is positive definite. Let and be the minimum eigenvalues of and , respectively. Then . We have to show that . To do this, let be the sequence generated by Algorithm 4.1 and write for each . Then, by Remark 4.1,
and by Theorem 3.1,
Therefore, for each ,
thanks to (4.8)-(4.10). Since
it follows that
thanks to (4.11). This implies that and completes the proof. □
5 Theorems under the Kantorovich’s condition and the γ-condition
If is a constant, then the L-average Lipschitz condition is reduced to the classical Lipschitz condition.
Let , , and let T be a mapping from G to . Then T is said to satisfy the L Lipschitz condition on if
holds for any and such that and , where .
Let and . The quadratic majorizing function h is reduced to
Let denote the sequence generated by Newton’s method with initial value for h, that is,
Assume that . Then h has two zeros and :
moreover, is monotonic increasing and convergent to , and satisfies that
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 [18].
Theorem 5.1 Suppose that satisfies the L-Lipschitz condition on and that . Then the sequence generated by Newton’s method (3.1) with initial point is well defined and converges to a zero of f. Moreover, the following assertions hold for each :
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 [18].
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: .
Furthermore, if is additionally positive definite and the following Lipschitz condition is satisfied:
Then is a local solution of (4.1).
Let k be a positive integer and assume further that is a -map. Define the map by
for each . In particular,
Let . Then, in view of the definition, one has that
In particular, for fixed ,
This implies that is linear with respect to and so is . Consequently, is a multilinear map from to because is arbitrary. Thus we can define the norm of by
For the remainder of the paper, we always assume that f is a -map from G to . Then, taking , we have
Thus, (2.13) is applied (with in place of for each ) to conclude the following formula:
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 [19] 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 .
Definition 5.1 Let be such that exists. f is said to satisfy the γ-condition at on if, for any with such that ,
As shown in Proposition 5.3, if f is analytic at , then f satisfies the γ-condition at .
Let and let L be the function defined by
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).
Proof Let and let be such that and . Write . Observe from (5.2) that
Combining this with the assumption yields that
Hence, satisfies the L-average Lipschitz condition on with L defined by (5.3). □
Corresponding to the function L defined by (5.3), and b in (2.16) are and , and the majorizing function given in (2.17) reduces to
Hence the condition is equivalent to . Let denote the sequence generated by Newton’s method with the initial value for h. Then the following proposition was proved in [27], see also [10] and [6].
Proposition 5.2 Assume that . Then the zeros of h are
and
Moreover, the following assertions hold:
where
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 [19].
Theorem 5.3 Suppose that
and that f satisfies the γ-condition at on . Then Newton’s method (3.1) with initial point is well defined, and the generated sequence converges to a zero of f. Moreover, if , then for each ,
where ν is given by (5.4).
Below, we always assume that f is analytic on G. For such that exists, we define
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 [19].
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 [19].
Corollary 5.1 Suppose that
Then Newton’s method (3.1) with initial point is well defined and the generated sequence converges to a zero of f. Moreover, if , then for each ,
where ν is given by (5.4).
References
Kantorovich LV, Akilov GP: Functional Analysis. Pergamon, Oxford; 1982.
Smale S: Newton’s method estimates from data at one point. In The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics. Edited by: Ewing R, Gross K, Martin C. Springer, New York; 1986:185–196.
Ezquerro JA, Hernández MA: Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 2002, 22: 187–205. 10.1093/imanum/22.2.187
Ezquerro JA, Hernández MA: On an application of Newton’s method to nonlinear operators with w -conditioned second derivative. BIT Numer. Math. 2002, 42: 519–530.
Gutiérrez JM, Hernández MA: Newton’s method under weak Kantorovich conditions. IMA J. Numer. Anal. 2000, 20: 521–532. 10.1093/imanum/20.4.521
Wang XH: Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 2000, 20(1):123–134. 10.1093/imanum/20.1.123
Zabrejko PP, Nguen DF: The majorant method in the theory of Newton-Kantorovich approximates and the Ptak error estimates. Numer. Funct. Anal. Optim. 1987, 9: 671–674. 10.1080/01630568708816254
Ferreira OP, Svaiter BF: Kantorovich’s theorem on Newton’s method in Riemannian manifolds. J. Complex. 2002, 18: 304–329. 10.1006/jcom.2001.0582
Dedieu JP, Priouret P, Malajovich G: Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 2003, 23: 395–419. 10.1093/imanum/23.3.395
Li C, Wang JH: Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the γ -condition. IMA J. Numer. Anal. 2006, 26: 228–251.
Wang JH, Li C: Uniqueness of the singular points of vector fields on Riemannian manifolds under the γ -condition. J. Complex. 2006, 22: 533–548. 10.1016/j.jco.2005.11.004
Li C, Wang JH: Convergence of Newton’s method and uniqueness of zeros of vector fields on Riemannian manifolds. Sci. China Ser. A 2005, 48: 1465–1478. 10.1360/04ys0147
Wang JH: Convergence of Newton’s method for sections on Riemannian manifolds. J. Optim. Theory Appl. 2011, 148: 125–145. 10.1007/s10957-010-9748-4
Li C, Wang JH: Newton’s method for sections on Riemannian manifolds: generalized covariant α -theory. J. Complex. 2008, 24: 423–451. 10.1016/j.jco.2007.12.003
Alvarez F, Bolte J, Munier J: A unifying local convergence result for Newton’s method in Riemannian manifolds. Found. Comput. Math. 2008, 8: 197–226. 10.1007/s10208-006-0221-6
Mahony RE: The constrained Newton method on a Lie group and the symmetric eigenvalue problem. Linear Algebra Appl. 1996, 248: 67–89.
Owren B, Welfert B: The Newton iteration on Lie groups. BIT Numer. Math. 2000, 40(2):121–145.
Wang JH, Li C: Kantorovich’s theorems for Newton’s method for mappings and optimization problems on Lie groups. IMA J. Numer. Anal. 2011, 31: 322–347. 10.1093/imanum/drp015
Li C, Wang JH, Dedieu JP: Smale’s point estimate theory for Newton’s method on Lie groups. J. Complex. 2009, 25: 128–151. 10.1016/j.jco.2008.11.001
Helgason S: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York; 1978.
Varadarajan VS Graduate Texts in Mathematics 102. In Lie Groups, Lie Algebras and Their Representations. Springer, New York; 1984.
DoCarmo MP: Riemannian Geometry. Birkhäuser Boston, Cambridge; 1992.
Wang XH: Convergence of Newton’s method and inverse function theorem in Banach spaces. Math. Comput. 1999, 68: 169–186. 10.1090/S0025-5718-99-00999-0
Helmke U, Moore JB Commun. Control Eng. Ser. In Optimization and Dynamical Systems. Springer, London; 1994.
Wang XH, Han DF: Criterion α and Newton’s method in weak condition. Chin. J. Numer. Math. Appl. 1997, 19: 96–105.
Wang XH: Convergence on the iteration of Halley family in weak conditions. Chin. Sci. Bull. 1997, 42: 552–555. 10.1007/BF03182614
Wang XH, Han DF: On the dominating sequence method in the point estimates and Smale’s theorem. Sci. Sin., Ser. A, Math. Phys. Astron. Tech. Sci. 1990, 33: 135–144.
Acknowledgements
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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He, J., Wang, J. & Yao, JC. Convergence criteria of Newton’s method on Lie groups. Fixed Point Theory Appl 2013, 293 (2013). https://doi.org/10.1186/1687-1812-2013-293
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DOI: https://doi.org/10.1186/1687-1812-2013-293