F(x)=0; x 2X: .... Newton's method for singular smooth equations using outer inverses ... (1.9) for singular nonsmooth equations that have not been previously ...
Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses 1 Xiaojun Chen School of Mathematics, University of New South Wales Sydney 2052, Australia Zuhair Nashed Department of Mathematical Sciences, University of Delaware Newark, DE 19716, USA Liqun Qi School of Mathematics, University of New South Wales Sydney 2052, Australia (January 1993, Revised June 1995)
ABSTRACT. We present a local convergence analysis of generalized Newton meth-
ods for singular smooth and nonsmooth operator equations using adaptive constructs of outer inverses. We prove that for a solution x of F (x) = 0, there exists a ball S = S (x ; r), r > 0 such that for any starting point x0 2 S the method converges to a solution x 2 S of ?F (x) = 0, where ? is a bounded linear operator that depends on the Frechet derivative of F at x0 or on a generalized Jacobian of F at x0 . Point x may be di erent from x when x is not an isolated solution. Moreover, we prove that the convergence is quadratic if the operator is smooth, and superlinear if the operator is locally Lipschitz. These results are sharp in the sense that they reduce in the case of an invertible derivative or generalized derivative to earlier theorems with no additional assumptions. The results are illustrated by a system of smooth equations and a system of nonsmooth equations, each of which is equivalent to a nonlinear complementarity problem.
Key words: Newton's method, convergence theory, nonsmooth analysis, outer inverses, nonlinear complementarity problems. AMS(MOS) subject classi cation. 65J15, 65H10, 65K10, 49M15. Abbreviated title: Newton's method for singular equations
This work is supported by the Australian Research Council and by the National Science Foundation Grant DMS-901526. 1
1
1. Introduction
Let X and Y be Banach spaces and let L(X; Y ) denote the set of all bounded linear operators on X into Y . Let F : X ! Y be a continuous function. We consider the nonlinear operator equation F (x) = 0; x 2 X: (1:1) When X = Y , it is well-known that if F is Frechet di erentiable and F 0 is locally Lipschitz and invertible at a solution x, then there exists a ball S (x; r); r > 0 such that for any x0 2 S (x; r), the Newton method xk+1 = xk ? F 0(xk )?1F (xk ) (1:2) is quadratically convergent to x . See, e.g., [9, 27, 34]. In the nonsmooth case, F 0(xk ) may not exist. The generalized Newton method proposes to use generalized Jacobians of F to play the role of F 0 in the Newton method (1.2) in the nite dimensional case. Let F be a locally Lipschitzian mapping from Rn into Rm. Then Rademacher's theorem implies that F is almost everywhere di erentiable. Let DF be the set where F is di erentiable. Denote
@B F (x) = f lim ! rF (xi )g: 2
xi
xi
x
DF
The generalized Jacobian of F at x 2 Rn in the sense of Clarke [8] is equal to the convex hull of @B F (x), @F (x) = conv@B F (x); which is a nonempty convex compact set. The Newton method for nonsingular nonsmooth equations using the generalized Jacobian is de ned by xk+1 = xk ? Vk?1F (xk ); Vk 2 @F (xk ): (1:3) A local superlinear convergence theorem is given in [33], where it is assumed that all V 2 @F (x ) are nonsingular. Qi [31] suggested a modi ed version of method (1.3) in the form xk+1 = xk ? Vk?1F (xk ); Vk 2 @B F (xk ) (1:4) and gave a local superlinear convergence theorem for method (1.4). His theorem reduced the nonsingularity requirement on all members of @F (x ) to all members of @B F (x). Another modi cation is an iteration function method introduced by Han, Pang and Rangaraj [13] using an iteration function G(; ) : Rn Rn ! Rn. If F has a one-sided directional derivative F (x + td) ? F (x) F 0(x; d) := lim (1:5) t#0 t 2
and G(x; d) = F 0(x; d), a variant of the iteration function method can be de ned by ( solve F (xk ) + F 0(xk ; d) = 0; set xk+1 = xk + d: (1:6) See also Pang [28] and Qi [31]. Methods (1.2), (1.3), (1.4) and (1.6) are very useful, but they are not applicable to the singular case. At each step in (1.2), (1.3) and (1.4), the inverse of a Jacobian or a generalized Jacobian is required; in (1.6) a nonlinear equation is solved at each step (in the singular case, it may have no solutions). Often the inverse cannot be guaranteed to exist; singularity occurs in many applications. For example, we consider the nonlinear complementarity problem(NCP): For a given f : Rn ! Rn, nd x 2 Rn such that
x 0; f (x) 0 and xT f (x) = 0: Mangasarian [19] formulated the NCP in the case when f is Frechet di erentiable as an equivalent system of smooth equations: F^i (x) = (fi(x) ? xi )2 ? fi(x)jfi (x)j ? xi jxij = 0; i = 1; 2; :::; n; (1:7) where f = (f1 ; :::; fn)T : Let sgn( ) =
(
1; ?1;
0; < 0;
and let ij denote the Kronecker function. It is easy to show that the Jacobian of F^ := (F^1 ; :::; F^n)T at x is given by: @ F^i (x) = 2f (x) @fi (x) (1 ? sgn(f (x))) + 2x (1 ? sgn(x )) ? 2f (x) ? 2x @fi (x) i i i ij i i ij i @xj @xj @xj i; j = 1; 2; :::; n: The Jacobian rF^ (x) is singular when there is some degeneracy, i.e. xi = fi(x) = 0 for some i. The NCP can also be formulated as a system of nonsmooth equations [28] : F~ (x) = min(f (x); x) = 0; (1:8) where the \min" operator denotes the componentwise minimum of two vectors. It is hard to guarantee that all members of @B F (x) are nonsingular when there is some nonsmoothness, i.e. xi = fi(x) and ei 6= rfi(x) for some i, where ei is the i-th row of the identity matrix I 2 Rnn. 3
In [5], Chen and Qi studied a parameterized Newton method:
xk+1 = xk ? (Vk + k I )?1 F (xk ); Vx 2 @B F (xk ); where k is a parameter to ensure the existence of the inverse of Vk + k I: The local superlinear convergence theorem in [5] requires all V 2 @B F (x ) to be nonsingular. In Newton-like methods for solving smooth and nonsmooth equations, e.g. quasiNewton methods and splitting methods, the Jacobian is often required to be nonsingular at a solution x to which the method is supposed to converge [4, 5, 6, 9, 15, 16, 27, 28, 32, 40]. Hence it is interesting to know what happens with the Newton methods when F 0(x) or some V 2 @B F (x ) are singular at x . In this case the solution set is locally a manifold of positive dimension, hence x is not an isolated solution. Let A 2 L(X; Y ). We denote the range and nullspace of A by R(A) and N (A), respectively. A linear operator A] : Y ! X is said to be an outer inverse of A if A] AA] = A] . In this paper, for X = Rn and Y = Rm , we consider a generalized Newton method xk+1 = xk ? Vk] F (xk ) (1:9) where Vk 2 @B F (xk ) and Vk] denotes an outer inverse of Vk . Newton's method for singular smooth equations using outer inverses xk+1 = xk ? F 0(xk )] F (xk ) (1:10) has been considered by Ben-Israel [2], Deu hard and Heindl [10] and Nashed [25] and more recently by Nashed and Chen [26]. Paper [26] presented a Kantorovichtype theorem (semilocal convergence) for Newton-like methods for singular smooth equations using outer inverses : if some conditions hold at the starting point x0 , method (1.10) converges to a solution of F 0(x0 )]F (x) = 0. This paper establishes new results on Newton's method for smooth and nonsmooth equations. In particular we consider the behavior of methods (1.9) and (1.10) when the singularity occurs at a solution x which is close to the starting point. In Section 2 we state the de nitions and properties of generalized gradients, semismooth functions and outer inverses. These results are used to analyze convergence of methods (1.9) and (1.10). In Section 3 by using a Kantorovich-type theorem, we give a locally quadratic convergence theorem for Newton's method (1.10) in the following sense: for a solution x of (1.1), there is a ball S (x ; r) with r > 0 such that for any x0 2 S (x; r); Newton's method (1.10) with F 0(xk )] = (I + F 0(x0)] (F 0(xk ) ? F 0(x0)))?1 F 0(x0)] converges quadratically to a solution x of F 0(x0 )]F (x) = 0. Here, x may be di erent from x , because of singularity, there is no guarantee for uniqueness of the solutions. This is a major di erence between singular and nonsingular equations. 4
In Section 4 by using a Mysovskii-type theorem we prove the superlinear convergence of method (1.9) for nonsmooth equations. Diculties in the analysis of method (1.9) for singular nonsmooth equations that have not been previously resolved in the literature arise from the fact that there are some singular elements Vx 2 @B F (x), so rank(Vx) are di erent and Vx] Vx 6= I . Previous results for nonsingular equations require that all Vx 2 @B F (x) have full rank and VxVx?1 = Vx?1Vx = I . We develop new techniques for considering singular nonsmooth equations. It is noteworthy that the solution to which our method converges (in both smooth and nonsmooth cases) need not be the original solution, and our method applies even when the solution set is locally a manifold of positive dimension. In Section 5 we illustrate the singularity issue by numerical examples. We give numerical results for computing three examples of the NCP by methods (1.9) and (1.10) via a system of smooth equations (1.7) and a system of nonsmooth equations (1.8), respectively.
2. De nitions and lemmas on outer inverses and semismooth functions
Outer inverses of linear operators play a pivotal role in the formulation and convergence analysis of the iterative methods studied in this paper. Their role is derived from projectional properties of outer inverses and more importantly from perturbation and stability analysis of outer inverses. The strategy is based on Banach-type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses (to a given bounded linear operator) admits selections that behave like bounded linear inverses, in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operators. This strategy was rst used in [26] to generate adaptive constructs of outer inverses that would lead to sharp convergence results. Lemmas 2.1-2.4 below summarize important perturbation bounds and projectional properties of outer inverses which are used in the convergence analysis. For detailed proofs and related properties and references, see [26]. For de nition and properties of generalized inverses in Banach spaces, see [24] or [25]. Lemma 2.1(Banach-type lemma for outer inverses). Let A 2 L(X; Y ) and let A] 2 L(Y; X ) be an outer inverse of A. Let B 2 L(X; Y ) be such that jjA](B ? A)jj < 1. Then B ] := (I +A](B ?A))?1 A] is a bounded outer inverse of B with N (B ] ) = N (A] ) and R(B ] ) = R(A] ). Moreover,
jjB ] ? A]jj jj1A?(jjBA?](BA)?jjjjAA)jjjj ]
and
]
jjB ]Ajj 1 ? jjA](1B ? A)jj : 5
Lemma 2.2. Let A 2 L(X; Y ): If A] is a bounded outer inverse of A, then the following topological direct sum decompositions hold:
X = R(A]) N (A] A) Y = N (A] ) R(AA] ): Lemma 2.3. Let A; B 2 L(X; Y ) and let A] and B ] 2 L(Y; X ) be outer inverses of A and B , respectively. Then B ](I ? AA] ) = 0 if and only if N (A] ) N (B ] ): Lemma 2.4. Let A 2 L(X; Y ) and let Ay be a bounded generalized inverse of A. Let B 2 L(X; Y ) satisfy the condition jjAy(B ? A)jj < 1, and de ne B ] := (I + Ay(B ? A))?1Ay. Then B ] is a generalized inverse of B if and only if dimN (B ) = dimN (A) and
codimR(B ) = codimR(A): Note that if A and B are Fredholm operators with the same index, then the two dimensionality conditions are equivalent. Thus Lemma 2.4 and Theorem 3.4 below are not restricted to nite dimensional spaces. A simple example of a Fredholm operator is an operator of the form I + K , where K is a compact operator. For nonsmooth problems, we consider functions which are semismooth. We now recall two de nitions and an important property related to this class of functions. De nition 2.5. A function F : Rn ! Rm is said to be B-di erentiable at a point x if F has a one-sided directional derivative F 0(x; h) at x (see (1.5)) and
F (x + h) ? F (x) ? F (x; h) = 0: lim h!0 khk 0
(2:1)
We may write (2.1) as F (x + h) = F (x) + F 0(x; h) + o(k h k): De nition 2.6. A function F : Rn ! Rm is semismooth at x if F is locally Lipschitz at x and 0 2 lim 0 fV h g V
0! #
@F (x+th
h
h;t
)
0
exists for every h 2 Rn. Lemma 2.7 [31]. If F : Rn ! Rm is semismooth at x, then F is directionally di erentiable at x and for any V 2 @F (x + h);
V h ? F 0(x; h) = o(jjhjj): Shapiro [36] showed that a locally Lipschitzian function F is B-di erentiable at x if and only if it is directionally di erentiable. Hence F is B-di erentiable at x if F is 6
semismooth at x. For a comprehensive analysis of the role of semismooth functions see [21, 31, 33].
3. Local convergence for smooth equations
S (x; r) denotes the open ball in X with center x and radius r; and S(x; r) is its closure. For a xed A 2 L(X; Y ), we denote the set of nonzero outer inverses of A by
(A) := fB 2 L(Y; X ) : BAB = B; B 6= 0g: In this section we give two local convergence theorems for method (1.10) by using the following Kantorovich-type theorem (semilocal convergence). Theorem 3.1. Let F : D X ! Y be Frechet di erentiable. Assume that there exist an x0 2 D, F 0(x0)] 2 (F 0(x0 )) and constants ; K > 0 such that for all x; y 2 D the following conditions hold:
jjF 0(x0 )] F (x0)jj (3:1) jjF 0(x0 )] (F 0(x) ? F 0(y))jj K jjx ? yjj (3:2) h := K 21 ; S (x0; t ) D; (3:3) p where t = (1 ? 1 ? 2h)=K: Then the sequence fxk g de ned by method (1.10) with F 0(xk )] = (I + F 0(x0 )](F 0(xk ) ? F 0(x0 )))?1F 0(x0 )] lies in S (x0 ; t) and converges to a solution x of F 0(x0 )] F (x) = 0. (See Theorem 3.1 and Corollary 3.1 in [26].) Theorem 3.2. Let F : D X ! Y be Frechet di erentiable and assume that F 0(x) satis es a Lipschitz condition
jjF 0(x) ? F 0(y)jj Ljjx ? yjj; x 2 D: (3:4) Assume that there exists an x 2 D such that F (x ) = 0: Let p > 0 be a positive number such that S (x; p1 ) D. Suppose that the following condition holds: (a) there is a F 0(x )] 2 (F 0(x )) such that jjF 0(x )] jj p and for any x 2
1 ) the set (F 0 (x)) contains an element of minimal norm. S (x ; 3Lp 1 such that for any Then there exists a ball S (x ; r) D with 0 < r < 3Lp x0 2 S (x ; r), the sequence fxk g de ned by method (1.10) with
F 0(x0 )] 2 argminfjjB jj : B 2 (F 0(x0 ))g
(3:5)
and with F 0(xk )] = (I + F 0(x0 )](F 0(xk ) ? F 0(x0 )))?1F 0(x0 )] converges quadratically to x 2 S (x0; Lp1 ) \ fR(F 0(x0)] ) + x0 g, which is a solution of F 0(x0 )] F (x) = 0: Here R(F 0(x0 )] ) + x0 := fx + x0 : x 2 R(F 0(x0 )])g: 7
Proof. Let
1 and = 4 : r = 3Lp 27Lp2 We rst prove that there exists a ball S (x; r) D; 0 < r r such that for any x0 2 S (x ; r); all conditions of Theorem 3.1 hold. Since F is continuous at x , there exists a ball S (x ; r) D; 0 < r r, such that for any x 2 S (x ; r), jjF (x)jj < : From (3.4), there is a F 0(x)] 2 (F 0(x )) such that jjF 0(x )] (F 0(x) ? F 0(x ))jj pLr < 1: By Lemma 2.1, we have that F 0(x)] = (I + F 0(x)] (F 0(x) ? F 0(x )))?1F 0(x )] is an outer inverse of F 0(x) and 0 )] jj jjF 0(x)] jj jj1F?(xpLr 1 ?ppLr =: : Hence for any x0 2 S (x; r), the outer inverse F 0(x0 )] 2 argminfjjB jj : B 2 (F 0(x0 ))g satis es jjF 0(x0 )]jj . Let K = 32 L: Then for x; y 2 D,
jjF 0(x0)] (F 0(x) ? F 0(y))jj jjF 0(x) ? F 0(y)jj K jjx ? yjj;
and
h = K jjF 0(x0)] F (x0)jj 23 L 2 9(1 ?2pLr)2 21 :
p
Furthermore, for any x 2 S (x0 ; t) with t = (1 ? 1 ? 2h)=K , we have 1 + 2 1 + 2(1 ? Lpr) 1 : jjx ? xjj jjx0 ? x jj + jjx0 ? xjj 3Lp 3L 3Lp 3Lp Lp This implies S (x0; t ) S (x; Lp1 ) D. Hence all conditions of Theorem 3.1 hold at x0 . Thus the sequence fxk g lies in S (x0 ; t) and converges to a solution x of F 0(x0 )] F (x) = 0: Now we prove that the convergence rate is quadratic. Since F 0(xk )] = (I +F 0(x0)] (F 0(xk )?F 0 (x0)))?1F 0(x0 )] ; by Lemma 2.1, R(F 0(x0 )]) = R(F 0(xk )]). By xk+1 ? xk = F 0(xk )] F (xk ) 2 R(F 0(xk )] ); 8
we have
xk+1 2 R(F 0(xk )] ) + xk = R(F 0(xk?1)] ) + xk = R(F 0(x0 )] ) + x0 and x 2 R(F 0(xk )]) + xk+1 for any k 0. This implies that
x 2 R(F 0(x0 )] ) + x0 = R(F 0(xk )]) + x0 and
F 0(xk )]F 0(xk )(x ? xk+1) = F 0(xk )] F 0(xk )(x ? x0 ) ? F 0(xk )]F 0(xk )(xk+1 ? x0 ) = x ? xk+1: From Lemma 2.3, we have F 0(xk )] = F 0(xk )] F 0(x0)F 0(x0 )] : Using F 0(x0 )] F (x) = 0 and N (F 0 (x0)] ) = N (F 0 (xk )]); we obtain F 0(xk )]F (x ) = 0 and
jjx ? xk+1jj = jjF 0(xk )] F 0(xk )(x ? xk+1 )jj = jjF 0(xk )] F 0(xk )(x ? xk + F 0(xk )] (F (xk ) ? F (x )))jj Z1 0 ] 0 = jjF (xk ) (F (xk )(x ? xk ) ? F 0(xk + t(xk ? x ))dt(x ? xk ))jj 0
Z1 0 ] 0 0 ] 0 = jjF (xk ) F (x0)jjjjF (x0 ) (F (xk ) ? F 0(xk + t(xk ? x ))dt)(x ? xk )jj 0 1 K 1 ? Kt 2 jjx ? xk jj2:
Hence xk ! x quadratically.
2
Lemma 3.3. Let A 2 L(X; Y ); A 6= 0, where X and Y are nite dimensional normed spaces. Then the in mum of jjB jj over (A) is attained. Proof. Let A be a xed nonzero linear operator from X into Y . For any nonzero outer inverse B of A, we have jjB jj = jjBAB jj jjB jj2jjAjj; hence jjB jj jjA1 jj : Let :=inf fjjB jj : B 2 (A)g. There exists fBk g (A) such that lim kBk k = ; and since fBk g is bounded, it has a limit point B . Then Bk ABk = Bk and jjBk jj jjA1 jj . Hence BAB = B and jjB jj = . Thus (A) contains an element of minimal operator norm.
2
Theorem 3.4. Let F satisfy the assumptions of Theorem 3.2 except that condition (a) is replaced by the following condition: 1 ); (b) the generalized inverse F 0(x )y exists, kF 0(x )yk p and for any x 2 S (x ; 3Lp dimN (F 0 (x)) = dimN (F 0(x )) 9
and
codimR(F 0(x)) = codimR(F 0(x )): Then the conclusion of Theorem 3.2 holds with
F 0(x0)] 2 fB : B 2 (F 0(x0 )); jjB jj jjF 0(x0 )yjjg:
(3:6)
Proof. Condition (a) of Theorem 3.2 ensures that for any x 2 S (x ; r); 0 < r 1=3Lp, the outer inverse F 0(x)] 2 argminfkB k : B 2 (F 0(x))g satis es kF 0(x)]k p=(1 ? Lpr): Now we show that under condition (b), for any x 2 S (x ; r); 0 < r 1=3Lp; the outer inverse F 0(x)] 2 fB : B 2 (F 0 (x)); jjB jj jjF 0(x)yjjg satis es kF 0(x)] k p=(1 ? Lpr): From (3.4),
jjF 0(x )y(F 0(x) ? F 0(x ))jj pjjF 0(x) ? F 0(x )jj pLjjx ? x jj pLr < 1: By Lemma 2.4,
F 0(x)y = (I + F 0(x)y(F 0(x) ? F 0(x )))?1F 0(x )y is the generalized inverse of F 0(x). By Lemma 2.1, ) jj p =: : jjF 0(x)yjj jj1F?(xpLr 1 ? pLr 0 y
Hence for any x0 2 S (x ; r), the outer inverse
F 0(x0 )] 2 fB : B 2 (F 0(x0 )); jjB jj jjF 0(x0 )yjjg satis es jjF 0(x0 )]jj . By the same argument in the proof of Theorem 3.2, we can show the conclusion of Theorem 3.2 holds with
F 0(x0 )] 2 fB : B 2 (F 0(x0 )); kB k kF (x0)ykg:
2
Remark 3.5. Let X = Rn and Y = Rm . Then condition (a) of Theorem 3.2 holds
automatically. Condition (b) of Theorem 3.4 holds if and only if F 0(x) is of a constant 1 ). In the case of in nite dimensional spaces, condition (a) depends rank in S (x; 3Lp on the norm being used. Operator extremal properties of various generalized inverses have been studied by Engl and Nashed [11]. Remark 3.6. Rall [34] assumed that F 0(x )?1 exists, jjF 0(x)?1 jj p;
jjF 0(x) ? F 0(y)jj Ljjx ? yjj 10
and S (x; Lp1 ) D, and proved that there is a ball S (x ; r) such that Kantorovich conditions hold at each x0 2 S (x; r). Under Rall's conditions, all conditions of Theorem 3.4 hold. Therefore, Theorem 3.4 reduces to Rall's theorem for nonsingular equations. In [39] Yamamoto and Chen compared three local convergence balls for Newton-like methods for nonsingular equations. Their results also can be generalized to singular equations using the technique in the proof of Theorem 3.2.
4. Local convergence for nonsmooth equations
In this section, we consider method (1.9) for singular nonsmooth equations with X = Rn and Y = Rm. The discussion in this section is presented in nite dimensional spaces since for technical reasons we wish to con ne ourselves to the notion of the generalized derivative of locally Lipschitzian mappings. Furthermore, because we could not restrict jjVx ? Vy jj by jjx ? yjj for nonsmooth operators, Lemma 2.1 could not be used to construct Vk from V0 . A Kantorovich-type theorem is dicult to establish for local analysis of singular nonsmooth equations. For overcoming the diculty, we use a Mysovskii-type theorem [27] to give a local convergence theorem for singular nonsmooth equations. First, we give a Mysovskii theorem(semilocal convergence) for singular nonsmooth equations. Theorem 4.1. Let F : R] m ! Rn be locally Lipschitz. Assume that there exist x0 2 D; V0 2 @B F (x0 ), V0 2 (V0 ) and constants > 0 and 2 (0; 1) such that for any Vx 2 @B F (x); x 2 D, there exists an outer inverse Vx] 2 (Vx) satisfying N (Vx] ) = N (V0] ) and that for this outer inverse the following conditions hold : jjV0] F (x0)jj ; jjVy](F (y) ? F (x) ? Vx(y ? x))jj jjy ? xjj; if y = x ? Vx]F (x): (4:1) Let S := S (x0 ; r) D with r = =(1 ? ). Then the sequence fxk g de ned by (1.9) with Vk] satisfying N (Vk] ) = N (V0]) lies in S = S(x0; r) and converges to a solution x of V0] F (x) = 0 in S. Proof. First we show that the sequence de ned by method (1.9) lies in S . For k = 1, we have jjx1 ? x0 jj = jjV0]F (x0 )jj = r(1 ? ); and thus x1 2 S . Suppose now x1 ; x2; :::; xk 2 S: Let Vk] be an outer inverse of Vk 2 @B F (xk ) such that N (Vk]) = N (Vk]?1) = N (V0] ). Then by Lemma 2.3, we have Vk] (I ? Vk?1Vk]?1) = 0 and thus jjxk+1 ? xk jj = jjVk] F (xk )jj = jjVk](F (xk ) ? Vk?1(xk ? xk?1) ? Vk?1Vk]?1F (xk?1))jj = jjVk](F (xk ) ? Vk?1(xk ? xk?1) ? F (xk?1))jj jjxk ? xk?1 jj k jjx1 ? x0 jj k = r k (1 ? ): 11
Hence
jjxk+1 ? x0 jj
k X j =0
jjxj+1 ? xj jj
k X j =0
r j (1 ? ) r:
This proves that fxk g S: Hence for any positive integers k and p,
jjxk+p+1 ? xk jj
kX +p j =k
jjxj+1 ? xj jj
kX +p j =k
r j (1 ? ) r k :
So the method (1.9) converges to a point x 2 S . Since F is Lipschitz on S, jjVk jj is uniformly bounded on S. Thus by Lemma 2.3,
jjV ] V V ]F (xk )jj jjV0] F (x)jj = klim jjV ] F (xk )jj = klim !1 0 !1 0 k k ] jjjjV V ] F (x )jj = lim jjV ] jjjjV (x klim jj V k k k k k+1 ? xk )jj = 0: 0 !1 k!1 0
Therefore, V0]F (x ) = 0.
2
Remark 4.2. Suppose that m = n and all Vx 2 @F (x), x 2 S are nonsingular. Then we have Vx?1 2 (Vx) and N (Vx?1 ) = N (V0?1): Moreover x is a solution of
F (x) = 0. Hence Theorem 4.1 generalizes Theorem 3.3 in [33] to singular equations. Moreover our assumptions are weaker than assumptions of Theorem 3.3 in [33] in the nonsingular case. Theorem 4.3. Let F : Rn ! Rm be locally Lipschitz. Let p be a positive constant. Assume that there exist a ? 2 Rnm and an x 2 D such that ?F (x ) = 0 and for any V 2 @B F (x), there is an outer inverse V] 2 (V) satisfying N (V] ) = N (?) and k V] k p: Then there exists a positive number r such that for any x 2 S (x ; r) and any Vx 2 @B F (x) there is an outer inverse Vx] 2 (Vx) such that N (Vx] ) = N (?). Moreover assume that (4.1) holds for this outer inverse. Then there is a 2 (0; r=2] such that for any x0 2 S (x ; ) the sequence fxk g de ned by (1.9) with Vk] 2 (Vk ) and N (Vk]) = N (?) lies in S (x; r) and converges to a solution x of ?F (x) = 0. Furthermore, if F is semismooth at x and R(Vk] ) = R(V0]); then the convergence rate is superlinear. Proof. First we claim that for ^ 2 (0; 1=p) there is a ball S (x ; r) D with r > 0 such that for any Vx 2 @B F (x), x 2 S (x; r), we have
k Vx ? V k< ^; for a V 2 @B F (x ): (4:2) If (4.2) is not true, then there is a sequence fyk : yk 2 DF g with yk ! x, such that k rF (yk ) ? V k ^; for all V 2 @B F (x): (4:3) 12
By passing to a subsequence, we may assume that frF (yk )g converges to a V 2 @B F (x): This contradicts (4.3). Hence (4.2) holds. Suppose that x 2 S (x ; r). Then there is a V 2 @B F (x) such that (4:4) k Vx ? V k ^ < p1 : From the assumptions of this theorem, there is a V] 2 (V ) which satis es jjV]jj p and N (V]) = N (?): Hence from Lemma 2.1, we have that Vx] = (I + V](Vx ? V ))?1V] is an outer inverse and N (Vx] ) = N (V]) = N (?); R(Vx] ) = R(V]); k Vx] V k 1 ?1 ^p =: :
Since F is continuous and kV] k is bounded, for any 2 (0; r(1 ? )=2 ], there exists a 2 (0; r=2) such that for any x 2 S (x ; ); jjV]F (x)jj < . Therefore for any x0 2 S (x ; ), V0 2 @B F (x0 ), there exists V 2 @B F (x ) such that (4.2) holds. Moreover there exist V0] 2 (V0) and V] 2 (V ) such that N (V0] ) = N (V]) = N (?) and jjV0] F (x0)jj = jjV0]VV] F (x0)jj jjV0] VjjjjV] F (x0)jj r(1 ? )=2: Since x0 2 S (x ; r=2), we have S (x0; r=2) S (x ; r): Hence all conditions of Theorem 4.1 hold with = r(1 ? )=2. By Theorem 4.1 for any x0 2 S (x; ), the sequence fxk g de ned by method (1.9) with Vk] satisfying (4.1) lies in S (x ; r) and converges to a solution x of V0]F (x) = 0. Since N (V0]) = N (?), we have ?F (x) = 0. Now, we prove the convergence rate is superlinear. Since xk 2 S (x ; r), there is a V] 2 (V ) such that N (Vk] ) = N (V]) and kVk ? Vk < 1=p: From Lemma 2.3, Vk] = Vk]V V] and kVk]k kVk] VkkV] k p; i.e. kVk] k is bounded. Since N (Vk] ) = N (?) and ?F (x) = 0, Vk]F (x ) = 0. Since R(Vk] ) = R(V0] ), xk+1 ? x 2 R(Vk]): Hence k xk+1 ? x k = jjVk] Vk (xk+1 ? x )jj = jjVk] Vk (xk ? Vk](F (xk ) ? F (x )) ? x)jj = jjVk] (Vk (xk ? x ) ? F (xk ) + F (x ))jj k Vk] k (k F (xk ) ? F (x) ? F 0(x ; xk ? x ) k + k Vk (xk ? x) ? F 0(x ; xk ? x ) k): 13
By (2.1) and Lemma 2.7, we have
k F (x) ? F (x) ? F 0(x ; x ? x ) k= o(k x ? x k) and This implies
k Vx(x ? x) ? F 0(x ; x ? x ) k= o(k x ? x k): k xk+1 ? x k= o(k xk ? x k):
Hence method (1.9) converges to x superlinearly.
2
Remark 4.4. Suppose that there is a V0] 2 (V0) satisfying N (V0]) = N (?). If there is a Vk 2 @B F (xk ) such that kV0](Vk ? V0 )k < 1; (4:5) then Vk] = (I + V0] (Vk ? V0))?1V0] is an outer inverse of Vk with N (Vk]) = N (V0] ) = ] ]
N (?) and R(Vk ) = R(V0 ). Because of the nonsmoothness, we do not have the bene t of a condition such as (3.4) to ensure (4.5). This is a major di erence between smooth and nonsmooth equations. Remark 4.5. Superlinear convergence results in [31, 33] assume that all V 2 @B F (x) are nonsingular. Under this assumption, x is the unique solution of F (x) = 0 in a neighbourhood N of x and it was only shown that kxk+1 ? x k = o(kxk ? x k) for a sequence fxk g N. In singular case, the set of solutions may be locally a manifold of positive dimension. There is no neighbourhood N of x such that method (1.9) converges to x for any close starting point x0 2 N. Condition (4.1) is imposed in Theorem 4.3 to guarantee the existence of a neighbourhood N such that method (1.9) converges to a solution of V0]F (x) = 0 for any starting point x0 2 N. The following corollary shows that condition (4.1) can be replaced by special choices of starting points. Corollary 4.6. Let p be a positive number, ? be an nm matrix and x be a solution of ?F (x) = 0. Suppose that F is semismooth at x and for all V 2 @B F (x ); there exists a V] 2 (V), such that N (V]) = N (?) and jjV]jj p: Then method (1.9) with Vk] satisfying N (Vk] ) = N (?), R(Vk] ) = R(V0]) and x0 2 R(V0] )+ x is convergent to x superlinearly in a neighbourhood of x . Proof. From the proof of Theorem 4.3, we have that there is a ball S (x; r), such that for any Vx 2 @B F (x); x 2 S (x ; r), there is an outer inverse Vx] 2 (Vx) satisfying N (Vx] ) = N (?) and jjVx] Vjj for a V 2 @B F (x ). Choosing x0 2 R(V0] ) + x ; we have xk+1 ? x 2 R(Vk]) since R(Vk]) = R(V0] ) and xk+1 ? xk 2 R(Vk]). Then we 14
can show the superlinear convergence by the last part of the proof of Theorem 4.3 (superlinear convergence). 2 Remark 4.7. If we assume that m = n and all V 2 @B F (x ) are nonsingular, then we can take ? = I 2 Rnn. Furthermore there is a neighbourhood N of x, such that for any x 2 N, all Vx 2 @B F (x) are nonsingular. Then for any xk 2 N, we can take Vk] = Vk?1 which satis es N (Vk] ) = N (?) = f0g, R(Vk]) = R(V0]) = Rn and xk 2 R(V0] ) + x: Hence Theorem 4.3 and Corollary 4.6 generalize the local convergence theorems given in [31, 33]. Remark 4.8. In Theorem 4.3 and Corollary 4.6, Vk] should be chosen to satisfy N (Vk] ) = N (V0] ) and R(Vk] ) = R(V0] ) at each step of (1.9). There exists an outer inverse Vk] such that N (Vk]) = N (V0] ) and R(Vk] ) = R(V0]) if and only if N (V0] ) and R(Vk V0]) are complementary subspaces of Rm. If such an outer inverse exists, it is unique [2; p.62]. Now we give a method for numerical construction of such an outer inverse based on V0] and Vk . Let s =rank(V0] ) and rank(Vk ) s. Let U be a matrix whose columns form a basis for R(V0]), and let W be a matrix whose rows form a basis for the orthogonal complement of N (V0] ). Then WVk U is an s s matrix with rank s, so it is invertible. Let Vk] := U (WVk U )?1 W: Then Vk] is an outer inverse of Vk with N (Vk]) = N (V0] ) and R(Vk]) = R(V0] ).
5. Examples and numerical experiments
In this section we give methods for constructing outer inverses which are needed in the theorems and illustrate our results with three examples from nonlinear complementarity problems. The rst example compares the theorems given in this paper with earlier results. The second example shows how outer inverses apply while generalized inverses fail for Newton's method. The third example tests the methods (1.9) and (1.10) for problems with di erent dimensions. The performance of algorithms is given by using Matlab 4.2c on a Sun 2000 workstation.
5.1 Calculation of outer inverses
Methods for constructing outer inverses of a given matrix or a linear operator are given in [2, 23, 24, 26]. For the case of an m n matrix A with rank r > 0, we have a method using singular value decomposition (SVD). Let A = V U T , where is a diagonal matrix of the same size as A and with nonnegative diagonal elements in decreasing order, V and U are m m and n n orthogonal matrices, respectively. Let > 0 be a computational error control, ]s=diag(v1; v2; :::; vs; 0; ::0) 2 Rnm where s min(m; n) and ( ?1 i;i > jj; i;i ; vi = 0; otherwise: Then U ]s V T is an outer inverse of A. 15
As we know orthogonal-triangular decomposition (QR) is less expensive than SVD. Here we give a new method to construct an outer inverse of A by using QR. Let A = QR be a factorization, where Q is an m m orthogonal matrix, R is an m n upper triangular matrix of the form ! R R 11 12 R= 0 0 and R11 is an r r matrix of rank r. Then A] = R] QT is an outer inverse of A, where ?1 0 ! R R] = 0 11 0 ; and R11 is an r r matrix of rank r r. Remark 5.1 Outer inverses of a matrix A are more stable than the generalized inverses when some singular values of A are close to zero, because we can choose ] and R ] such that their elements are bounded. For details of the perturbation analysis that demonstrates stability of certain selections of outer inverses, see [22, 24, 26]. Remark 5.2 For the smooth case, we need not construct an outer inverse at each step but only at the starting point. In practice, method (1.10) is implemented in the following form: x1 = x0 ? F 0(x0)] F (x0) and for k 1, we let xk+1 = xk + d, where d is the unique solution of the linear system (I + F 0(x0 )](F 0(xk ) ? F 0(x0 )))d = ?F 0 (x0 )]F (xk ): Obviously, if m = n and F 0(xk ) is invertible, then the method reduces to F 0(xk )d = ?F (xk ).
5.2. Numerical examples
As we stated in Section 1, a nonlinear complementarity problem (NCP) can be formulated as a system of smooth equations by (1.7) and also as a system of nonsmooth equations by (1.8). We can solve the smooth equations (1.7) by method (1.10) and the nonsmooth equations (1.8) by method (1.9). The singularity occurs very often in solving these two systems. It is interesting to see how to overcome the singularity by methods (1.9) and (1.10) and Theorems 3.2 and 4.3. Example 1. We consider the following example [12]. Let f (x) = (1 ? x2; x1 )T : The solution set of the associated NCP is W = f(0; ); j 0 1g. For each x 2 W , the Jacobian of F^ de ned by (1.7) is ! ? 2(1 ? ) 0 0 F^ (x ) = ?2 0 16
which is singular. Hence previous local convergence theorems of Newton's method [9, 27, 34] are not applicable for this example. Now we apply Theorem 3.2 to this problem. We take x = (0; 0:5). Then ! a ? 1 ? a 0 ] ^ F (x ) = a ?1 ? a 2 (F 0(x )) for any a 2 (?1; 1). We take a = ?1, L = 4, and p = 1. Then we can show that there is a ball S (x ; r) with 0 < r < 0:5, such that jjF^ 0(x )]jj1 p and for any x 2 S (x; r); jjF^ 0(x) ? F^ 0(y)jj1 Ljjx ? yjj1. Hence all conditions of Theorem 3.2 hold. Now we consider the nonsmooth equation (1.8). For any x 2 W , ! 1 0 V = 1 0 2 @B F~ (x ): This implies F is not strongly BD-regular at all solutions in the sense of [31, 33]. Hence the local convergence theorems in [31, 33] are not applicable for this example. However, we can choose an outer inverse as ! 1 0 ] V = 0 0 2 (V): Take x = (0:0; 0:0). Then F is nonsmooth at x . There is a ball S (x ; r) with 0 < r < 0:5 such that for any x 2 S (x ; r), the generalized Jacobian is ! ! 1 0 1 0 V (x) = 1 0 or V (x) = 0 1 : (For determination of a Vx 2 @B F (x) see [7, 31].) Hence we can take outer inverses Vx] = V]. Note that this is a linear complementarity problem, all conditions of Theorem 4.3 hold. Furthermore, for any x0 = ( ; ) 2 S (x ; r), x1 = x0 ? V0] F~ (x0 ) = (0; ). If 0, then (0; ) 2 W . Example 2. The generalized inverse Ay of a matrix A is an outer inverse, but Ay may not be a good outer inverse for Newton's method. This example given by one of the referees illustrates that conditions of Theorem 4.3 and Corollary 4.6 fail when we use generalized inverses. However these conditions hold for a number of outer inverses. Consider the piecewise linear equation
F (x1 ; x2) = min(2x1 + x2 ? 2; ?2x1 + x2 ? 2): 17
The solution of F (x) = 0 is the union of the two rays:
f(x1; x2 ) : x1 0; x2 = ?2x1 + 2g [ f(x1 ; x2) : x1 0; x2 = 2x1 + 2g: This particular function, though nonsmooth, is well behaved in that its set of zeros is a 1-dimensional manifold, just as the solution of a linear equation in two variable is (usually) a line. For x1 0; let V = V1 = [2; 1] 2 @B F (x) and " # 2 = 5 y V1 = 1=5 : Similarly for x1 > 0, let V = V2 = [?2; 1] 2 @B F (x), " # ? 2 = 5 y V2 = 1=5 : It can be seen that for any staring point x0 = (x01 ; x02 ) such that x02 ?(1=2)x01 + 2 and x02 (1=2)x01 + 2, for instance x0 = (0; 0), the Newton's method de ned by
xk+1 = xk ? ViyF (xk );
where i = 1 if xk1 0 and i = 2 otherwise, converges linearly but not superlinearly to x = (0; 2). The convergence is only linear, because although N (V1y) = N (V2y) = f0g, we have R(V1y) 6= R(V2y). We also see kV2y(V1 ? V2)k = 8=5 > 1. Thus Lemma 2.1 cannot be applied; likewise (4.1) fails to hold in this case. Now we consider the use of outer inverses. It is easy to verify !
(V1 ) = f ; 2 + = 1; j j + j j 6= 0g and
!
(V2) = f ; ?2 + = 1; j j + j j 6= 0g: For any V2] 2 (V2 ) with < 1=4; kV2](V1 ? V2)k2 < 1: By Lemma 2.1, V1] = (I + V2] (V1 ? V2 ))?1V2] is an outer inverse of V1 with N (V1] ) = N (V2]) and R(V1] ) = R(V2] ). For instance, we choose !
?1=5 : 2 = 3=5
V] Then
V] 1
= (I + V ](V 2
?1 ] 1 ? V2 )) V2 =
18
!
?1 : 3
Table 1
n
(1.10) for smooth eq. k jjF (xk )]F (xk )jj cputime 50 13 2:83 10?15 31.85 100 12 1:25 10?14 144.15 ?10 200 8 3:10 10 586.97 ?14 350 14 4:91 10 4:92 103
(1.9) for nonsmooth eq. k jjVk]F (xk )jj cputime 5 1:0 10?16 3.97 4 1:0 10?16 15.18 4 1:0 10?16 137.20 4 1:0 10?16 577.95
Furthermore for the starting point x0 = (0; 0),
x1 = x0 ? V1]F (x0 ) =
!
?2 : 6
is a solution of F (x) = 0. Example 3. To compare methods (1.9) and (1.10), we randomly generate a NCP with f (x) = Ax2 + Bx + c; where A and B are n n matrices, c 2 Rn is a vector and x2 = (x2i ) 2 Rn. We rst randomly generate singular matrices A and B , and a nonnegative vector x which has some zero elements. Then we choose c such that x is a solution of a NCP with f (x) = Ax2 + Bx + c: The problem is randomly generated but with known solution characteristic and singularity, so we can test the eciency of methods (1.9) and (1.10). Table 1 summarizes computational results with di erent n. Also we choose n = 100, x0 = x +random vector and show jjF 0(xk )]F (xk )jj; jjVk] F (xk )jj and convergence rate jjF (xk+1)jj=jjxk+1 ? xk jj in Figure 1.
Concluding Remarks.
In this paper, we discussed local convergence of Newton's method for singular smooth and nonsmooth equations, respectively. These results generalize and extend earlier results on nonsingular smooth and nonsmooth equations. Singularity occurs in many areas of optimization and numerical analysis. Pang-Gabriel [29] mentioned that the singularity badly a ected the convergence of the NE/SQP method for the nonlinear complementarity problem in a number of numerical examples. The results in this paper present a strategy to treat singularity and to guarantee convergence of Newton's method and related iterative methods. Some of our results are also stronger than earlier results in the nonsingular case since they involve weaker assumptions.
Acknowledgements We are grateful to the referees for their constructive comments. 19
5
method (1.9)
2
convergence rate
||V(x)#F(x)||
4 3 2 1 0 1
2
3
1.5 1 0.5 0 1
4
method(1.9)
2
iteration k method (1.10)
600
60 40 20 0 0
5
10 iteration k
4
iteration k
convergence rate
||F (x)#F(x)||
80
3
400
200
0 0
15
method (1.10)
5
10 iteration k
15
Figure 1: Computational results for Example 3.
REFERENCES 1. A. Ben-Israel, A Newton-Raphson method for the solution of equations, J. Math. Anal. Appl., 15(1966), pp. 243-253. 2. A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley, New York, 1974. 3. B. Chen and P. Harker, A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14(1993), pp. 11681190. 4. X. Chen, On the convergence of Broyden-like methods for nonlinear equations with nondi erentiable terms, Ann. Inst. Statist. Math., 42(1990), pp. 387-401. 5. X. Chen and L. Qi, Parameterized Newton method and Broyden-like method for solving nonsmooth equations, Comp. Optim. Appl., 3(1994), pp. 157-179. 20
6. X. Chen and T. Yamamoto, On the convergence of some quasi-Newton methods for nonlinear equations with nondi erentiable operators, Computing, 48(1992), pp. 87-94. 7. X. Chen and T. Yamamoto, Newton-like methods for solving underdetermined nonlinear equations, J. Comp. Appl. Math., 55(1995), pp. 311-324. 8. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983. 9. J. E. Dennis, Jr and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cli s, NJ, 1983. 10. P. Deu hard and G. Heindl, Ane invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal., 16(1979), pp. 1-10. 11. H. W. Engl and M. Z. Nashed, New extremal characterizations of generalized inverses of linear operators, J. Math. Anal. Appl., 82(1981), pp. 566-586. 12. M. Ferris and S. Lucidi, Globally convergent methods for nonlinear equations, Preprint of Computer Sciences Department, University of Wisconsin-Madison, # 1030 (1991). 13. S. P. Han, J. S. Pang and N. Rangaraj, Globally convergent Newton methods for nonsmooth equations, Math. Oper. Res., 17(1992), pp. 586-607. 14. W. M. Hauler, A Kantorovich-type convergence analysis for the Gauss-Newton method, Numer. Math., 48(1986), pp. 119-125. 15. M. Heinkenschloss, C. T. Kelley and H. T. Tran, Fast algorithms for nonsmooth compact xed point problems, SIAM J. Numer. Anal., 29(1992), pp. 17691792. 16. C. M. Ip and J. Kyparisis, Local convergence of quasi -Newton methods for B-di erentiable equations, Math. Programming, 56(1992), pp. 71-89. 17. M. Kojima and S. Shindo, Extensions of Newton and quasi-Newton methods to systems of PC 1 equations, J. Oper. Res. Soc. Japan, 29(1986), pp. 352-374. 18. B. Kummer, Newton's method for non-di erentiable functions, in J. Guddat, B. Bank, H. Hollatz, P. Kall, D. Klatte, B. Kummer, K. Lommatzsch, L. Tammer, M. Vlach and K. Zimmerman, eds., Advances in Mathematical Optimization, Akademi-Verlag, Berlin, 1988, pp. 114-125. 21
19. O. L. Mangasarian, Equivalence of the complementarity problem to a system of nonlinear equations, SIAM J. Appl. Math., 31(1976), pp. 89-92. 20. O. L. Mangasarian and M. V. Solodov, Nonlinear complementarity as unconstrained and constrained minimization, Math. Programming, 62(1993), pp. 277-297. 21. R. Miin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15(1977), pp. 191-207. 22. R. H. Moore and M. Z. Nashed, Approximations to generalized inverses of linear operators, SIAM J. Appl. Math., 27(1974), pp. 1-16. 23. M. Z. Nashed, ed., Generalized Inverses and Applications, Academic Press, New York, 1976. 24. M. Z. Nashed, Generalized inverse mapping theorems and related applications of generalized inverses, In: Nonlinear equations in abstract spaces (V. Lakshmikantham, ed.), Academic Press, New York, 1978, pp. 217-252. 25. M. Z. Nashed, Inner, outer, and generalized inverses in Banach and Hilbert spaces, Numer. Funct. Anal. Optim., 9(1987), pp. 261-325. 26. M. Z. Nashed and X. Chen, Convergence of Newton-like methods for singular operator equations using outer inverses, Numer. Math., 66(1993), pp. 235-257. 27. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. 28. J-S. Pang, Newton's method for B-di erentiable equations, Math. Oper. Res., 15(1990), pp. 311-341. 29. J-S. Pang and S. A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem, Math. Programming, 60(1993), pp. 295-337. 30. J-S. Pang and L. Qi, Nonsmooth equations: motivation and algorithms, SIAM J. Optimization, 3(1993), pp. 443-465. 31. L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18(1993), pp. 227-244. 32. L. Qi and X. Chen, A globally convergence successive approximation method for severely nonsmooth equations, SIAM J. Control Optim., 33(1995), pp. 402418. 22
33. L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Programming, 58(1993), pp. 353-367. 34. L. B. Rall, A note on the convergence of Newton's method, SIAM J. Numer. Anal., 1(1974), pp. 34-36. 35. S. M. Robinson, Newton's method for a class of nonsmooth functions, SetValued Analysis, 2(1994), pp. 291-305. 36. A. Shapiro, On concepts of directional di erentiability, J. Optim. Theory Appl., 66(1990), pp. 477-487. 37. T. Yamamoto, On some convergence theorems for the Gauss-Newton method for solving nonlinear least squares problem, Proceedings of the Summer Symposium on Computation in Mathematical Sciences-Fusion of Symbolic and Numerical Computations, IIAS-SIS Fujitsu Ltd., 1987, pp. 94-103. 38. T. Yamamoto, Uniqueness of the solution in a Kantorovich-type theorem of Hauler for the Gauss-Newton method, Japan J. Appl. Math., 6(1989),pp. 77-81. 39. T. Yamamoto and X. Chen, Ball-convergence theorems and error estimates for certain iterative methods for nonlinear equations, Japan J. Appl. Math., 7(1990), pp. 131-143. 40. N. Yamashita and M. Fukushima, Modi ed Newton methods for solving semismooth reformulations of monotone complementarity problems, Math. Programming, to appear.
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