Applied Mathematics E-Notes, 12(2012), 44-52 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
A Multipoint Iterative Method For Semistable
Solutions
Steeve Burnet, Célia Jean-Alexis and Alain Pietrusy
Received 28 June 2011
Abstract
This paper deals with variational inclusions of the form : 0 2 '(z) + F (z)
where ' is a single-valued function admitting a second order Fréchet derivative
and F is a set-valued map from Rq to the closed subsets of Rq . In order to approximate a solution z of the previous inclusion, we use an iterative scheme based
on a multipoint method. We obtain, thanks to some semistability properties of
z, local superquadratic or cubic convergent sequences.
1
Introduction
This paper is devoted to the study of a multipoint iterative method for approximating
a solution of the variational inclusion
0 2 '(z) + F (z);
(1)
where ' is a single-valued function and F is a set-valued map.
Variational inclusions are an abstract model of a wide variety of variational problems including linear and non-linear complementarity problems, systems of non-linear
equations, variational inequalities. In the last decade, several iterative methods for
solving the inclusion (1) have been introduced. These methods consist in generating
an iterative sequence (zn ) obtained by subsequently solving implicit subproblems of the
form 0 2 A(zn ; zn+1 ) + F (zn+1 ) where A denotes some approximation of the mapping
'.
Dontchev [7, 8] associates to (1) a Newton-type method based on a partial linearization. Inspired and motivated by his works, various authors proved the convergence of
some methods based on di¤erent techniques such as one or second order Taylor’s expansion, interpolation formula and a multipoint formula given in [23]. For more details
on these methods, the reader could refer to [6, 12, 13, 14, 15, 17, 20]. Let us point out
that all these methods have been studied when a metric regularity property is satis…ed
for the set-valued map (' + F ) 1 or one of its approximation. For more details on this
property, the reader could refer to [1, 2, 9, 10, 11, 18, 19, 22].
Mathematics Subject Classi…cations: 49J53, 47H04, 65K10.
LAMIA, EA4540, Université des Antilles et de la Guyane, Département de Mathématiques et Informatique, Campus de Fouillole, 97159 Pointe-à-Pitre, France
y Laboratoire
44
Burnet et al.
45
In this paper, the function ' is de…ned from Rq to Rq and F is a set-valued map from
R to the closed subsets of Rq : Our aim is to study inclusion (1) using an assumption
which is directly connected to a solution : the semistability concept. This concept has
been introduced by Bonnans [3] for variational inequalities. A solution z of a variational
inclusion is said to be semistable if, given a small perturbation on the left-hand side,
a solution z of the perturbed variational inclusion that is su¢ ciently close to z is such
that the distance of z to z is of the order of the magnitude of the perturbation.
Studies using such property has recently been made in [4, 5] for methods based on
the second order Fréchet derivative. Following these works, we consider the relation
q
0 2 '(zk ) +
N
X
ai Mki (zk+1
zk ) + F (zk+1 );
(2)
i=1
where Mki is a q q matrix satisfying some conditions and we prove the convergence
of this procedure under the semistability property. We can observe that if Mki =
'0 zk + i (zk+1 zk ) , we get the method proposed in [6] and if N = 2; a1 = a2 = 21 ;
1 = 0 and 2 = 1; we obtain the method introduced in [4].
The rest of the paper is organized as follows. In section 2, we collect a number of
basic de…nitions regarding semistability of solutions and regularity for set-valued maps
that we will need afterwards. Then, in section 3, we study the behaviour of the method
(2) and we give a classical problem which could be treated with our method.
In this paper, for simplicity reasons, all the norms are denoted by k:k.
2
Background Materials
Here, we recall the concept of semistability introduced by Bonnans in [3].
DEFINITION 1. A solution z of (1) is said to be semistable if c1 > 0 and c2 > 0
exist such that, for all (z; ) 2 Rq Rq , solution of 2 '(z) + F (z); and kz zk c1 ,
then kz zk c2 k k.
Note that a su¢ cient condition for semistability is the strong regularity of Robinson
[21]. Recently, Izmailov and Solodov in [16] used this concept in order to study the
convergence of Inexact Josephy-Newton method for solving generalized equations. We
will also need the following Hölder-type property.
DEFINITION 2. Let ' : Rq ! Rq be a function. One says that ' satis…es a
Hölder-type condition on a neighborhood of z if
9K > 0;
Note that when
3
2]0; 1]; k'(x)
'(y)k
K kx
yk ; 8 x; y 2 :
= 1, we have the Lipschitz condition for '.
Convergence Analysis
Our purpose in this section is to provide an iterative procedure for solving (1) and to
show how the semistability property can be an e¢ cient tool to estimate the rate of
convergence of the method (2).
46
A Multipoint Iterative Method for Semistable Solutions
First of all, we consider the approximation
(u; v) = '(u) +
N
X
of ' de…ned by
ai '0 u +
i (v
u) (v
u)
(3)
i=1
and then we introduce the algorithm:
given any starting point z0 in some neighborhood of z which is a solution of (1),
for k = 0; 1; : : :, while zk does not satisfy (1), choose k (:) an approximation of
PN
(zk ; ) de…ned by k (z) = '(zk ) + i=1 ai Mki (z zk );
compute a solution zk+1 of
02
k (z)
+ F (z):
(4)
The main result of this study reads as follows.
THEOREM 1. Let z be a semistable solution of (1) and let (zk ) be a sequence
computed by (2) which converges towards z. We suppose that '00 is a locally Lipschitz
function. Then :
(i) if
(zk ; zk+1 )
cally.
(ii) if
(zk ; zk+1 )
2
then (zk ) converges superquadrati-
3
then (zk ) converges cubically.
k (zk+1 )
= o kzk+1
zk k
k (zk+1 )
= O kzk+1
zk k
To prove Theorem 1, we will need the following lemma:
LEMMA 1. Let ' : Rq ! Rq be a function admitting a second order Fréchet
derivative which is L-Lipschitz on and let be de…ned by (3). One has the following
inequality
N
L LX
3
+
jai j 2i kv uk :
(5)
k'(v)
(u; v)k
6
2 i=1
PROOF. We have
k'(v)
(u; v)k =
'(v)
'(u)
N
X
ai '0 u +
i (v
u) (v
u)
i=1
=
'(v)
'(u)
1
+ '00 (u)(v
2
'0 (u)(v
2
u)
N
X
u)
ai '0 u +
'(u)
i (v
u)2 + '0 (u)(v
u) (v
i=1
Let
A1 = '(v)
1 00
' (u)(v
2
'0 (u)(v
u)
1 00
' (u)(v
2
u)2
u) :
u)
Burnet et al.
47
and
1
u) + '00 (u)(v
2
A2 = '0 (u)(v
Since '00 is L-Lipschitz on
A2
=
=
N
X
ai
'0 (u)
ai
Z
ai '0 u +
L
6
N
X
u)2
=
i
=
N
X
ai
'0 u +
i (v
'00 u +
i t(v
u)
i=1
1
u) (v
1
'00 u +
jai j
2
i L kv
i (v
u) (v
1
u) + '00 (u)(v
2
(v
i t(v
u) (v
0
i=1
N
X
i
Z
uk : Moreover,
3
uk
1
L
2
tdt
0
N
X
i=1
u)2
1
u)dt + '00 (u)(v
2
Z 1
N
X
u)2 dt +
ai i
'00 (u)(v
u)(v
2
i
jai j
kv
u)2
u)2 dt
0
i=1
Z
u)
i=1
0
i=1
u) :
u) (v
3
kv
ai '0 u +
i=1
N
X
i (v
i=1
, we obtain A1
1
u) + '00 (u)(v
2
'0 (u)(v
N
X
u)2
3
uk :
And then
N
k'(v)
L LX
+
jai j
6
2 i=1
(u; v)k
2
i
kv
3
uk :
PROOF of THEOREM 1. We write (4) as
rk 2 '(zk+1 ) + F (zk+1 );
(6)
with
rk :=
k
+ (zk+1 ; zk );
where k = (zk ; zk+1 )
(zk+1 ; zk ) = '(zk+1 )
k (zk+1 ) and
2
(5), we get k (zk+1 ; zk )k = o(kzk+1 zk k ): Then krk k = o(kzk+1
Since z is semistable, we get kzk+1 zk = O(krk k): Then
kzk+1
zk = o(kzk+1
2
zk + 2 kzk+1
zk kzk
zk + kzk
(zk ; zk+1 ): Using
2
zk k ):
2
zk );
i.e.,
0
=
=
lim
k!1
kzk+1
kzk+1
zk + 2 kzk+1
1
kzk+1
zk + 2 kzk
lim
k!1
2
zk
zk kzk
zk +
zk + kzk
kzk zk2
kzk+1 zk
:
2
zk
48
A Multipoint Iterative Method for Semistable Solutions
Since zk ! z; the latter relation implies that limk!1
kzk+1
kzk zk2
kzk+1 zk
= +1; i.e.,
2
zk = o(kzk
zk )
which proves (i).
Similarly for (ii), with (5), we get
3
krk k = O(kzk+1
zk k ):
Thanks to the semistability of z, we obtain
kzk+1
3
zk = O(kzk+1
kzk+1 zk k
kzk zk
Since (zk ) converges superquadratically then
kzk+1
zk k ):
zk = O(kzk
! 1, and then
3
zk );
which completes the proof.
Let us extend this study to the case where the second order Fréchet derivative '00
of the function ' satis…es a Hölder-type condition.
LEMMA 2. Let ' : Rq ! Rq be a function admitting a second order Fréchet
derivative and let
be de…ned by (3). If '00 satis…es a Hölder-type condition with
constants and L on , one has, for all u; v 2 , the following inequality
k'(v)
L
L
+
(1 + )(2 + ) 1 +
(u; v)k
N
X
i=1
jai j
1+
i
kv
uk
2+
:
(7)
PROOF. The proof is similar to the proof of Lemma 1. We write
k'(v)
(u; v)k
'(u)
'0 (u)(v
A1 + A2
with
A1 = '(v)
1 00
' (u)(v
2
u)
u)2
and
A2 = '0 (u)(v
1
u) + '00 (u)(v
2
u)2
N
X
ai '0 u +
i (v
Since '00 satis…es a Hölder-type condition with constants
A1
u) (v
u) :
i=1
L
kv
(1 + )(2 + )
2+
uk
and L on
, we obtain
:
Moreover,
A2
N
X
i=1
jai j
1+
i
L kv
2+
uk
Z
0
1
t dt
L
1+
N
X
i=1
jai j
1+
i
kv
uk
2+
:
Burnet et al.
49
Then
k'(v)
L
L
+
(1 + )(2 + ) 1 +
(u; v)k
N
X
i=1
jai j
1+
i
kv
2+
uk
:
THEOREM 2. Let z be a semistable solution of (1) and let (zk ) be a sequence
computed by (2) which converges towards z. If '00 satis…es a Hölder-type condition
with constants and L then :
(i) if
(zk ; zk+1 )
(ii) if (zk ; zk+1 )
ically.
1+
k (zk+1 )
= o kzk+1
zk k
k (zk+1 )
= O kzk+1
zk k
then (zk ) converges superlinearly;
2+
then (zk ) converges superquadrat-
PROOF. The proof is similar to that of Theorem 1, but using Lemma 2 instead of
Lemma 1.
As an illustration of our results let us consider the standard following nonlinear
programming problem :
minimize f (x)
(8)
subject to
gi (x) 0;
gj (x) = 0;
8i 2 I
8i 2 J
where I; J are a partition of f1; :::; pg; f : Rn ! R is twice continuously di¤erentiable
on Rn while the functions gi : R ! R; i = 1; :::; p are di¤erentiable on Rn and are
twice di¤erentiable in a neighborhood of a solution x of (8).
To problem (8) is associated the …rst order optimality condition (in which 2 Rp ):
rf (x) + rg(x)
=0
gi (x) 0; 8i 2 I; gj (x) = 0; 8j 2 J;
0;
I
i gi (x)
= 0;
8i 2 I:
(9)
As observed in [3], we may embed (9) into (2) in putting z = (x; );
'(x; ) =
rf (x) + rg(x)
g(x)
and F (x; ) = N (x; ) = f0g N ( ) where = f 2 Rp ;
N ( ) denotes the normal cone to at the point i.e.
N ( )=
; if
2
=
f 2 Rp ; J = 0;
I
0;
i
= 0 if
I
i
0; 8i = 1; :::; pg and
> 0; 8i 2 Ig
:
The corresponding variational inequality can be written in the following way:
rf (x) + rg(x)
g(x) 2 N ( )
=0
:
(10)
50
A Multipoint Iterative Method for Semistable Solutions
Thanks to a result given in Bonnans [3] and using a similar reasoning, we obtain
that the semistability of (10) is equivalent to :
8
= 0;
< (y; ) = 0 is the unique solution of H(x; )y + rg(x)
y 2 C; I 0 0;
=
0
if
g
(x)
<
0;
8i
2
I;
(11)
i
i
:
0
rg
(x)y
=
0;
8i
2
I
i
i
where
H(x; ) = r2 f (x) +
p
X
i=1
ir
2
gi (x);
I = fi 2 I; gi (x) = 0g;
I 0 = fi 2 I;
I + = fi 2 I;
i
i
= 0g;
> 0g
and
C = fy 2 Rn ; rgI (x)y
0; rgJ (x)y = 0; rgI + (x)y = 0g:
For more details, the reader can refer to [3].
Now, let us apply the method (2) presented in this paper to (10). The subproblem
to be solved at the step k is :
8
>
k+1 = 0
< rf (xk ) +pH(xk ; k )(xk+1 xk ) + rg(xk )
X
k
ai rg(xk + i (xk+1 xk ))(xk+1 xk ) 2 N ( k+1 ):
>
: g(x ) +
i=1
As the evaluation of rg(xk ) is already necessary in order to evaluate '(xk ; k ); the
only part of the Jacobian that perhaps needs to be approximated is H(xk ; k ): We then
obtain the following algorithm:
given any starting point (x0 ;
if (xk ;
(xk+1 ;
k)
0)
2 Rn
Rp
is not solution of (10), choose Mk ; an n
solution of
n matrix, compute the
k+1 )
8
>
k+1 = 0
< rf (xk ) +pMk (xk+1 xk ) + rg(xk )
X
ai rg(xk + i (xk+1 xk ))(xk+1 xk ) 2 N (
>
: g(xk ) +
k+1 ):
i=1
When Mk = H(xk ; k ); by applying Theorem 1 and the relation (11),we obtain the
convergence of the sequence computed by (2) with Mki = rg(xk + i (xk+1 xk )).
Acknowledgment. We would like to thank the referee for valuable suggestions
that enabled us to improve the presentation of this paper.
Burnet et al.
51
References
[1] J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems,
Math. Oper. Res., 9(1984), no. 1, 87–111.
[2] J. P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhauser, Boston, 1990.
[3] F. Bonnans, Local analysis of Newton-type methods for variational inequalities
and nonlinear programming, Appl. Math. Optim., 29(1994), 161–186.
[4] S. Burnet, C. Jean-Alexis and A. Piétrus, An iterative method for semistable
solutions, RACSAM, 105(1)(2011), 133–138.
[5] S. Burnet and A. Piétrus, Local analysis of a cubically convergent method for
variational inclusions, Appl. Mat., 38(2)(2011), 183–191.
[6] C. Cabuzel and A. Piétrus, Solving variational inclusions by a method obtained
using a multipoint iteration formula, Rev. Mat. Comput., 22(1)(2009), 63–74.
[7] A. L. Dontchev, Local analysis of a Newton-type method based on partial linearization, Lectures in Appl. Math., 32, Amer. Math. Soc., Providence (1996).
[8] A. L. Dontchev, Local convergence of the Newton method for generalized equations. C. R. Acad. Sci. Paris Seris I Math., 322(4)(1996), 327–331.
[9] A. L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity, J. Convex Anal., 13(2)(2006), 2, 281–297.
[10] A. L. Dontchev and R. T. Rockafellar, Characterizations of strong regularity for
variational inequalities over polyhedral convex sets, SIAM J. Optim., 6(4)(1996),
1087–1105.
[11] A. L. Dontchev and R. T. Rockafellar, Regularity and conditioning of solutions
mappings in variational analysis, Set-Valued Anal., 12(1-2)(2004), 79–109.
[12] M. H. Geo¤roy, S. Hilout and A. Piétrus, Acceleration of convergence in Dontchev’s
iterative method for solving variational inclusions, Serdica Math. J., 29(1)(2003),
45–54.
[13] M. H. Geo¤roy, S. Hilout and A. Piétrus, Stability of a cubically convergent
method for generalized equations, Set-Valued Analysis, 14(1)(2006), 41–54.
[14] M. H. Geo¤roy, C. Jean-Alexis and A. Piétrus, A Hummel-Seebeck type method
for variational inclusions, Optimization, 58(4)(2009), 389–399.
[15] M. H. Geo¤roy and A. Piétrus, A superquadratic method for solving generalized
equations in the Holder case, Ricerce Mat., 52(2)(2003), 231–240.
[16] A. F. Izmailov and M. V. Solodov, Inexact Josephy-Newton framework for generalized equations and its applications to local analysis of Newtonian methods for
constrained optimization, Comput. Optim. Appl., 46(2)(2010), 347–368.
52
A Multipoint Iterative Method for Semistable Solutions
[17] C. Jean-Alexis, A cubic method without second order derivative for solving variational inclusions, C. R. Acad. Bulgare Sci., 59(12)(2006), 1213–1218.
[18] B. S. Mordukhovich, Complete characterization of openness metric regularity and
Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc., 340(1)(1993),
1–35.
[19] B. S. Mordukhovich, Stability theory for parametric generalized equations
and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc.,
343(2)(1994), 609–657.
[20] A. Piétrus, Generalized equations under mild di¤erentiability conditions, Rev. R.
Acad. Cienc. Exactas Fis. Nat., 94(1)(2000), 15–18.
[21] S. M. Robinson, Strongly regular generalized equations, Math. Oper. Res.,
5(1)(1980), 43–62.
[22] R. T. Rockafellar and J. B. Wets, Variational Analysis, Springer-Verlag, Berlin,
1998.
[23] J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing
Company, New-York, 1985.