Iterative proximal regularization method for finding a saddle point in the semicoercive Signorini...

1932

ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2006, Vol. 46, No. 11, pp. 1932–1939. ©

MAIK “Nauka

/Interperiodica” (Russia), 2006.Original Russian Text © G. Woo, S. Kim, R.V. Namm, S.A. Sachkoff, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46,No. 11, pp. 2024–2031.

Iterative Proximal Regularization Method for Finding a Saddle Point in the Semicoercive

Signorini Problem

G. Woo

a

, S. Kim

a

, R. V. Namm

b

, S. A. Sachkoff

b

a

Changwon National University, Changwon, 641773, South Korea

b

Pacific Ocean State University, Tikhookeanskaya ul. 136, Khabarovsk, 680035 Russiae-mail: [email protected]

Received May 2, 2006

Abstract

—An algorithm for seeking a saddle point for the semicoercive variational Signorini inequalityis studied. The algorithm is based on an iterative proximal regularization of a modified Lagrangian func-tional.

DOI:

10.1134/S0965542506110091

Keywords:

Signorini problem for a variational inequality, modified Lagrangian functional, Uzawa algo-rithm, proximal regularization

1. INTRODUCTIONApproximate methods for solving variational inequalities in mechanics based on classical duality

schemes assume, as a rule, that the functionals to be minimized are strongly convex. For semicoercive vari-ational inequalities, the strong convexity of functionals holds only on finite-codimensional subspaces of theoriginal space. Therefore, duality schemes with the classical Lagrange functional are inapplicable. A wayout is to use modified Lagrangian functionals (see [1]).

In this work, the Uzawa algorithm based on the method of iterative proximal regularization of a modifiedLagrangian functional is studied as applied to finding a saddle point in the semicoercive Signorini problem.

2. ITERATIVE PROXIMAL REGULARIZATION IN THE SEMICOERCIVE SIGNORINI PROBLEMConsider the problem

(1)

where

Ω ⊂

n

is a bounded domain with a sufficiently smooth boundary

Γ

,

f

∈

L

2

(

Ω

)

is a given function,

and

γ

w

∈

(

Γ

)

is the trace of

w

∈

(

Ω

)

on

Γ

.

In what follows, we assume that

which provides the existence and uniqueness of a solution to problem (1).Let

(

L

2

(

Γ

))

+

be the cone of functions in

L

2

(

Γ

)

that are nonnegative on

Γ

.Consider the Lagrangian functional

which is defined on

(

Ω

)

×

L

2

(

Γ

)

.

J v( ) 12--- ∇v 2 Ωd

Ω∫ fv Ω min,d

Ω∫–=

v G∈ w W21 Ω( ) : γw 0 a.e. on Γ≥∈ ,=

W21/2 W2

1

f Ωd

Ω∫ 0,<

L v l,( ) J v( ) lγv Γd

Γ∫–

12--- ∇v 2 Ωd

Ω∫ fv Ωd

Ω∫– lγv Γ,d

Γ∫–= =

W21

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS

Vol. 46

No. 11

2006

ITERATIVE PROXIMAL REGULARIZATION METHOD 1933

Definition 1.

A point

(

v

*,

l

*)

∈

(

Ω

)

×

(

L

2

(

Γ

))

+

is called a saddle point of

L

(

v

,

l

) if the following two-sided inequality is satisfied:

It was shown in [1] that, if the solution u* to semicoercive problem (1) belongs to (Ω), the Lagrangianfunctional has the unique saddle point (u*, ∂u*/∂n), where n is the outward unit normal vector to Γ.

In this work, the uniqueness of the saddle point is not assumed.

The modified Lagrangian functional [1] is defined as

where r > 0 = const.

We consider an iterative method for solving problem (1) that combines the modified Lagrangian func-tional with proximal regularization.

Given an arbitrary starting point (u0, l0) ∈ (Ω) × (Γ), the sequence (uk, lk) is generated as fol-lows:

(i) At the (k + 1)th iteration, the functional

is constructed and the point uk + 1 ∈ (Ω) is determined by the condition

(2)

where = (v), δk > 0, and < ∞.

(ii) The dual variable lk + 1 is corrected by the formula

Simultaneously with lk + 1, we introduce the point µk + 1 = (lk – rγ )+, k = 0, 1, 2, ….

If M(v, lk) is taken instead of Lk(v) at step (i), we obtain the well-known Uzawa algorithm for searching

for a saddle point with the modified Lagrangian functional [1]. The regularizing addition ||v – uk

provides the strong convexity of the minimized functionals Lk(v). This guarantees that the auxiliary prob-lems

(3)

have a unique solution, which can be found by efficient optimization methods. Such an algorithm for solvingfinite-dimensional convex optimization problems was considered in [2, 3].

Let the symbol (·, ·)0 denote the scalar product in L2(Ω). Let the scalar products in L2(Ω) × L2(Γ) and

(Ω) × L2(Γ) be defined as

W21

L v* l,( ) L v* l*,( ) L v l*,( ) v l,( )∀ W21 Ω( ) L2 Γ( )( )+.×∈≤ ≤

W22

M v l,( ) J v( ) 12r----- max 0 l rγv–, ( )2 l2–[ ] Γ,d

Γ∫+=

W21 W2

1/2

Lk v( ) M v lk,( ) 12--- v uk– L2 Ω( )

2v∀ W2

1 Ω( )∈+=

W21

uk 1+ uk 1+– W21 Ω( ) δk,≤

uk 1+ Lv W2

1 Ω( )∈÷argmin? δkk 1=

∞∑

lk 1+ lk rγ uk 1+–( )+max 0 lk rγ uk 1+–, .≡=

uk 1+

12--- ||L2 Ω( )

2

Lk v lk,( ) min, v W21 Ω( ),∈

W21

v 1 t1,( ) v 2 t2,( ),⟨ ⟩0 v 1 v 2,( )01r--- t1 t2,⟨ ⟩ L2 Γ( ),+=

v 1 t1,( ) v 2 t2,( ),⟨ ⟩1 v 1 v 2,( )W2

1 Ω( )

1r--- t1 t2,⟨ ⟩ L2 Γ( ),+=

1934

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 46 No. 11 2006

WOO et al.

and the corresponding norms have the form

Let zk = (uk, lk) and wk = ( , µk).

Theorem 1. Assume that the set of saddle points of the Lagrangian functional L(v, l) is nonempty. Then

the sequence wk generated by algorithm (i), (ii) is bounded in (Ω) × L2(Γ), and the sequence is

compact in (Ω).

Proof. The existence of the points wk and zk (k = 1, 2, …) is obvious. Let us show that the following ine-quality is satisfied for any k:

(4)

where z* = (v*, l*) is an arbitrary saddle point of L(v, l).

The condition that z* is a saddle point of L(v, l) is equivalent to the relations

(5)

(6)

Conditions (6) are equivalent to the equality

(7)

(see [4, 5]).

Inequality (5) is equivalent to the equality

where a(u, v) = · ∇vdΩ .

This relation and (7) imply

(8)

Obviously, the point satisfies the equality

(9)

Let v = in (8) and v = v* in (9). Summing these equalities, we obtain

v t,( ) 02

v L2 Ω( )2 1

r--- t L2 Γ( )

2 ,+=

v t,( ) 12

vW2

1 Ω( )2 1

r--- t L2 Γ( )

2 .+=

uk

W21 uk

W21

z* zk– 02

wk 1+ z*– 02

wk 1+ zk– 02,+≥

L v* l*,( ) L v l*,( ) v∀ W21 Ω( ),∈≤

γv* 0, l* 0, γv* l*⋅≥ ≥ 0 a.e. on Γ.=

l* l* rγv*–( )+=

a v* v v*–,( ) f v v*–,( )0– l*γ v v*–( ) Γd

Γ∫– 0 v∀ W2

1 Ω( ),∈=

∇uΩ∫

a v* v v*–,( ) f v v*–,( )0– l* rγv*–( )+γ v v*–( ) Γd

Γ∫– 0 v∀ W2

1 Ω( ).∈=

uk 1+

uk 1+ uk– v uk 1+–,( )0 a uk 1+v uk 1+–,( ) f v uk 1+–,( )0– lk rγ uk 1+–( )+γ v uk 1+–( ) Γd

Γ∫–+ 0=

v∀ W21 Ω( ).∈

uk 1+

uk 1+ uk– v* uk 1+–,( )0 a uk 1+v*– uk 1+

v*–,( ) l* rγu*–( )+ lk rγ uk 1+–( )+–[ ]γ uk 1+

v*–( ) Γd

Γ∫+=

= a uk 1+v*– uk 1+

v*–,( ) – µk 1+ l*–( )γ uk 1+v*–( ) Γd

Γ∫



i.e.,

(10)

While deriving (10), we used the well-known inequality

(see [4]). Then we have

Multiplying the second equality by 1/r and adding it to the first equality yields

This relation and (10) imply the formula

i.e., (4) is proved.

= a uk 1+v*– uk 1+

v*–,( ) 1r--- µk 1+ l*–( )γ ruk 1+– rv*+( ) Γd

Γ∫+

= a uk 1+v*– uk 1+

v*–,( ) 1r--- µk 1+ l*–( ) lk rγ uk 1+– lk– l*– rγv* l*+ +( ) Γd

Γ∫+

= a uk 1+v*– uk 1+

v*–,( ) 1r--- µk 1+ l*–( ) lk rγ uk 1+–( ) lk–( ) l* – rγv*( ) – l*( )–[ ] Γd

Γ∫+

= a uk 1+v*– uk 1+

v*–,( )1r--- µk 1+ l*–( ) lk rγ uk 1+–( ) l* rγv*–( )–[ ] Γd

Γ∫ 1

r--- µk 1+ l*–( ) –lk l*+( ) Γd

Γ∫+ +

≥ a uk 1+v

k– uk 1+v*–,( ) 1

r--- µk 1+ l*–( ) lk rγ uk 1+–( )+

l* rγv*–( )+–[ ] Γd

Γ∫+

+1r--- µk 1+ l*–( ) –lk l*+( ) Γd

Γ∫ a uk 1+

v*– uk 1+v*–,( )=

+1r--- µk 1+ l*–( ) µk 1+ l*–( ) Γd

Γ∫ 1

r--- µk 1+ l*–( ) –lk l*+( ) Γd

Γ∫+

= a uk 1+v*– uk 1+

v*–,( ) 1r--- µk 1+ l*–( ) µk 1+ l*– lk– l*+( ) Γd

Γ∫+

= a uk 1+v*– uk 1+

v*–,( ) 1r--- µk 1+ l*–( ) µk 1+ lk–( ) Γ;d

Γ∫+

uk 1+ uk– v* uk 1+–,( )01r--- µk 1+ l*–( ) lk µk 1+–( ) Γd

Γ∫+ a uk 1+

v*– uk 1+v*–,( ) 0.≥ ≥

x+ y+–( ) x y–( ) x+ y+–( )2≥

ukv*– L2 Ω( )

2uk uk 1+–( ) uk 1+

v*–( )+ L2 Ω( )2

=

= uk uk 1+– L2 Ω( )2

2 uk uk 1+– uk 1+v*–,( )0 uk 1+

v*– L2 Ω( )2

,+ +

lk l*– L2 Γ( )2 µk 1+ lk– L2 Γ( )

2l* µk 1+– L2 Γ( )

22 µk 1+ lk– l* µk 1+–,⟨ ⟩ L2 Γ( ).+ +=

ukv*– L2 Ω( )

2 1r--- lk l*– L2 Ω( )

2+ uk uk 1+– L2 Ω( )

22 uk uk 1+– uk 1+

v*–,( )0 uk 1+v*– L2 Ω( )

2+ +=

+1r--- µk 1+ lk– L2 Γ( )

2 1r--- l* µk 1+– L2 Γ( )

2 2r--- µk 1+ lk– l* µk 1+–,⟨ ⟩ L2 Γ( ).+ +

ukv*– L2 Ω( )

2 1r--- lk l*– L2 Γ( )

2+ uk uk 1+– L2 Ω( )

2 1r--- µk 1+ lk– L2 Γ( )

2uk 1+

v*– L2 Ω( )2 1

r--- l* µk 1+– L2 Γ( )

2;+ + +≥

1936


WOO et al.

The formulas for lk + 1 and µk + 1 and relation (2) imply

(11)

where C > 0 is the constant for the embedding of (Ω) into L2(Γ).

Therefore,

(12)

Relation (4) implies

Inequality (12) yields

Thus, we have

………………………………………………

After summing, we obtain

(13)

Therefore, zk – z* is a bounded sequence in L2(Ω) × L2(Γ). Then (12) implies that the sequence wk isbounded in L2(Ω) × L2(Γ). Let us show that ||zk – z*||0 is a converging sequence. Assume that

Inequality (13) for k = kn yields

If n tends to infinity, we have, for p = 1, 2, …,

or

i.e., – z*||0 exists.

Relation (4) implies

lk 1+ µk 1+– L2 Γ( ) lk rγ uk 1+–( )+lk rγ uk 1+–( )+

– L2 Γ( )=

≤ lk rγ uk 1+– lk– rγ uk 1++ L2 Γ( ) r γ uk 1+ γ uk 1+– L2 Γ( ) rC uk 1+ uk 1+– W21 Ω( ) rCδk,≤ ≤=

W21

zk 1+ wk 1+– 12 δk

2 1r---r2C2δk

2+≤ 1 rC2+( )δk2.=

wk 1+ z*– 0 zk z*– 0.≤

zk 1+ z*– 0 zk 1+ wk 1+– 0 wk 1+ z*– 0+ 1 rC2+ δk zk z*– 0, k+≤ ≤ 1 2 …., ,=

zk 1+ z*– 0 1 rC2+ δk zk z*– 0,+≤

zk 2+ z*– 0 1 rC2+ δk 1+ zk 1+ z*– 0,+≤

zk p+ z*– 0 1 rC2+ δk p 1–+ zk p 1–+ z*– 0.+≤

zk p+ z*– 0 1 rC2+ δk i 1–+

i 1=

p

∑ zk z*– 0+ 1 rC2+ δk i 1–+

i 1=

∞

∑ zk z*– 0.+≤ ≤

lim zk z*– 0 zkn z*– 0.

n ∞→lim=

zkn p+

z*– 0 1 rC2+ δkn i 1–+

i 1=

∞

∑ zkn z*– 0.+≤

zkn p+

z*– 0n ∞→lim zk z*– 0,

k ∞→lim≤

zm z*– 0m ∞→lim zk z*– 0,

k ∞→lim≤

||zm

m ∞→lim

wk 1+ zk– 02

zk z*– 02

wk 1+ z*– 02

– zk z*– 02

wk 1+ zk 1+– 0 zk 1+ z*– 0–( )2.–≤ ≤



Taking into account (12), we have

(14)

as k ∞. Then (10) implies

(15)

Therefore, wk is bounded in (Ω) × L2(Γ). Then the sequence is compact in L2(Ω). Applying

(15) once more, we find that is compact in (Ω). The theorem is proved.

Theorem 2. Let the assumption of Theorem 1 be satisfied and the solutions (k = 1, 2, …) to auxiliaryproblems (3) satisfy the following conditions:

(i) ∈ (Ω), k = 1, 2, …,

(ii) ≤ C, C > 0 = const.

Then the sequence zk converges to the unique saddle point (z*, l*) of the functional L(v, l).Here, C denotes different constants.Proof. Relation (15) implies that

(16)

Since

we have

This, together with inequality (14), yields

(17)

Since the solution to problem (3) belongs to (Ω), it is simultaneously the solution to the bound-ary value problem

Thus, ∂ /∂n = µk + 1 and assumptions (i) and (ii) of the theorem guarantee that µk is compact in

L2(Γ). Therefore, the sequence wk is compact in (Ω) × L2(Γ).

Let = c* ≡ (a*, b*).

Since || – c*||1 ≤ || – ||1 + || – c*||1, it follows from (12) that – c*||1 = 0. Let us show

that c* is a saddle point of L(v, l). We have

Taking into account (17), this yields

wk 1+ zk– 0k ∞→lim 0=

a uk 1+v*– uk 1+

v*–,( )k ∞→lim 0.=

W21 uk

uk W21

uk

uk W22

ukW2

2 Ω( )

a uk 1+v*– uk 1+

v*–,( )k ∞→lim 0.=

a uk 1+ uk– uk 1+ uk–,( ) a uk 1+v*– uk 1+

v*–,( ) 2a uk 1+v*– v* uk–,( )0 a v* uk– v* uk–,( ),+ +=

a uk 1+ uk– uk 1+ uk–,( )k ∞→lim 0.=

wk 1+ zk– 1k ∞→lim 0.=

uk 1+ W22

∆u– u+ f uk in Ω,+=

∂u∂n------– lk rγu–( )+

on Γ.–=

uk 1+

W21

wkτ

τ ∞→lim

zkτ z

kτ wkτ w

kτ ||zkτ

τ ∞→lim

wkτ 1+

c*– 1 wkτ 1+

zkτ– 1 z

kτ c*– 1.+≤

wkτ 1+

c*– 1τ ∞→lim 0.=

1938


WOO et al.

Thus, – c*||1 = 0. Let τ tend to infinity in the formula = ( – rγ )+. Then =

= c* in (Ω) × L2(Γ) implies

This equality is equivalent to the conditions

(18)

(see [4, 5]).Equality (9) is equivalent to the relation

Let k = kτ and τ tend to infinity. Taking into account (17), we then obtain

or

Therefore,

(19)

Conditions (18) and (19) mean that c* = (a*, b*) is a saddle point of L(v, l).

The existence of – z*||0 for any saddle point z* and, in particular, for c* was shown earlier.

Since – c*||0 = 0, – c*||0 = 0. This, together with condition (16), implies

i.e., the whole sequence zk converges to c* in (Ω) × L2(Γ).

Now it remains to prove the uniqueness of the saddle point. For this purpose, we use a line of reasoningsimilar to that employed in [6].

Conditions (i) and (ii) of the theorem imply that is weakly compact in (Ω). Let be a weak

limit of the subsequence in (Ω). The sequence , together with , converges strongly to

a* in (Ω). Therefore, converges weakly to a* in (Ω).

Since the dual of (Ω) is embedded into the dual of (Ω), the uniqueness of the weak limit implies

that = a* and, therefore, a* ∈ (Ω). Then, as was shown in [1], the Lagrangian functional L(v, l) hasthe unique saddle point c* = (u*, ∂u*/∂n); i.e., a* = u* and b* = ∂u*/∂n.

The theorem is proved.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 04-01-97003.

||zkτ 1+

τ ∞→lim l

kτ 1+lkτ u

kτ 1+z

kτ

τ ∞→lim

zkτ 1+

τ ∞→lim W2

1

b* b* rγa*–( )+.=

γa* 0, b* 0, γa*b*≥ ≥ 0=

uk 1+ uk– v,( )0 a uk 1+v,( ) f v,( )0– lk rγ uk 1+–( )+γv Γd

Γ∫–+ 0 v∀ W2

1 Ω( ).∈=

a a* v,( ) f v,( )0– b* rγa*–( )+γv Γd

Γ∫– 0 v∀ W2

1 Ω( ),∈=

a a* v,( ) f v,( )0– b*γv Γd

Γ∫– 0 v∀ W2

1 Ω( ).∈=

L a* b*,( ) L v b*,( ) v∀ W21 Ω( ).∈≤

||zm

m ∞→lim

||zkτ

m ∞→lim ||zm

m ∞→lim

zm c*– 1m ∞→lim 0;=

W21

uk W22 a

uki W2

2 uki u

ki

W21 u

ki W21

W21 W2

2

a W22



REFERENCES1. G. Woo, R. V. Namm, and S. A. Sachkoff, “An Iterative Method Based on a Modified Lagrangian Functional for

Finding a Saddle Point in the Semicoercive Signorini Problem,” Zh. Vychisl. Mat. Mat. Fiz. 46, 26–36 (2006)[Comput. Math. Math. Phys. 46, 23–33 (2006)].

2. A. S. Antipin, “Convex Programming Method Involving a Symmetric Modification of the Lagrange Functional,”Ekon. Mat. Metody 12, 1164–1173 (1976).

3. R. T. Rockafellar, “Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Pro-gramming,” Math. Operat. Res. 1 (2), 97–116 (1976).

4. F. P. Vasil’ev, Numerical Methods for Solving Extremum Problems (Nauka, Moscow, 1980) [in Russian].5. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer-Verlag, New York, 1984).

6. R. V. Namm and A. G. Podgaev, “On Regularity of Solutions to Semicoercive Variational Inequalities,”Dal’nevost. Mat. Zh. 3, 210–215 (2002).

W22

Iterative proximal regularization method for finding a saddle point in the semicoercive Signorini...

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