JJ II

(1)

volume 3, issue 3, article 34, 2002.

Received 15 February, 2002;

accepted 22 February, 2002.

Communicated by:Th. M. Rassias

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

PROJECTION ITERATIVE SCHEMES FOR GENERAL VARIATIONAL INEQUALITIES

MUHAMMAD ASLAM NOOR, YI JU WANG AND NAIHUA XIU

Etisalat College of Engineering, Sharjah,

United Arab Emirates EMail:noor@ece.ac.ae

School of Mathematics and Computer Science, Nanjing Normal University

Nanjing, Jiangsu, 210097, The People’s Republic of China.

AND

Institute of Operations Research, Qufu Normal University, Qufu Shandong, 273165, The People’s Republic of China.

Department of Applied Mathematics, Northern Jiaotong University, Beijing, 100044,

The People’s Republic of China.

EMail:nhxiu@center.njtu.edu.cn

c

2000Victoria University ISSN (electronic): 1443-5756 010-02

(2)

Norms of certain operators on weighted`pspaces and Lorentz

sequence spaces

M. Aslam Noor, Y. Wang and N. Xiu

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of18

Abstract

In this paper, we propose some modified projection methods for general variational inequalities. The convergence of these methods requires the monotonicity of the underlying mapping. Preliminary computational experience is also reported.

2000 Mathematics Subject Classification:90C30

Key words: General variational inequalities, Projection method, Monotonicity.

The research of Yiju Wang and Naihua Xiu is partially supported by N.S.F. of China (Grants No. Q99A11 and 19971002,10171055).

1. Introduction

Let K be a nonempty closed convex set in Euclidean space Rⁿ. For given nonlinear operators T, g : Rⁿ → Rⁿ, consider the problem of finding vector u^∗ ∈Rⁿsuch thatg(u^∗)∈Kand

(1.1) hT(u^∗), g(u)−g(u^∗)i ≥0, ∀g(u)∈K.

This problem is called general variational inequality (GVI) which was introduced by Noor in [10]. General variational inequalities have important applications in many fields including economics, operations research and nonlinear analysis, see, e.g., [5], [10] – [15] and the references therein.

Ifg(u) ≡ u, then the general variational inequality (1.1) reduces to finding vectoru^∗ ∈K such that

(1.2) hT (u^∗), u−u^∗i ≥0, ∀u∈K,

which is known as the classical variational inequality and was introduced and studied by Stampacchia [18] in 1964. For the recent state-of-the-art, see e.g., [1] – [22].

IfK^∗∗ = {u∈ Rⁿ| hu, vi ≥ 0, ∀v ∈ K}is a polar cone of a convex cone K inRⁿ, then problem (1.1) is equivalent to findingu^∗ ∈Rⁿsuch that

(1.3) g(u)∈K, T(u)∈K^∗∗, hg(u), T(u)i= 0,

which is known as the general complementarity problem. Ifg(u) =u−m(u), wheremis a point-to-set mapping, then problem (1.3) is called quasi (implicit)

(4)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of18

complementarity problem. Forg(u) =u,problem (1.3) is known as the generalized complementarity problem.

For general variational inequality, Noor [10] gave a fixed point equation re- formulation, Pang and Yao [15] established some sufficient conditions for the existence of the solutions and investigated their stability, and He [5] proposed an inexact implicit method. In this paper, we consider a projection method for solving GVI under the assumptions that the solution set is nonempty and the underlying mapping is monotone in a generalized sense.

(5)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of18

2. Preliminaries

For nonempty closed convex setK ⊂Rⁿand any vectoru∈Rⁿ, the orthogonal projection ofuontoK, i.e.,arg min{||v−u|| |v ∈K}, is denoted byP_K(u). In the following, we state some well known properties of the projection operator.

Lemma 2.1. [23]. Let K be a closed convex subset of Rⁿ, for any u ∈ Rⁿ, v ∈K, then

hP_K(u)−u, v−P_K(u)i ≥0.

From Lemma2.1, it follows that the projection operatorPK is nonexpansive.

Invoking Lemma 2.1, one can prove that the general variational inequality (1.1) is equivalent to the fixed-point problem For GVI, this result is due to Noor [10].

Lemma 2.2. [10]. A vector u^∗ ∈ Rⁿ withg(u^∗) ∈ K is a solution of GVI if and only ifg(u^∗) =P_K(g(u^∗)−ρT (u^∗))for someρ >0.

Based on this fixed-point formulation, various projection type iterative methods for solving general variational inequalities have been suggested and ana- lyzed, see [5], [10] – [15].

In this paper, we suggest another projection method which needs two projections at each iteration and its convergence requires the following assumptions.

Assumption 1.

(i) The solution set of GVI, denoted byK^∗, is nonempty.

(ii) MappingT :Rⁿ→Rⁿisg-monotone, i.e.,

hT(u)−T(v), g(u)−g(v)i ≥0, ∀u, v∈Rⁿ.

(6)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of18

(iii) Mappingg :Rⁿ →Rⁿis nonsingular, i.e., there exists a positive constant µsuch that

||g(u)−g(v)|| ≥µ||u−v||, ∀u, v ∈Rⁿ.

Note that for g ≡ I, g-monotonicity of mapping T reduces to the usual definition of monotone. Furthermore, every solvable monotone variational inequality of form (1.2) satisfies the above assumptions.

Throughout this paper, we define the residue vectorR_ρ(u)by the following relation

Rρ(u) :=g(u)−PK(g(u)−ρT(u)).

Invoking Lemma 2.2, one can easily conclude that vector u^∗ is a solution of GVI if and only ifu^∗ is a root of the following equation:

R_ρ(u) = 0, for someρ≥0.

(7)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of18

3. Algorithms and Convergence

The basic idea of our method is as follows. First, take an initial pointu⁰ ∈ Rⁿ such thatg(u⁰)∈ K and compute the projection residue. If it is a zero vector, then stop; otherwise, take the negative projection residue as a direction and per- form a line search along this direction to get a new point; after constructing a

“descent direction” related to the current point and the new point, the next iterative point can be obtained by using a projection. Repeat this process until the projection residue is a zero vector. So the algorithm needs only two projections at each iteration.

Now, we formally describe our method for solving the GVI problem.

Algorithm 1.

Initial step: Chooseu⁰ ∈Rⁿsuch thatg(u⁰)∈K, select anyσ, γ ∈(0,1), ρ∈(0,+∞), letk := 0.

Iterative step: Forg u^k

∈K,takew^k∈Rⁿsuch that g w^k

:=P_K g u^k

−ρT u^k . If

R_ρ u^k

= 0, then stop. Otherwise, computev^k∈Rⁿsuch thatg v^k

:=g u^k

−ηkRρ u^k

,whereηk =γ^m^k

withm_kbeing the smallest nonnegative integermsatisfying

(3.1) ρ

T u^k

−T v^k

, R_ρ u^k

≤σ

R_ρ u^k

2.

Computeu^k+1 by solving the following equation g u^k+1

=P_K g u^k

+α_kd_k , whered_k=− η_kR_ρ u^k

+η_kT u^k

+ρT v^k , α_k = ^(1−σ)η^kk^R^ρ(^u^k)k²

kdkk² .

(8)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of18

Remark 3.1. We analyze the step-size rule given in (3.1). If Algorithm1 ter- minates with Rρ u^k

= 0, then u^k is a solution of GVI. Otherwise, by non- singularity ofg and continuity ofT andg,η_ksatisfying (3.1) exists.

Remark 3.2. In Algorithm1, several implicit equations ofg must be solved at each iteration. Ifg ≡I,thenv^k = (1−ηk)u^k+ηkw^k.

Remark 3.3. We recall the searching directions appear in existing projection- type methods for solving VI of form (1.2). They are

(i) the direction−T(¯u^k)by Korpelevich [9], whereu¯^k=P_K u^k−α_kT u^k

; (ii) the direction−

u^k−u¯^k−αk

T u^k

−T u¯^k by Solodov and Tseng [17], Tseng [20], Sun [19] and He [6].

(iii) the direction−

u^k−u¯^k+T u¯^k by Noor [13].

(iv) the direction−T v^k

by Iusem and Svaiter [7] and Solodov and Svaiter [16].

(v) the direction− η_kr u^k

+T v^k

by Wang, Xiu and Wang [22].

In our algorithm, wheng ≡I, the searching direction reduces to

− η_kr u^k

+η_kT u^k

+ρT v^k .

It is a combination of the projection residue and T, and differs from the above five types of directions.

(9)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of18

Now, we discuss the convergence of Algorithm1. From the iterative procedure, we know thatg u^k

, g v^k

, g w^k

∈Kfor allk. For anyg(u^∗)∈K^∗, by Assumption(ii), we have

(3.2)

ρT u^k

, g u^k

−g(u^∗)

≥0.

From Lemma2.1, we know that g u^k

−ρT u^k

−g w^k

, g w^k

−g(u^∗)

≥0,

which can be written as g u^k

−ρT u^k

−g w^k

, g w^k

−g u^k +

g u^k

−g w^k

−ρT u^k

, g u^k

−g(u^∗)

≥0.

Combining with inequality (3.2), we obtain (3.3)

R_ρ u^k

, g u^k

−g(u^∗)

≥

R_ρ u^k

2−ρ T u^k

, R_ρ u^k .

So

g u^k

−g(u^∗),−d_k

= g u^k

−g(u^∗), η_kR_ρ u^k

+η_kT u^k

+ρT v^k

= g u^k

−g(u^∗), η_kR_ρ u^k +

g u^k

−g(u^∗), η_kT u^k +

g u^k

−g(u^∗), ρT v^k

(10)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of18

≥ηk

Rρ u^k

2−ρηk

T u^k

, Rρ u^k +

g u^k

−g v^k

, ρT v^k

=η_k

R_ρ u^k

2−ρη_k T u^k

, R_ρ u^k +η_k

R_ρ u^k

, ρT v^k

=η_k

R_ρ u^k

2−ρη_k T u^k

−T v^k

, R_ρ u^k

≥η_k

R_ρ u^k

2−ση_k

R_ρ u^k

2

= (1−σ)η_k

R_ρ u^k

2,

where the first inequality uses (3.3) and the g-monotonicity of T, the second inequality follows from inequality (3.1).

For anyα >0, one has P_K g u^k

+αd_k

−g(u^∗)

2

≤ g u^k

−g(u^∗) +αd_k

2

= g u^k

−g(u^∗)

2+α²kd_kk²+ 2α

d_k, g u^k

−g(u^∗)

≤ g u^k

−g(u^∗)

2+α²kd_kk²−2α(1−σ)η_k

R_ρ u^k

2,

where the first inequality uses non-expansiveness of projection operator.

Based on the above analysis, we show that Algorithm 1 converges under Assumptions(i)–(iii).

Theorem 3.1. Under Assumptions(i)–(iii), if Algorithm1generates an infinite sequence{u^k}, then{u^k}globally converges to a solutionu^∗of GVI.

Proof. Letα := α = ^(1−σ)η^k^||R^ρ(^u^k)^||²

in the aforementioned inequalities, we

(11)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of18

obtain

g(u^k+1)−g(u^∗) ≤

g u^k

−g(u^∗)

2− (1−σ)²η²_k

R_ρ u^k

4

||dk||² .

So

g u^k

−g(u^∗)

is a non-increasing sequence, and

g u^k is a bounded sequence. Sincegis nonsingular, we conclude that{u^k}is a bounded sequence.

Short discussion leads to that{d_k}is bounded. So, there exists an infinite subset N₁ such that

k∈Nlim1,k→∞

R_ρ u^k = 0 or an infinite subsetN2such that

k∈Nlim₂,k→∞ηk = 0.

If lim

k∈N1,k→∞

R_ρ u^k

= 0, we know that any cluster u˜ of {u^k : k ∈ N₁} is a solution of GVI. Since

g u^k

−g(u^∗)

is non-increasing, if we take u^∗ = ˜u, then we know that{g u^k

}globally converges tog(˜u)and thus{u^k} globally converges tou˜from Assumption(iii).

If lim

k∈N2,k→∞η_k = 0, letv¯^k ∈Rⁿsuch thatg v^k

=g u^k

− ^η^k

γRρ(^u^k). From the linear searching procedure ofη_k, we have

ρ T u^k

−T(¯v^k), Rρ u^k

> σ

Rρ u^k

2, for sufficiently largek∈N2. Therefore,

ρ T u^k

−T(¯v^k) > σ

Rρ u^k

, for sufficiently largek ∈N2.

(12)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of18

This, plus lim

k∈N2,k→∞

η_k

γ = 0, yields lim

k∈N2,k→∞

R_ρ u^k

= 0. Similar discussion leads to that any cluster of

u^k :k ∈N₂ is a solution to GVI. Replacing u^∗ by this cluster point yields the desired result.

If we replace ρ with ρ_k in Algorithm 1, then we obtain the following im- proved algorithm to GVI.

Algorithm 2.

Initial step: Chooseu⁰ ∈Rⁿsuch thatg(u⁰)∈K, select anyσ, γ ∈(0,1), η₋₁ = 1,θ >0.

Letk= 0.

Iterative step: Forg u^k

∈K,defineρ_k= min{θηk−1,1}, and takew^k∈Rⁿ such thatg w^k

=PK(g u^k

−ρkT u^k ).

IfR_ρ_k u^k

= 0, then stop. Otherwise, takev^k∈Rⁿin the following way: g v^k

= (1−η_k)g u^k

+η_kg w^k , whereη_k =γ^m^k, withm_k being the smallest

nonnegative integermsatisfying ρ_khT u^k

−T v^k

, R_ρ_k u^k

i ≤σ

R_ρ_k u^k

2. Computeu^k+1 by solving the following equation:

g(u^k+1) =P_K(g u^k

+α_kd_k) whered_k =− η_kR_ρ_k u^k

+η_kT u^k

+ρ_kT v^k , α_k = ^(1−σ)η^kk^Rρk(^u^k)k²

kd_kk² .

The convergence of Algorithm2can be proved similarly.

(13)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of18

4. Preliminary Computational Experience

In the following, we present some numerical experiments for Algorithms1and 2. For these algorithms, we used

r x^k, ρk

≤10⁻⁸as stopping criteria.

Throughout the computational experiments, the parameters used were set as σ = 0.5, γ = 0.8. All computational results were undertaken on a PC-II by MATLAB.

Example 4.1. This example is a quadratic subproblem of the trust region ap- proach for solving medium-size nonlinear programming problem:

min 1

2x^>Hx+c^>x|x∈C

.

This problem is equivalent to VI(F, C) withF(x) =Hx+c. the data is chosen as: H = V W V, where V = I −2_||v||^vv^>2 is a Householder matrix and W = diag(σ_i)withσ_i = cos_n+1^iπ + 1000. The vectorsvandccontain pseudo-random numbers:

v₁ = 13846, v_i = (42108vi−1+ 13846)mod46273, i= 2, . . . , n;

c₁ = 13846, c_i = (45287ci−1 + 13846)mod46219, i= 2, . . . , n.

For this test problems, the domain set C = {x∈Rⁿ| ||x|| ≤10⁵}. Ta- ble 1 gives the numerical results for this example with starting point x⁰ = (0,0, . . . ,0)^T for different dimensionsn.

(14)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of18

Dimension Algorithm1(ρ= 1) Algorithm2(θ = 400)

n= 10 73 56

n= 20 75 58

n= 50 78 58

n= 80 81 60

n = 100 84 60

n = 200 97 60

Table 1: Numbers of iterations for Example4.1

Example 4.2. This example is a general variational inequality with g(x) = Ax+qandF(x) = x, where

A=







4 −2 0 · · · 0 1 4 −2 · · · 0 0 1 4 · · · 0 ... ... ... ... ... 0 0 0 · · · −2 0 0 0 · · · 4





 , q =





 1 1 1 ... 1 1





 .

For this test problems, the domain setC = {x ∈ Rⁿ|0 ≤ x_i ≤ 1, fori = 1,2,· · ·n}. Table2gives the results for this example with starting pointx⁰ =

−A⁻¹qfor different dimensionsn.

(15)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of18

Dimension Algorithm1(ρ= 1) Algorithm2(100 ≤θ≤400)

n= 10 492 492

n= 20 489 489

n= 50 484 484

n= 80 481 481

n = 100 480 480

n = 200 476 476

Table 2: Numbers of iterations for Example4.2

From Table1and Table2, one observes that Algorithms1and2work quite well for these examples, respectively, and there is not much difference to the choice of parameterρk in the second algorithm, especially for Example4.2.

(16)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of18

References

[1] F. GIANNESSIANDA. MAUGERI, Variational inequalities and network equilibrium problems, Plenum Press, New York, 1995.

[2] R. GLOWINSKI, Numerical Methods for Nonlinear Variational Prob- lems, Springer-Verlag, Berlin, 1984.

[3] R. GLOWINSKI, J.L. LIONSANDR. TREMOLIERES, Numerical Anal- ysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

[4] P.T. HARKER AND J.S. PANG, Finite-dimensional variation inequality and nonlinear complementarity problems: A survey of theory, algorithm and applications, Math. Programming, 48 (1990), 161–220.

[5] B.S. HE, Inexact implicit methods for monotone general variational in- equalities, Math. Programming, 86 (1999), 199–217.

[6] B.S. HE, A class of projection and contraction methods for monotone vari- ational inequalities, Appl. Math. Optim., 35 (1997), 69–76.

[7] A.N. IUSEM AND B.F. SVAITER, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309–321.

[8] D. KINDERLEHRERANDG. STAMPACCHIA, An Introduction to Vari- ational Inequalities and their Applications, Academic Press, New York, 1980.

[9] G.M. KORPELEVICH, The extragradient method for finding saddle

(17)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of18

[10] M.A. NOOR, General variational inequalities, Appl. Math. Lett., 1(2) (1988), 119–121.

[11] M.A. NOOR, Some recent advances in variational inequalities, Part I, ba- sic concepts, New Zealand J. Math, 26 (1997), 53–80.

[12] M.A. NOOR, Some recent advances in variational inequalities, Part II, other concepts, New Zealand J. Math, 26 (1997), 229–255.

[13] M.A. NOOR, Some algorithms for general monotone mixed variational inequalities, Math. Comput. Modeling, 19(7) (1999),1–9.

[14] M.A. NOOR, Modified projection method for pseudomonotone variational inequalities, Appl. Math. Lett., 15 (2002).

[15] J.S. PANG AND J. C. YAO, On a generalization of a normal map and equation, SIAM J. Control and Optim., 33 (1995),168–184.

[16] M.V. SOLODOV ANDB.F. SVAITER, A new projection method for vari- ational inequality problems, SIAM J. Control and Optim., 37 (1999), 765–

776.

[17] M.V. SOLODOV AND P. TSENG, Modified projection-type methods for monotone variational inequalities, SIAM J. Control and Optim., 34 (1996), 1814–1830.

[18] G. STAMPACCHIA, Formes bilineaires coercitives sur les ensembles con- vexes, C.R. Acad. Sci. Paris, 258 (1964), 4413–4416.

[19] D. SUN, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl., 91 (1996), 123–140.

(18)

sequence spaces

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of18

[20] P. TSENG, A modified forward-backward splitting method for maximal monotone mapping, SIAM J. Control and Optim., 38 (2000), 431–446.

[21] Y.J. WANG, N.H. XIU AND C.Y. WANG, A new version of extragradi- ent method for variational inequality problems, Comput. Math. Appl., 43 (2001), 969–979.

[22] Y.J. WANG, N.H. XIU AND C.Y. WANG, Unified framework of extragradient-type methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 111(3) (2001), 643–658.

[23] E.H. ZARANTONELLO, Projections on convex sets in Hilbert space and spectral theory, contributions to nonlinear functional analysis, (E. H.

Zarantonello, Ed.), Academic Press, New York, 1971.