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VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
INSTITUTE OF MATHEMATICS
PHAM DUY KHANH
SOLUTION METHODS FOR
PSEUDOMONOTONE VARIATIONAL INEQUALITIES
Speciality: Applied Mathematics
Speciality code: 62 46 01 12
SUMMARY
DOCTORAL DISSERTATION IN MATHEMATICS
HANOI - 2015
The dissertation was written on the basis of the author’s research works carried
at Institute of Mathematics, Vietnam Academy of Science and Technology.
Supervisors:
1. Prof. Dr. Hab. Nguyen Dong Yen
2. Dr. Trinh Cong Dieu
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To be defended at the Jury of Institute of Mathematics, Vietnam Academy
of Science and Technology:
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The dissertation is publicly available at:
• The National Library of Vietnam
• The Library of Institute of Mathematics
Introduction
Monotone operators have been studied since the early 1960s. F. Browder

systematically employed the monotonicity of operators to study various prob-
lems related to nonlinear elliptic partial differential equations. Independently,
P. Hartman and G. Stampacchia studied variational inequalities (VIs for
brevity) with monotone operators. Until now, monotone VIs continue to be a
subject of the concern of many researchers. Different solution methods have
been proposed for monotone VIs: the projection method, the Tikhonov reg-
ularization method, the proximal point method, the extragradient method,
etc.
The concept of pseudomonotone operator introduced by S. Karamardian
(1976) is an important generalization of monotone operator. Inspired by this
paper, S. Karamardian and S. Schaible (1990) introduced several general-
ized monotonicity concepts such as strict pseudomonotonicity, strong pseu-
domonotonicity, and quasimonotonicity. For each type of generalized mono-
tonicity of operators, these authors established a relation to the corresponding
type of generalized convexity of functions. It turns out that pseudomonotone
operator is a special case of quasimonotone operator. In the last decade, solu-
tion existence and solution methods for pseudomonotone and quasimonotone
VIs have been studied in many books and papers. The two-volume book of
F. Facchinei and J.S. Pang (2003) and the handbook edited by N. Hadjisav-
vas, S. Koml´osi, and S. Schaible (2005) are among the most cited references
in this field.
Facchinei and Pang (2003) raised a question about the convergence of the
Tikhonov regularization method (TRM) for pseudomonotone VIs. With the
aid of a solution existence theorem based on the degree theory and some
interesting arguments, N. Thanh Hao (2006) solved the question in the af-
firmative. Namely, she proved that if the original problem has a solution,
then the Tikhonov regularized problem has a compact nonempty solution
set which diameter tends to zero, and any selection of the solution mapping
1
converges to the least-norm solution of the original problem. The results of

Facchinei and Pang on solution existence of pseudomonotone VIs have been
extended by B.T. Kien, J C. Yao, and N.D. Yen (2008) to VIs and set-valued
VIs in reflexive Banach spaces. The results of Thanh Hao on the convergence
of the TRM for pseudomonotone VIs have been developed to VIs in Hilbert
spaces by N.N. Tam, J C. Yao, and N.D. Yen (2008).
For monotone VIs, the convergence of the iterative sequence generated by
the proximal point algorithm (PPA) and the applicability of the algorithm
(in the exact form as proposed by B. Martinet (1970), or in the inexact form
as proposed by R.T. Rockafellar (1976)) are a novel research theme in this
direction. For pseudomonotone VIs in Hilbert spaces, N.N. Tam, J C. Yao,
and N.D. Yen (2008) have obtained some new results on the convergence of
the exact PPA and inexact PPA.
The auxiliary problems of the TRM and of the PPA, applied to pseu-
domonotone VIs, may not be pseudomonotone, or may remain without any
solution (if one considers the infinite-dimensional Hilbert space setting). In
addition, if the auxiliary problems have a solution then they may have mul-
tiple solutions. These phenomena indicate that the auxiliary problems can
be more difficult than the original one. A natural question arises: If there is
any algorithm that can solve pseudomonotone VIs in an effective way? The
extragradient method (EGM) proposed by G.M. Korpelevich (1976) is such
an algorithm.
The thesis has five chapters.
Chapter 1 recalls some basic notions like variational inequality problem,
complementarity problem, monotonicity, pseudomonotonicity, and metric pro-
jection. Several fundamental solution methods for monotone VIs are also
presented.
Chapter 2 deals with some questions related to applying the TRM for pseu-
domonotone VIs. Solution uniqueness for the regularized problems is studied
in two cases: unconstrained VIs and linear complementarity problems. The
pseudomonotonicity of the regularized mappings of an affine mapping defined

on a polyhedral convex set is investigated.
Chapter 3 presents a modified projection method for solving strongly pseu-
domonotone VIs. Strong convergence and error estimates for the iterative
sequences are investigated in two versions of the method: the stepsizes are
chosen arbitrarily from a given fixed closed interval and the stepsizes form
2
a non-summable decreasing sequence of positive real numbers. In addition,
an interesting class of strongly pseudomonotone infinite-dimensional VIs is
considered.
Chapter 4 is devoted to a modified EGM for solving pseudomonotone VIs
in Hilbert spaces. The convergence and convergence rate of the iterative
sequences generated by this method are studied.
Chapter 5 proposes a new EGM for solving strongly pseudomonotone VIs
in Hilbert spaces. A detailed analysis of the iterative sequences’ convergence
and of the range of applicability of the method is provided.
The results of the thesis were reported by the author at
- Seminar of Department of Numerical Analysis and Scientific Computing
of Institute of Mathematics, Vietnam Academy of Science and Technology,
Hanoi;
- Summer School “Variational Analysis and Applications”, Institute of
Mathematics, Vietnam Academy of Science and Technology, Hanoi (June
20–25, 2011);
- The 8
th
Vietnam-Korea Workshop “Mathematical Optimization Theory
and Applications”, University of Dalat (December 8–10, 2011);
-The VMS-SMF Joint Congress at University of Hue (August 20–24, 2012).
3
Chapter 1
Preliminaries

The concepts of variational inequality, complementarity problem, metric pro-
jection, together with three basic solution methods for variational inequalities
(the Tikhonov regularization method, the proximal point algorithm, the ex-
tragradient method) are described in this chapter.
1.1 Variational Inequalities and Complementarity Prob-
lems
Let K be a nonempty subset of a real Hilbert space (H, ., .) and let F :
K → H be a single-valued mapping. The variational inequality defined by
K and F which is denoted by VI(K, F ) is the problem of finding a vector
u
∗
∈ K such that
F (u
∗
), u − u
∗
 ≥ 0, ∀u ∈ K. (1.1)
The set of solutions to this problem is denoted by Sol(K, F ).
The complementarity problem given by a convex cone K and a mapping
F : K → H is the problem of finding a vector u
∗
∈ H with
u
∗
∈ K, F(u
∗
) ∈ K
∗
, F(u
∗

), u
∗
 = 0, (1.2)
where
K
∗
:= {d ∈ H : d, u ≥ 0 ∀u ∈ K}
is the dual cone of K. Problem (1.2) is abbreviated to CP(K, F ).
If u ∈ K and F (u) ∈ K
∗
then u is called a feasible vector of CP(K, F ). If
the problem CP(K, F ) has a feasible vector, it is said to be feasible. When
H = IR
n
, F is an affine mapping, i.e., F(u) = Mu + q with M ∈ IR
n×n
,
4
q ∈ IR
n
, and K = IR
n
+
(in this case K
∗
= IR
n
+
), CP(K, F ) becomes the linear
complementarity problem LCP(M, q):

u
∗
≥ 0, Mu
∗
+ q ≥ 0, Mu
∗
+ q, u
∗
 = 0. (1.3)
Here the inequalities are taken componentwise. The solution set of this prob-
lem is denoted by Sol(M, q).
1.2 Monotonicity and Generalized Monotonicity
A mapping F : K ⊂ H → H is said to be
(a) strongly monotone if there exists γ > 0 such that
F (u) − F (v), u − v ≥ γu −v
2
∀u, v ∈ K;
(b) strongly pseudomonotone if there exists γ > 0 such that, for all u, v ∈ K,
F (u), v − u ≥ 0 =⇒ F (v), v − u ≥ γu − v
2
;
(c) monotone if F (u) − F (v), u −v ≥ 0 for all u, v ∈ K;
(d) pseudomonotone if, for all u, v ∈ K,
F (u), v − u ≥ 0 =⇒ F (v), v − u ≥ 0;
(e) quasimonotone if, for all u, v ∈ K,
F (u), v − u > 0 =⇒ F (v), v − u ≥ 0.
1.3 Metric Projection
Let K ⊂ H be a closed convex set. Then for each u ∈ H, there is a unique
v ∈ K such that
u − v = inf

w∈K
u − w. (1.4)
The unique vector v satisfying (1.4) is called the metric projection of u on
K. It is denoted by P
K
(u).
Several basic properties of the metric projection are recalled in the thesis.
5
1.4 The Tikhonov Regularization Method
Consider the problem VI(K, F) in a real Hilbert space H. Denote the identity
mapping of H by I, and put F
ε
= F + εI for every ε > 0. To solve VI(K, F ),
one solves the sequence of problems VI(K, F
ε
k
) where {ε
k
} is a sequence
of positive real numbers converging to zero and F
ε
k
= F + ε
k
I. For each
k ∈ IN, one selects a solution u
k
∈ Sol(K, F
ε
k

) and compute the limit lim
k→∞
u
k
.
When such limit exists, we may hope that the obtained vector is a solution
of VI(K, F ). To terminate the computation process after a finite number of
steps and to obtain an approximate solution of VI(K, F ), one has to introduce
a stopping criterion. For example, we can terminate the computation when
u
k
− u
k−1
 ≤ θ where θ > 0 is a constant.
Two basic convergence theorems for the Tikhonov regularization are re-
called in the thesis.
1.5 The Proximal Point Algorithm
Choose a point u
0
∈ H and a sequence {ρ
k
} of positive real numbers satisfying
the condition ρ
k
≥ ρ > 0 for all k ∈ IN. If u
k−1
has been defined, one can
choose u
k
as any solution of the auxiliary problem VI(K, F

(k)
) where
F
(k)
(u) = ρ
k
F (u) + u − u
k−1
, u ∈ K, (1.5)
that is u
k
∈ K and
ρ
k
F (u
k
) + u
k
− u
k−1
, v − u
k
 ≥ 0, ∀v ∈ K.
(Suppose that Sol(K, F
(k)
) is nonempty.) If the iterative scheme yields a
sequence {u
k
}, then one computes the limit lim
k→∞

u
k
in the norm topology
or in the weak topology of H. When the limit exists, one may hope that
the obtained element belongs to the solution set of VI(K, F ). To terminate
the computation process after a finite number of steps and to obtain an
approximate solution of VI(K, F), one introduces a stopping criterion. For
example, one can terminate the computation when u
k
− u
k−1
 ≤ θ where
θ > 0 is a constant.
Basic convergence theorems for the proximal point algorithm are recalled
in the thesis.
6
1.6 The Extragradient Method
The extragradient method executes two projections per iteration. Suppose
that F is Lipschitz continuous on K with the Lipschitz constant L > 0, that
is
F (u) − F (v) ≤ Lu − v, ∀u, v ∈ K. (1.6)
Algorithm 1.1
Data: u
0
∈ K and α ∈ (0, 1/L).
Step 0: Set k = 0.
Step 1: If u
k
= P
K

(u
k
− αF (u
k
)), stop.
Step 2: Compute
¯u
k
= P
K
(u
k
− αF (u
k
)),
¯u
k+1
= P
K
(u
k
− αF (¯u
k
));
set k ← k + 1 and go to Step 1.
Two convergence theorems for the extragradient method are recalled in
the thesis.
7
Chapter 2
On the Tikhonov Regularization

Method and the Proximal Point
Algorithm for Pseudomonotone
Problems
This chapter presents our partial solutions for the some open questions about
the solution uniqueness of the regularized problem of a pseudomonotone VI
and the preservation of the pseudomonotonicity under the regularization.
2.1 Open Questions on Pseudomonotone Variational
Inequalities
Open questions. If K ⊂ IR
n
is a nonempty closed convex set, F : K → IR
n
is a continuous pseudomonotone mapping, and the problem VI(K, F ) has a
solution, then there exists ε
1
> 0 such that the mapping F
ε
= F + εI is
pseudomonotone for each ε ∈ (0, ε
1
)? Is there any ε
2
> 0 such that the
problem VI(K, F
ε
) has a unique solution for every ε ∈ (0, ε
2
)?
2.2 Solution Uniqueness of the Regularized Problems
Let K be a subset of IR

n
. A mapping F : K → IR
n
is said to be pseudoaffine
on K if F and −F are both pseudomonotone.
8
Theorem 2.1 A mapping F : IR
n
→ IR
n
is pseudoaffine if and only if there
exists a skew matrix M ∈ IR
n×n
, i.e., M
T
= −M, a vector q ∈ IR
n
, and a
positive function g : IR
n
→ IR such that
F (u) = g(u)(Mu + q), ∀u ∈ IR
n
.
Theorem 2.2 Suppose that F(u) = g(u)(Mu + q) with g : IR
n
→ IR being a
positive and continuously differentiable function, M ∈ IR
n×n
a skew symmet-

ric and nonsingular matrix, and q ∈ IR
n
an arbitrarily given vector. Then
there exists ¯ε > 0 such that the regularized problem VI(K, F
ε
) has a unique
solution for all ε ∈ (0, ¯ε).
If detM = 0 and M
T
= −M, then n must be an even number. This shows
that the assumptions of Theorem 2.2 are rather strict. It is natural to find
some ways to enlarge the applicability of Theorem 2.2.
Theorem 2.3 Suppose that F : IR
n
→ IR
n
is a map of the form
F (u) = g(u)(Mu + q),
where g : IR
n
→ IR is a positive and continuously differentiable function,
M ∈ IR
n×n
is a positive semidefinite nonsingular matrix, and q ∈ IR
n
. Then
there exists ¯ε > 0 such that the regularized problem VI(K, F
ε
) has a unique
solution for all ε ∈ (0, ¯ε).

Consider the linear complementarity problem of the form (1.3).
Theorem 2.4 Suppose that (1.3) is feasible and the mapping F(u) = Mu+q
is pseudomonotone on IR
n
+
. Then the regularized problem LCP(M
ε
, q), where
M
ε
= M + εI, has a unique solution for any ε ∈ (0, +∞).
2.3 Pseudomonotonicity of the Regularized Mappings
Theorem 2.5 Let K ⊂ IR be a closed convex subset and F (u) = au + b be
an affine map. Then F is pseudomonotone on K if and only if one of the
following five cases occurs:
(a) K has a unique element;
(b) K = IR and a ≥ 0;
(c) K = [α, +∞) for some α ∈ IR, and either a ≥ 0 or a < 0 and aα+b < 0;
9
(d) K = (−∞, β] for some β ∈ IR, and either a ≥ 0 or a < 0 and aβ +b > 0;
(e) K = [α, β], for some α, β ∈ IR with α < β, and either a ≥ 0 or a < 0
and aα + b < 0, or a < 0 and aβ + b > 0.
Corollary 2.1 Let K be a closed convex set in IR and F(u) = au + b be an
affine map. If F is pseudomonotone on K then there exists ¯ε > 0 such that
F
ε
(u) = (a + ε)u + b is pseudomonotone on K for all ε ∈ (0, ¯ε).
The pseudomonotonicity preservation of F
ε
for small enough ε > 0 (pro-

vided that F is pseudomonotone), which was described in Corollary 2.1 for
the case K ⊂ IR, is no longer valid if K is a nonempty closed convex subset
of IR
n
, n ≥ 2.
Theorem 2.6 Suppose that F (u) = Mu + q is an affine map, where
M = diag(λ
1
, λ
2
, . . . , λ
n
), q = (q
1
, q
2
, . . . , q
n
)
T
(2.1)
are respectively a diagonal matrix and a vector in IR
n
. Then F is pseu-
domonotone on IR
n
+
if and only if one of the following conditions holds:
(i) λ
i

≥ 0 for every i ∈ {1, 2, . . . , n};
(ii) There exists a unique i ∈ {1, 2, . . . , n} such that

λ
i
< 0, q
i
< 0,
λ
j
= 0, q
j
= 0 ∀j ∈ {1, 2, . . . , n} \{i}.
(2.2)
There is a class of pseudomonotone affine mappings whose regularized op-
erators F
ε
are not pseudomonotone for all sufficiently small ε > 0.
Theorem 2.7 Let F (u) = Mu + q, with M = diag(λ
1
, λ
2
, . . . , λ
n
) being a
diagonal matrix and q = (q
1
, q
2
, . . . , q

n
)
T
being a vector in IR
n
. If F is merely
pseudomonotone on IR
n
+
(i.e., F is pseudomonotone but not monotone on
IR
n
+
), then there exists ¯ε > 0 such that F
ε
(u) = F (u) +εu is not pseudomono-
tone on IR
n
+
for all ε ∈ (0, ¯ε).
2.4 Some Remarks on the Proximal Point Algorithm
in the Affine Case
Observe that if F is Lipschitz continuous on K with a Lipschitz constant
L > 0, then for any u
k
∈ IR
n
and ε > L the regularized operator
F
k,ε

(u) := F (u) + ε(u − u
k
)
10
is strongly monotone.
For the PPA, we have
F
(k)
(u) = ρ
k
F (u) + u − u
k−1
= ρ
k
F (u) + ε(u − u
k−1
), (2.3)
where ε = 1. Hence, if L is a Lipschitz constant of F on K then ρ
k
L is a Lip-
schitz constant of ρ
k
F (.) on K, so the above fact on the strong monotonicity
of the regularized operator tells us that if ε = 1 > ρ
k
L (i.e., 0 < ρ
k
< L
−1
),

then F
(k)
(.) is strongly monotone on K with the constant α
k
:= 1 − ρ
k
L.
The following advantage of PPA in comparison with the TRM is clear:
the auxiliary problem VI(K, F
(k)
), for every k ∈ IN, is strongly monotone
if F is merely Lipschitzian (no monotonicity is required!), provided that the
coefficient ρ
k
is such that 0 < ρ
k
< L
−1
.
Since affine operators are obviously Lipschitzian, the PPA can be effective
applied for pseudomonotone affine VIs. Namely, if F (u) = Mu + q with
M ∈ IR
n×n
\{0} and q ∈ IR
n
then F
(k)
(.) is strongly monotone on K with the
constant α
k

:= 1 − ρ
k
M for any ρ
k
satisfies 0 < ρ
k
< M
−1
. Therefore,
the iterative sequence given by the PPA, where 0 < ρ ≤ ρ
k
< M
−1
for
all k, converges, provided that the original problem has a solution. In other
words, the PPA can solve any pseudomonotone affine VI.
11
Chapter 3
A Modified Projection Method
The projection method is a fundamental solution method for strongly mono-
tone variational inequalities. We now consider a modified projection method
for strongly pseudomonotone variational inequalities.
3.1 Algorithm
Consider the problem VI(K, F ) under the assumption that K ⊂ H is a
nonempty closed convex set.
Algorithm 3.1
Data: u
0
∈ K and {λ
k

} ⊂ (0, +∞).
Step 0: Set k = 0.
Step 1: If u
k
= P
K
(u
k
− λ
k
F (u
k
)) then stop.
Step 2: Compute u
k+1
= P
K
(u
k
− λ
k
F (u
k
)) and replace k by k + 1; go to
Step 1.
If the computation terminates at a step k, then we put u
k

= u
k

for all
k

≥ k + 1. Thus, for a given sequence of variable stepsizes {λ
k
} ⊂ (0, +∞),
Algorithm 3.1 produces for each initial point u
0
∈ K a unique iterative se-
quence {u
k
}.
Proposition 3.1 Suppose that F is strongly pseudomonotone on K with a
constant γ and Lipschitz continuous on K with a constant L. Let {u
k
} be a
sequence generated by Algorithm 3.1. If u
∗
is a unique solution of VI(K, F )
then
[1 + λ
k
(2γ −λ
k
L
2
)]u
k+1
− u
∗


2
≤ u
k
− u
∗

2
∀k ∈ IN. (3.1)
12
3.2 Modified Projection with A Priori Constants
Iterative sequences generated by Algorithm 3.1 for strongly pseudomonotone
VIs are linearly convergent, provided that the given problem has a solution.
Theorem 3.1 Let F be strongly pseudomonotone on K with a constant γ
and Lipschitz continuous on K with a constant L. Suppose that
0 < a ≤ λ
k
≤ b <
2γ
L
2
∀k ∈ IN, (3.2)
where a, b are some positive constants. Let {u
k
} be the sequence generated by
Algorithm 3.1. If u
∗
is the unique solution of VI(K, F ) then the sequence {u
k
}

converges linearly to u
∗
. Moreover, the priori and posteriori error estimates
u
k+1
− u
∗
 ≤
µ
k+1
1 − µ
u
1
− u
0
 (3.3)
and
u
k+1
− u
∗
 ≤
µ
1 − µ
u
k+1
− u
k
 (3.4)
hold for all k ∈ IN. Here

µ :=
1

1 + a(2γ − bL
2
)
∈ (0, 1). (3.5)
As an illustration, we now apply Theorem 3.1 to a class of strongly pseu-
domonotone infinite-dimensional variational inequality problems. To the best
of our knowledge, this is the first time a class of strongly pseudomonotone
operators, not just a single problem, is given explicitly.
Example 3.1 Let H = 
2
, the Hilbert space whose elements are the 2-
summable sequences of real scalars, i.e.,
H = {u = (u
1
, u
2
, . . . , u
i
, . . .) :
∞

i=1
|u
i
|
2
< +∞}.

The inner product and the norm on H are given by setting
u, v =
∞

i=1
u
i
v
i
and u =

∞

i=1
u
i

1/2
for any u = (u
1
, u
2
, . . . , u
i
, . . .) and v = (v
1
, v
2
, . . . , v
i

, . . .) in H.
Let α, β ∈ IR be such that β > α >
β
2
> 0 and define
K
α
= {u ∈ H : u ≤ α}, F
β
(u) = (β − u)u.
13
Here α and β are parameters. We have Sol(K
α
, F
β
) = {0}. The operator F
β
is Lipschitz continuous with the Lipschitz constant L := β + 2α and strongly
pseudomonotone with the modulus γ := β − α on K
α
. Moreover, F
β
are
neither strongly monotone nor monotone on K
α
. To see this, it suffices to
choose u = (
β
2
, 0, . . . , 0, . . .), v = (α, 0, . . . , 0, . . .) ∈ K

α
and note that
F
β
(u) − F
β
(v), u − v =

β
2
− α

3
< 0.
Pick any u
0
∈ K
α
and set λ
k
= λ for all k ∈ IN, where
λ ∈ (0,
2γ
L
2
) =

0,
2(β −α)
(β + 2α)

2

is taken arbitrarily. From Theorem 3.1 it follows that the sequence {u
k
} gen-
erated by Algorithm 3.1 converges linearly to 0, which is the unique solution
of the problem VI(K
α
, F
β
). Moreover, in view of (3.3) and (3.4),
u
k+1
− 0 ≤
µ
k+1
1 − µ
u
1
− u
0
 and u
k+1
− 0 ≤
µ
1 − µ
u
k+1
− u
k


for all k ∈ IN, where
µ =
1

1 + λ[2(β − α) − λ(β + 2α)
2
]
.
3.3 Modified Projection without A Priori Constants
Theorem 3.2 Let F be strongly pseudomonotone on K with a constant γ
and Lipschitz continuous on K with a constant L. Suppose that {λ
k
} is a
sequence of positive scalars with
∞

k=0
λ
k
= +∞, lim
k→∞
λ
k
= 0. (3.6)
Let {u
k
} be a sequence generated by Algorithm 3.1. If VI(K, F ) has a unique
solution u
∗

, then the sequence {u
k
} converges in norm to u
∗
. Moreover, there
exists an index k
0
∈ IN such that, for each k ≥ k
0
, one has λ
k
(2γ −λ
k
L
2
) > 0
and
u
k+1
− u
∗
 ≤
1


k
i=k
0
[1 + λ
i

(2γ −λ
i
L
2
)]
u
k
0
− u
∗
. (3.7)
To analyze the conditions (3.2) and (3.6) given respectively in Theorem
3.1 and Theorem 3.2, we can consider two examples.
14
Example 3.2 Put K = IR and F(u) = u. It is clear that F is Lipschitz
continuous, strongly monotone on K, and Sol(K, F )={0}. Choose u
0
= 1 ∈
K and λ
k
= (k + 2)
−2
for all k ∈ IN. Since lim
k→∞
λ
k
= 0 and
∞

k=0

λ
k
< +∞,
both the conditions (3.2) and (3.6) are violated. The iterative sequence {u
k
}
produced by Algorithm 3.1 for u
0
= 1 is given by
u
k+1
= P
K
(u
k
− λ
k
F (u
k
)) = u
k
− λ
k
u
k
= (1 − λ
k
)u
k
.

Hence,
u
k+1
=
k

i=0
(1 − λ
i
) =
k

i=0

1 −
1
(i + 2)
2

∀k ∈ IN.
This shows that the sequence {u
k
} is decreasing and bounded from below.
So {u
k
} is convergent. Note that
u
k+1
=
k


i=0

1 −
1
(i + 2)
2

=
k

i=0
(i + 1)(i + 3)
(i + 2)
2
=
k + 3
2(k + 2)
.
Letting k → ∞, we obtain lim
k→∞
u
k
=
1
2
. This means that {u
k
} does not
converge to the unique solution of VI(K, F ) under our consideration.

We have seen that, in Theorem 3.1 and Theorem 3.2, the conditions (3.2)
and (3.6) cannot be dropped.
Finally, let us show that the sequence {u
k
} considered in Theorem 3.2 may
not converge linearly to the unique solution of VI(K, F ).
Example 3.3 Let K, F be the same as in Example 3.2 and u
0
∈ IR\{0}. Let
{λ
k
} ⊂ (0, +∞) be satisfying (3.6) and λ
k
= 1 for all k ∈ IN. The iteration
formula in Algorithm 3.1 now becomes
u
k+1
= P
K
(u
k
− λ
k
F (u
k
)) = u
k
− λ
k
u

k
= (1 − λ
k
)u
k
.
Since lim
k→∞
λ
k
= 0 and u
k
= 0 for all k ∈ IN, we have
lim
k→∞
u
k+1
− 0
u
k
− 0
= lim
k→∞
|1 − λ
k
| = 1
Hence one cannot find any µ ∈ (0, 1) such that the inequality
u
k+1
− 0 ≤ µu

k
− 0
holds for every k ∈ IN. This means that {u
k
} does not converge linearly to
the unique solution of VI(K, F ).
15
Chapter 4
A Modified Extragradient Method
4.1 Algorithm
Consider the problem VI(K, F ) where K is a nonempty convex subset of a
real Hilbert space H and F is Lipschitz continuous for some constant L > 0.
The modified EGM for solving VI(K, F ) can be described as follows.
Algorithm 4.1
Data: u
0
∈ K and {α
k
}
∞
k=0
⊂ [a, b], where 0 < a ≤ b <
1
L
.
Step 0: Set k = 0.
Step 1: If u
k
= P
K

(u
k
− α
k
F (u
k
)) then stop.
Step 2: Set
¯u
k
= P
K
(u
k
− α
k
F (u
k
)),
u
k+1
= P
K
(u
k
− α
k
F (¯u
k
)),

(4.1)
and replace k by k + 1; go to Step 1.
4.2 Convergence of the Iterative Sequences
Theorem 4.1 Let F be a pseudomonotone mapping and let {u
k
} be a se-
quence generated by the Algorithm 4.1. If Sol(K, F ) is nonempty, then {u
k
}
is a bounded sequence and lim
k→∞
u
k
−¯u
k
 = 0. Moreover, if there exists a sub-
sequence {u
k
i
} ⊂ {u
k
} converging strongly to some ˆu ∈ K, then ˆu ∈ Sol(K, F )
and the whole sequence {u
k
} converges strongly to ˆu.
Theorem 4.2 Let F be a pseudomonotone mapping and let {u
k
} be a se-
quence generated by Algorithm 4.1. If {u
k

} converges strongly to some u
∗
,
16
then u
∗
is a solution of VI(K, F ).
Theorem 4.3 Suppose that F is a pseudomonotone mapping and {u
k
} is a
sequence generated by the Algorithm 4.1. If {u
k
} is bounded and it has a
strongly convergent subsequence, then the whole sequence converges strongly
to a solution of VI(K, F ).
We are interested in finding a lower bound for the constant a and an upper
bound for the constant b in Algorithm 4.1 and in Theorems 4.1-4.3.
Example 4.1 Consider the problem VI(K, F ) with K = IR and F(u) = u.
We can easily check that F is Lipschitz continuous with Lipschitz constant
L = 1, monotone, and Sol(K, F ) = {0}. Let u
0
= 1 ∈ K and let the real
sequence {α
k
} ⊂ (0, 1) be defined by setting α
k
= 2
−(k+1)
for all k ∈ IN :=
{0, 1, 2, . . .}. The iterative sequence in (4.1) is given by

u
k+1
=
k+1

i=1
(1 − 2
−i
+ 4
−i
) ∀k ∈ IN.
Then {u
k
} converges to ¯u /∈ Sol(K, F ).
Example 4.1 shows that the exact lower bound for a is 0.
Example 4.2 Let K, F, u
0
be the same as in Example 4.1 and let α
k
=
1 − 2
−(k+1)
for all k ∈ IN. Observe that α
k
→ 1 = 1/L as k → ∞, where
L = 1 is the Lipschitz constant of F . The iterative sequence in (4.1) is given
by
u
k+1
=

k+1

i=1
(1 − 2
−i
+ 4
−i
) ∀k ∈ IN.
Then {u
k
} converges to ¯u /∈ Sol(K, F ).
Example 4.2 shows that the exact upper bound for b is 1/L.
Theorem 4.4 Suppose that F is a monotone mapping and {u
k
} is a sequence
generated by Algorithm 4.1. If Sol(K, F ) is nonempty, then there exists ˆu in
Sol(K, F ) such that {u
k
} converges weakly to ˆu.
4.3 Convergence Rate of the Iterative Sequences
Suppose that a sequence {u
k
} in H converges in norm to u
∗
∈ H. We say
that
17
(a) {u
k
} converges to u

∗
with R-linear convergence rate if
limsup
k→∞
u
k
− u
∗

1/k
< 1,
(b) {u
k
} converges to u
∗
with Q-linear convergence rate if there exists a con-
stant µ ∈ (0, 1) such that, for all sufficiently large k,
u
k+1
− u
∗
 ≤ µu
k
− u
∗
.
Note that Q-linear convergence rate implies R-linear convergence rate. To
see that R-linear convergence does not imply Q-linear convergence in general,
one can considers the next simple example.
Example 4.3 Let {u

k
} ⊂ IR be the sequence of real numbers defined by
setting u
k
= 2
−k
if k is even and u
k
= 3
−k
if k is odd. Since limsup
k→∞
u
k
−
0
1/k
=
1
2
, {u
k
} converges to 0 with R-linear convergence rate. However {u
k
}
does not converge to 0 with Q-linear convergence rate since
limsup
k→∞
u
k+1

− 0
u
k
− 0
= +∞.
The problem VI(K, F ) is said to satisfy Tseng’s regularity assumption if
it has a solution and there exist real numbers δ, η > 0 such that
d(u, Sol(K, F )) ≤ ηu − P
K
(u − F (u)) (4.2)
for all u ∈ K with the property
u − P
K
(u − F (u)) ≤ δ. (4.3)
Theorem 4.5 Let F be a pseudomonotone mapping. Suppose that the prob-
lem VI(K, F ) satisfies Tseng’s regularity condition and {u
k
} is an iterative
sequence produced by Algorithm 4.1. Then, {u
k
} converges in norm to an
element of Sol(K, F ) with R-linear convergence rate.
If Tseng’s regularity assumption is violated, then the iterative sequence
{u
k
} may not converge in norm to any solution of VI(K, F ) with R-linear
convergence rate. The following example clarifies this remark.
Example 4.4 Consider the problem VI(K, F ) with K = [0, 1] ⊂ IR and
F (u) = u
2

for every u ∈ IR. Then F is Lipschitz continuous on K with
Lipschitz constant L = 2, monotone on K, and Sol(K, F ) = {0}. For every
u ∈ K, note that
d(u, Sol(K, F )) = u and u − P
K
(u − F (u)) = u
2
.
18
Hence one cannot find any pair {η, δ} of positive real numbers such that
(4.2) holds for every u ∈ K satisfying (4.3). This means that our problem
VI(K, F ) does not satisfy Tseng’s regularity assumption. Choose u
0
= 1 ∈ K
and α
k
= 1/4 ∈ (0, 1/2) for all k ∈ IN. The sequence {u
k
} produced by (4.1)
is given by
u
k+1
= −(1/64)(u
k
)
4
+ (1/8)(u
k
)
3

− (1/4)(u
k
)
2
+ u
k
∀k ∈ IN.
Using induction, we can prove that u
k
≥
1
k+1
for all k ∈ IN. Hence
limsup
k→∞
u
k

1/k
≥ limsup
k→∞

1
k + 1

1/k
= 1.
Therefore, although {u
k
} converges to the unique solution u

∗
= 0 of VI(K, F ),
the convergence rate is not R-linear.
Theorem 4.6 Suppose that F is a strongly pseudomonotone mapping and
VI(K, F ) has a solution u
∗
. Then, the sequence {u
k
} generated by Algorithm
4.1 converges in norm to u
∗
with Q-linear convergence rate.
19
Chapter 5
A New Extragradient Method
The idea of using stepsizes which form a non-summable diminishing sequence
of positive real numbers to deal with a strongly pseudomonotone problem
VI(K, F ) came to us from the theory of solution methods for nonsmooth
convex optimization problems.
Algorithm 5.1
Data: Let u
0
∈ K and {α
k
}
∞
k=0
⊂ IR
+
be such that

∞

k=0
α
k
= +∞, lim
k→∞
α
k
= 0. (5.1)
Step 0: Set k = 0.
Step 1: If u
k
= u
k
= P
K
(u
k
− α
k
F (u
k
)) then stop.
Step 2: Compute
¯u
k
= P
K
(u

k
− α
k
F (u
k
)),
u
k+1
= P
K
(u
k
− α
k
F (¯u
k
)),
(5.2)
and replace k by k + 1; go to Step 1.
If the computation terminates at a step k, then one puts u
k

= u
k
for all
k

≥ k + 1. Thus, Algorithm 5.1 produces an infinite iterative sequence.
5.1 Convergence of the Iterative Sequences
Theorem 5.1 If F : K → H is Lipschitz continuous and strongly pseu-

domonotone on K, and if VI(K, F ) has a unique solution u
∗
, then the se-
quence {u
k
} generated by Algorithm 5.1 converges in norm to u
∗
. Moreover,
20
there exists an index k
0
∈ IN such that γα
k
< 1 for all k ≥ k
0
and
u
k+1
− u
∗
 ≤




k

j=k
0
(1 − γα

j
)u
k
0
− u
∗
, (5.3)
where γ > 0 is a strong pseudomonotonicity constant of F . In addition,
lim
k→∞




k

j=k
0
(1 − γα
j
) = 0. (5.4)
5.2 Further Analysis
The strong pseudomonotonicity assumption on F and the two conditions in
(5.1) are essential for the validity of the assertion of Theorem 5.1. To see this,
let us start with analyzing the condition
∞

k=0
α
k

= +∞ described in (5.1).
Example 5.1 Put K = IR and F(u) = u. It is clear that F is Lipschitz
continuous, strongly monotone on K, and Sol(K, F )={0}. Choose u
0
= 1 ∈
K and define the sequence {α
k
} by setting α
k
= (k + 1)
−2
for all k ∈ IN.
Since
∞

k=0
α
k
< +∞, the first condition in (5.1) is violated. The sequence
{u
k
} produced by (5.2) is given by
u
k+1
=
k

j=0
(1 − α
j

+ α
2
j
) =
k

j=0

1 −
1
(j + 1)
2
+
1
(j + 1)
4

∀k ∈ IN.
Note that lim
k→∞
u
k
= u
∗
for some u
∗
≥
1
2
. So {u

k
} does not converge to
the unique solution of the problem VI(K, F ) under our consideration. Thus
we have seen that the assumption
∞

k=0
α
k
= +∞ cannot be dropped in the
formulation of Theorem 5.1.
The second condition in (5.1) is analyzed in the next example.
Example 5.2 Let K, F, u
0
be the same as in Example 5.1 and let α
k
= 1
for all k ∈ IN. Here
∞

k=0
α
k
= +∞, but {α
k
} does not converge to 0. By
the calculations done in Example 5.1 we get u
k
= 1 for all k ∈ IN. So {u
k

}
does not converge to the unique solution of VI(K, F ). We have seen that the
condition lim
k→∞
α
k
= 0 cannot be omitted in the formulation of Theorem 5.1.
21
Finally, let us show that the strong pseudomonotonicity assumption on F
in Theorem 5.1 is essential.
Example 5.3 Put K = IR
2
and F (u) = (−u
2
, u
1
)
T
for all u = (u
1
, u
2
)
T
∈ K.
It is clear that F is Lipschitz continuous and monotone on K, and Sol(K, F )
= {(0, 0)
T
}. Let u
0

= (u
0
1
, u
0
2
)
T
be any point in K \ {(0, 0)
T
} and let
α
k
=
1
k + 1
∀k ∈ IN.
This sequence satisfies (5.1). To see that F is not strongly pseudomonotone
on K with any constant γ > 0, it suffices to choose u = (1, 0)
T
, v = (2, 0)
T
and note that F (u), v − u = 0, but F (v), v − u = 0. The sequence {u
k
}
in (5.2) is given by u
0
= (u
0
1

, u
0
2
)
T
and







u
k+1
1
=

1 −
1
(k + 1)
2

u
k
1
+
1
k + 1
u

k
2
u
k+1
2
=

1 −
1
(k + 1)
2

u
k
2
−
1
k + 1
u
k
1
(5.5)
for k = 0, 1, 2, . . Observe that lim
k→∞
u
k
 = µu
0
, where
µ = lim

k→∞




k

j=0

1 −
1
(j + 1)
2
+
1
(j + 1)
4

≥
√
2
2
. (5.6)
Hence {u
k
} does not converge to the unique solution of VI(K, F ). Interest-
ingly, the set of the cluster points of the sequence {u
k
} is the circle
S := {u ∈ IR

2
: u = µu
0
}.
Example 5.3 shows that Algorithm 5.1, which works well for strongly pseu-
domonotone VIs, cannot serve as an adequate solution method for pseu-
domonotone VIs.
22
General Conclusions
The main results of this dissertation include:
- Partial solutions for the open questions stated in a paper by N. Thanh Hao
(2006) about the solution uniqueness of the Tikhonov regularized problem of
a pseudomonotone variational inequality and the preservation of the pseu-
domonotonicity under the regularization;
- A modified extragradient method for solving infinite-dimensional varia-
tional inequalities together with a convergence analysis;
- A new extragradient method for strongly pseudomonotone variational
inequalities, accompanied by a detailed analysis of the iterative sequences’
convergence and of the range of applicability of the method;
- Two modified projection methods for strongly pseudomonotone varia-
tional inequalities which have a strong convergence;
- Several theorems on convergence rates of the iterative sequences.
Further investigations on the following topics would be of interest:
- Verification of the pseudomonotonicity of an affine operator on a given
polyhedral convex set;
- Effective solution methods for quasimonotone variational inequalities;
- Extragradient methods for continuous pseudomonotone variational in-
equalities.
23