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Levitin–Polyak well-posedness of variational inequalities
Nonlinear Analysis 72 (2010) 373–381
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Levitin–Polyak well-posedness of variational inequalitiesI Rong Hu a,b , Ya-ping Fang a,∗ a
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
b
Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, PR China
article
info
Article history: Received 27 November 2007 Accepted 15 June 2009 MSC: 49J40 49K40 90C31
abstract In this paper we consider the Levitin–Polyak well-posedness of variational inequalities. We derive a characterization of the Levitin–Polyak well-posedness by considering the size of Levitin–Polyak approximating solution sets of variational inequalities. We also show that the Levitin–Polyak well-posedness of variational inequalities is closely related to the Levitin–Polyak well-posedness of minimization problems and fixed point problems. Finally, we prove that under suitable conditions, the Levitin–Polyak well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Variational inequality Minimization problem Fixed point problem Levitin–Polyak well-posedness Uniqueness
1. Introduction The importance of well-posedness has been widely recognized in the theory of optimization problems. The first concept of well-posedness is due to Tykhonov [1] for a global minimization problem, which has been known as Tykhonov wellposedness. The Tykhonov well-posedness of a global minimization problem requires the existence and uniqueness of minimizer, and the convergence of every minimizing sequence toward the unique minimizer. The concept of Tykhonov well-posedness can also be used in a constrained minimization problem, but only in an abstract way. The Tykhonov wellposedness of a constrained minimization problem requires that every minimizing sequence should lie in the constraint set. In many practical situations, the minimizing sequence produced by a numerical optimization method usually fails to be feasible but gets closer and closer to the constraint set. Such a sequence is called a generalized minimizing sequence for constrained minimization problems. To take care of such a case, Levitin and Polyak [2] strengthened the concept of Tykhonov wellposedness by requiring the existence and uniqueness of minimizer, and the convergence of every generalized minimizing sequence toward the unique minimizer, which has been known as Levitin–Polyak (for short, LP) well-posedness. In the literature, there are a very large number of results concerned with Tykhonov well-posedness, LP well-posedness and their generalizations for minimization problems. For details, we refer the readers to [1–8]. On the other hand, various variational inequalities have been studied extensively (see e.g. [9–13]). As known, under convexity and differentiability assumptions a minimization problem is equivalent to a variational inequality problem. This fact motivates some researchers to study the well-posedness of various variational inequalities [4,14–21]. It is worth
I This work was partially supported by the National Natural Science Foundation of China (10826064, 10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Youth Science Foundation of Sichuan University (07069). ∗ Corresponding author. E-mail address: [email protected] (Y.-p. Fang).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.071
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mentioning that the first notion of well-posedness for a variational inequality is due to Lucchetti and Patrone [4]. The concept of well-posedness has been also generalized to other variational problems such as fixed point problems [22,23,19,24], saddle point problems [25] and equilibrium problems [26–32]. For a minimization problem, its Tykhonov well-posedness can be characterized by the behavior of approximating solution sets, see for instance the Furi–Vignoli criterion [33]. Another classical result is that under suitable conditions, the Tykhonov well-posedness of a minimization problem is equivalent to the uniqueness and existence of its solution. Analogous results also hold for the LP well-posedness of a constrained minimization problem. In recent years, some authors established some analogous results for the well-posedness (in the sense of Tykhonov) of various variational inequalities [14,17,19]. In this paper we further consider the LP well-posedness of variational inequalities. We derive a characterization of LP well-posedness by considering the size of Levitin–Polyak approximating solution sets of variational inequalities. We show that the LP well-posedness of variational inequalities is closely related to the LP well-posedness of minimization problems and fixed point problems. At last we obtain that under suitable conditions, the LP well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution. 2. Preliminaries Let K be a nonempty subset of a Euclidian space Rn and F : Rn → Rn be a function. The classical variational inequality associated with (F , K ) is formulated as follows: VI (F , K ) :
find x ∈ K such that hF (x), x − yi ≤ 0,
∀y ∈ K ,
which has been studied intensively (see e.g. [9–11]). In terms of a gap function introduced by Auslender [34], the variational inequality can be transformed into a minimization problem. Recall that the Auslender gap function for VI (F , K ) is defined by g (v) = suphF (v), v − wi, w∈K
∀v ∈ Rn .
(1)
A point x∗ ∈ K solves VI (F , K ) if and only if x∗ solves the following minimization problem: MP (g , K ) :
min g (x),
x∈K
and g (x ) = 0. On the other hand, when f : Rn → R is differentiable and convex on an open set containing K , MP (f , K ) is equivalent to VI (∇ f , K ), where ∇ f denotes the gradient operator of f . In terms of the projection operator, the variational inequality is also equivalent to a fixed point problem. Recall that the metric projection PK (z ) of z on K is defined by: ∗
PK (z ) = argmin{kz − xk : x ∈ K },
∀z ∈ Rn .
It is known that u = PK (z ) if and only if
hu − z , v − ui ≥ 0,
∀v ∈ K .
The projection operator PK is nonexpansive, i.e., kPK (x) − PK (y)k ≤ kx − yk for all x, y ∈ Rn . By means of PK , VI (F , K ) is equivalent to the following fixed point problem: FP (PK (I − λF ), K ) :
find x ∈ K such that x = PK (I − λF )(x),
where λ > 0 is a constant. Summarizing the above results, we have the following lemma: Lemma 2.1 (See e.g. [9–11,34]). Let K be a nonempty, closed and convex subset of Rn and F : Rn → Rn . Then the following conclusions are equivalent: (i) x solves VI (F , K ). (ii) x solves MP (g , K ) and g (x) = 0, where g : Rn → R is the gap function defined by (1). (iii) x solves FP (PK (I − λF ), K ), where λ > 0 is a constant. In addition, if f : Rn → R is differentiable and convex on an open set containing K , then x solves MP (f , K ) if and only if x solves VI (∇ f , K ). In what follows we recall some concepts. Definition 2.1. A mapping F : Rn → Rn is said to be (i) monotone if
hF (x) − F (y), x − yi ≥ 0,
∀x, y ∈ Rn .
(ii) strictly monotone if for any x, y ∈ Rn with x 6= y,
hF (x) − F (y), x − yi > 0.
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Definition 2.2. A mapping F : Rn → Rn is said to be hemicontinuous if for any x, y ∈ Rn , the function t 7→ hF (x + t (y − x)), y − xi from [0, 1] into R is continuous at 0+ . Definition 2.3. Let X and Y be two topological spaces. A mapping G : X → Y is said to be uniformly continuous if for any neighborhood V of 0 in Y , there exists a neighborhood U of 0 in X such that G(x) − G(y) ∈ V for all x, y ∈ U. Obviously, the uniform continuity of a function F : Rn → Rn implies its hemicontinuity. Lemma 2.2 (See [9–11]). Let K be a nonempty, closed and convex subset of Rn and x ∈ K a given point, and let F : Rn → Rn be monotone and hemicontinuous. Then
hF (x), x − yi ≤ 0,
∀y ∈ K
if and only if
hF (y), x − yi ≤ 0,
∀y ∈ K .
Lemma 2.3 (See [9,10]). Let K be a nonempty, bounded, closed and convex subset of Rn and let F : Rn → Rn be strictly monotone and hemicontinuous. Then there exists a unique x∗ ∈ K such that
hF (x∗ ), x∗ − yi ≤ 0,
∀y ∈ K .
3. LP approximating solutions and LP well-posedness In this section we introduce the concept of LP approximating sequences for a variational inequality. In terms of LP approximating sequences, we generalize the concept of LP well-posedness to a variational inequality. We study the LP wellposedness of variational inequalities by considering the size of LP approximating solution sets. Definition 3.1 ([14]). A sequence {xn } ⊂ Rn is called an approximating sequence for VI (F , K ) if xn ∈ K for all n ∈ N and there exists a sequence 0 < n → 0 such that
hF (xn ), xn − yi ≤ n ,
∀y ∈ K , ∀n ∈ N .
Definition 3.2 ([14]). VI (F , K ) is said to be well-posed (in the sense of Tykhonov) if VI (F , K ) has a unique solution and every approximating sequence converges to the unique solution. Definition 3.3. A sequence {xn } ⊂ Rn is called an LP approximating sequence for VI (F , K ) if there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that xn + wn ∈ K for all n ∈ N and
hF (xn ), xn − yi ≤ n ,
∀y ∈ K , ∀n ∈ N .
Definition 3.4. We say that VI (F , K ) is LP well-posed if VI (F , K ) has a unique solution and every LP approximating sequence converges to the unique solution. Remark 3.1. (i) Every approximating sequence is an LP approximating sequence. (ii) The LP well-posedness implies the well-posedness of VI (F , K ). To characterize the LP well-posedness, we consider the following set of LP approximating solutions of VI (F , K ):
Ω () = {x ∈ Rn : d(x, K ) ≤ ,
hF (x), x − yi ≤ , ∀y ∈ K },
∀ ≥ 0.
It is easy to see that
Ω () = {x ∈ Rn : d(x, K ) ≤ ,
g (u) ≤ }.
In what follows we always suppose that u∗ is a fixed solution of VI (F , K ). Define
α() = sup{ku − u∗ k : u ∈ Ω ()},
∀ ≥ 0.
By definition, α() is the radius of the smallest closed ball centered at u∗ containing Ω (). The following result shows that the LP well-posedness of VI (F , K ) can be characterized by the behavior of α . Theorem 3.1. VI (F , K ) is LP well-posed if and only if α() → 0 as → 0. Proof. Let VI (F , K ) be LP well-posed. Then u∗ ∈ K is the unique solution of VI (F , K ). Assume by contradiction that α() 6→ 0 as → 0. Then there exist δ and 0 < n → 0 such that
α(n ) > δ > 0.
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By the definition of α , there exists un ∈ Ω (n ) such that
kun − u∗ k > δ. It follows from un ∈ Ω (n ) that d(un , K ) ≤ n < n +
1 n
and hF (un ), un − vi ≤ n ,
∀v ∈ K .
Then there exists vn ∈ K such that 1
kun − vn k < n + . n
Set
wn = vn − un . It follows that un + w n ∈ K
and
wn → 0.
Thus {un } is an LP approximating sequence for VI (F , K ). By the LP well-posedness of VI (F , K ), we get kun − u∗ k → 0, a contradiction. Conversely assume that α() → 0 as → 0. We first show that u∗ ∈ K is the unique solution of VI (F , K ). Indeed, if u¯ (6= u∗ ) is another solution of VI (F , K ). By definition, α() ≥ ku∗ − u¯ k > 0 for all ≥ 0, a contradiction. Let {un } be an LP approximating sequence for VI (F , K ). Then there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K
and
hF (un ), un − vi ≤ n ,
∀v ∈ K .
It is easy to see that un ∈ Ω (n ) for sufficiently large n. Set tn = kun − u∗ k. By the definition of α ,
α(n ) ≥ tn = kun − u∗ k. Taking α(n ) → 0 into account, we get kun − u∗ k → 0 as n → ∞. So VI (F , K ) is LP well-posed.
4. Links with LP well-posedness of minimization problems In this section we shall investigate the relationship between the LP well-posedness of variational inequalities and the LP well-posedness of minimization problems. Let K be a nonempty subset of Rn and let f : Rn → R be a functional. Consider the following minimization problem: MP (f , K ) :
min f (x),
x ∈ K.
In what follows we recall some concepts. Definition 4.1 (See [1–3]). A sequence {un } ⊂ Rn is said to be a minimizing sequence for MP (f , K ) if un ∈ K for all n ∈ N and f (un ) converges to infv∈K f (v). A sequence {un } ⊂ Rn is said to be a generalized minimizing sequence for MP (f , K ) if there exists 0 < n → 0 such that d(un , K ) ≤ n
and f (un ) ≤ inf f (v) + n , v∈K
∀n ∈ N .
Definition 4.2 (See [1–3]). MP (f , K ) is said to be well-posed (resp. LP well-posed) if MP (f , K ) has a unique solution and every minimizing (resp. generalized minimizing) sequence converges to the unique solution. When f is convex and differentiable on an open set containing K , it has been shown in [4,14] that under certain conditions the Tykhonov well-posedness of MP (f , K ) is equivalent to the well-posedness of VI (∇ f , K ). Next we shall establish analogous results for LP well-posedness. Theorem 4.1. Let K be a nonempty, closed and convex subset of Rn and let f : Rn → R be convex and differentiable on an open set containing K . If MP (f , K ) is LP well-posed, then VI (∇ f , K ) is LP well-posed. Conversely if K is bounded and VI (∇ f , K ) is Tykhonov well-posed, then MP (f , K ) is LP well-posed. Proof. Let MP (f , K ) be LP well-posed. Then MP (f , K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of VI (∇ f , K ). Let {un } be an LP approximating sequence for VI (∇ f , K ). Then there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K
(2)
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and
h∇ f (un ), un − vi ≤ n ,
∀v ∈ K .
(3)
From (2) we get
kun − kk = kun + wn − k + wn k ≤ kun + wn − kk + kwn k,
∀k ∈ K .
This yields d(un , K ) = inf kun − kk ≤ inf kun + wn − kk + kwn k = kwn k. k∈K
k∈K
Since f is convex, it follows from (3) that f (un ) − f (v) ≤ h∇ f (un ), un − vi ≤ n ,
∀v ∈ K ,
which yields f (un ) ≤ inf f (v) + n . v∈K
Set n = max{kwn k, n }. It follows that 0
d(un , K ) ≤ n0
f (un ) ≤ inf f (v) + n0 .
and
v∈K
Thus {un } is a generalized minimizing sequence for MP (f , K ). By the LP well-posedness of MP (f , K ), un converges to u∗ . So VI (∇ f , K ) is LP well-posed. Conversely, let VI (∇ f , K ) be Tykhonov well-posed. Then VI (∇ f , K ) has a unique solution u∗ which is the unique solution of MP (f , K ). Let {un } be a generalized minimizing sequence for MP (f , K ). Then there exists 0 < n → 0 such that d(un , K ) ≤ n
(4)
f (un ) ≤ inf f (v) + n .
(5)
and v∈K
By (4), there exists xn ∈ K such that 1
kxn − un k < n + . n
Take
w n = x n − un .
(6)
Since f is convex and differentiable on an open set containing K , it follows from Proposition 2.2.6 of [35] that f is Lipschitz continuous. Since wn = xn − un → 0, there exists 0 < δn → 0 such that
kf (xn ) − f (un )k ≤ δn .
(7)
By (5) and (7), f (xn ) ≤ inf f (v) + (n + δn ).
(8)
v∈K
By Ekeland Theorem [36], there exists x¯ n ∈ K such that
kxn − x¯ n k ≤
p n + δn
(9)
and
h∇ f (¯xn ), x¯ n − xi ≤
p
n + δn k¯xn − xk,
∀x ∈ K .
(10)
Since K is bounded, from (10) we get
p h∇ f (¯xn ), x¯ n − xi ≤ L n + δn ,
∀x ∈ K ,
(11)
where L denotes the diameter of K . Thus {¯xn } is an approximating sequence for VI (∇ f , K ). By the Tykhonov well-posedness of VI (∇, K ), x¯ n converges to u∗ . It follows from (6) and (9) that
kun − u∗ k ≤ kun − xn k + kxn − x¯ n k + k¯xn − u∗ k → 0. Therefore, MP (f , K ) is LP well-posed.
Next we establish the relationship between the LP well-posedness of variational inequalities and the LP well-posedness of minimization problems by means of gap functions.
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Theorem 4.2. Let K be a nonempty subset of Rn and F : Rn → Rn . Then VI (F , K ) is LP well-posed if and only if MP (g , K ) is LP well-posed and the optimal value of MP (g , K ) is zero, where g : Rn → R is the gap function defined by (1). Proof. Let VI (F , K ) be LP well-posed. Then VI (F , K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of MP (g , K ) with g (u∗ ) = 0. Let {un } be a generalized minimizing sequence for MP (g , K ), i.e., there exists 0 < n → 0 such that d(un , K ) ≤ n
(12)
g (un ) ≤ inf g (v) + n = n .
(13)
and v∈K
By (12), there exists vn ∈ K such that 1
kun − vn k < n + . n
Take wn = vn − un . It follows that un + w n ∈ K
and
wn → 0.
(14)
From (13) we further get
hF (un ), un − vi ≤ n ,
∀v ∈ K .
(15)
By (14) and (15), {un } is an LP approximating sequence for VI (F , K ). By the LP well-posedness of VI (F , K ), un converges to u∗ . Thus MP (g , K ) is LP well-posed. Conversely, suppose that MP (g , K ) is LP well-posed and the optimal value of MP (g , K ) is zero. By Lemma 2.1, MP (g , K ) has a unique solution u∗ which is the unique solution of VI (F , K ). Let {un } be an LP approximating sequence for VI (F , K ), i.e., there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K
(16)
and
hF (un ), un − vi ≤ n ,
∀v ∈ K .
(17)
It follows from (16) that d(un , K ) = inf kun − kk k∈K
≤ inf {kun + wn − kk + kwn k} = kwn k. k∈K
From (17) we get g (un ) ≤ inf g (v) + n . v∈K
Therefore {un } is a generalized minimizing sequence for MP (g , K ). Since MP (g , K ) is LP well-posed, un converges to u∗ . So VI (F , K ) is LP well-posed. 5. Links with LP well-posedness of fixed point problems In this section we shall investigate the relationship between the LP well-posedness of variational inequalities and the LP well-posedness of fixed point problems. We also prove that under certain conditions, the LP well-posedness of a variational inequality is equivalent to the existence and uniqueness of its solution. Let K be a nonempty subset of Rn and T : Rn → Rn . The fixed point problem associated with (T , K ) is formulated as follows: FP (T , K ) :
find x ∈ K such that T (x) = x.
We first recall some concepts. Definition 5.1 (See [22,23]). A sequence {xn } ⊂ K is called an approximating sequence for FP (T , K ) if kxn − T (xn )k → 0 as n → ∞. We say that FP (T , K ) is well-posed if FP (T , K ) has a unique solution and every approximating sequence for FP (T , K ) converges to the unique solution. Now we introduce the concepts of LP approximating sequences and LP well-posedness for fixed point problems. Definition 5.2. A sequence {xn } ⊂ Rn is called an LP approximating sequence for FP (T , K ) if there exists wn ∈ Rn with wn → 0 such that xn + wn ∈ K and kxn − T (xn )k → 0 as n → ∞. We say that FP (T , K ) is LP well-posed if FP (T ) has a unique solution and every LP approximating sequence for FP (T , K ) converges to the unique solution. Clearly, the LP well-posedness of FP (T , K ) implies its well-posedness.
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Theorem 5.1. Let K be a nonempty, closed and convex subset of Rn and let F : Rn → Rn be a monotone and uniformly continuous function. Then FP (PK (I − λF ), K ) is LP well-posed whenever VI (F , K ) has a unique solution, where λ > 0 is a constant. Proof. Suppose that VI (F , K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of FP (PK (I − λF ), K ). Let {un } be an LP approximating sequence for FP (PK (I − λF ), K ), i.e., there exists wn ∈ Rn with wn → 0 such that un + wn ∈ K
and
kun − PK (un − λF (un ))k → 0.
Take xn = PK (un − λF (un )).
(18)
It follows that
k un − x n k → 0
(19)
and
hxn − un + λF (un ), xn − vi ≤ 0,
∀v ∈ K ,
which yields
hF (un ), xn − vi ≤
1
λ
hun − xn , xn − vi,
∀v ∈ K .
(20)
If {un } is bounded, then {xn } is bounded from (19). Without loss of generality, we can suppose that xn → x∗ ∈ K . Consequently we get un → x∗ . From (18) we get x∗ = PK (I −λF (x∗ )) and so x∗ = u∗ . Thus FP (PK (I −λF ), K ) is LP well-posed. If {un } is unbounded, then {un + wn } is an unbounded sequence of K . Without loss of generality, we can suppose that
kun + wn k → ∞,
tn =
1
kun + wn − u∗ k
∈ (0, 1).
Take zn = u∗ + tn (un + wn − u∗ ). Clearly {zn } is a bounded sequence of K . Without loss of generality, we can suppose that zn → z ∈ K \ {u∗ }. For any v ∈ K , it follows that
hF (v), z − vi = hF (v), z − zn i + hF (v), zn − u∗ i + hF (v), u∗ − vi = hF (v), z − zn i + hF (v), u∗ − vi + tn hF (v), un + wn − u∗ i = hF (v), z − zn i + (1 − tn )hF (v), u∗ − vi + tn hF (v), wn i + tn hF (v), un − xn i + tn hF (v), xn − vi.
(21)
Since F is monotone, from (20) we get
hF (v), xn − vi ≤ hF (xn ), xn − vi = hF (xn ) − F (un ), xn − vi + hF (un ), xn − vi ≤ hF (xn ) − F (un ), xn − vi +
1
λ
hun − xn , xn − vi,
∀v ∈ K .
(22)
Further we have
hF (v), u∗ − vi ≤ hF (u∗ ), u∗ − vi ≤ 0,
∀v ∈ K
(23)
since u∗ is the unique solution of VI (F , K ). It follows from (21)–(23) that
hF (v), z − vi ≤ hF (v), z − zn i + tn hF (v), wn i + tn hF (v), un − xn i 1
+ tn hF (xn ) − F (un ), xn − vi + tn hun − xn , xn − vi, λ Since F is uniformly continuous and xn − un → 0, we get
∀v ∈ K .
kF (xn ) − F (un )k → 0. Since kun − xn k → 0, it is easy to see that tn (xn − v) is bounded. It follows from (24) and (25) that
hF (v), z − vi ≤ 0,
∀v ∈ K ,
(24)
(25)
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which together with Lemma 2.2 implies that
hF (z ), z − vi ≤ 0,
∀v ∈ K .
Thus z is a solution of VI (F , K ), a contradiction. Thus FP (PK (I − λF ), K ) is LP well-posed. Theorem 5.2. Let K be a nonempty, closed and convex subset of Rn and let F : Rn → Rn be a hemicontinuous and monotone function. Then VI (F , K ) is LP well-posed whenever FP (PK (I − λF ), K ) has a unique solution. Proof. Suppose that FP (PK (I − λF ), K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of VI (F , K ). Let {un } be an LP approximating sequence for VI (F , K ), i.e., there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K and
hF (un ), un − vi ≤ n ,
∀v ∈ K .
Since F is monotone, we get
hF (v), un − vi ≤ hF (un ), un − vi ≤ n ,
∀v ∈ K .
(26)
If {un } is unbounded, then {un + wn } is unbounded. Without loss of generality, we can suppose that kun + wn k → ∞. Take tn =
1
and
k un + w n − u∗ k
zn = u∗ + tn (un + wn − u∗ ).
Without loss of generality we can suppose that tn ∈ (0, 1) and zn → z ∈ K \ {u∗ }. For any v ∈ K , it follows from (26) that
hF (v), z − vi = = = ≤
hF (v), z − zn i + hF (v), zn − u∗ i + hF (v), u∗ − vi hF (v), z − zn i + tn hF (v), un + wn − u∗ i + hF (v), u∗ − vi hF (v), z − zn i + (1 − tn )hF (v), u∗ − vi + tn hF (v), un − vi + tn hF (v), wn i hF (v), z − zn i + (1 − tn )hF (v), u∗ − vi + tn n + tn hF (v), wn i.
Letting n → ∞ in the above inequality, we get
hF (v), z − vi ≤ hF (v), u∗ − vi,
∀v ∈ K .
(27)
Since u is the unique solution of VI (F , K ), it follows from Lemma 2.2 that ∗
hF (v), u∗ − vi ≤ 0,
∀v ∈ K .
This together with (27) yields
hF (v), z − vi ≤ 0,
∀v ∈ K .
Again from Lemma 2.2, we have
hF (z ), z − vi ≤ 0,
∀v ∈ K .
Therefore z is a solution of VI (F , K ), a contradiction. Thus we can suppose that {un } is bounded. Without loss of generality, we can suppose un → u¯ . Since un + wn ∈ K and wn → 0, we get u¯ ∈ K . It follows from (26) that
hF (v), u¯ − vi ≤ 0,
∀v ∈ K ,
which together with Lemma 2.2 yields
hF (¯u), u¯ − vi ≤ 0,
∀v ∈ K .
Since u is the unique solution of VI (F , K ), we have u¯ = u∗ . Thus VI (F , K ) is LP well-posed. ∗
As a consequence of Theorems 5.1 and 5.2 and Lemma 2.1, we have the following result: Theorem 5.3. Let K be a nonempty, closed and convex subset of Rn and let F : Rn → Rn be a monotone and uniformly continuous function. Then the following conclusions are equivalent: (i) VI (F , K ) is LP well-posed. (ii) FP (PK (I − λF ), K ) is LP well-posed, where λ > 0 is a constant.
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(iii) VI (F , K ) has a unique solution. (iv) FP (PK (I − λF ), K ) has a unique solution. The following result gives a class of functions which guarantee the LP well-posedness of variational inequalities. Theorem 5.4. Let K be a nonempty, bounded, closed and convex subset of Rn and let F : Rn → Rn be strictly monotone and uniformly continuous. Then VI (F , K ) is LP well-posed. Proof. The conclusion follows directly from Theorem 5.3 and Lemma 2.3.
Acknowledgement The authors would like to express their deep thanks to one anonymous referee for his/her helpful comments and suggestions which led to the improvement of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Levitin–Polyak well-posedness of variational inequalitiesI Rong Hu a,b , Ya-ping Fang a,∗ a
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
b
Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, PR China
article
info
Article history: Received 27 November 2007 Accepted 15 June 2009 MSC: 49J40 49K40 90C31
abstract In this paper we consider the Levitin–Polyak well-posedness of variational inequalities. We derive a characterization of the Levitin–Polyak well-posedness by considering the size of Levitin–Polyak approximating solution sets of variational inequalities. We also show that the Levitin–Polyak well-posedness of variational inequalities is closely related to the Levitin–Polyak well-posedness of minimization problems and fixed point problems. Finally, we prove that under suitable conditions, the Levitin–Polyak well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Variational inequality Minimization problem Fixed point problem Levitin–Polyak well-posedness Uniqueness
1. Introduction The importance of well-posedness has been widely recognized in the theory of optimization problems. The first concept of well-posedness is due to Tykhonov [1] for a global minimization problem, which has been known as Tykhonov wellposedness. The Tykhonov well-posedness of a global minimization problem requires the existence and uniqueness of minimizer, and the convergence of every minimizing sequence toward the unique minimizer. The concept of Tykhonov well-posedness can also be used in a constrained minimization problem, but only in an abstract way. The Tykhonov wellposedness of a constrained minimization problem requires that every minimizing sequence should lie in the constraint set. In many practical situations, the minimizing sequence produced by a numerical optimization method usually fails to be feasible but gets closer and closer to the constraint set. Such a sequence is called a generalized minimizing sequence for constrained minimization problems. To take care of such a case, Levitin and Polyak [2] strengthened the concept of Tykhonov wellposedness by requiring the existence and uniqueness of minimizer, and the convergence of every generalized minimizing sequence toward the unique minimizer, which has been known as Levitin–Polyak (for short, LP) well-posedness. In the literature, there are a very large number of results concerned with Tykhonov well-posedness, LP well-posedness and their generalizations for minimization problems. For details, we refer the readers to [1–8]. On the other hand, various variational inequalities have been studied extensively (see e.g. [9–13]). As known, under convexity and differentiability assumptions a minimization problem is equivalent to a variational inequality problem. This fact motivates some researchers to study the well-posedness of various variational inequalities [4,14–21]. It is worth
I This work was partially supported by the National Natural Science Foundation of China (10826064, 10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Youth Science Foundation of Sichuan University (07069). ∗ Corresponding author. E-mail address: [email protected] (Y.-p. Fang).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.071
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mentioning that the first notion of well-posedness for a variational inequality is due to Lucchetti and Patrone [4]. The concept of well-posedness has been also generalized to other variational problems such as fixed point problems [22,23,19,24], saddle point problems [25] and equilibrium problems [26–32]. For a minimization problem, its Tykhonov well-posedness can be characterized by the behavior of approximating solution sets, see for instance the Furi–Vignoli criterion [33]. Another classical result is that under suitable conditions, the Tykhonov well-posedness of a minimization problem is equivalent to the uniqueness and existence of its solution. Analogous results also hold for the LP well-posedness of a constrained minimization problem. In recent years, some authors established some analogous results for the well-posedness (in the sense of Tykhonov) of various variational inequalities [14,17,19]. In this paper we further consider the LP well-posedness of variational inequalities. We derive a characterization of LP well-posedness by considering the size of Levitin–Polyak approximating solution sets of variational inequalities. We show that the LP well-posedness of variational inequalities is closely related to the LP well-posedness of minimization problems and fixed point problems. At last we obtain that under suitable conditions, the LP well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution. 2. Preliminaries Let K be a nonempty subset of a Euclidian space Rn and F : Rn → Rn be a function. The classical variational inequality associated with (F , K ) is formulated as follows: VI (F , K ) :
find x ∈ K such that hF (x), x − yi ≤ 0,
∀y ∈ K ,
which has been studied intensively (see e.g. [9–11]). In terms of a gap function introduced by Auslender [34], the variational inequality can be transformed into a minimization problem. Recall that the Auslender gap function for VI (F , K ) is defined by g (v) = suphF (v), v − wi, w∈K
∀v ∈ Rn .
(1)
A point x∗ ∈ K solves VI (F , K ) if and only if x∗ solves the following minimization problem: MP (g , K ) :
min g (x),
x∈K
and g (x ) = 0. On the other hand, when f : Rn → R is differentiable and convex on an open set containing K , MP (f , K ) is equivalent to VI (∇ f , K ), where ∇ f denotes the gradient operator of f . In terms of the projection operator, the variational inequality is also equivalent to a fixed point problem. Recall that the metric projection PK (z ) of z on K is defined by: ∗
PK (z ) = argmin{kz − xk : x ∈ K },
∀z ∈ Rn .
It is known that u = PK (z ) if and only if
hu − z , v − ui ≥ 0,
∀v ∈ K .
The projection operator PK is nonexpansive, i.e., kPK (x) − PK (y)k ≤ kx − yk for all x, y ∈ Rn . By means of PK , VI (F , K ) is equivalent to the following fixed point problem: FP (PK (I − λF ), K ) :
find x ∈ K such that x = PK (I − λF )(x),
where λ > 0 is a constant. Summarizing the above results, we have the following lemma: Lemma 2.1 (See e.g. [9–11,34]). Let K be a nonempty, closed and convex subset of Rn and F : Rn → Rn . Then the following conclusions are equivalent: (i) x solves VI (F , K ). (ii) x solves MP (g , K ) and g (x) = 0, where g : Rn → R is the gap function defined by (1). (iii) x solves FP (PK (I − λF ), K ), where λ > 0 is a constant. In addition, if f : Rn → R is differentiable and convex on an open set containing K , then x solves MP (f , K ) if and only if x solves VI (∇ f , K ). In what follows we recall some concepts. Definition 2.1. A mapping F : Rn → Rn is said to be (i) monotone if
hF (x) − F (y), x − yi ≥ 0,
∀x, y ∈ Rn .
(ii) strictly monotone if for any x, y ∈ Rn with x 6= y,
hF (x) − F (y), x − yi > 0.
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Definition 2.2. A mapping F : Rn → Rn is said to be hemicontinuous if for any x, y ∈ Rn , the function t 7→ hF (x + t (y − x)), y − xi from [0, 1] into R is continuous at 0+ . Definition 2.3. Let X and Y be two topological spaces. A mapping G : X → Y is said to be uniformly continuous if for any neighborhood V of 0 in Y , there exists a neighborhood U of 0 in X such that G(x) − G(y) ∈ V for all x, y ∈ U. Obviously, the uniform continuity of a function F : Rn → Rn implies its hemicontinuity. Lemma 2.2 (See [9–11]). Let K be a nonempty, closed and convex subset of Rn and x ∈ K a given point, and let F : Rn → Rn be monotone and hemicontinuous. Then
hF (x), x − yi ≤ 0,
∀y ∈ K
if and only if
hF (y), x − yi ≤ 0,
∀y ∈ K .
Lemma 2.3 (See [9,10]). Let K be a nonempty, bounded, closed and convex subset of Rn and let F : Rn → Rn be strictly monotone and hemicontinuous. Then there exists a unique x∗ ∈ K such that
hF (x∗ ), x∗ − yi ≤ 0,
∀y ∈ K .
3. LP approximating solutions and LP well-posedness In this section we introduce the concept of LP approximating sequences for a variational inequality. In terms of LP approximating sequences, we generalize the concept of LP well-posedness to a variational inequality. We study the LP wellposedness of variational inequalities by considering the size of LP approximating solution sets. Definition 3.1 ([14]). A sequence {xn } ⊂ Rn is called an approximating sequence for VI (F , K ) if xn ∈ K for all n ∈ N and there exists a sequence 0 < n → 0 such that
hF (xn ), xn − yi ≤ n ,
∀y ∈ K , ∀n ∈ N .
Definition 3.2 ([14]). VI (F , K ) is said to be well-posed (in the sense of Tykhonov) if VI (F , K ) has a unique solution and every approximating sequence converges to the unique solution. Definition 3.3. A sequence {xn } ⊂ Rn is called an LP approximating sequence for VI (F , K ) if there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that xn + wn ∈ K for all n ∈ N and
hF (xn ), xn − yi ≤ n ,
∀y ∈ K , ∀n ∈ N .
Definition 3.4. We say that VI (F , K ) is LP well-posed if VI (F , K ) has a unique solution and every LP approximating sequence converges to the unique solution. Remark 3.1. (i) Every approximating sequence is an LP approximating sequence. (ii) The LP well-posedness implies the well-posedness of VI (F , K ). To characterize the LP well-posedness, we consider the following set of LP approximating solutions of VI (F , K ):
Ω () = {x ∈ Rn : d(x, K ) ≤ ,
hF (x), x − yi ≤ , ∀y ∈ K },
∀ ≥ 0.
It is easy to see that
Ω () = {x ∈ Rn : d(x, K ) ≤ ,
g (u) ≤ }.
In what follows we always suppose that u∗ is a fixed solution of VI (F , K ). Define
α() = sup{ku − u∗ k : u ∈ Ω ()},
∀ ≥ 0.
By definition, α() is the radius of the smallest closed ball centered at u∗ containing Ω (). The following result shows that the LP well-posedness of VI (F , K ) can be characterized by the behavior of α . Theorem 3.1. VI (F , K ) is LP well-posed if and only if α() → 0 as → 0. Proof. Let VI (F , K ) be LP well-posed. Then u∗ ∈ K is the unique solution of VI (F , K ). Assume by contradiction that α() 6→ 0 as → 0. Then there exist δ and 0 < n → 0 such that
α(n ) > δ > 0.
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By the definition of α , there exists un ∈ Ω (n ) such that
kun − u∗ k > δ. It follows from un ∈ Ω (n ) that d(un , K ) ≤ n < n +
1 n
and hF (un ), un − vi ≤ n ,
∀v ∈ K .
Then there exists vn ∈ K such that 1
kun − vn k < n + . n
Set
wn = vn − un . It follows that un + w n ∈ K
and
wn → 0.
Thus {un } is an LP approximating sequence for VI (F , K ). By the LP well-posedness of VI (F , K ), we get kun − u∗ k → 0, a contradiction. Conversely assume that α() → 0 as → 0. We first show that u∗ ∈ K is the unique solution of VI (F , K ). Indeed, if u¯ (6= u∗ ) is another solution of VI (F , K ). By definition, α() ≥ ku∗ − u¯ k > 0 for all ≥ 0, a contradiction. Let {un } be an LP approximating sequence for VI (F , K ). Then there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K
and
hF (un ), un − vi ≤ n ,
∀v ∈ K .
It is easy to see that un ∈ Ω (n ) for sufficiently large n. Set tn = kun − u∗ k. By the definition of α ,
α(n ) ≥ tn = kun − u∗ k. Taking α(n ) → 0 into account, we get kun − u∗ k → 0 as n → ∞. So VI (F , K ) is LP well-posed.
4. Links with LP well-posedness of minimization problems In this section we shall investigate the relationship between the LP well-posedness of variational inequalities and the LP well-posedness of minimization problems. Let K be a nonempty subset of Rn and let f : Rn → R be a functional. Consider the following minimization problem: MP (f , K ) :
min f (x),
x ∈ K.
In what follows we recall some concepts. Definition 4.1 (See [1–3]). A sequence {un } ⊂ Rn is said to be a minimizing sequence for MP (f , K ) if un ∈ K for all n ∈ N and f (un ) converges to infv∈K f (v). A sequence {un } ⊂ Rn is said to be a generalized minimizing sequence for MP (f , K ) if there exists 0 < n → 0 such that d(un , K ) ≤ n
and f (un ) ≤ inf f (v) + n , v∈K
∀n ∈ N .
Definition 4.2 (See [1–3]). MP (f , K ) is said to be well-posed (resp. LP well-posed) if MP (f , K ) has a unique solution and every minimizing (resp. generalized minimizing) sequence converges to the unique solution. When f is convex and differentiable on an open set containing K , it has been shown in [4,14] that under certain conditions the Tykhonov well-posedness of MP (f , K ) is equivalent to the well-posedness of VI (∇ f , K ). Next we shall establish analogous results for LP well-posedness. Theorem 4.1. Let K be a nonempty, closed and convex subset of Rn and let f : Rn → R be convex and differentiable on an open set containing K . If MP (f , K ) is LP well-posed, then VI (∇ f , K ) is LP well-posed. Conversely if K is bounded and VI (∇ f , K ) is Tykhonov well-posed, then MP (f , K ) is LP well-posed. Proof. Let MP (f , K ) be LP well-posed. Then MP (f , K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of VI (∇ f , K ). Let {un } be an LP approximating sequence for VI (∇ f , K ). Then there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K
(2)
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377
and
h∇ f (un ), un − vi ≤ n ,
∀v ∈ K .
(3)
From (2) we get
kun − kk = kun + wn − k + wn k ≤ kun + wn − kk + kwn k,
∀k ∈ K .
This yields d(un , K ) = inf kun − kk ≤ inf kun + wn − kk + kwn k = kwn k. k∈K
k∈K
Since f is convex, it follows from (3) that f (un ) − f (v) ≤ h∇ f (un ), un − vi ≤ n ,
∀v ∈ K ,
which yields f (un ) ≤ inf f (v) + n . v∈K
Set n = max{kwn k, n }. It follows that 0
d(un , K ) ≤ n0
f (un ) ≤ inf f (v) + n0 .
and
v∈K
Thus {un } is a generalized minimizing sequence for MP (f , K ). By the LP well-posedness of MP (f , K ), un converges to u∗ . So VI (∇ f , K ) is LP well-posed. Conversely, let VI (∇ f , K ) be Tykhonov well-posed. Then VI (∇ f , K ) has a unique solution u∗ which is the unique solution of MP (f , K ). Let {un } be a generalized minimizing sequence for MP (f , K ). Then there exists 0 < n → 0 such that d(un , K ) ≤ n
(4)
f (un ) ≤ inf f (v) + n .
(5)
and v∈K
By (4), there exists xn ∈ K such that 1
kxn − un k < n + . n
Take
w n = x n − un .
(6)
Since f is convex and differentiable on an open set containing K , it follows from Proposition 2.2.6 of [35] that f is Lipschitz continuous. Since wn = xn − un → 0, there exists 0 < δn → 0 such that
kf (xn ) − f (un )k ≤ δn .
(7)
By (5) and (7), f (xn ) ≤ inf f (v) + (n + δn ).
(8)
v∈K
By Ekeland Theorem [36], there exists x¯ n ∈ K such that
kxn − x¯ n k ≤
p n + δn
(9)
and
h∇ f (¯xn ), x¯ n − xi ≤
p
n + δn k¯xn − xk,
∀x ∈ K .
(10)
Since K is bounded, from (10) we get
p h∇ f (¯xn ), x¯ n − xi ≤ L n + δn ,
∀x ∈ K ,
(11)
where L denotes the diameter of K . Thus {¯xn } is an approximating sequence for VI (∇ f , K ). By the Tykhonov well-posedness of VI (∇, K ), x¯ n converges to u∗ . It follows from (6) and (9) that
kun − u∗ k ≤ kun − xn k + kxn − x¯ n k + k¯xn − u∗ k → 0. Therefore, MP (f , K ) is LP well-posed.
Next we establish the relationship between the LP well-posedness of variational inequalities and the LP well-posedness of minimization problems by means of gap functions.
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Theorem 4.2. Let K be a nonempty subset of Rn and F : Rn → Rn . Then VI (F , K ) is LP well-posed if and only if MP (g , K ) is LP well-posed and the optimal value of MP (g , K ) is zero, where g : Rn → R is the gap function defined by (1). Proof. Let VI (F , K ) be LP well-posed. Then VI (F , K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of MP (g , K ) with g (u∗ ) = 0. Let {un } be a generalized minimizing sequence for MP (g , K ), i.e., there exists 0 < n → 0 such that d(un , K ) ≤ n
(12)
g (un ) ≤ inf g (v) + n = n .
(13)
and v∈K
By (12), there exists vn ∈ K such that 1
kun − vn k < n + . n
Take wn = vn − un . It follows that un + w n ∈ K
and
wn → 0.
(14)
From (13) we further get
hF (un ), un − vi ≤ n ,
∀v ∈ K .
(15)
By (14) and (15), {un } is an LP approximating sequence for VI (F , K ). By the LP well-posedness of VI (F , K ), un converges to u∗ . Thus MP (g , K ) is LP well-posed. Conversely, suppose that MP (g , K ) is LP well-posed and the optimal value of MP (g , K ) is zero. By Lemma 2.1, MP (g , K ) has a unique solution u∗ which is the unique solution of VI (F , K ). Let {un } be an LP approximating sequence for VI (F , K ), i.e., there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K
(16)
and
hF (un ), un − vi ≤ n ,
∀v ∈ K .
(17)
It follows from (16) that d(un , K ) = inf kun − kk k∈K
≤ inf {kun + wn − kk + kwn k} = kwn k. k∈K
From (17) we get g (un ) ≤ inf g (v) + n . v∈K
Therefore {un } is a generalized minimizing sequence for MP (g , K ). Since MP (g , K ) is LP well-posed, un converges to u∗ . So VI (F , K ) is LP well-posed. 5. Links with LP well-posedness of fixed point problems In this section we shall investigate the relationship between the LP well-posedness of variational inequalities and the LP well-posedness of fixed point problems. We also prove that under certain conditions, the LP well-posedness of a variational inequality is equivalent to the existence and uniqueness of its solution. Let K be a nonempty subset of Rn and T : Rn → Rn . The fixed point problem associated with (T , K ) is formulated as follows: FP (T , K ) :
find x ∈ K such that T (x) = x.
We first recall some concepts. Definition 5.1 (See [22,23]). A sequence {xn } ⊂ K is called an approximating sequence for FP (T , K ) if kxn − T (xn )k → 0 as n → ∞. We say that FP (T , K ) is well-posed if FP (T , K ) has a unique solution and every approximating sequence for FP (T , K ) converges to the unique solution. Now we introduce the concepts of LP approximating sequences and LP well-posedness for fixed point problems. Definition 5.2. A sequence {xn } ⊂ Rn is called an LP approximating sequence for FP (T , K ) if there exists wn ∈ Rn with wn → 0 such that xn + wn ∈ K and kxn − T (xn )k → 0 as n → ∞. We say that FP (T , K ) is LP well-posed if FP (T ) has a unique solution and every LP approximating sequence for FP (T , K ) converges to the unique solution. Clearly, the LP well-posedness of FP (T , K ) implies its well-posedness.
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Theorem 5.1. Let K be a nonempty, closed and convex subset of Rn and let F : Rn → Rn be a monotone and uniformly continuous function. Then FP (PK (I − λF ), K ) is LP well-posed whenever VI (F , K ) has a unique solution, where λ > 0 is a constant. Proof. Suppose that VI (F , K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of FP (PK (I − λF ), K ). Let {un } be an LP approximating sequence for FP (PK (I − λF ), K ), i.e., there exists wn ∈ Rn with wn → 0 such that un + wn ∈ K
and
kun − PK (un − λF (un ))k → 0.
Take xn = PK (un − λF (un )).
(18)
It follows that
k un − x n k → 0
(19)
and
hxn − un + λF (un ), xn − vi ≤ 0,
∀v ∈ K ,
which yields
hF (un ), xn − vi ≤
1
λ
hun − xn , xn − vi,
∀v ∈ K .
(20)
If {un } is bounded, then {xn } is bounded from (19). Without loss of generality, we can suppose that xn → x∗ ∈ K . Consequently we get un → x∗ . From (18) we get x∗ = PK (I −λF (x∗ )) and so x∗ = u∗ . Thus FP (PK (I −λF ), K ) is LP well-posed. If {un } is unbounded, then {un + wn } is an unbounded sequence of K . Without loss of generality, we can suppose that
kun + wn k → ∞,
tn =
1
kun + wn − u∗ k
∈ (0, 1).
Take zn = u∗ + tn (un + wn − u∗ ). Clearly {zn } is a bounded sequence of K . Without loss of generality, we can suppose that zn → z ∈ K \ {u∗ }. For any v ∈ K , it follows that
hF (v), z − vi = hF (v), z − zn i + hF (v), zn − u∗ i + hF (v), u∗ − vi = hF (v), z − zn i + hF (v), u∗ − vi + tn hF (v), un + wn − u∗ i = hF (v), z − zn i + (1 − tn )hF (v), u∗ − vi + tn hF (v), wn i + tn hF (v), un − xn i + tn hF (v), xn − vi.
(21)
Since F is monotone, from (20) we get
hF (v), xn − vi ≤ hF (xn ), xn − vi = hF (xn ) − F (un ), xn − vi + hF (un ), xn − vi ≤ hF (xn ) − F (un ), xn − vi +
1
λ
hun − xn , xn − vi,
∀v ∈ K .
(22)
Further we have
hF (v), u∗ − vi ≤ hF (u∗ ), u∗ − vi ≤ 0,
∀v ∈ K
(23)
since u∗ is the unique solution of VI (F , K ). It follows from (21)–(23) that
hF (v), z − vi ≤ hF (v), z − zn i + tn hF (v), wn i + tn hF (v), un − xn i 1
+ tn hF (xn ) − F (un ), xn − vi + tn hun − xn , xn − vi, λ Since F is uniformly continuous and xn − un → 0, we get
∀v ∈ K .
kF (xn ) − F (un )k → 0. Since kun − xn k → 0, it is easy to see that tn (xn − v) is bounded. It follows from (24) and (25) that
hF (v), z − vi ≤ 0,
∀v ∈ K ,
(24)
(25)
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which together with Lemma 2.2 implies that
hF (z ), z − vi ≤ 0,
∀v ∈ K .
Thus z is a solution of VI (F , K ), a contradiction. Thus FP (PK (I − λF ), K ) is LP well-posed. Theorem 5.2. Let K be a nonempty, closed and convex subset of Rn and let F : Rn → Rn be a hemicontinuous and monotone function. Then VI (F , K ) is LP well-posed whenever FP (PK (I − λF ), K ) has a unique solution. Proof. Suppose that FP (PK (I − λF ), K ) has a unique solution u∗ ∈ K . By Lemma 2.1, u∗ is the unique solution of VI (F , K ). Let {un } be an LP approximating sequence for VI (F , K ), i.e., there exist wn ∈ Rn with wn → 0 and 0 < n → 0 such that un + w n ∈ K and
hF (un ), un − vi ≤ n ,
∀v ∈ K .
Since F is monotone, we get
hF (v), un − vi ≤ hF (un ), un − vi ≤ n ,
∀v ∈ K .
(26)
If {un } is unbounded, then {un + wn } is unbounded. Without loss of generality, we can suppose that kun + wn k → ∞. Take tn =
1
and
k un + w n − u∗ k
zn = u∗ + tn (un + wn − u∗ ).
Without loss of generality we can suppose that tn ∈ (0, 1) and zn → z ∈ K \ {u∗ }. For any v ∈ K , it follows from (26) that
hF (v), z − vi = = = ≤
hF (v), z − zn i + hF (v), zn − u∗ i + hF (v), u∗ − vi hF (v), z − zn i + tn hF (v), un + wn − u∗ i + hF (v), u∗ − vi hF (v), z − zn i + (1 − tn )hF (v), u∗ − vi + tn hF (v), un − vi + tn hF (v), wn i hF (v), z − zn i + (1 − tn )hF (v), u∗ − vi + tn n + tn hF (v), wn i.
Letting n → ∞ in the above inequality, we get
hF (v), z − vi ≤ hF (v), u∗ − vi,
∀v ∈ K .
(27)
Since u is the unique solution of VI (F , K ), it follows from Lemma 2.2 that ∗
hF (v), u∗ − vi ≤ 0,
∀v ∈ K .
This together with (27) yields
hF (v), z − vi ≤ 0,
∀v ∈ K .
Again from Lemma 2.2, we have
hF (z ), z − vi ≤ 0,
∀v ∈ K .
Therefore z is a solution of VI (F , K ), a contradiction. Thus we can suppose that {un } is bounded. Without loss of generality, we can suppose un → u¯ . Since un + wn ∈ K and wn → 0, we get u¯ ∈ K . It follows from (26) that
hF (v), u¯ − vi ≤ 0,
∀v ∈ K ,
which together with Lemma 2.2 yields
hF (¯u), u¯ − vi ≤ 0,
∀v ∈ K .
Since u is the unique solution of VI (F , K ), we have u¯ = u∗ . Thus VI (F , K ) is LP well-posed. ∗
As a consequence of Theorems 5.1 and 5.2 and Lemma 2.1, we have the following result: Theorem 5.3. Let K be a nonempty, closed and convex subset of Rn and let F : Rn → Rn be a monotone and uniformly continuous function. Then the following conclusions are equivalent: (i) VI (F , K ) is LP well-posed. (ii) FP (PK (I − λF ), K ) is LP well-posed, where λ > 0 is a constant.
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(iii) VI (F , K ) has a unique solution. (iv) FP (PK (I − λF ), K ) has a unique solution. The following result gives a class of functions which guarantee the LP well-posedness of variational inequalities. Theorem 5.4. Let K be a nonempty, bounded, closed and convex subset of Rn and let F : Rn → Rn be strictly monotone and uniformly continuous. Then VI (F , K ) is LP well-posed. Proof. The conclusion follows directly from Theorem 5.3 and Lemma 2.3.
Acknowledgement The authors would like to express their deep thanks to one anonymous referee for his/her helpful comments and suggestions which led to the improvement of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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