1. Introduction
The variational inequality problem (VIP) is one of classical mathematical problems. Many models of problems in engineering and physics are constructed by partial differential equations with some suitable boundary conditions and primal conditions and are described by different kinds of variational inequality problems. Consider the variational inequality problem
which take
F as the objective function and
S as the domain: seek
satisfying
where
,
is a vector-valued function,
represents the inner product,
means the set of real numbers.
is consist of all optimal solution
,
. We assume the solution set
of
is nonempty.
Finite convergence of the feasible solution sequence produced by any algorithm for VIP has been widely concerned and researched for a long time. Early in previous studies, Rockafellar [
1], Polyak [
2], and Ferris [
3] successively put forward the weak sharp minima and strong non-degeneracy of the solution sets of mathematical programming problems, and prove that any one of them is the sufficient condition for finite convergence of the proximal point algorithm [
1,
4,
5,
6] and some important iterative algorithms [
7,
8,
9,
10,
11,
12]. It is worth noting that the finite convergence of the feasible solution sequence produced by the algorithms above, depending on not only the weak sharp minima or strong non-degeneracy of the solution set, but also the specific structural features of the algorithms. Thus, it has more universal significance to study further the conditions for the finite convergence—which does not depend on the specific algorithm—under the condition of weak sharp minima or strong non-degeneracy. Burke and Moré [
8] propose the necessary and sufficient condition for the fact that the feasible solution sequence is finitely convergent at the stationary point when the sequence converges to a strongly non-degenerate point for smooth programming problem (Corollary 3.5 in [
8]). In order to generalize the study into variational inequality, Marcotte and Zhu [
13] give the notion of weak sharpness for its solution set, then generalize the study into the continuous pseudo-monotone
variational inequality and yield the necessary and sufficient condition for finite convergence of a feasible solution sequence (Theorem 5.2 in [
13]) under the solution set being weak sharpness, a feasible solution sequence satisfying two assumption conditions, and the feasible solution set being compact. Xiu and Zhang [
6] improve Theorem 5.2 in [
13] by removing the conditions for the pseudo-monotone
and the feasible solution set being compact, and yielded the same results (Theorem 3.2 in [
6]).
However, in many cases, the weak sharpness and strong non-degeneracy for the solution set of variational inequality are not true (see Example 1 in this paper). Therefore, in order to give finite convergence of a feasible solution sequence under more relaxed conditions, we establish the notion of augmented weak sharpness relative to a feasible solution sequence for a variational inequality problem. The new notion is a substantive extension of the weak sharpness of the solution set of monotone variational inequality. We give a necessary and sufficient condition for the finite convergence of a feasible solution sequence, under the feasible solution set of VIP is augmented weak sharp. The result is an extension of published results (see the corollaries in
Section 3 in this paper). In addition, the notion of augmented weak sharpness which we establish also provides weaker sufficient conditions for the finite convergence of many optimization algorithms.
The paper is organized as follows. After some elementary definitions and notations, in
Section 2, we introduce the notions of weak sharpness and strong non-degeneracy of the solution set, we establish the notion of augmented weak sharpness, and prove that augmented weak sharpness is a necessary condition for the weak sharpness and strong non-degeneracy of the solution set to VIP. In
Section 3, we give the necessary and sufficient conditions for the finite convergence of a feasible solution sequence, under the condition that the solution set is augmented weak sharp on feasible solution sequence.
Next, we introduce the notions and symbols used in this paper.
Suppose is an infinite subsequence, and . We define
According to the above notions, it follows that
Suppose
is a subset,
denotes the interior of
C.
,
, the tangent cone of
C at point
is defined as [
14]
The regular normal cone of C at point is
In general meaning, the normal cone of C at point is defined as
The polar cone of C is
According to Proposition 6.5 in [
15], we have
When
C is convex, according to Theorem 6.9 in [
15], we have
Suppose ; the projection of x on a closed set C is
Note by the above definition that when C is a closed convex set, is a single-valued mapping. However, when C is a closed set, may be a set-valued mapping.
The distance between x and C is given by
If C is a closed set, we have
Suppose the subdifferential of function at point is a non-empty set. Then, the projection subdifferential of at point x is
If
is continuously differentiable in a neighborhood of point
, according to Practice 8.8 in [
15], we have
; i.e., the projection subdifferential is projection gradient
We say that converges finitely to C if there exists such that for all .
2. The Augmented Weak Sharpness of Solution Sets
In this section, we first introduce the notions of the weak sharpness and strong non-degeneracy of solution sets for variational inequality (VIP) in some existing literature. Then, we give the definition of the augmented weak sharpness of solution sets relative to feasible solution sequence for VIP, and clarify, by examples, the new notion is an extension of the weak sharpness and strong non-degeneracy of the solution set of monotone variational inequality.
Now we give the notions of weak sharpness and strong non-degeneracy of solution sets for
(see [
16]).
Definition 1. The solution set is called a set of weak sharpness if The point satisfying Definition 1 is called weak sharp point.
In the general case,
is not necessarily a convex set. When
is non-convex, according to Proposition 6.5 in [
15],
is a closed convex cone, but
is only a closed cone, and there is
Thus, there is
So, we give the following definition of weak sharpness. It is more relaxed than Definition 1.
Definition 2. The solution set is called a set of weak sharpness if Next, we introduce the notion of strong non-degeneracy.
Definition 3. The solution set is a set of strong non-degeneracy, if The point satisfying (2) is called a strongly non-degenerate point. As described earlier, in many cases, the solution set is not weak sharp or strongly non-degenerate. In order to overcome this defect, we now introduce the notion of the augmented weak sharpness of a solution set relative to a feasible solution sequence for VIP.
Definition 4. Suppose that solution set is closed, . We say is a set of augmented weak sharpness relative to the feasible solution sequence if one of the following is true:
- (1)
is a finite set;
- (2)
When is an infinite sequence, there exist a set-valued mapping such that
- (a)
there is a constant satisfying - (b)
for and , there is
The following simple example shows that the solution set is not weak sharp, but it is augmented weak sharp relative to the feasible solution sequence.
Example 1. Consider where This is a non-monotone variational inequality problem. According to (3) and (4), is not a strongly non-degenerate set, because is not a strongly non-generate point. Next, we prove that is an augmented weak sharp set relative to , satisfying the following conditions.
- (i)
- (ii)
Suppose is an infinite sequence. Let , introducing set-valued mapping By (4) and (5), it maintains of Definition 4. According to , the accumulation points of the bounded sequence can only be or . Without loss of generality, suppose is one of its accumulation points, so there is an infinite sequence such that By (5)–(7) , if is large enough, we have Then, according to (7), we have So it maintains of Definition 4.
Next, we will prove that the augmented weak sharpness of the solution set is an extension of the weak sharpness and strong non-degeneracy.
Theorem 1. For , F is monotonous, is a closed set. If is a set of weak sharpness, then is a set of augmented weak sharpness relative to any one .
Proof of Theorem 1. As
is a monotonous variational inequality problem, then its solution set
is a closed convex set, so
, and
is single valued mapping. Let
. Suppose
} is an infinite sequence, then let
According to the hypothesis,
is a set of weak sharp minima, so according to (
1) , we know
in Definition 4 is true. Additionally, because
is monotonous, it maintains
of Definition 4. The proof is complete. ☐
Remark 1. Theorem 1 shows that the augmented weak sharpness of the solution sets for are an extension of the weak sharp minima of the solution sets of monotone variational inequality.
Theorem 2. For , suppose is a closed set, If is bounded and any one of its accumulation points satisfies . Then, is a set of augmented weak sharpness relative to .
Proof of Theorem 2. Suppose
is an infinite sequence. According to the hypotheses,
is bounded, so
must have an accumulation point
, such that
; that is, a constant
exists, such that
Let
,
, by
, which maintains
of Definition 4.
It maintains of Definition 4. The proof is complete. ☐
Theorem 3. For , suppose is continuous on S. If is bounded and any one of its accumulation points is a strong non-degenerate point, then is a set of augmented weak sharpness relative to .
Proof of Theorem 3. Suppose
is an infinite sequence. Then, by the assumption
must have an accumulation point
. Suppose
satisfies
According to the hypothesis,
is a strong non-degenerated point and
is continuous; by Proposition 5.1 in [
14],
is an isolated point of
, so we have
Thus, by (
2) and (
11), there exists a constant
, such that
Now we define a set-valued mapping
as follows
By (
12) and (
13), we have
It maintains of Definition 4.
Then, because
is an isolated point of
, by (
10) for all large enough
, there is
. So, by (
10) and (
13) and continuity of
, we have
It maintains of Definition 4. The proof is complete. ☐
The theorems above show that the augmented weak sharpness of the solution sets for VIP is an extension of the weak sharp and the weak sharpness.