Abstract
We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.
1. Introduction and Preliminaries
The concept well-posedness is important in both theory and methodology for optimization problems. An initial, already classical concept of well-posedness for unconstrained optimization problem is due to Tykhonov in [1]. Let be a real-valued functional on Banach space . The problem of minimizing on is said to be well-posed if there exists a unique minimizer, and every minimizing sequence converges to the unique minimizer. Soon after, Levitin and Polyak [2] generalized the Tykhonov well-posedness to the constrained optimization problem, which has been known as the Levitin-Polyak well-posedness. It is clear that the concept of well-posedness is motivated by the numerical methods producing optimizing sequences for optimization problems. Unfortunately, these concepts generally cannot establish appropriate continuous dependence of the solution on the data. In turn, they are not suitable for the numerical methods when the objective functional is approximated by a family or a sequence of functionals. For this reason, another important concept of well-posedness for optimization problem, which is called the well-posedness by perturbations or extended well-posedness, has been introduced and studied by [3–6]. Also, many other notions of well-posedness have been introduced and studied for optimization problem. For details, we refer to [7] and the reference therein.
The concept well-posedness also has been generalized to other related problems, especially to the variational inequality problem. Lucchetti and Patrone [8] first introduced the well-posedness for a variational inequality, which can be regarded as an extension of the Tykhonov well-posedness of optimization problem. Since then, many authors were devoted to generalizing the concept of well-posedness for the optimization problem to various variational inequalities. In [9], Huang et al. introduced several types of (generalized) Levitin-Polyak well-posednesses for a variational inequality problem with abstract and functional constraint and gave some criteria, characterizations, and their relations for these types of well-posednesses. Recently, Fang et al. [10] generalized the concept of well-posedness by perturbations, introduced by Zelezzi for a minimization problem, to a generalized mixed variational inequality problem in Banach space. They established some metric characterizations of well-posedness by perturbations and discussed its links with well-posedness by perturbations of corresponding inclusion problem and the well-posedness by perturbations of corresponding fixed point problem. Also they derived some conditions under which the well-posedness by perturbations of the mixed variational inequality is equivalent to the existence and uniqueness of its solution. For further more results on the well-posedness of variational inequalities, we refer to [8–15] and the references therein.
When the corresponding energy functions are not convex, the mathematical model describing many important phenomena arising in mechanics and engineering is no longer variational inequality but a new type of inequality problem that is called hemivariational inequality, which was first introduced by Panagiotopoulos [16] as a generalization of variational inequality. A more generalized variational formulation which is called variational-hemivariational inequality is presented to model the problems subject to constraints because the setting of hemivariational inequalities cannot incorporate the indicator function of a convex closed subset. Due to the fact that the potential is neither convex nor smooth generally, the hemivariational inequalities have been proved very efficient to describe a variety of mechanical problems using the generalized gradient of Clarke for nonconvex and nondifferentiable functions [17], such as unilateral contact problems in nonlinear elasticity, obstacles problems, and adhesive grasping in robotics (see, e.g., [18–20]). So, in recent years all kinds of hemivariational inequalities have been studied [21–30] and the study of hemivariational inequalities has emerged as a new and interesting branch of applied mathematics. However, there are very few researchers extending the well-posedness to hemivariational inequality. In 1995, Goeleven and Mentagui [23] first defined the well-posedness for hemivariational inequalities. Recently, Xiao et al. [31] generalized the concept of well-posedness to hemivariational inequalities. They established some metric characterizations of the well-posed hemivariational inequality, derived some conditions under which the hemivariational inequality is strongly well-posed in the generalized sense, and proved the equivalence between the well-posedness of hemivariational inequality and the well-posedness of a corresponding inclusion problem. Moreover, Xiao and Huang [32] studied the well-posedness of variational-hemivariational inequalities and generalized some related results.
In the present paper, we generalize the well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequality. We establish some metric characterizations of the well-posedness by perturbations for variational-hemivariational inequality and prove the equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.
We suppose in what follows that is a real reflexive Banach space with its dual , and is the duality between and . We denote the norms of Banach space and by and , respectively. Let be a mapping, let be a locally Lipschitz functional, let be a proper, convex, and lower semicontinuous functional, and let be some given element. Denote by the domain of functional , that is, The functional is called proper if its domain is nonempty. The variational-hemivariational inequality associated with is specified as follows: where denotes the generalized directional derivative in the sense of Clarke of a locally Lipschitz functional at in the direction (see [17]) given by The variational-hemivariational inequality which includes many problems as special cases has been studied intensively. Some special cases of VHVI are as follows: (i)if , then VHVI reduces to hemivariational inequality: (ii)if , then VHVI is equivalent to the following mixed variational inequality: (iii)if ,?? and , then VHVI reduces to the global minimization problem:
Let and denote the subgradient of convex functional in the sense of convex analysis (see [33]) and the Clarke's generalized gradient of locally Lipschitz functional (see [17]), respectively, that is, About the subgradient in the sense of convex analysis, the Clarke's generalized directional derivative and the Clarke's generalized gradient, we have the following basic properties (see, e.g., [17, 19, 33, 34]).
Proposition 1.1. Let be a Banach space and a convex and proper functional. Then one has the following properties of : (i) is convex and -closed;(ii)if is continuous at , then is nonempty, convex, bounded, and -compact;(iii)if is Gâteaux differentiable at , then , where is the Gâteaux derivative of at .
Proposition 1.2. Let be a Banach space, and let , be two convex functionals. If there is a point at which is continuous, then the following equation holds:
Proposition 1.3. Let be a Banach space, , , and let be a locally Lipschitz functional defined on . Then (1)the function is finite, positively homogeneous, subadditive, and then convex on ,(2) is upper semicontinuous as a function of , as a function of alone, is Lipschitz continuous on ,(3),(4) is a nonempty, convex, bounded, and -compact subset of ,(5)for every , one has
Suppose that is a parametric normed space with norm , is a closed ball with positive radius, and is a given point. We denote the perturbed mappings of , , as and , , respectively, which have the property that for any , is a locally Lipschitz functional in , is proper, convex, and lower semicontinuous in , and Then the perturbed Clarke's generalized directional derivative and the perturbed Clarke's generalized gradient corresponding to the perturbed locally Lipschitz functional are, respectively, specified as The perturbed subgradient corresponding to the perturbed convex functional is Based on the above-perturbed mappings, the perturbed problem of is given by
In the sequel, we recall some important definitions and useful results.
Definition 1.4 (see [35]). Let be a nonempty subsets of . The measure of noncompactness of the set is defined by where diam means the diameter of set .
Definition 1.5 (see [35]). Let and be two given subsets of . The excess of over is defined by where is the distance function generated by , that is, The Hausdorff metric between and is defined by Let be a sequence of nonempty subset of . One says that converges to in the sense of Hausdorff metric if . It is easy to see that if and only if for all selection . For more details on this topic, the reader should refer to [35]. The following theorem is crucial to our main results.
Theorem 1.6 (see [36]). Let be nonempty, closed, and convex, let be nonempty, closed, convex, and bounded, let be proper, convex, and lower semicontinuous, and let be arbitrary. Assume that, for each , there exists such that Then, there exists such that
2. Well-Posedness by Perturbations of with Metric Characterizations
In this section, we generalize the concept of well-posedness by perturbations to the variational-hemivariational inequality and establish its metric characterizations.
Definition 2.1. Let ??with??. A sequence is said to be an approximating sequence corresponding to for if there exists a nonnegative sequence with as such that and
Definition 2.2. is said to be strongly (resp., weakly) well-posed by perturbations if has a unique solution in , and for any with , every approximating sequence corresponding to converges strongly (resp., weakly) to the unique solution.
Remark 2.3. Strong well-posedness by perturbations implies weak well-posedness by perturbations, but the converse is not true in general.
Definition 2.4. is said to be strongly (resp., weakly) well-posed by perturbations in the generalized sense if has a nonempty solution set in , and for any ??with??, every approximating sequence corresponding to has some subsequence which converges strongly (resp., weakly) to some point of solution set .
Remark 2.5. Strong well-posedness by perturbations in the generalized sense implies weak well-posedness by perturbations in the generalized sense, but the converse is not true in general.
To derive the metric characterizations of well-posedness by perturbations for , we define the following approximating solution set of : For any ,
where denotes the closed ball centered at with radius . For any and any set , we define the following two functions which are specified as follows:
It is easy to see that is the smallest radius of the closed ball centered at containing , and is the excess of approximating solution set over .
Based on the two functions and , we now give some metric characterizations of well-posedness by perturbations for the .
Theorem 2.6. is strongly well-posed by perturbations if and only if there exists a solution for and as .
Proof. “Necessity”: suppose that is strongly well-posed by perturbations. Then for all since there is a unique solution belonging to by the strong well-posedness by perturbations for . We now need to prove as . Assume by contradiction that does not converge to as , then there exist a constant and a nonnegative sequence with such that
By the definition of function , there exists such that
Since , there exists some such that
It is obvious that as and so is an approximating sequence corresponding to for . Therefore, by the strong well-posedness by perturbations for , we can get which is a contradiction to (2.4).
“Sufficiency”: suppose that has a solution and as . First, we claim that is a unique solution for . In fact, if has another solution with , it follows from the definition of that and belong to for all , which together with the definition of implies that
which is a contradiction to the assumption as . Now, let with and be an approximating sequence corresponding to for . Then there exists a nonnegative sequence with such that
Taking and , it easy to see that as and . Since is the unique solution for , also belongs to . And so, it follows from the definition of that
which implies that is strongly well-posed by perturbations. This completes the proof of Theorem 2.6.
Theorem 2.7. is strongly well-posed by perturbations in the generalized sense if and only if the solution set of is nonempty and compact, and as .
Proof. “Necessity”: suppose that is strongly well-posed by perturbations in the generalized sense. Then has nonempty solution set by the definition of strong well-posedness by perturbations in the generalized sense of . Let be any sequence in . It is obvious that is an approximating sequence corresponding to constant sequence for . Again by the strong well-posedness by perturbations in the generalized sense of , has a subsequence which converges strongly to some point of , which implies that the solution set of is compact. Now we show that as . Assume by contradiction that as , then there exist a constant and a nonnegative sequence with and such that
Since , there exists such that
Clearly, as . This together with the above inequality implies that is an approximating consequence corresponding to for . It follows from the strongly well-posedness by perturbations in the generalized sense for that there is a subsequence of which converges to some point of . This is contradiction to (2.10) and so as .
“Sufficiency”: we suppose that the solution set of is nonempty compact and as . Let be any sequence with and an approximating sequence corresponding to for , which implies that
Taking , it is easy to see that and . It follows that
Since the solution set of is compact, there exists such that
Again from the compactness of solution set , has a subsequence converging strongly to some point . It follows from (2.14) that
which implies that converges strongly to . Thus, is strongly well-posed by perturbations in the generalized sense. This completes the proof of Theorem 2.7.
The strong well-posedness by perturbations in the generalized sense for can also be characterized by the behavior of noncompactness measure of its approximating solution set.
Theorem 2.8. Let be a finite-dimensional space. Suppose that (i), the perturbed mapping of , is continuous with respect to ,(ii), the perturbed functional of , is lower semicontinuous with respect to and continuous with respect to for any given ,(iii), the perturbed functional of , is locally Lipschitz with respect to for any , and its Clarke's generalized directional derivative is continuous with respect to .
Then, is strongly well-posed by perturbations in the generalized sense if and only if
Proof. From the metric characterization of strongly well-posedness by perturbations in the generalized sense for in Theorem 2.7, we can easily prove the necessity. In fact, since is strongly well-posed by perturbations in the generalized sense, it follows from Theorem 2.7 that the solution set of is nonempty compact and as . Then, we can easily get from the compactness of and the fact for all that for all and
Now we prove the sufficiency. First, we claim that is closed for all . In fact, let and . Then there exists such that
Without loss of generality, we can suppose that since is finite dimensional. By taking at both sides of above inequality, it follows from the assumptions (i)–(iii) and the upper semicontinuity of with respect to that
Thus, and so is closed.
Second, we prove that
It is obvious that since the solution set for all . Conversely, let , and let be a nonnegative sequence with as . Then for any , , and so there exists such that
Since and , it is clear that . By letting in the above inequality, we get from the continuity of , , and in assumptions that
Thus, and so .
Now, we suppose that
From the definition of approximating solution set , is increasing with respect to . Then by applying the Kuratowski theorem on page 318 in [35], we have from (2.20) that is nonempty compact and
Therefore, by Theorem 2.7, is strongly well-posed by perturbations in the generalized sense.
Example 2.9. Let be a finite-dimensional space with norm , let be a closed ball in , and let be a given point in . We supposed that the perturbed mappings , , of the mapping , , are, respectively, specified as follows: where , , are three positive numbers. It is obvious that is continuous with respect to due to the continuity of the mapping , and is lower semicontinuous with respect to and continuous with respect to for any given because the functional is proper convex and lower semicontinuous. Also, the perturbed functional is locally Lipschitz with respect to since is locally Lipschitz. Furthermore, it is easy to check that the perturbed Clarke's generalized directional derivative corresponding to the perturbed function can be specified as which implies that is continuous with respect to . Thus, the assumptions in Theorem 2.8 are satisfied, and so the is strongly well-posed by perturbations in the generalized sense if and only if (2.16) holds.
3. Links with Well-Posedness by Perturbations for Corresponding Inclusion Problem
In this section, we recall some concepts of well-posedness by perturbations for inclusion problems, which are introduced by Lemaire et al. [4], and investigate the relations between the well-posedness by perturbations for and the well-posedness by perturbations for the corresponding inclusion problem.
In what follows, we always let be a set-valued mapping from real reflexive Banach space to its dual space . The inclusion problem associated with mapping is defined by whose corresponding perturbed problem is specified as where is the perturbed set-valued mapping such that .
Definition 3.1 (see [4]). Let be a sequence in with . A sequence is said to be an approximating sequence corresponding to for inclusion problem if for all and , or equivalently, there exists a sequence such that as .
Definition 3.2 (see [4]). One says that inclusion problem is strongly (resp., weakly) well-posed by perturbations if it has a unique solution, and for any with , every approximating sequence corresponding to converges strongly (resp., weakly) to the unique solution of .
Definition 3.3 (see [4]). One says that inclusion problem is strongly (resp., weakly) well-posed by perturbations in the generalized sense if the solution set of is nonempty, and for any with , every approximating sequence corresponding to has a subsequence converging strongly (resp., weakly) to some point of solution set for .
In order to obtain the relations between the strong (resp., weak) well-posedness by perturbations for variational-hemivariational inequality and the strong (resp., weak) well-posedness by perturbations for the corresponding inclusion problem, we first give the following important lemma which establishes the equivalence between the variational-hemivariational inequality and the corresponding inclusion problem. Although the lemma is a corollary of Lemma 4.1 in [32] with , we also give proof here for its importance and the completeness of our paper.
Lemma 3.4. Let be a mapping from Banach space to its dual , let be a locally Lipschitz functional, let be a proper, convex, and lower semicontinuous functional, and let be a given element in dual space . Then is a solution of if and only if is a solution of the following inclusion problem:
Proof. “Sufficiency”: assume that is a solution of inclusion problem . Then there exist and such that
By multiplying at both sides of above equation (3.4), we obtain from the definitions of the Clarke's generalized gradient for locally Lipschitz functional and the subgradient for convex functional that
which implies that is a solution of .
“Necessity”: conversely, suppose that is a solution of . Then,
From the fact that
we get that there exists a such that
By virtue of Proposition 1.3, is a nonempty, convex, and bounded subset in which implies that is nonempty, convex, and bounded in . Since is a proper, convex, and lower semicontinuous functional, it follows from (3.8) and Theorem 1.6 with that there exists , which is independent on , such that
For the sake of simplicity in writing, we denote . Then by (3.9), we have
that is, . Thus, it follows from that
which implies that is a solution of the inclusion problem . This completes the proof of Lemma 3.4.
Remark 3.5. The corresponding perturbed problem of inclusion problem is specified as
Now we prove the following two theorems which establish the relations between the strong (resp., weak) well-posedness by perturbations for variational-hemivariational inequality and the strong (resp., weak) well-posedness by perturbations for the corresponding inclusion problem .
Theorem 3.6. Let be a mapping from Banach space to its dual , let be a locally Lipschitz functional, let be a proper, convex, and lower semicontinuous functional, and let be a given element in dual space . The variational-hemivariational inequality is strongly (resp., weakly) well-posed by perturbations if and only if the corresponding inclusion problem is strongly (resp., weakly) well-posed by perturbations.
Proof. “Necessity”: assume that is strongly (resp., weakly) well-posed by perturbations, which implies that there is a unique solution of . Clearly, the existence and uniqueness of solution for inclusion problem is obtained easily by Lemma 3.4. Let be a sequence with and an approximating sequence corresponding to for . Then there exists a sequence such that as . And so, there exist and such that
From the definition of the perturbed Clarke's generalized gradient corresponding to the perturbed locally Lipschitz functional and the definition of the perturbed subgradient corresponding to the perturbed convex functional , we obtain by multiplying at both sides of above equation (3.13) that
Letting , we obtain from (3.14) and the fact as that is an approximating sequence corresponding to for . Therefore, it follows from the strong (resp., weak) well-posedness by perturbations for that converges strongly (resp., weakly) to the unique solution . Thus, the inclusion problem is strongly (resp., weakly) well-posed.
“Sufficiency”: conversely, suppose that inclusion problem is strongly (resp., weakly) well-posed by perturbations. Then has a unique solution , which implies that is the unique solution of by Lemma 3.4. Let be a sequence with and an approximating sequence corresponding to for . Then there exists a nonnegative sequence with as such that
By the same arguments in proof of Lemma 3.4, there exists a such that
and the set is nonempty, convex, and bounded in . Then, it follows from (3.16) and Theorem 1.6 with , which is proper convex and lower semicontinuous, that there exists such that
For the sake of simplicity in writing, we denote . Then it follows from (3.17) that
Define functional as follows:
where are two functional on defined by
Clearly, the functionals and are convex and continuous on , and so is proper, convex, and lower semicontinuous because is proper, convex, and lower semicontinuous with respect to . Furthermore, it follows from (3.18) that is a global minimizer of on . Thus, the zero element in , we also denote to be , belongs to the subgradient which is specified as follows due to Proposition 1.2:
It is easy to calculate that
and so there exists a with such that
Let , then due to as . This together with (3.23) and implies that
Therefore, is an approximating sequence corresponding to for . Since inclusion problem is strongly (resp., weakly) well-posed by perturbations, converges strongly (resp., weakly) to the unique solution . Therefore, variational-hemivariational inequality is strongly (resp., weakly) well-posed. This completes the proof of Theorem 3.6.
Theorem 3.7. Let be a mapping from Banach space to its dual , let be a locally Lipschitz functional, let be a proper, convex, and lower semicontinuous functional, and let be a given element in dual space . The variational-hemivariational inequality is strongly (resp., weakly) well-posed by perturbations in the generalized sense if and only if the corresponding inclusion problem is strongly (resp., weakly) well-posed by perturbations in the generalized sense.
Proof. The proof of Theorem 3.7 is similar to Theorem 3.6, and so we omit it here.
Acknowledgments
This work was supposed by the National Natural Science Foundation of China (11101069 and 81171411), the Fundamental Research Funds for the Central Universities (ZYGX2009J100), and the Open Fund (PLN1104) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).