Some Results about the Isolated Calmness of a Mixed Variational Inequality Problem

It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on. And convex programming problem is closely related to variational inequality problem.The so-called alternative direction ofmultiplier method (ADMM) is an importance class of numerical methods for solving convex programming problem. When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required. The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point. This paper is to study the isolated calmness of the inverse KKT mapping onto the mixed variational inequality problem with nonlinear term defined by norm function and indicator function of a convex polyhedral set, respectively. We also consider the isolated calmness of the inverse KKTmapping onto classical variational inequality problem with equality and inequality constrains under strictMangasarian-Fromovitz constraint qualification condition.The results obtained here are new and very interesting.


Introduction
In this paper, let us consider the following mixed variational inequality problem: find  ∈   such that ⟨ () ,  − ⟩ +  () −  () ≥ 0, ∀ ∈   , where  :   →   is a twice continuously differentiable mapping and  :   →  ∪ {+∞} is a proper convex lower semicontinuous functional.Problem (1) is denoted by MVIP(, ).Formulation of such problem was originally considered and studied by Brézis [1] in Hilbert space.It has been shown that a large class of obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, physical, mathematical, engineering, and applied sciences can be studied in the unified and general framework of the mixed variational inequality problem.Some researchers considered and studied the existence of solutions and the numerical algorithms of the MVIP; for example, see [2] and the references therein.
On the other hand, many important optimization problems can be reformulated as the mixed variational inequality problem (1) to study.For example, if () = ∇(), where :   →  is a continuously differentiable convex function, then (1) is related to the following convex optimization problem: min  () +  () where  :   →  is a proper l.s.c.Convex function,  :   →   with  = ( 1 , . . .,   ) and each   is a continuous convex function.Let us denote the feasible set of (2) by  = { ∈   :  = ,  () ≤ 0} . ( When there exists a vector  0 ∈   such that where   is the indicator function of .Obviously, f is a proper lower semicontinuous convex function and if  is a solution to problem (2), then Noting that we have from [3] that Noting that [ +   ]() = () +   (), we obtain from ( 6) and ( 8) that which implies from the definition of subdifferential that and this is an alternative expression for inequality (1) when  = .
It is well known that optimization problem model (2) has many applications arising from matrix completion, image processing, and statistical learning.The so-called alternative direction of multiplier method (ADMM) is an importance class of numerical methods for solving problem (2).When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required; see Han and Yuan (2013) [4], Hong and Luo (2017) [5], and Han et al. (2015) [6].The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping bolds at the given KKT point; see Han et al. [6] and this observation motivates us to study the stability of problem (1).
The stability on variational inequalities is an important topic in optimization theory.For nonlinear programming, the strong regularity of the Karush-Kuhn-Tucker (KKT) system is completely characterized by Robinson (1980) [7] and Jongen et al. (1990) [8].The equivalence of the strong regularity with the Aubin property for affine variational inequalities over convex polyhedral sets was established by Dontchev and Rockafellar (1996) [9].Recently, results have been obtained for some matrix optimization problems.For instance, Y. Zhang and L. Zhang [10] proved the isolated calmness of the KKT mapping under the strict Robinson constraint qualification (SRCQ) and the second-order sufficient optimality condition for the nonlinear SDP problem.Han et al. [6] gave a characterization of isolated calmness of the KKT mapping for the convex composite quadratic semidefinite programming problems.Liu and Pan [11] extended the results in [6,10] to the isolated calmness of the KKT mapping for the matrix optimization problem involving the epigraph cone of Ky Fan -norm.Ding et al. [12] showed for a class of conic programming problems, under the Robinson constraint qualification, that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second-order sufficient condition hold.
Recently, Mordukhovich and Sarabi (2017) [13] established a set of conditions to characterize the robust isolated calmness of the KKT mapping for a type composite optimization problem; in particular, they adopted noncritical multiplier to characterize the isolated calmness.In our paper, the problem considered is a variational inequality, not a composite optimization problem; and we use second-order sufficient optimality conditions and the strict Robinson constraint qualification to ensure the isolated calmness of the solution mapping.In addition, we mainly discuss the isolated calmness of the inverse KKT mapping on problem (1) with nonlinear term  defined by norm function and the indicator function of a convex polyhedral set, respectively.And we also consider the isolated calmness of the inverse KKT mapping on classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition.

Preliminaries
In this section, we first recall some related definitions and lemma, which are needed in the subsequent discussions.
Let  be a finite dimensional real Euclidean space and  :   ⇒  be a set-valued mapping and (, ) ∈ gph .Definition 1.The set-valued mapping  is said to be isolated calm at  for  if there exist a constant  > 0 and neighborhoods  of  and  of  such that where  ⊆  is the unit ball in .
The graphical derivative is a convenient tool for investigating the isolated calmness property.
Definition 2 (see [14]).The graphical derivative of  at  for  is a set-valued mapping DS( | ) :   ⇒  defined by In the proof of the main theorem about the isolated calmness, we use the following basic characterization of the isolated calmness of the set-valued mapping  at  for .

The Isolated Calmness of MVIP
In this section, we consider MVIP(, ) (1), where  :   →   is a continuously differentiable mapping and  :   →  is a proper, lower semicontinuous and convex function.It is easy to know that MVIP(, ) (1) is equivalent to the following inclusion: For an extended real-valued function , the Moreau-Yosida regularization of , denoted by   , is defined by It follows from [14] that, when  is a proper lower semicontinuous convex function,    is continuous differentiable and where    is the proximal mapping of  and defined by The following lemma shows that ( 13) can be expressed as a nonsmooth equation.As it is an obvious result from the definition of the proximal mapping, we omit its proof.
From Lemma 3, we can easily obtain the following result.
Theorem 5. Assume that the proximal mapping    is directionally differentiable.Then   is isolated clam at (0, ) if and only if In the following, we consider two cases of MVIP: in the first case () = ‖‖ 1 = ∑  =1 |  |, and in the second case () =   (), where  is a convex polyhedral set.In particular, we consider the nonlinear complementarity problem, which corresponds to  =   + in the second case.
Corollary 8.The mapping   is isolated calm at (0,  0 ) if and only if Proof.Noting that, when  =   + , we have We obtain the result from Proposition 7.

VIP with Equality and Inequality Constraints
In this section, we consider the following variational inequality problem (for short, denoted by VIP): find  ∈  such that ⟨ () ,  − ⟩ ≥ 0, ∀ ∈ , where with ℎ :   →   and  :   →   being twice continuously differentiable.
Let  be a solution to VIP (37).We need the following assumption.