Gap Functions and Algorithms for Variational Inequality Problems

We solve several kinds of variational inequality problems through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis. By generalized gap functions and generalized D-gap functions, we give global bounds for the set-valued mixed variational inequality problems. And through gap function, we equivalently transform the generalized variational inequality problem into a constraint optimization problem, give the steepest descent method, and show the convergence of the method.


Introduction
Variational inequality problem (VIP) provides us with a simple, natural, unified, and general frame to study a wide class of equilibrium problems arising in transportation system analysis [1,2], regional science [3,4], elasticity [5], optimization [6], and economics [7].Canonical VIP can be described as follows: find a point  ∈  ⊂ R  such that ⟨ () ,  − ⟩ ≥ 0, ∀ ∈ , where  is a nonempty closed convex subset of R  ,  is a mapping from R  into itself, and ⟨⋅, ⋅⟩ denotes the inner product in R  .In recent years, considerable interest has been shown in developing various, useful, and important extensions and generalizations of VIP, both for its own sake and for its applications, such as general variational inequality problem (GVIP) [8] and set-valued (mixed) variational inequality problem (SMVIP) [9].There are significant developments of these problems related to multivalued operators, nonconvex optimization, iterative methods, and structural analysis.More recently, much attention has been given to reformulate the VIP as an optimization problem.And gap functions, which can constitute an equivalent optimization problem, turn out to be very useful in designing new globally convergent algorithms and in analyzing the rate of convergence of some iterative methods.Various gap functions for VIP have been suggested and proposed by many authors in [8,[10][11][12][13] and the references therein.Error bounds are functions which provide a measure of the distance between a solution set and an arbitrary point.Therefore, error bounds play an important role in the analysis of global or local convergence analysis of algorithms for solving VIP.
For the VIP defined in (1), the authors in [10] provided an equivalent optimization problem formulation through regularized gap function   :  → R defined by where  is a parameter.The authors proved that  is the solution of problem (1) if and only if  is global minimizer of function   () in  and   () = 0.In order to expand the definition of regularized gap function, the authors in [14] gave the definition of generalized regularized gap function defined by where  is an abstract function which satisfies conditions ranked as follows: (C1)  is continuous differentiable on  × ; ( Note that ∇ 2 is the partial of  with respect to the second component and conditions (C1)-(C5) can make sense.One can refer to [10,14] and so forth for more details.Many gap functions have been explored during the past two decades as it is shown in [10][11][12][13][14][15][16] and the references therein.Motivated by their work, in this paper, we solve some classes of VIP through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis.We consider generalized gap functions and generalized D-gap functions for SMVIP and give global bounds for the problem through the two functions, respectively.And for GVIP, we equivalently transform it into a constraint optimization problem through gap function, introduce the steepest descent method, and show the convergence of the method.

Preliminaries
Let  be a real Hilbert space whose inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖ ⋅ ‖, respectively.Let  be a nonempty closed convex set in  and let 2  be the family of all nonempty compact subsets of .
Note that when  = 0, the original problem (7) reduces to a set-valued variational inequality problem; when  = 0 and  is a single-valued operator, problem (7) is the right problem discussed in (1).
Recall that the multivalued operator  :  ⊂  → 2  is said to be strongly monotone with modulus  > 0 on  if And  is said to be Lipschtiz continuous on a nonempty bounded set  ⊂ , if there exists a positive constant  such that where (⋅, ⋅) is the Hausdorff metric on  defined by Let  : And   is a uniform approximation if  is independent of .
A matrix  ∈ R × is a  0 -matrix if each of its principal minors is nonnegative.
We need the following lemmas.The parameters involved in the lemmas can be found in the following sections.

Gap Functions and Error Bounds for SMVIP
In this section, by introducing appropriate gap functions, we give global error bound for SMVIP.Firstly, we need the following propositions.
Proposition 8. Let  be a nonempty closed convex set in  and let  be strictly convex in .Then  has only one minimum in .
Proof.We use proof by contradiction to show the desired result.Let  1 ,  2 ∈  be two minimal points of ; that is, ( 1 ) = ( 2 ) = min ().Since  is strictly convex, one obtains that This implies that there exists a point , which is a contradiction.This completes the proof.
By the generalized D-gap function, we have the following error bound for SMVIP(, ).

Steepest Descent Method for GVIP
In this section, by introducing appropriate generalized gap function, the original GVIP(, ) in ( 6) can be changed into an optimization problem with restrictions.When one designs algorithms to solve the optimization problem, the gradient of objective function is unavoidable.We try to design a new algorithm, constructing a class of descent direction, to solve the optimization problem.In the following, we set  to be R  .And we introduce the following generalized gap function for GVIP(, ): () = max where   () is a minimal point for −Ψ  (, ⋅),  is a positive parameter, and  satisfies conditions (C1)-(C5) stated above.
And similar to the discussion in [10,11], we also give the following two assumptions: From Lemmas 5-7, we obtain that the original GVIP ( 6) is equivalent to the following optimization problem: min ..∈   () . ( For problem (46), we give the following algorithm.
Step 1.If   () ≤ , then we can end the circulation.