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.

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 [

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) [

For the VIP defined in (

Many gap functions have been explored during the past two decades as it is shown in [

Let

Let

For single-valued operator

Recall that the multivalued operator

Let

A matrix

We need the following lemmas. The parameters involved in the lemmas can be found in the following sections.

If abstract function

If abstract function

If abstract function

If abstract function

If abstract function

If abstract function

Let abstract function

In this section, by introducing appropriate gap functions, we give global error bound for SMVIP. Firstly, we need the following propositions.

Let

We use proof by contradiction to show the desired result. Let

Let

If abstract function

From the definition of

On the one hand, if

On the other hand, if

Based on the above discussion, one can obtain the following global error bound.

If abstract function

Since

If abstract function

By Lemma

Now, we introduce generalized D-gap function

If abstract function

From the definition of

From Proposition

If

From Proposition

On the one hand, if

On the other hand, if

By the generalized D-gap function, we have the following error bound for

Let

From Lemma

In this section, by introducing appropriate generalized gap function, the original

From Lemmas

Let

To begin, we show that

Now, we are in a position to show the global convergence result for Algorithm

Let

Let

The authors would like to thank the referees for the helpful suggestions. This work is supported by the National Natural Science Foundation of China, Contact/Grant nos. 11071109 and 11371198, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Foundation for Innovative Program of Jiangsu Province Contact/Grant no. CXZZ12_0383.