Gap Functions and Algorithms for Variational Inequality Problems

Abstract

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.

1. 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 𝑥∈𝐾⊂ℝ^𝑛 such that ⟨𝑇(𝑥),𝑦−𝑥⟩≥0,∀𝑦∈𝐾, (1) where 𝐾 is a nonempty closed convex subset of ℝ^𝑛, 𝑇 is a mapping from ℝ^𝑛 into itself, and ⟨⋅,⋅⟩ denotes the inner product in ℝ^𝑛.

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, 10–13] 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 𝐺_𝛼∶𝐻→ℝ defined by 𝐺_𝛼(𝑥)=max_𝑦∈𝐾(𝑥)𝛼⟨𝐹,𝑥−𝑦⟩−2𝑥−𝑦², (2) where 𝛼 is a parameter. The authors proved that 𝑥 is the solution of problem (1) if and only if 𝑥 is global minimizer of function 𝐺_𝛼(𝑥) in 𝐾 and 𝐺_𝛼(𝑥)=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 𝐺_𝛼(𝑥)=max_𝑦∈𝐾⟨𝐹(𝑥),𝑥−𝑦⟩−𝛼𝜙𝑥,𝑦, (3) where 𝜙 is an abstract function which satisfies conditions ranked as follows:(C1)𝜙 is continuous differentiable on 𝐻×𝐻;(C2)𝜙 is nonnegative on 𝐻×𝐻;(C3)𝜙 is uniformly strongly convex on 𝐻; that is, there exists a positive number 𝜆 such that 𝜙𝑥,𝑦₁−𝜙𝑥,𝑦₂≥⟨∇₂𝜙𝑥,𝑦₂,𝑦₁−𝑦₂⟩+𝜆𝑦₁−𝑦₂²,∀𝑥,𝑦₁,𝑦₂∈𝐻; (4)(C4)𝜙(𝑥,𝑦)=0⇔𝑥=𝑦;(C5)∇₂(𝑥,𝑦) is uniformly Lipschtiz continuous on 𝐻; that is, there exists a constant 𝐿^′>0 such that ∇₂𝜙𝑥,𝑦₁−∇₂𝜙𝑥,𝑦₂≤𝐿^′𝑦₁−𝑦₂,∀𝑥,𝑦₁,𝑦₂∈𝐻. (5)Note that ∇₂ is the partial of 𝜙 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 [10–16] 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.

2. Preliminaries

Let 𝐻 be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐾 be a nonempty closed convex set in 𝐻 and let 2^𝐻 be the family of all nonempty compact subsets of 𝐻.

Let 𝐹,𝑓∶𝐻→𝐻 be nonlinear operators. The GVIP can be described as follows: Find 𝑥∈𝐻, 𝑓(𝑥)∈𝐾 such that ⟨𝐹(𝑥),𝑓𝑦−𝑓(𝑥)⟩≥0,∀𝑦∈𝐻,𝑓𝑦∈𝐾. (6)

For single-valued operator 𝑓∶𝐻→ℝ∪{+∞}, which is proper convex and lower semicontinuous, and for given multivalued operator 𝑇∶𝐻→2^𝐻, the SMVIP can be described as follows: Find 𝑥∈𝐾,𝑤∈𝑇(𝑥) such that ⟨𝑤,𝑦−𝑥⟩+𝑓𝑦−𝑓(𝑥)≥0,∀𝑦∈𝐾. (7) Note that when 𝑓=0, the original problem (7) reduces to a set-valued variational inequality problem; when 𝑓=0 and 𝑇 is a single-valued operator, problem (7) is the right problem discussed in (1).

Recall that the multivalued operator 𝑇∶𝐾⊂𝐻→2^𝐻 is said to be strongly monotone with modulus 𝛽>0 on 𝐾 if 𝑤−𝑤^′,𝑥−𝑥^′≥𝛽𝑥−𝑥^′2,∀(𝑥,𝑤),𝑥^′,𝑤^′∈graph(𝑇). (8) And 𝑇 is said to be Lipschtiz continuous on a nonempty bounded set 𝐵⊂𝐾, if there exists a positive constant 𝐿 such that 𝐻𝑇(𝑥),𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐵, (9) where 𝐻(⋅,⋅) is the Hausdorff metric on 𝐵 defined by 𝐻𝑇(𝑥),𝑇𝑦=maxsup_{(𝑥)𝑟∈𝑇}inf_{𝑦𝑠∈𝑇}‖𝑟−𝑠‖,sup_{𝑠∈𝑇(𝑦)}inf_{𝑟∈𝑇(𝑥)}‖𝑟−𝑠‖,∀𝑥,𝑦∈𝐵. (10)

Let 𝐹∶ℝ^𝑛→ℝ^𝑛. Then 𝐹 is a 𝑃₀-function if max_{1≤𝑖≤𝑛,𝑥𝑖≠𝑦𝑖}(𝑥_𝑖−𝑦_𝑖)(𝐹_𝑖(𝑥)−𝐹_𝑖(𝑦))≥0, for all 𝑥,𝑦∈ℝ^𝑛 and 𝑥≠𝑦. Assume 𝐹_𝜇(⋅)∶ℝ^𝑛→ℝ^𝑛(𝜇>0). 𝐹_𝜇 is called smoothing approximation function of 𝐹, if there exists a positive constant 𝑘 such that 𝐹_𝜇(𝑥)−𝐹(𝑥)≤𝑘𝑢,∀𝑢>0,𝑥∈ℝ^𝑛. (11) And 𝐹_𝜇 is a uniform approximation if 𝑘 is independent of 𝑥.

A matrix 𝑀∈ℝ^𝑛×𝑛 is a 𝑃₀-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.

Lemma 1 (see [11]). If abstract function 𝜙 satisfies condition (C1), then the following holds: ⟨∇₂𝜙𝑥,𝑦₁−∇₂𝜙𝑥,𝑦₂,𝑦₁−𝑦₂⟩≥2𝜆𝑦₁−𝑦₂²,∀𝑦₁,𝑦₂∈𝐻; (12) that is, ∇₂(𝑥,⋅) is strong monotone in 𝐻, and by (C5), one obtains that 2𝜆≤𝐿^′.

Lemma 2 (see [17]). If abstract function 𝜙 satisfies conditions (C1)–(C4), then ∇₂𝜙𝑥,𝑦=0⟺𝑥=𝑦. (13)

Lemma 3 (see [18]). If abstract function 𝜙 satisfies conditions (C1)–(C5) and 𝜆 and 𝐿^′ are the corresponding coefficients defined above, then one has 𝜆𝑥−𝑦²≤𝜙𝑥,𝑦≤𝐿^′−𝜆𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (14)

Lemma 4 (see [19]). If abstract function 𝜙 satisfies conditions (C1)–(C4), then 𝐺_𝛼(𝑥)≥𝛼𝜆‖𝑥−𝜋_𝛼(𝑥)‖². Moreover, when 𝐺_𝛼(𝑥)=0, 𝑥 is a solution of SMVIP.

Lemma 5 (see [10]). If abstract function 𝜙 satisfies conditions (C1)–(C4), then 𝑔_𝛼(𝑥) is differentiable and ∇𝑔_𝛼(𝑥)=∇𝑔(𝑥)𝐹(𝑥)+∇𝐹(𝑥)𝑔(𝑥)−𝑦_𝛼(𝑥)−𝛼∇_𝑥𝜙𝑔(𝑥),𝑦_𝛼(𝑥). (15)

Lemma 6 (see [10, 19]). If abstract function 𝜙 satisfies conditions (C1)–(C4), then 𝑔_𝛼 is nonnegative, and 𝑔_𝛼(𝑥)=0⇔𝑥 is a solution of GVIP.

Lemma 7 (see [10]). Let abstract function 𝜙 satisfy conditions (C1)–(C4). If ∇𝑔_𝛼(𝑥)=0 and ∇𝐹(𝑥) is positive definite, then 𝑥 is a solution of 𝐺𝑉𝐼𝑃(𝐹,𝑓).

3. 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 𝐶 be a nonempty closed convex set in 𝐻 and let 𝑓 be strictly convex in 𝐶. Then 𝑓 has only one minimum in 𝐶.

Proof. We use proof by contradiction to show the desired result. Let 𝑥₁,𝑥₂∈𝐶 be two minimal points of 𝑓; that is, 𝑓(𝑥₁)=𝑓(𝑥₂)=min𝑓(𝑥). Since 𝑓 is strictly convex, one obtains that 𝑓𝛼𝑥₁+(1−𝛼)𝑥₂<𝛼𝑓𝑥₁+(1−𝛼)𝑓𝑥₂=𝑓𝑥₁,∀𝛼∈(0,1). (16) This implies that there exists a point 𝑥₃=𝛼𝑥₁+(1−𝛼)𝑥₂∈𝐶, such that 𝑓(𝑥₃)<𝑓(𝑥₁), which is a contradiction. This completes the proof.

Let 𝑇, 𝑓, and 𝜙 be defined as above and let 𝐾 be a nonempty closed convex set in 𝐻. Now, we can introduce generalized gap function 𝐺_𝛼 of SMVIP(𝑇,𝐾) defined as follows: 𝐺_𝛼(𝑥)=max_𝑦∈𝐻Ψ_𝛼𝑥,𝑦=max_𝑦∈𝐻𝑤,𝑥−𝑦+𝑓(𝑥)−𝑓𝑦−𝛼𝜙𝑥,𝑦,∀𝑥,𝑦∈𝐻,𝛼>0. (17) From uniform convex of 𝜙(𝑥,⋅), one obtains that −Ψ_𝛼(𝑥,⋅) is also uniform convex in 𝐻. By Proposition 8, there exists a minimal point 𝜋_𝛼(𝑥) of 𝜙(𝑥,⋅) in 𝐻, such that 𝐺_𝛼(𝑥)=⟨𝑤,𝑥−𝜋_𝛼(𝑥)⟩+𝑓(𝑥)−𝑓𝜋_𝛼(𝑥)−𝛼𝜙𝑥,𝜋_𝛼(𝑥). (18)

Proposition 9. If abstract function 𝜙 satisfies conditions (C1)–(C4) and 𝑓∶𝐻→ℝ∪{+∞} is proper convex and lower semicontinuous, then for all 𝛼>0, 𝑥=𝜋_𝛼(𝑥)⇔𝑥 is a solution of 𝑆𝑀𝑉𝐼𝑃(𝑇,𝐾).

Proof. From the definition of 𝜋_𝛼(𝑥), one has 0∈𝜕−Ψ𝑥,𝜋_𝛼(𝑥)=𝑤+𝜕𝑓𝜋_𝛼(𝑥)+𝛼∇₂𝜙𝑥,𝜋_𝛼(𝑥). (19) By the definition of subgradient, we have 𝑓𝑦≥𝑓𝜋_𝛼(𝑥)−⟨𝑤+𝛼∇₂𝜙𝑥,𝜋_𝛼,𝑦−𝜋_𝛼(𝑥)⟩, (20) which is equivalent to ⟨𝑤,𝑦−𝜋_𝛼(𝑥)⟩+𝑓𝑦−𝑓𝜋_𝛼(𝑥)≥𝛼⟨−∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑦−𝜋_𝛼(𝑥)⟩. (21)
On the one hand, if 𝑥=𝜋_𝛼(𝑥), from Lemma 2, one obtains ∇₂𝜙(𝑥,𝜋_𝛼(𝑥))=0, and so does 𝛼⟨−∇₂𝜙(𝑥,𝜋_𝛼(𝑥)),𝑦−𝜋_𝛼(𝑥)⟩=0. So, from (21), we have ⟨𝑤,𝑦−𝜋_𝛼(𝑥)⟩+𝑓𝑦−𝑓𝜋_𝛼(𝑥)≥0, (22) which implies that 𝑥 is a solution of SMVIP(𝑇,𝐾).
On the other hand, if 𝑥 is a solution of SMVIP(𝑇,𝐾), take 𝑦=𝜋_𝛼(𝑥) in (7), then we have ⟨𝑤,𝜋_𝛼(𝑥)−𝑥⟩+𝑓𝜋_𝛼(𝑥)−𝑓(𝑥)≥0. (23) From condition (C3), one has 𝜙(𝑥,𝑥)−𝜙𝑥,𝜋_𝛼(𝑥)≥∇₂𝜙𝑥,𝜋𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)+𝜆𝑥−𝜋_𝛼(𝑥)². (24) And by conditions (C2) and (C4), 𝜙(𝑥,𝑥)−𝜙𝑥,𝜋_𝛼(𝑥)≤0. (25) So we have ⟨∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)⟩+𝜆𝑥−𝜋_𝛼(𝑥)²≤0. (26) Combining (23) with (26), we have 𝑥=𝜋_𝛼(𝑥). This completes the proof.

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

Theorem 10. If abstract function 𝜙 satisfies conditions (C1)–(C5), 𝑓 is closed convex, and 𝑇 is strong monotone and Lipschtiz continuous with respect to the solution 𝑥 of 𝑆𝑀𝑉𝐼𝑃(𝑇,𝐾), then one has ‖𝑥−𝑥‖≤𝐿+𝛼𝐿^′𝛽𝑥−𝜋_𝛼(𝑥), (27)where 𝐿 and 𝐿^′ can be found in (5) and (9), respectively.

Proof. Since 𝑥 is a solution of SMVIP(𝑇,𝐾), take 𝑤∈𝑇(𝑥), then we obtain ⟨𝑤,𝑦−𝑥⟩+𝑓𝑦−𝑓(𝑥)≥0. (28) Let 𝑦=𝜋_𝛼(𝑥), for all 𝑥∈𝐻. Then inequality (28) reduces to ⟨𝑤,𝜋_𝛼(𝑥)−𝑥⟩+𝑓𝜋_𝛼(𝑥)−𝑓(𝑥)≥0. (29) Take 𝑦=𝑥, 𝑤∈𝑇(𝑥) in (21) such that ‖𝑤−𝑤‖≤𝐻(𝑇(𝑥),𝑇(𝑥)). Then inequality (21) changes to ⟨𝑤,𝑥−𝜋_𝛼(𝑥)⟩+𝑓(𝑥)−𝑓𝜋_𝛼(𝑥)≥𝛼⟨−∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)⟩. (30) Combining (29) and (30), we have ⟨𝑤−𝑤,𝜋_𝛼(𝑥)−𝑥⟩≤𝛼⟨∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)⟩. (31) And note that 𝛼⟨∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)⟩=𝛼∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝑥+𝛼⟨∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)⟩=𝛼∇₂𝜙𝑥,𝜋_𝛼(𝑥)−∇₂𝜙(𝑥,𝑥),𝑥−𝑥−𝛼⟨∇₂𝜙(𝑥,𝑥)−∇₂𝜙𝑥,𝜋_𝛼(𝑥),𝑥−𝜋_𝛼(𝑥)⟩≤𝛼∇₂𝜙𝑥,𝜋_𝛼(𝑥)−∇₂𝜙(𝑥,𝑥)‖𝑥−𝑥‖−2𝛼𝜆𝑥−𝜋_𝛼(𝑥)²≤𝛼𝐿^′𝑥−𝜋_𝛼(𝑥)‖𝑥−𝑥‖−2𝛼𝜆𝑥−𝜋_𝛼(𝑥)². (32) From (8), one has 𝛽‖𝑥−𝑥‖²≤⟨𝑤−𝑤,𝑥−𝑥⟩≤⟨𝑤−𝑤,𝑥−𝜋_𝛼(𝑥)⟩+⟨𝑤−𝑤,𝜋_𝛼(𝑥)−‖𝑥⟩≤𝐿𝑥−𝑥‖𝑥−𝜋_𝛼(𝑥)+𝛼𝐿^′𝑥−𝜋_𝛼(𝑥)‖𝑥−𝑥‖≤𝐿+𝛼𝐿^′‖𝑥−𝑥‖𝑥−𝜋_𝛼(𝑥), (33) so we have ‖𝑥−𝑥‖≤𝐿+𝛼𝐿^′𝛽𝑥−𝜋_𝛼(𝑥). (34) This completes the proof.

Theorem 11. If abstract function 𝜙 satisfies conditions (C1)–(C5) and 𝑇 is strong monotone for the solution 𝑥 of SMVIP and is Lipschtiz continuous with module 𝐿, then 𝐺_𝛼 has global error bound with respect to SMVIP; that is, ‖𝑥−𝑥‖≤𝐿+𝐿^′𝛽√𝛼𝜆𝐺_𝛼(𝑥). (35)

Proof. By Lemma 4 and Theorem 10, one obtains 𝐺_𝛼(𝑥)≥𝛼𝜆𝑥−𝜋_𝛼(𝑥)²,‖𝑥−𝑥‖≤𝐿+𝛼𝐿^′𝛽𝑥−𝜋_𝛼(𝑥). (36) So we can obtain 𝐺_𝛼(𝑥)≥𝛼𝜆𝛽²𝐿+𝛼𝐿^′2‖𝑥−𝑥‖², (37) which implies that ‖𝑥−𝑥‖≤𝐿+𝛼𝐿^′𝛽√𝛼𝜆𝐺_𝛼(𝑥). (38) This completes the proof.

Now, we introduce generalized D-gap function 𝐻_𝛼𝛾 for SMVIP which is defined by 𝐻_𝛼𝛾(𝑥)=𝐺_𝛼(𝑥)−𝐺_𝛾(𝑥)=max_𝑦∈𝐻Ψ_𝛼𝑥,𝑦−max_𝑦∈𝐻Ψ_𝛾=𝑥,𝑦𝑤,𝜋_𝛾(𝑥)−𝜋_𝛼(𝑥)+𝑓𝜋_𝛾(𝑥)−𝑓𝜋_𝛼(𝑥)+𝛽𝜙𝑥,𝜋_𝛾(𝑥)−𝛼𝜙𝑥,𝜋_𝛼(𝑥), (39) where 𝜋_𝛼(𝑥) and 𝜋_𝛾(𝑥) are minimal points for −Ψ_𝛼(𝑥,⋅) and −Ψ_𝛾(𝑥,⋅) in 𝐻, respectively, and 0<𝛼<𝛾. For 𝐻_𝛼𝛾(𝑥), we can conclude the following result.

Proposition 12. If abstract function 𝜙 satisfies condition (C3), then one has 𝛾−𝛼𝜙𝑥,𝜋_𝛾(𝑥)≤𝐻_𝛼𝛾(𝑥)≤𝛾−𝛼𝜙𝑥,𝜋_𝛼(𝑥). (40)

Proof. From the definition of 𝐻_𝛼𝛾(𝑥), one obtains that 𝐻_𝛼𝛾(𝑥)=max_𝑦∈𝐻Ψ_𝛼𝑥,𝑦−max_𝑦∈𝐻Ψ_𝛾𝑥,𝑦=Ψ_𝛼𝑥,𝜋_𝛼(𝑥)−Ψ_𝛾𝑥,𝜋_𝛾(𝑥)≥Ψ_𝛼𝑥,𝜋_𝛾(𝑥)−Ψ_𝛾𝑥,𝜋_𝛾(𝑥)=𝑤,𝑥−𝜋_𝛾(𝑥)−𝛼𝜙𝑥,𝜋_𝛾(𝑥−)𝑤,𝑥−𝜋_𝛾(𝑥)+𝛾𝜙𝑥,𝜋_𝛾(𝑥)=𝛾−𝛼𝜙𝑥,𝜋_𝛾(𝑥). (41)𝐻_𝛼𝛾(𝑥)≤(𝛾−𝛼)𝜙(𝑥,𝜋_𝛼(𝑥)) can be proved similarly. This completes the proof.

From Proposition 12, one has the following.

Proposition 13. If 𝜙 satisfies conditions (C1)–(C4), then 𝐻_𝛼𝛾(𝑥) is nonnegative, and 𝐻_𝛼𝛾(𝑥)=0⇔𝑥 is a solution of 𝑆𝑀𝑉𝐼𝑃(𝑇,𝐾).

Proof. From Proposition 12 and nonnegative property of 𝜙(⋅,⋅), we have that 𝐻_𝛼𝛾(𝑥) is nonnegative.
On the one hand, if 𝐻_𝛼𝛾(𝑥)=0, then by conditions (C2) and (C4), one has 𝑥=𝜋_𝛼(𝑥). Then by Proposition 9, we conclude that 𝑥 is a solution of SMVIP(𝑇,𝐾).
On the other hand, if 𝑥 is a solution of SMVIP(𝑇,𝐾), by Proposition 9, one obtains that 𝑥=𝜋_𝛼(𝑥). From condition (C4), one has 𝜙(𝑥,𝜋_𝛼(𝑥))=0. And since 𝐻_𝛼𝛾(𝑥) is nonnegative, we have 𝐻_𝛼𝛾(𝑥)=0. This completes the proof.

By the generalized D-gap function, we have the following error bound for SMVIP(𝑇,𝐾).

Theorem 14. Let 𝜙 satisfy conditions (C1)–(C5). 𝑇 is strong monotone for the solution 𝑥 of SMVIP and is Lipschtiz continuous with module 𝐿; then 𝐻_𝛼𝛾(𝑥) has global error bound with respect to SMVIP; that is, ‖𝑥−𝑥‖≤𝐿+𝐿^′𝛽𝜆𝛾−𝛼𝐻_𝛼𝛾(𝑥). (42)

Proof. From Lemma 3, Theorem 10, and Proposition 13, we have 𝐻_𝛼𝛾(𝑥)≥𝛾−𝛼𝜙𝑥−𝜋_𝛾(𝑥≥𝜆)𝛾−𝛼𝑥−𝜋_𝛾(𝑥)²𝛽≥𝜆𝛾−𝛼𝐿+𝛼𝐿^′2‖𝑥−𝑥‖², (43) which implies that ‖𝑥−𝑥‖≤𝐿+𝐿^′𝛽𝜆𝛾−𝛼𝐻_𝛼𝛾(𝑥). (44) This completes the proof.

4. Steepest Descent Method for GVIP

In this section, by introducing appropriate generalized gap function, the original GVIP(𝐹,𝑓) 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 𝐻 to be ℝ^𝑛. And we introduce the following generalized gap function for GVIP(𝐹,𝑓): 𝑔_𝛼(𝑥)=max_{𝑔(𝑦)∈𝐾}Ψ_𝛼𝑥,𝑦=max_{𝑔𝑦∈𝐾}𝐹(𝑥)(𝑥)𝑦𝑓(𝑥)𝑦=(𝑥)(𝑥),𝑓−𝑓−𝛼𝜙,𝑓⟨𝐹,𝑓−𝑦_𝛼(𝑥)⟩−𝛼𝜙𝑓(𝑥),𝑦_𝛼(𝑥), (45) where 𝑦_𝛼(𝑥) is a minimal point for −Ψ_𝛼(𝑥,⋅), 𝛼 is a positive parameter, and 𝜙 satisfies conditions (C1)–(C5) stated above. For 𝑔_𝛼, we have the following useful results [14]:(A1)𝑔_𝛼(𝑥) is nonnegative in 𝐾;(A2)𝑔_𝛼(𝑥)=0 for some 𝑥∈𝐾⇔𝑥 is a solution of VIP;(A3)𝑦_𝛼(𝑥) is the only minimizer of Ψ_𝛼(𝑥,⋅) in 𝐾. And similar to the discussion in [10, 11], we also give the following two assumptions:(a)∇𝐹(𝑥) is positive definite for all 𝑥∈𝐾;(b)∇_𝑥𝜙(𝑥,𝑦)=−∇_𝑦𝜙(𝑥,𝑦).

From Lemmas 5–7, we obtain that the original GVIP (6) is equivalent to the following optimization problem: min_{𝑠.𝑡.𝑥∈𝐾}𝑔_𝛼(𝑥). (46) For problem (46), we give the following algorithm.

Algorithm 15.
Step  0. Choose an initial value 𝑥⁰∈𝐾, 𝜀,𝑡∈(0,1), and put 𝑘=0.
Step  1. If 𝑔_𝛼(𝑥)≤𝜀, then we can end the circulation.
Step  2. Compute 𝑦_𝛼(𝑥^𝑘), and let 𝑑^𝑘=𝑦_𝛼𝑥^𝑘−𝑓𝑥^𝑘. (47)
Step  3. Let 𝑚_𝑘 be the minimal nonnegative integer 𝑚, such that 𝑔_𝛼𝑥^𝑘+𝑡^𝑚𝑑^𝑘≤𝑔_𝛼𝑥^𝑘−𝑡^2𝑚𝑑^𝑘2. (48)
Step  4. Let 𝑓(𝑥^𝑘+1)=𝑓(𝑥^𝑘)+𝑡^𝑚𝑘𝑑^𝑘, 𝑘=𝑘+1; go to Step 1.

Proposition 16. Let {𝑥^𝑘} be a sequence generated by Algorithm 15. If {𝑥^𝑘} are not the solutions of 𝐺𝑉𝐼𝑃(𝐹,𝑓), then ∇𝑔_𝛼𝑥^𝑘𝑇𝑑^𝑘<0; (49) that is, 𝑑^𝑘 is the descent direction of 𝑔_𝛼 at 𝑥^𝑘, where 𝑑^𝑘 is defined in (47).

Proof. To begin, we show that 𝑓(𝑥^𝑘)∈𝐾, for all positive integer 𝑘. From Algorithm 15, one obtains that 𝑓(𝑥⁰)∈𝐾. We prove this result by induction. Assume 𝑓(𝑥^𝑘)∈𝐾; we only need to show that 𝑓(𝑥^𝑘+1)∈𝐾. Since 𝑥^𝑘,𝑦_𝛼(𝑥^𝑘)∈𝐾, 𝑡_𝑘∈(0,1), and 𝐾 is convex, we have 𝑓𝑥^𝑘+1=𝑓𝑥^𝑘+𝑡_𝑘𝑑^𝑘=1−𝑡_𝑘𝑓𝑥^𝑘+𝑡_𝑘𝑦_𝛼𝑥^𝑘∈𝐾. (50) For simplicity, 𝑦_𝛼(𝑥^𝑘), 𝑥^𝑘 are replaced by 𝑦_𝛼, 𝑥, respectively. From Lemma 5, one has ∇𝑔_𝛼(𝑥)^𝑇𝑑=𝑦_𝛼𝐹(𝑥)+∇𝐹(𝑥)𝑔(𝑥)−𝑦_𝛼−𝛼∇_𝑥𝜙𝑔(𝑥),𝑦_𝛼^𝑇𝑑=𝑔(𝑥)−𝑦_𝛼^𝑇∇𝐹(𝑥)𝑦_𝛼(𝑥+(𝑥)𝐹(𝑥)−𝑔)∇𝑔−𝛼∇_𝑥𝜙𝑔(𝑥),𝑦_𝛼^𝑇𝑦_𝛼−𝑔(𝑥). (51) Since (𝑔(𝑥)−𝑦_𝛼)^𝑇∇𝐹(𝑥)(𝑦_𝛼−𝑔(𝑥))<0, we only need to show that {∇𝑔(𝑥)𝐹(𝑥)−𝛼∇_𝑥𝜙(𝑔(𝑥),𝑦_𝛼)}^𝑇(𝑦_𝛼−𝑔(𝑥))≤0. Since 𝑦_𝛼 is the unique minimizer of −Ψ(𝑥,⋅) in 𝐾, we have ⟨−∇_𝑦Ψ𝑥,𝑦_𝛼,𝑢−𝑦_𝛼⟩=∇𝑔(𝑥)𝐹(𝑥)+𝛼∇_𝑦𝜙𝑥,𝑦_𝛼,𝑢−𝑦_𝛼≥0,∀𝑢∈𝐾. (52) Let 𝑢=𝑥∈𝐾 in (52). One has ∇𝑔(𝑥)𝐹(𝑥)+𝛼∇_𝑦𝜙𝑥,𝑦_𝛼^𝑇𝑦_𝛼−𝑥≤0. (53) From assumption (b), we have ∇𝑔(𝑥)𝐹(𝑥)−𝛼∇_𝑥𝜙𝑥,𝑦_𝛼^𝑇𝑦_𝛼−𝑥≤0. (54) This completes the proof.

Now, we are in a position to show the global convergence result for Algorithm 15.

Theorem 17. Let {𝑥^𝑘} be a sequence generated by Algorithm 15, and let 𝑥^⋆ be the cluster point of {𝑥^𝑘}. Then 𝑥^⋆ is a solution of 𝐺𝑉𝐼𝑃(𝐹,𝑓).

Proof. Let {𝑥^𝑘}_𝐾 be a subsequence which converges to 𝑥^⋆. If 𝑔_𝛼(𝑥^⋆)=0, then from Lemma 6, 𝑥^⋆ is a solution of GVIP. If 𝑔_𝛼(𝑥^⋆)≠0, from the continuous property, one obtains that {𝑦_𝛼(𝑥^𝑘)}_𝐾→𝑦_𝛼(𝑥^⋆) which implies that 𝑑^𝑘⟶𝑦_𝛼𝑥^⋆−𝑓𝑥^⋆. (55) Now, we begin to show that the cluster point 𝑑^𝑥⋆ of {𝑑^𝑘}_𝐾 is zero. We use proof by contradiction. Assume 𝑑^⋆=𝑦_𝛼(𝑥^⋆)−𝑥^⋆≠0. On the one hand, from Proposition 16, one has that ∇𝑔_𝛼𝑥^⋆𝑇<0. (56) On the other hand, from Proposition 16, we obtain that {𝑔_𝛼(𝑥^𝑘)} is monotonically decreasing and bounded; that is, the sequence {𝑔_𝛼(𝑥^𝑘)} is convergent. From step 3 of Algorithm 15, one has 0≤𝑡^2𝑚𝑘𝑑^𝑘2≤𝑔_𝛼𝑥^𝑘−𝑔_𝛼𝑥^𝑘+1⟶0,as𝑘⟶∞. (57) Hence, we have lim_𝑘→∞𝑡^2𝑚𝑘‖𝑑^𝑘‖²=0; that is, lim_𝑘→∞𝑡^2𝑚𝑘=0𝑑^⋆≠0. (58) Without loss of generality, we assume 𝑡_𝑘∈(0,1), for all 𝑘. Then one cannot find the minimal nonnegative integer 𝑚_𝑘; that is, 𝑔_𝛼𝑥^𝑘+𝑡^𝑚𝑘−1𝑑^𝑘>𝑔_𝛼𝑥^𝑘−𝑡^𝑚𝑘2𝑑^𝑘2,∀𝑘, (59) Or, equally, 𝑔_𝛼𝑥^𝑘+𝑡^𝑚𝑘−1𝑑^𝑘−𝑔_𝛼𝑥^𝑘𝑡^𝑚𝑘>−𝑡^𝑚𝑘𝑑^𝑘2,∀𝑘. (60) Let 𝑘→∞, from (58), and 𝑔_𝛼 be continuous and differentiable; we can obtain ∇𝑔_𝛼𝑥^⋆𝑇≥0. (61) Inequalities (56) and (61) are at odds. This completes the proof.

Acknowledgments

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.