Citation: S. S. Chang, Salahuddin, M. Liu, X. R. Wang, J. F. Tang. Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings[J]. AIMS Mathematics, 2021, 6(2): 1800-1815. doi: 10.3934/math.2021108
[1] | Saudia Jabeen, Bandar Bin-Mohsin, Muhammad Aslam Noor, Khalida Inayat Noor . Inertial projection methods for solving general quasi-variational inequalities. AIMS Mathematics, 2021, 6(2): 1075-1086. doi: 10.3934/math.2021064 |
[2] | Bancha Panyanak, Chainarong Khunpanuk, Nattawut Pholasa, Nuttapol Pakkaranang . A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators. AIMS Mathematics, 2023, 8(4): 9692-9715. doi: 10.3934/math.2023489 |
[3] | Feng Ma, Bangjie Li, Zeyan Wang, Yaxiong Li, Lefei Pan . A prediction-correction based proximal method for monotone variational inequalities with linear constraints. AIMS Mathematics, 2023, 8(8): 18295-18313. doi: 10.3934/math.2023930 |
[4] | Doaa Filali, Mohammad Dilshad, Mohammad Akram . Generalized variational inclusion: graph convergence and dynamical system approach. AIMS Mathematics, 2024, 9(9): 24525-24545. doi: 10.3934/math.20241194 |
[5] | Muhammad Aslam Noor, Khalida Inayat Noor, Bandar B. Mohsen . Some new classes of general quasi variational inequalities. AIMS Mathematics, 2021, 6(6): 6406-6421. doi: 10.3934/math.2021376 |
[6] | Nagendra Singh, Sunil Kumar Sharma, Akhlad Iqbal, Shahid Ali . On relationships between vector variational inequalities and optimization problems using convexificators on the Hadamard manifold. AIMS Mathematics, 2025, 10(3): 5612-5630. doi: 10.3934/math.2025259 |
[7] | Xiuzhi Yang, G. Farid, Waqas Nazeer, Muhammad Yussouf, Yu-Ming Chu, Chunfa Dong . Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions. AIMS Mathematics, 2020, 5(6): 6325-6340. doi: 10.3934/math.2020407 |
[8] | Shahram Rezapour, Maryam Iqbal, Afshan Batool, Sina Etemad, Thongchai Botmart . A new modified iterative scheme for finding common fixed points in Banach spaces: application in variational inequality problems. AIMS Mathematics, 2023, 8(3): 5980-5997. doi: 10.3934/math.2023301 |
[9] | Sevda Sezer, Zeynep Eken . The Hermite-Hadamard type inequalities for quasi $ p $-convex functions. AIMS Mathematics, 2023, 8(5): 10435-10452. doi: 10.3934/math.2023529 |
[10] | Jurancy Ereú, Luz E. Marchan, Liliana Pérez, Henry Rojas . Some results on the space of bounded second κ-variation functions. AIMS Mathematics, 2023, 8(9): 21872-21892. doi: 10.3934/math.20231115 |
In 2014, Li et al. [1] suggested a new class of inverse mixed variational inequality in Hilbert spaces that has simple problem of traffic network equilibrium control, market equilibrium issues as applications in economics and telecommunication network problems. The concept of gap function plays an important role in the development of iterative algorithms, an evaluation of their convergence properties and useful stopping rules for iterative algorithms, see [2,3,4,5]. Error bounds are very important and useful because they provide a measure of the distance between a solution set and a feasible arbitrary point. Solodov [6] developed some merit functions associated with a generalized mixed variational inequality, and used those functions to achieve mixed variational error limits. Aussel et al. [7] introduced a new inverse quasi-variational inequality (IQVI), obtained local (global) error bounds for IQVI in terms of certain gap functions to demonstrate the applicability of IQVI, and provided an example of road pricing problems, also see [8,9]. Sun and Chai [10] introduced regularized gap functions for generalized vector variation inequalities (GVVI) and obtained GVVI error bounds for regularized gap functions. Wu and Huang [11] implemented generalized f-projection operators to deal with mixed variational inequality. Using the generalized f-projection operator, Li and Li [12] investigated a restricted mixed set-valued variational inequality in Hilbert spaces and proposed four merit functions for the restricted mixed set valued variational inequality and obtained error bounds through these functions.
Our goal in this paper is to present a problem of generalized vector inverse quasi-variational inequality problems. They propose three gap functions, the residual gap function, the regularized gap function, and the global gap function, and obtain error bounds for generalized vector inverse quasi-variational inequality problem using these gap functions and generalized f-projection operator under the monotonicity and Lipschitz continuity of underlying mappings.
Throughout this article, R+ denotes the set of non-negative real numbers, 0 denotes the origins of all finite dimensional spaces, ‖⋅‖ and ⟨⋅,⋅⟩ denotes the norms and the inner products in finite dimensional spaces, respectively. Let Ω,F,P:Rn→Rn be the set-valued mappings with nonempty closed convex values, Ni:Rn×Rn→Rn(i=1,2,⋯,m) be the bi-mappings, B:Rn→Rn be the single-valued mappings, and fi:Rn→R(i=1,2,⋯,m) be real-valued convex functions. We put
f=(f1,f2,⋯,fm),N(⋅,⋅)=(N1(⋅,⋅),N2(⋅,⋅),⋯,Nm(⋅,⋅)), |
and for any x,w∈Rn,
⟨N(x,x),w⟩=(⟨N1(x,x),w⟩,⟨N2(x,x),w⟩,⋯,⟨Nn(x,x),w⟩). |
In this paper, we consider the following generalized vector inverse quasi-variational inequality for finding ˉx∈Ω(ˉx), ˉu∈F(ˉx) and ˉv∈P(ˉx) such that
⟨N(ˉu,ˉv),y−B(ˉx)⟩+f(y)−f(B(ˉx))∉−intRm+,∀y∈Ω(ˉx), | (2.1) |
and solution set is denoted by ℧.
Special cases:
(i) If P is a zero mapping and N(⋅,⋅)=N(⋅), then (2.1) reduces to the following problem for finding ˉx∈Ω(ˉx) and ˉu∈F(ˉx) such that
⟨N(ˉu),y−B(ˉx)⟩+f(y)−f(B(ˉx))∉−intRm+,∀y∈Ω(ˉx), | (2.2) |
studied in[13] and solution set is denoted by ℧1.
(ii) If F is single valued mapping, then (2.2) reduces to the following vector inverse mixed quasi-variational inequality for finding ˉx∈Ω(ˉx) such that
⟨N(ˉx),y−B(ˉx)⟩+f(y)−f(B(ˉx))∉−intRm+,∀y∈Ω(ˉx), | (2.3) |
studied in [14] and solution set is denoted by ℧2.
(iii) If C⊂Rn is a nonempty closed and convex subset, B(x)=x and Ω(x)=C for all x∈Rn, then (2.3) collapses to the following generalized vector variational inequality for finding ˉx∈C such that
⟨N(ˉx),y−x⟩+f(y)−f(ˉx)∉−intRm+,∀y∈C, | (2.4) |
which is considered in [10].
(iv) If f(x)=0 for all x∈Rn, then (2.4) reduces to vector variational inequality for finding ˉx∈C such that
⟨N(ˉx),y−x⟩∉−intRm+,∀y∈C, | (2.5) |
studied in [15].
(v) If Rm+=R+ then (2.5) reduces to variational inequality for finding ˉx∈C such that
⟨N(ˉx),y−x⟩≥0,∀y∈C, | (2.6) |
studied in [16].
Definition 2.1 [7] Let G:Rn→Rn and g:Rn→Rn be two maps.
(i) (G,g) is said to be a strongly monotone if there exists a constant μg>0 such that
⟨G(y)−G(x),g(y)−g(x)⟩≥μg‖y−x‖2,∀x,y∈Rn; |
(ii) g is said to be Lg-Lipschitz continuous if there exists a constant Lg>0 such that
‖g(x)−g(y)‖≤Lg‖x−y‖,∀x,y∈Rn. |
For any fixed γ>0, let G:RnטΩ→(−∞,+∞] be a function defined as follows:
G(φ,x)=‖x‖2−2⟨φ,x⟩+‖φ‖2+2γf(x),∀φ∈Rn,x∈˜Ω, | (2.7) |
where ˜Ω⊂Rn is a nonempty closed and convex subset, and f:Rn→R is convex.
Definition 2.2 [11] We say that ℷf˜Ω:Rn→2˜Ω is a generalized f-projection operator if
ℷf˜Ωφ={w∈˜Ω:G(φ,w)=infy∈˜ΩG(φ,y)},∀φ∈Rn. |
Remark 2.3 If f(x)=0 for all x∈˜Ω, then the generalized f-projection operator ℷf˜Ω is equivalent to the following metric projection operator:
P˜Ω(φ)={w∈˜Ω:‖w−φ‖=infy∈˜Ω‖y−φ‖},∀φ∈Rn. |
Lemma 2.4 [1,11] The following statements hold:
(i) For any given φ∈Rn,ℷf˜Ωφ is nonempty and single-valued;
(ii) For any given φ∈Rn,x=ℷf˜Ωφ if and only if
⟨x−φ,y−x⟩+γf(y)−γf(x)≥0,∀y∈˜Ω; |
(iii) ℷf˜Ω:Rn→Ω is nonexpansive, that is,
‖ℷf˜Ωx−ℷf˜Ωy‖≤‖x−y‖,∀x,y∈Rn. |
Lemma 2.5 [17] Let m be a positive number, B⊂Rn be a nonempty subset such that
‖d‖≤mfor alld∈B. |
Let Ω:Rn→Rn be a set-valued mapping such that, for each x∈Rn, Ω(x) is a closed convex set, and let f:Rn→R be a convex function on Rn. Assume that
(i) there exists a constant τ>0 such that
D(Ω(x),Ω(y))≤τ‖x−y‖,x,y∈Rn, |
where D(⋅,⋅) is a Hausdorff metric defined on Rn;
(ii) 0∈⋂w∈RnΩ(w);
(iii) f is ℓ-Lipschitz continuous on Rn. Then there exists a constant κ=√6τ(m+γℓ) such that
‖ℷfΩ(x)z−ℷfΩ(x)z‖≤κ‖x−y‖,∀x,y∈Rn,z∈B. |
Lemma 2.6 A function r:Rn→R is said to be a gap function for a generalized vector inverse quasi-variational inequality on a set ˜S⊂Rn if it satisfies the following properties:
(i) r(x)≥0for anyx∈˜S; (ii) r(ˉx)=0,ˉx∈˜S if and only if ˉx is a solution of (2.1).
Definition 2.7 Let B:Rn→Rn be the single-valued mapping and N:Rn×Rn→Rn be a bi-mapping.
(i) (N,B) is said to be a strongly monotone with respect to the first argument of N and B, if there exists a constant μB>0 such that
⟨N(y,⋅)−N(x,⋅),B(y)−B(x)⟩≥μB‖y−x‖2,∀x,y∈Rn; |
(ii) (N,B) is said to be a relaxed monotone with respect to the second argument of N and B, if there exists a constant ζB>0 such that
⟨N(⋅,y)−N(⋅,x),B(y)−B(x)⟩≥−ζB‖y−x‖2,∀x,y∈Rn; |
(iii) N is said to be σ-Lipschitz continuous with respect to the first argument with constant σ>0 and ℘-Lipschitz continuous with respect to the second argument with constant ℘>0 such that
‖N(x,ˉx)−N(y,ˉy)‖≤σ‖x−y‖+℘‖ˉx−ˉy‖,∀x,ˉx,y,ˉy∈Rn. |
(iv) B is said to be ℓ-Lipschitz continuous if there exists a constant ℓ>0 such that
‖B(x)−B(y)‖≤ℓ‖x−y‖,∀x,y∈Rn. |
Example 2.8 The variational inequality (2.6) can be solved by transforming it into an equivalent optimization problem for the so-called merit function r(⋅;τ):X=Rn→R∪{+∞} defined by
r(x;τ)=sup{⟨N(ˉx),y−x⟩X−τ‖ˉx−x‖2X|x∈C} for ˉx∈C, |
where τ is a nonnegative parameter. If X is finite dimensional, the function r(⋅;0) is usually called the gap function for τ=0, and the function r(⋅;τ) for τ>0 is called the regularized gap function.
Example 2.9 Assume that N:Rn→Rn be a given mapping and C a closed convex set in Rn. Let ⋎ and ⋏ be given scalar satisfying ⋎>⋏>0 then (2.6) has a D-gap function if
N⋎⋏(x)=N⋎(x)−N⋏(x),∀x∈Rn |
where D stands for difference.
In this section, we discuss the residual gap function for generalized vector inverse quasi-variational inequality problem by using the strong monotonicity, relaxed monotonicity, Hausdorff Lipschitz continuity and prove error bounds related to the residual gap function. We define the residual gap function for (2.1) as follows:
rγ(x)=min1≤i≤m{‖B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]‖},x∈Rn,u∈F(x),v∈P(x),γ>0. | (3.1) |
Theorem 3.1 Suppose that F,P:Rn→Rn are set-valued mappings and Ni:Rn×Rn→Rn(i=1,2,⋯,m) are the bi-mappings. Assume that B:Rn→Rn is single-valued mapping, then for any γ>0,rγ(x) is a gap function for (2.1) on Rn.
Proof. For any x∈Rn,
rγ(x)≥0. |
On the other side, if
rγ(ˉx)=0, |
then there exists 0≤i0≤m such that
B(ˉx)=ℷfi0Ω(ˉx)[B(ˉx)−γNi0(ˉu,ˉv)],∀ˉu∈F(ˉx),ˉv∈P(ˉx). |
From Lemma 2.4, we have
⟨B(ˉx)−[B(ˉx)−γNi0(ˉu,ˉv)],y−B(ˉx)⟩+γf(y)−γf(B(ˉx))≤0,∀y∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx) |
and
⟨Ni0(ˉu,ˉv),y−B(ˉx)⟩+f(y)−f(B(ˉx))≤0,∀y∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx). |
It gives that
⟨N(ˉu,ˉv),y−B(ˉx)⟩+f(y)−f(B(ˉx))∉−intRm+,∀y∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx). |
Thus, ˉx is a solution of (2.1).
Conversely, if ˉx is a solution of (2.1), there exists 1≤i0≤m such that
⟨Ni0(ˉu,ˉv),y−B(ˉx)⟩+fi0(y)−fi0(B(ˉx))≥0,∀y∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx). |
By using the Lemma 2.4, we have
B(ˉx)=ℷfi0Ω(ˉx)[B(ˉx)−γNi0(ˉu,ˉv)],∀ˉu∈F(ˉx),ˉv∈P(ˉx). |
This means that
rγ(ˉx)=min1≤i≤m{‖B(ˉx)−ℷfiΩ(ˉx)[B(ˉx)−γNi(ˉu,ˉv)]‖}=0. |
The proof is completed.
Next we will give the residual gap function rγ, error bounds for (2.1).
Theorem 3.2 Let F,P:Rn→Rn be D-ϑF-Lipschitz continuous and D-ϱP-Lipschitz continuous mappings, respectively. Let Ni:Rn×Rn→Rn(i=1,2,⋯,m) be σi-Lipschitz continuous with respect to the first argument and ℘i-Lipschits continuous with respect to the second argument, and B:Rn→Rn be ℓ-Lipschitz continuous, and (Ni,B) be strongly monotone with respect to the first argument of Ni and B with positive constant μBi, and relaxed monotone with respect to the second argument of Ni and B with positive constant ζBi. Let
m⋂i=1(℧i)≠∅. |
Assume that there exists κi∈(0,μBi−ζBiσiϑF+ϱP℘i) such that
‖ℷfiΩ(x)z−ℷfiΩ(y)z‖≤κi‖x−y‖,∀x,y∈Rn,u∈F(x),v∈P(x),z∈{w∣w=B(x)−γNi(u,v)}. | (3.2) |
Then, for any x∈Rn and μBi>ζBi+κi(σiϑF+℘iϱP),
γ>κiℓμBi−ζBi−κi(σiϑF+℘iϱP), |
d(x,℧)≤γ(σiϑF−℘iϱP)+ℓγ(μBi−ζBi−κi(σiϑF+℘iϱP))−κiℓrγ(x), |
where
d(x,℧)=infˉx∈℧‖x−ˉx‖ |
denotes the distance between the point x and the solution set ℧.
Proof. Since
m⋂i=1(℧i)≠∅. |
Let ˉx∈Ω(ˉx) be the solution of (2.1) and thus for any i∈{1,⋯,m}, we have
⟨Ni(ˉu,ˉv),y−B(ˉx)⟩+fi(y)−fi(B(ˉx))≥0,∀y∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx). | (3.3) |
From the definition of ℷfiΩ(ˉx)[B(x)−γNi(u,v)], and Lemma 2.4, we have
⟨ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−(B(x)−γNi(u,v)),y−ℷfiΩ(ˉx)[B(x)−γNi(u,v)]⟩+γfi(y)−γfi(ℷfiΩ(ˉx)[B(x)−γNi(u,v)])≥0,∀y∈Ω(ˉx),u∈F(x),v∈P(x). | (3.4) |
Since
ˉx∈m⋂i=1(℧i),andB(ˉx)∈Ω(ˉx). |
Replacing y by B(ˉx) in (3.4), we get
⟨ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−(B(x)−γNi(u,v)),B(ˉx)−ℷfiΩ(ˉx)[B(x)−γNi(u,v)]⟩+γfi(B(ˉx))−γfi(ℷfiΩ(ˉx)[B(x)−γNi(u,v)])≥0,∀u∈F(x),v∈P(x). | (3.5) |
Since
ℷfiΩ(ˉx)[B(x)−γNi(u,v)]∈Ω(ˉx), |
from (3.3), it follows that
⟨γNi(ˉu,ˉv),ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−B(ˉx)⟩+γfi(ℷfiΩ(ˉx)[B(x)−γNi(u,v)])−γfi(B(ˉx))≥0. | (3.6) |
Utilizing (3.5) and (3.6), we have
⟨γNi(ˉu,ˉv)−γNi(u,v)−ℷfiΩ(ˉx)[B(x)−γNi(u,v)]+B(x),ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−B(ˉx)⟩≥0, |
which implies that
⟨γNi(ˉu,ˉv)−γNi(u,v),ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−B(x)⟩−⟨γNi(ˉu,ˉv)−γNi(u,v),B(ˉx)−B(x)⟩ |
+⟨B(x)−ℷfiΩ(ˉx)[B(x)−γNi(u,v)],ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−B(x)⟩ |
+⟨B(x)−ℷfiΩ(ˉx)[B(x)−γNi(u,v)],B(x)−B(ˉx)⟩≥0. |
Since F is D-ϑF-Lipschitz continuous, P is D-ϱP-Lipschits continuous and Ni is σi-Lipschitz continuous with respect to the first argument and ℘i-Lipschitz continuous with respect to the second argument, we have
‖ˉu−u‖≤D(F(ˉx),F(x))≤ϑF‖ˉx−x‖;‖ˉv−v‖≤D(P(ˉx),P(x))≤ϱP‖ˉx−x‖;‖Ni(ˉu,ˉv)−Ni(u,v)‖≤σi‖ˉu−u‖+℘i‖ˉv−v‖. | (3.7) |
Again, for i=1,2,⋯,m, (Ni,B) are strongly monotone with respect to the first argument of Ni and B with a positive constant μBi,, and relaxed monotone with respect to the second argument of Ni and B with a positive constant ζBi, we have
⟨γNi(ˉu,ˉv)−γNi(u,v),ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−B(x)⟩−‖B(x)−ℷfiΩ(ˉu)[B(x)−γNi(u,v)]‖2 |
+⟨B(x)−ℷfiΩ(ˉx)[B(x)−γNi(u,v)],B(x)−B(ˉx)⟩≥γμBi‖x−ˉx‖2−γζBi‖x−ˉx‖2. |
By adding ℷfiΩ(x)[B(x)−γNi(u,v)] and using the Cauchy-Schwarz inequality along with the triangular inequality, we have
‖γNi(ˉu,ˉv)−γNi(u,v)‖{‖ℷfiΩ(ˉx)[B(x)−γNi(u,v)]−ℷfiΩ(x)[B(x)−γNi(u,v)]‖ |
+‖ℷfiΩ(x)[B(x)−γNi(u,v)]−B(x)‖} |
+‖B(x)−B(ˉx)‖{‖B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]‖+‖ℷfiΩ(x)[B(x)−γNi(u,v)] |
−ℷfiΩ(ˉx)[B(x)−γNi(u,v)]‖}≥γμBi‖x−ˉx‖2−γζBi‖x−ˉx‖2. |
Using the (3.7) and condition (3.2), we have
(σiϑF+℘iϱP)γ‖ˉx−x‖{κi‖ˉx−x‖+‖ℷfiΩ(x)[B(x)−γNi(u,v)]−B(x)‖} |
+ℓ‖x−ˉx‖{‖B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]‖+κi‖x−ˉx‖}≥γ(μBi−ζBi)‖x−ˉx‖2. |
Hence, for any x∈Rn and i∈{1,2,⋯,m}, μBi>ζBi+κi(σiϑF+℘iϱP),
γ>κiℓμBi−ζBi−κi(σiϑF+℘iϱP), |
we have
‖x−ˉx‖≤γ(σiϑF+℘iϱP)+ℓγ(μBi−ζBi−κi(σiϑF+℘iϱP))−κiℓ‖B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]‖,∀u∈F(x),v∈P(x). |
This implies
‖x−ˉx‖≤γ(σiϑF+℘iϱP)+ℓγ(μBi−ζBi−κi(σiϑF+℘iϱP))−κiℓmin1≤i≤m{‖B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]‖} |
which means that
d(x,℧)≤‖x−ˉx‖≤γ(σiϑF+℘iϱP)+ℓγ(μBi−ζBi−κi(σiϑF+℘iϱP))−κiℓrγ(x). |
The proof is completed.
The regularized gap function for (2.1) is defined for all x∈Rn as follows:
ϕγ(x)=min1≤i≤msupy∈Ω(x),u∈F(x),v∈P(x){⟨Ni(u,v),B(x)−y⟩+fi(B(x))−fi(y)−12γ‖B(x)−y‖2} |
where γ>0 is a parameter.
Lemma 4.1 We have
ϕγ(x)=min1≤i≤m{⟨Ni(u,v),Riγ(x)⟩+fi(B(x))−fi(B(x)−Riγ(x))−12γ‖Riγ(x)‖2}, | (4.1) |
where
Riγ(x)=B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)],∀x∈Rn,u∈F(x),v∈P(x) |
and if
x∈B−1(Ω) |
and
B−1(Ω)={ξ∈Rn|B(ξ)∈Ω(ξ)}, |
then
ϕγ(x)≥12γrγ(x)2. | (4.2) |
Proof. For given x∈Rn,u∈F(x),v∈P(x) and i∈{1,2,⋯,m}, set
ψi(x,y)=⟨Ni(u,v),B(x)−y⟩+fi(B(x))−fi(y)−12γ‖B(x)−y‖2,y∈Rn. |
Consider the following problem:
gi(x)=maxy∈Ω(x)ψi(x,y). |
Since ψi(x,⋅) is a strongly concave function and Ω(x) is nonempty closed convex, the above optimization problem has a unique solution z∈Ω(x). Evoking the condition of optimality at z, we get
0∈Ni(u,v)+∂fi(z)+1γ(z−B(x))+NΩ(x)(z), |
where NΩ(x)(z) is the normal cone at z to Ω(x) and ∂fi(z) denotes the subdifferential of fi at z. Therefore,
⟨z−(B(x)−γNi(u,v)),y−z⟩+γfi(y)−γfi(z)≥0,∀y∈Ω(x),u∈F(x),v∈P(x) |
and so
z=ℷfiΩ(x)[B(x)−γNi(u,v)],∀u∈F(x),v∈P(x). |
Hence gi(x) can be rewritten as
gi(x)=⟨Ni(u,v),B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]⟩+fi(B(x))−fi(ℷfiΩ(x)[B(x)−γNi(u,v)]) |
−12γ‖B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)]‖2,∀u∈F(x),v∈P(x). |
Letting
Riγ(x)=B(x)−ℷfiΩ(x)[B(x)−γNi(u,v)],∀u∈F(x),v∈P(x), |
we get
gi(x)=⟨Ni(u,v),Riγ(x)⟩+fi(B(x))−fi(B(x)−Riγ(x))−12γ‖Riγ(x)‖2,∀u∈F(x),v∈P(x), | (4.3) |
(4.4) |
and so
ϕγ(x)=min1≤i≤m{⟨Ni(u,v),Riγ(x)⟩+fi(B(x))−fi(B(x)−Riγ(x))−12γ‖Riγ(x)‖2}. |
From the definition of projection ℷfiΩ(x)[B(x)−γNi(u,v)], we have
⟨ℷfiΩ(x)[B(x)−γNi(u,v)]−B(x)+γNi(u,v),y−ℷfiΩ(x)[B(x)−γNi(u,v)]⟩+γfi(y)−γfi(ℷfiΩ(x)[B(x)−γNi(u,v)])≥0,∀u∈F(x),v∈P(x). | (4.5) |
For any x∈B−1(Ω), we have
B(x)∈Ω(x). |
Therefore, putting y=B(x) in (4.5), we get
⟨γNi(u,v)−Riγ(x),Riγ(x)⟩+γfi(B(x))−γfi(B(x)−Riγ(x))≥0,∀u∈F(x),v∈P(x), |
that is,
⟨Ni(u,v),Riγ(x)⟩+fi(B(x))−fi(B(x)−Riγ(x))≥1γ⟨Riγ(x),Riγ(x)⟩=1γ‖Riγ(x)‖2. | (4.6) |
From the definition of rγ(x) and (4.1), we get
ϕγ(x)≥12γrγ(x)2. |
The proof is completed.
Theorem 4.2 For γ>0,ϕγ is a gap function for (2.1) on the set
B−1(Ω)={ξ∈Rn|B(ξ)∈Ω(ξ)}. |
Proof. From the definition of ϕγ, we have
ϕγ(x)≥min1≤i≤m{⟨Ni(u,v),B(x)−y⟩+fi(B(x))−fi(y)−12γ‖B(x)−y‖2},for ally∈Ω(x),u∈F(x),v∈P(x). | (4.7) |
Therefore, for any x∈B−1(Ω), putting y=B(x) in (4.7), we have
ϕγ(x)≥0. |
Suppose that ˉx∈B−1(ξ) with ϕγ(ˉx)=0. From (4.2), it follows that
rγ(ˉx)=0, |
which implies that ˉx is the solution of (2.1).
Conversely, if ˉx is a solution of (2.1), there exists 1≤i0≤m such that
⟨Ni0(ˉu,ˉv),B(ˉx)−y⟩+fi0(B(ˉx))−fi0(y)≤0,∀y∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx), |
which means that
min1≤i≤m{supy∈Ω(ˉx),ˉu∈F(ˉx),ˉv∈P(ˉx){⟨Ni(ˉu,ˉv),B(ˉx)−y⟩+fi(B(ˉx))−fi(y)−12γ‖B(ˉx)−y‖2}}≤0. |
Thus,
ϕγ(ˉx)≤0. |
The preceding claim leads to
ϕγ(ˉx)≥0 |
and it implies that
ϕγ(ˉx)=0. |
The proof is completed.
Since ϕγ can act as a gap function for (2.1), according to Theorem 4.2, investigating the error bound properties that can be obtained with ϕγ is interesting. The following corollary is obtained directly by Theorem 3.2 and (3.5).
Corollary 4.3 Let F,P:Rn→Rn be D-ϑF-Lipschitz continuous and D-ϱP-Lipschitz continuous mappings, respectively. Let Ni:Rn×Rn→Rn(i=1,2,⋯,m) be σi-Lipschitz continuous with respect to the first argument and ℘i-Lipschitz continuous with respect to the second argument, B:Rn→Rn be ℓ-Lipschitz continuous, and (Ni,B) be strongly monotone with respect to the first argument of N and B with respect to the constant μBi>0, and relaxed monotone with respect to the second argument of N and B with respect to the constant ζBi>0. Let
m⋂i=1(℧i)≠∅. |
Assume that there exists κi∈(0,μBi−ζBiϑFσi+℘iϱP) such that
‖ℷfiΩ(x)z−ℷfiΩ(y)z‖≤κi‖x−y‖,∀x,y∈Rn,u∈F(x),v∈P(x)∀z∈{w∣w=B(x)−γNi(u,v)}. |
Then, for any x∈B−1(Ω) and any
γ>κiℓμBi−ζBi−κi(ϑFσi+℘iϱP), |
d(x,℧)≤γ(ϑFσi+℘iϱP)+ℓγ(μBi−ζBi−κi(ϑFσi+℘iϱP))−κiℓ√2γϕγ(x). |
The regularized gap function ϕγ does not provide global error bounds for (2.1) on Rn. In this section, we first discuss the D-gap function, see [6] for (2.1), which gives Rn the global error bound for (2.1).
For (2.1) with ⋏>⋎>0, the D-gap function is defined as follows:
G⋏⋎(x)=min1≤i≤m{supy∈Ω(x),u∈F(x),v∈P(x){⟨Ni(u,v),B(x)−y⟩+fi(B(x))−fi(y)−12⋏‖B(x)−y‖2}−supy∈Ω(x)u∈F(x),v∈P(x){⟨Ni(u,v),B(x)−y⟩+fi(B(x))−fi(y)−12⋎‖B(x)−y‖2}}. |
From (4.1), we know G⋏⋎ can be rewritten as
G⋏⋎(x)=min1≤i≤m{⟨Ni(u,v),Ri⋏(x)⟩+fi(B(x))−fi(B(x)−Ri⋏(x))−12⋏‖Ri⋏(x)‖2−(⟨Ni(u,v),Ri⋎(x)⟩+fi(B(x))−fi(B(x)−Ri⋎(x))−12⋎‖Ri⋎(x)‖2)}, |
where
Ri⋏(x)=B(x)−ℷfiΩ(x)[B(x)−⋏Ni(u,v)] |
and
Ri⋎(x)=B(x)−ℷfiΩ(x)[B(x)−⋎Ni(u,v)],∀x∈Rn,u∈F(x),v∈P(x). |
Theorem 5.1 For any x∈Rn,⋏>⋎>0, we have
12(1⋎−1⋏)r2⋎(x)≤G⋏⋎(x)≤12(1⋎−1⋏)r2⋏(x). | (5.1) |
Proof. From the definition of G⋏⋎(x), it follows that
G⋏⋎(x)=min1≤i≤m{⟨Ni(u,v),Ri⋏(x)−Ri⋎(x)⟩−fi(B(x)−Ri⋏(x))−12⋏‖Ri⋏(x)‖2+fi(B(x)−Ri⋎(x))+12⋎‖Ri⋎(x)‖2},∀u∈F(x),v∈P(x). |
For any given i∈{1,2,⋯,m}, we set
gi⋏⋎(x)=⟨Ni(u,v),Ri⋏(x)−Ri⋎(x)⟩−fi(B(x)−Ri⋏(x))−12⋏‖Ri⋏(x)‖2+fi(B(x)−Ri⋎(x))+12⋎‖Ri⋎(x)‖2,∀u∈F(x),v∈P(x). | (5.2) |
From ℷfiΩ(x)[B(x)−⋎Ni(u,v)]∈Ω(x), by Lemma 2.4, we know
⟨ℷfiΩ(x)[B(x)−⋏Ni(u,v)]−(B(x)−⋏Ni(u,v)),ℷfiΩ(x)[B(x)−⋎Ni(u,v)]−ℷfiΩ(x)[B(x)−⋏Ni(u,v)]⟩ |
+⋏fi(ℷfiΩ(x)[B(x)−⋎Ni(u,v)])−⋏fi(ℷfiΩ(x)[B(x)−⋏Ni(u,v)])≥0,∀u∈F(x),v∈P(x) |
which means that
⟨⋏Ni(u,v)−Ri⋏(x),Ri⋏(x)−Ri⋎(x)⟩+⋏fi(B(x)−Ri⋎(x))−⋏fi(B(x)−Ri⋏(x))≥0. | (5.3) |
Combining (5.2) and (5.3), we get
gi⋏⋎(x)≥1⋏⟨Ri⋏(x),Ri⋏(x)−Ri⋎(x)⟩−12⋏‖Ri⋏(x)‖2+12⋎‖Ri⋎(x)‖2=12⋏‖Ri⋏(x)−Ri⋎(x)‖2+12(1⋎−1⋏)‖Ri⋎(x)‖2. | (5.4) |
Since
ℷfiΩ(x)[B(x)−⋏Ni(u,v)]∈Ω(x), |
from Lemma 2.4, we have
⟨ℷfiΩ(x)[B(x)−⋎Ni(u,v)]−(B(x)−⋎Ni(u,v)),ℷfiΩ(x)[B(x)−⋏Ni(u,v)]−ℷfiΩ(x)[B(x)−⋎Ni(u,v)]⟩ |
+⋎fi(ℷfiΩ(x)[B(x)−⋏Ni(u,v)])−⋎fi(ℷfiΩ(x)[B(x)−⋎Ni(u,v)])≥0,∀u∈F(x),v∈P(x). |
Hence
⟨⋎Ni(u,v)−Ri⋎(x),Ri⋎(x)−Ri⋏(x)⟩+⋎fi(B(x)−Ri⋏(x)) |
−⋎fi(B(x)−Ri⋎(x))≥0,∀u∈F(x),v∈P(x) |
and so
1⋎⟨Ri⋎(x),Ri⋏(x)−Ri⋎(x)⟩≥⟨Ni(u,v),Ri⋏(x)−Ri⋎(x)⟩−fi(B(x)−Ri⋏(x))+fi(B(x)−Ri⋎(x)). |
It will require and (5.3),
gi⋏⋎(x)≤1⋎⟨Ri⋎(x),Ri⋏(x)−Ri⋎(x)⟩−12⋏‖Ri⋏(x)‖2+12⋎‖Ri⋎(x)‖2=−12⋎‖Ri⋏(x)−Ri⋎(x)‖2+12(1⋎−1⋏)‖Ri⋏(x)‖2. | (5.5) |
From (5.4) and (5.5), for any i∈{1,2,⋯,m}, we get
12(1⋎−1⋏)‖Ri⋎(x)‖2≤gi⋏⋎(x)≤12(1⋎−1⋏)‖Ri⋏(x)‖2. |
Hence
12(1⋎−1⋏)min1≤i≤m{‖Ri⋎(x)‖2}≤min1≤i≤m{gi⋏⋎(x)}≤12(1⋎−1⋏)min1≤i≤m{‖Ri⋏(x)‖2}, |
and so
12(1⋎−1⋏)r2⋎(x)≤G⋏⋎(x)≤12(1⋎−1⋏)r2⋏(x). |
The proof is completed.
Now we are in position to prove that G⋏⋎ in the set Rn is a global gap function for (2.1).
Theorem 5.2 For 0<⋎<⋏, G⋏⋎ is a gap function for (2.1) on Rn.
Proof. From (5.2), we have
G⋏⋎(x)≥0,∀x∈Rn. |
Suppose that ˉx∈Rn with
G⋏⋎(ˉx)=0, |
then (5.2) implies that
r⋎(ˉx)=0. |
From Theorem 3.1, we know ˉx is a solution of (2.1).
Conversely, if ˉx is a solution of (2.1), than from Theorem 3.1, it follows that
r⋏(ˉx)=0. |
Obviously, (5.2) shows that
G⋏⋎(ˉx)=0. |
The proof is completed.
Use Theorem 3.2 and (5.2), we immediately get a global error bound in the set Rn for (2.1).
Corollary 5.3 Let F,P:Rn→Rn be D-ϑF-Lipschitz continuous and D-ϱP-Lipschitz continuous mappings, respectively. Let Ni:Rn×Rn→Rn(i=1,2,⋯,m) be σi-Lipschitz continuous with respect to the first argument and ℘i-Lipschitz continuous with respect to the second argument, and B:Rn→Rn be ℓ-Lipschitz continuous. Let (Ni,B) be the strongly monotone with respect to the first argument of Ni and B with constant μBi and relaxed monotone with respect to the second argument of N and B with modulus ζBi. Let
m⋂i=1(℧i)≠∅. |
Assume that there exists κi∈(0,μBi−ζBiϑFσi+℘iϱP) such that
‖ℷfiΩ(x)z−ℷfiΩ(y)z‖≤κi‖x−y‖,∀x,y∈Rn,u∈F(x),v∈P(x),z∈{w∣w=B(x)−⋎Ni(u,v)}. |
Then, for any x∈Rn and
⋎>κiℓμBi−ζBi−κi(ϑFσi+℘iϱP), |
d(x,℧i)≤⋎(ϑFσi+ϱP℘i)+ℓ⋎(μBi−ζBi−κi(ϑFσi+ϱP℘i))−κiℓ√2⋏⋎⋏−⋎G⋏⋎(x). |
One of the traditional approaches to evaluating a variational inequality (VI) and its variants is to turn into an analogous optimization problem by notion of a gap function. In addition, gap functions play a pivotal role in deriving the so-called error bounds that provide a measure of the distances between the solution set and feasible arbitrary point. Motivated and inspired by the researches going on in this direction, the main purpose of this paper is to further study the generalized vector inverse quasi-variational inequality problem (1.2) and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of (1.2).
The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article.
This work was supported by the Scientific Research Fund of Science and Technology Department of Sichuan Provincial (2018JY0340, 2018JY0334) and the Scientific Research Fund of SiChuan Provincial Education Department (16ZA0331).
The authors declare that they have no competing interests.
[1] |
X. Li, X. S. Li, N. J. Huang, A generalized f-projection algorithm for inverse mixed variational inequalities, Optim. Lett., 8 (2014), 1063-1076. doi: 10.1007/s11590-013-0635-4
![]() |
[2] | G. Y. Chen, C. J. Goh, X. Q. Yang, On gap functions for vector variational inequalities. In: F. Giannessi, (ed.), Vector variational inequalities and vector equilibria: mathematical theories, Kluwer Academic Publishers, Boston, 2000. |
[3] |
X. Q. Yang, J. C. Yao, Gap functions and existence of solutions to set-valued vector variational inequalities, J. Optim. Theory Appl., 115 (2002), 407-417. doi: 10.1023/A:1020844423345
![]() |
[4] | Salahuddin, Regularization techniques for Inverse variational inequalities involving relaxed cocoercive mapping in Hilbert spaces, Nonlinear Anal. Forum, 19 (2014), 65-76. |
[5] |
S. J. Li, H. Yan, G. Y. Chen, Differential and sensitivity properties of gap functions for vector variational inequalities, Math. Methods Oper. Res., 57 (2003), 377-391. doi: 10.1007/s001860200254
![]() |
[6] |
M. V. Solodov, Merit functions and error bounds for generalized variational inequalities, J. Math. Anal. Appl., 287 (2003), 405-414. doi: 10.1016/S0022-247X(02)00554-1
![]() |
[7] |
D. Aussel, R. Gupta, A. Mehra, Gap functions and error bounds for inverse quasi-variational inequality problems, J. Math. Anal. Appl., 407 (2013), 270-280. doi: 10.1016/j.jmaa.2013.03.049
![]() |
[8] |
B. S. Lee, Salahuddin, Minty lemma for inverted vector variational inequalities, Optimization, 66 (2017), 351-359. doi: 10.1080/02331934.2016.1271799
![]() |
[9] | J. Chen, E. Kobis, J. C. Yao, Optimality conditions for solutions of constrained inverse vector variational inequalities by means of nonlinear scalarization, J. Nonlinear Var. Anal., 1 (2017), 145-158. |
[10] |
X. K. Sun, Y. Chai, Gap functions and error bounds for generalized vector variational inequalities, Optim. Lett., 8 (2014), 1663-1673. doi: 10.1007/s11590-013-0685-7
![]() |
[11] |
K. Q. Wu, N. J. Huang, The generalised f-projection operator with an application, Bull. Aust. Math. Soc., 73 (2006), 307-317. doi: 10.1017/S0004972700038892
![]() |
[12] |
C. Q. Li, J. Li, Merit functions and error bounds for constrained mixed set-valued variational inequalities via generalized f-projection operators, Optimization, 65 (2016), 1569-1584. doi: 10.1080/02331934.2016.1163555
![]() |
[13] | S. S. Chang, Salahuddin, L. Wang, G. Wang, Z. L. Ma, Error bounds for mixed set-valued vector inverse quasi variational inequalities, J. Inequal Appl., 2020:160 (2020), 1-16. |
[14] |
Z. B. Wang, Z. Y. Chen, Z. Chen, Gap functions and error bounds for vector inverse mixed quasi-variational inequality problems, Fixed Point Theory Appl., 2019:14 (2019), 1-14. doi: 10.17654/FP014010001
![]() |
[15] | F. Giannessi, Vector variational inequalities and vector equilibria: mathematical theories, Kluwer Academic Publishers/Boston/London, Dordrecht, 2000. |
[16] | Salahuddin, Solutions of vector variational inequality problems, Korean J. Math., 26 (2018), 299-306. |
[17] |
X. Li, Y. Z. Zou, Existence result and error bounds for a new class of inverse mixed quasi-variational inequalities, J. Inequal. Appl., 2016 (2016), 1-13. doi: 10.1186/s13660-015-0952-5
![]() |
1. | Aviv Gibali, , Error bounds and gap functions for various variational type problems, 2021, 115, 1578-7303, 10.1007/s13398-021-01066-8 | |
2. | Soumitra Dey, Simeon Reich, A dynamical system for solving inverse quasi-variational inequalities, 2023, 0233-1934, 1, 10.1080/02331934.2023.2173525 | |
3. | Safeera Batool, Muhammad Aslam Noor, Khalida Inayat Noor, Merit functions for absolute value variational inequalities, 2021, 6, 2473-6988, 12133, 10.3934/math.2021704 | |
4. | Ting Liu, Wensheng Jia, Samuel Nicolay, An Approximation Theorem and Generic Convergence of Solutions of Inverse Quasivariational Inequality Problems, 2022, 2022, 2314-8888, 1, 10.1155/2022/2539961 |