Abstract
In this paper, we consider the numerical method for solving finitedimensional quasivariational inequalities with both equality and inequality constraints. Firstly, we present a semismooth equation reformulation to the KKT system of a finitedimensional quasivariational inequality. Then we propose a semismooth Newton method to solve the equations and establish its global convergence. Finally, we report some numerical results to show the efficiency of the proposed method. Our method can obtain the solution to some problems that cannot be solved by the method proposed in (Facchinei et al. in Comput. Optim. Appl. 62:85–109, 2015). Besides, our method outperforms than the interior point method proposed in (Facchinei et al. in Math. Program. 144:369–412, 2014).
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1 Introduction
We consider the finitedimensional quasivariational inequality QVI(K, F): Find a vector \(x^{*}\in K(x^{*})\) such that
where \(F:R^{n}\rightarrow R^{n}\) is a point to point mapping and \(K:R^{n}\rightrightarrows R^{n}\) is a point to set mapping with closed and convex images. Throughout the paper, we assume that F belongs to \(C^{1}\) and for each \(x\in R^{n}\), the feasible set mapping K is given by
where \(g:R^{n}\times R^{n}\rightarrow R^{m_{1}}\) belongs to \(C^{2}\) and \(g_{i}(\cdot ,x)\) is convex on \(R^{n}\) for each \(i=1,2,\ldots ,m_{1}\) and for all \(x\in R^{n}\), \(h:R^{n}\times R^{n}\rightarrow R^{m_{2}}\) belongs to \(C^{2}\) and \(h_{j}(\cdot ,x)\) is affine on \(R^{n}\) for each \(j=1,2,\ldots ,m_{2}\) and for all \(x\in R^{n}\). When the set \(K(x)\) is independent of x, (1.1) reduces to the famous variational inequality (VI). For VI, we refer the reader to [13] and the references therein.
QVI (1.1), which was first introduced by Bensoussan and Lions [2, 3], has important applications in many fields such as generalized Nash games, mechanics, economics, statistics, transportation and biology; see for example [1, 6, 10, 12] and the references therein. One interesting topic on QVI is to develop the efficient algorithms for the solution of QVI. Since QVI is nonsmooth and nonconvex, it is difficult to design effective methods for QVI, and by now, compared with VI, the numerical methods are still scarce. In this paper, we mainly focus on the numerical method based on the KKT conditions of QVI. This area attracts many people’s attention and much progress has been made. In [12] an interior point approach was proposed to solve QVI and the convergence was established for several classes of interesting QVIs. Reference [8, 9] developed a so called LPNewton method and the method can be successfully applied to nonsmooth systems of equations with nonisolated solutions. Reference [21] developed an efficient regularized smoothing Newtontype algorithm for QVI. The proposed algorithm takes the advantage of newly introduced smoothing functions and a nonmonotone line search strategy. [10] proposed a semismooth Newton method for QVI. They obtained global convergence and locally superlinear/quadratic convergence result for some important classes of quasivariational inequality problems. The numerical results show that the method performs well.
There are many ways to compute a numerical solution of the nonlinear complementarity problems (NCP), such as linearized projected relaxation methods [13], the modulusbased matrix splitting method [24] and the penalty method [7, 23, 25]. In the past two decades, the nonsmoothequationbased method has been thoroughly studied to solve NCP; see for example [5, 14–19] and the references therein. A common way to reformulate the complementarity system is to use the so called NCPfunction. A function \(\phi :R^{2}\rightarrow R\) is called an NCPfunction if it satisfies
For example, the famous Fischer–Burmeister (FB) function takes the form
By the use of the NCPfunction, nonlinear complementarity problem can be easily converted into a system of nonlinear equations. Most existing NCPfunctions are generally nondifferentiable in the sense of Fréderivative but semismooth in the sense of Mifflin [20] and Qi and Sun [22]. In [17], the authors proposed a nonsmooth equation reformulation to the NCP. Their reformulation enjoys a nice property that it is continuous differentiable everywhere except at the solution. In this paper, we present a semismooth equation reformulation to the KKT system of a quasivariational inequality and propose a semismooth Newton method to solve the equations.
The paper is organized as follows. In the next section, we describe a semismooth equation reformulation to the KKT system of a quasivariational inequality, present the semismooth Newton method and establish the global convergence for the method. In Sect. 3, we compare the proposed method with some other methods on problems list in [11].
In the following, we introduce some notations that will be used in this paper. For a continuously differentiable function \(F:R^{n}\rightarrow R^{n}\), we write \(\mathit{JF}(x)\) for the Jacobian of F at a point \(x\in R^{n}\), whereas \(\nabla F(x)\) denotes the transposed Jacobian of F. Given a smooth mapping \(g:R^{n}\times R^{n}\rightarrow R^{m}\), \((y,x)\mapsto g(y,x)\), \(\nabla _{y}g(y,x)\) denotes the transpose of the partial Jacobian of g with respect to the yvariables. If F is locally Lipschitz continuous around x, then \(\partial F(x)\) denotes Clarke’s generalized Jacobian of F at x. For a vector \(x\in R^{n}\) and a subset \(I\subset \{1,2,\ldots ,n\}\), we write \(x_{I}\) for the subvector consisting of the elements \(x_{i}\), \(i\in I\). For a matrix \(A\in R^{n\times n}\) and two subsets \(I, J\subset \{1,2,\ldots ,n\}\), the symbol \(A_{IJ}\) stands for the submatrix with entries \(a_{ij}\) for \(i\in I\), \(j\in J\). The symbol \(\operatorname{diag}(a_{11},a_{22},\ldots ,a_{nn})\) stands for a diagonal matrix with diagonal elements \(a_{11},a_{22},\ldots ,a_{nn}\).
2 Semismooth equation reformulation and semismooth Newton method
Firstly, we give the following definition that will be used.
Definition 2.1
([22])
A function \(F:R^{n}\rightarrow R^{n}\) is semismooth at a point \(x\in R^{n}\) if it is locally Lipschitzian at x and
exists for any \(d\in R^{n}\), where \(\partial F(x)\) is the generalized Jacobian of F at x. F is strongly semismooth at \(x\in R^{n}\) if for any \(d\rightarrow \mathbf{0} \) and any \(V\in \partial F(x+d)\),
where \(F'(x;d)\) denotes the directional derivative of F at x along the direction d.
A point x is called a KKT point of QVI (1.1) if there exist Lagrange multipliers \(\lambda \in R^{m_{1}}\) and \(\nu \in R^{m_{2}}\) such that
Similar to Theorem 1 of [12], we find that \(x^{*}\in K(x^{*})\) is a solution of (1.1) if there exist \(\lambda ^{*}\in R^{m_{1}}\) and \(\nu ^{*}\in R^{m_{2}}\) such that \((x^{*},\lambda ^{*},\nu ^{*})\) satisfies the KKT conditions (2.1). Moreover, if \(x^{*}\in K(x^{*})\) is a solution of (1.1) and some suitable constraint qualification holds at \(x^{*}\), then there exist \(\lambda ^{*}\in R^{m_{1}}\) and \(\nu ^{*}\in R^{m_{2}}\) such that \((x^{*},\lambda ^{*},\nu ^{*})\) satisfies the KKT conditions (2.1). Based on the above relationship, our aim is to develop a numerical method for solving the KKT conditions (2.1). For convenience, let
and then (2.1) can be rewritten as
where the \(w\in R^{m_{1}}\) are slack variables.
It is not easy to solve (2.2) directly since the fourth formula is a complementarity system. We replace the complementarity system by an NCPfunction [17], which is called the smoothed form of FB function:
It is clear that, for each \(\varepsilon \ne 0\), \(\phi (u,v,\varepsilon )\) is continuously differentiable. We use it to construct an almost smooth equation reformulation to the fourth formula.
Let \(\Phi _{FB}(\lambda ,w)=(\phi _{1}^{FB}(\lambda _{1},w_{1}),\ldots , \phi _{m_{1}}^{FB}(\lambda _{m_{1}},w_{m_{1}}))^{T}\) and \(S(\lambda ,w)=(S_{1}(\lambda ,w),\ldots ,S_{m_{1}}(\lambda ,w))^{T}\), where for each \(i=1,2,\ldots ,m_{1}\), the elements \(\phi _{i}^{FB}(\lambda _{i},w_{i})\) and \(S_{i}(\lambda ,w)\) are given by
and
respectively, where \(0<\mu <\frac{(\sqrt{2}+1)^{2}}{m_{1}}\) is a parameter, \(\\cdot \\) is the Euclidean norm, and
It is obvious that, for each \(i=1,2,\ldots ,m_{1}\), \(S_{i}(\lambda ,w)\) is differentiable everywhere except at the degenerate point \((\lambda ,w)\) which satisfies \(\theta (\lambda ,w)=0\) and \(\lambda _{i}=w_{i}=0\) for some \(i=1,2,\ldots ,m_{1}\). Moreover, we can obtain from Theorem 2.3 of [17] that \(S(\lambda ,w)=0\) is equivalent to \(\lambda \ge 0\), \(w\ge 0\), \(\lambda ^{T} w=0\). This means that \((x^{*},\lambda ^{*},\nu ^{*})\) is a KKT point of the QVI if and only if \((x^{*},\lambda ^{*},\nu ^{*},w^{*})\) with \(w^{*}=p(x^{*})\) is a solution of the nonsmooth system of equations
Associated with the system of \(H(x,\lambda ,\nu ,w)=0\), we consider its natural merit function
where we set \(z:=(x,\lambda ,\nu ,w)\).
By a direct calculation, we find that the gradient \(\nabla \theta (\lambda ,w)\) of \(\theta (\cdot ,\cdot )\) at \((\lambda ,w)\) can be expressed as follows:
where
and
which means that
Here, \(\partial _{\lambda _{i}} \phi _{i}^{FB}(\lambda _{i},w_{i})\) denotes the partial generalized gradient of \(\phi _{i}^{FB}(\cdot ,w_{i})\) at \(\lambda _{i}\) and \(\partial _{w_{i}} \phi _{i}^{FB}(\lambda _{i},w_{i})\) denotes the partial generalized gradient of \(\phi _{i}^{FB}(\cdot ,w_{i})\) at \(w_{i}\), respectively. In particular, if \(\theta (\lambda ,w)=0\), then \(\nabla \theta (\lambda ,w)=0\).
If \(\theta (\lambda ,w)\ne 0\), we can get by a direct calculation
where
Otherwise, we have \(\lambda _{i}\geq 0\), \(w_{i}\geq 0\) and \(\lambda _{i}w_{i}=0\) for any i, which means that, if \(\lambda _{i}^{2}+w_{i}^{2}\neq 0\), then
and, if \(\lambda _{i}^{2}+w_{i}^{2}=0\), then the element in \(\partial _{C}S_{i}(\lambda ,w)\) takes the form
where \(\bar{v}_{\lambda _{i}}\in \partial _{\lambda _{i}} \phi _{i}^{FB}( \lambda _{i},w_{i})\), \(\bar{v}_{w_{i}}\in \partial _{w_{i}} \phi _{i}^{FB}(\lambda _{i},w_{i})\), \(\u\=1\), and
Therefore, the partial generalized derivatives \(S(\lambda ,w)\) can be expressed in the form of
and
where \(a_{i}^{2}+b_{i}^{2}+c_{i}^{2}\le 1\), \(u\in R^{m_{1}}\) satisfies \(\u\=1\), E is a matrix whose elements are one, \(V_{\lambda }\) and \(V_{w}\) are diagonal matrices whose diagonal elements belong to \(\partial _{\lambda _{i}} \phi _{i}^{FB}(\lambda _{i},w_{i})\) and \(\partial _{w_{i}} \phi _{i}^{FB}(\lambda _{i},w_{i})\), respectively.
On the basis of the above calculations, we have the following proposition.
Proposition 2.2
Let the mapping H be defined by (2.4). Then the following statements hold:

(a)
If F is continuously differentiable and g, h are twice continuously differentiable, then H is semismooth and
$$ \partial H(x,\lambda ,\nu ,w)\subseteq \left \{ \begin{pmatrix} J_{x}L(x,\lambda ,\nu )& \nabla _{y} g(x,x) & \nabla _{y} h(x,x) & 0 \\ J q(x)&0&0&0 \\ J p(x)&0&0&I \\ 0&U_{\lambda }&0& U_{w} \end{pmatrix} \right \} , $$where \(U_{\lambda }\), \(U_{w}\) is defined by (2.7) and (2.8), respectively.

(b)
If, in addition, JF, \(\nabla ^{2}g_{i}\) (\(i=1,\ldots ,m_{1}\)) and \(\nabla ^{2}h_{j}\) (\(j=1,\ldots ,m_{2}\)) are locally Lipschitz, then H is strongly semismooth.

(c)
Let the merit function Ψ be defined by (2.5). If F is continuously differentiable and g, h are twice continuously differentiable, then Ψ is continuously differentiable, and its gradient is given by
$$ \nabla \Psi (z)=V^{T}H(z) $$for an arbitrary element \(V\in \partial H(z)\).
Remark 2.1
Consider \(\operatorname{QVI}(\tilde{K}, F)\), where
That is, there are no equality constraints in QVI (1.1). Similarly, we can formulate the above problem in terms of the nonsmooth system of equations
where \(\tilde{L}(x,\lambda ):=F(x)+\nabla _{y} g(x,x)\lambda \). Similar to the Proposition 2.2, if F is continuously differentiable and g is twice continuously differentiable, then H̃ is semismooth and
where \(U_{\lambda }\) and \(U_{w}\) are the same as in Proposition 2.2.
Now, we present the semismooth Newton method for (1.1).
Algorithm 1
(Semismooth Newton Method)

Step 0. Choose \(z^{0}=(x^{0},\lambda ^{0},\nu ^{0},w^{0})\in R^{n}\times R^{m_{1}} \times R^{m_{2}}\times R^{m_{1}}\), \(\rho >0\), \(\beta \in (0,1)\), \(\sigma \in (0,\frac{1}{2})\), \(p>2\), \(\varepsilon \ge 0\), and set \(k:=0\).

Step 1. If \(\\nabla \Psi (z^{k})\\le \varepsilon \), stop.

Step 2. Choose an arbitrary element \(V_{k}\in \partial H(z^{k})\), and compute \(d^{k}\) as a solution of the linear system of equations
$$ V_{k}d=H\bigl(z^{k}\bigr). $$(2.11)If either this system is not solvable or the sufficient decrease condition
$$ \nabla \Psi \bigl(z^{k}\bigr)^{T}d^{k} \le \rho \bigl\Vert d^{k} \bigr\Vert ^{p}$$(2.12)is not satisfied, then take \(d^{k}:=\nabla \Psi (z^{k})\).

Step 3. Compute a stepsize \(t_{k}\) as the maximum of the numbers \(\beta ^{l_{k}}\), \(l_{k}=0,1,2,\ldots \) , such that the following Armijo condition holds:
$$ \Psi \bigl(z^{k}+t_{k}d^{k} \bigr)\le \Psi \bigl(z^{k}\bigr)+\sigma t_{k}\nabla \Psi \bigl(z^{k}\bigr)^{T} d^{k}.$$(2.13) 
Step 4. Set \(z^{k+1}:=z^{k}+t_{k}d^{k}\), \(k\leftarrow k+1\), and go to Step 1.
End.
Below, we establish the following global convergence theorem for Algorithm 1.
Theorem 2.3
Let \(\{z^{k}\}=\{(x^{k},\lambda ^{k},\nu ^{k},w^{k})\}\) be a sequence of iterates generated by Algorithm 1. Then every accumulation point of the sequence \(\{z^{k}\}\) is a stationary point of the merit function Ψ.
Proof
We prove it by contradiction. Firstly, if for an infinite set of indices N, \(d^{k}=\nabla \Psi (z^{k})\) for all \(k\in N\), then, by [4] Proposition 1.16, we see that any limit point \(z^{*}\) of \({z^{k}}\) satisfies \(\nabla \Psi (z^{*})\).
In the following, we suppose the direction is always given by (2.11). Suppose \(\{z^{k}\}\rightarrow z^{*}\) and \(\nabla \Psi (z^{*})\ne 0\), by (2.11), we have
Noting that \(\V_{k}\\) cannot be 0, otherwise \(H(z^{k})=0\) and \(z^{k}\) would be a stationary point. Hence, we have
If for some subsequence N, \(\{d^{k}\}_{N}\rightarrow 0\), we have by (2.14), \(\{H(z^{k})\}_{N}\rightarrow 0\), and \(z^{*}\) is a solution of the QVI (1.1). Hence, there exists a \(m>0\) such that \(\d^{k}\\ge m\). Noting that \(\{\nabla \Psi (z^{k})\}_{N}\) is bounded and \(p>2\), there exists \(M>0\) such that \(\d^{k}\\le M\). Otherwise, it would contradict (2.12).
By (2.13) and \(\{z^{k}\}\) is a bounded sequence, \(\Psi (z^{k})\) is bounded from below and \(\{\Psi (z^{k+1})\Psi (z^{k})\}\rightarrow 0\), which implies
Suppose, subsequencing if necessary, we have \(\{\beta ^{l_{k}}\}\rightarrow 0\). By (2.13), we have
By \(m\le \d^{k}\\le M\), we can assume, subsequencing if necessary, that \(\{d^{k}\}\rightarrow \bar{d}\ne 0\). By passing to the limit in (2.16), we get
On the other hand, by (2.12), we have \(\nabla \Psi (z^{k})^{T}\bar{d}\le \rho \\bar{d}\^{p}<0\), which contradicts (2.17). Hence \(\beta ^{l_{k}}\) is bounded away from 0. (2.15) and (2.12) imply that \(\{d^{k}\}\rightarrow 0\), thus contradicting \(0< m\le \d^{k}\\), so that \(\nabla \Psi (z^{*})=0\). This completes the proof. □
Remark 2.2
The method proposed in [10] only considers the case of inequality constraints, while our method can solve QVI with both equality and inequality constraints. Besides, as we will see in the next section, our method can solve some problems in QVILIB [11], which cannot be solved by the method proposed by [10].
3 Numerical experiments
In this section, we report the results obtained by Algorithm 1 on problems list in QVILIB. All the computations in this paper were done using Matlab 2014a on a computer with 8.00 GB RAM and 2.5 GHz CPU. We solved all 55 test problems whose detailed description can be found in [11]. For each problem we list

the xpart of the starting point (the number reported is the value of all components of the xpart of the starting point);

the number of iterations;

the number of evaluations of Ψ;

the value of \(Y(x,\lambda ,\nu )\) at the termination.
In order to perform the linear algebra involved, we used Matlab’s linear system solver mldivide. If any entry of the solution given by mldivide is a NaN or it is equal to ±∞ or the sufficient decrease condition is not satisfied, then an anti gradient direction is used. We take \(\mu =10^{5}\), \(\beta =0.5\), \(\rho =10^{10}\), \(\sigma =0.01\) and \(p=2.1\). We choose \(\lambda ^{0}=0\), \(\nu ^{0}=0\) and \(w^{0}=0\) for all problems. For (2.6), we choose \(a_{i}=b_{i}=c_{i}=0\) when \((\lambda _{i},w_{i})= (0,0)\) and \(\theta =0\). Our aim is mainly to verify the reliability of the method, and compare the iteration numbers with the results presented in [10]. In order to perform a fair computation with the results in [10], we choose the same stopping criterion, i.e., let
and choose the termination criterion to be \(Y(x^{k},\lambda ^{k},\nu ^{k})\le 10^{4}\). The iteration is also stopped if the number of iterations exceeds 500 or the stepsize \(t_{k}\) computed at Step 3 is less than 10^{−6}.
We denote Algorithm 2.2 proposed in [10] by SSN, and compare our method with SSN. The results are list in Table 1. From Table 1, for problems that can be solved by SSN, they can also be solved by our method with almost the same iteration numbers except problems Box2B, Box3A, KunR11, KunR12, KunR21 and KunR22. However, our method can solve the problems RHS1A1, RHS1B1, RHS2A1, RHS2B1 and Wal3, which cannot be solved by SSN.
We also compare our method with the interior point method (denoted by IP) proposed in [12] from the iteration number and CPU time. For IP, we use the same parameters presented in [12] and the results are list in Table 2. From Table 2, we can see that our method is much more effective than IP for most problems.
We also consider other problems in QVILIB which are not test in Tables 1 and 2, including the QVIs with equality constraints, that is, Problems LunSS1 to Scrim12 in Table 3. As we can see from the table, Algorithm 1 can solve over half of those problems effectively.
We tried to make some modifications to the algorithm for those problems that cannot be solved. Specifically, when calculating the Jacobian of \(\tilde{H}(\tilde{z})\), we use \(\mathit{JF}(x)\) to approximate JL̃. For now, we cannot prove the convergence of the modified algorithm. However, it is interesting to find that the modified algorithm can find a solution for some problems, such as MoveSet3A1, MoveSet3A2, MoveSet3B1 and MoveSet3B2. The results are presented in Table 4.
Figures 1–4 display the performance of our method on the problems BiLin1A, Movset4A1, OutZ40 and Wal2. The vertical axis in those figures represents the value of Y and the horizontal axis represents the iteration number. As we can see from the figures, with the increase of the iteration numbers, the value of Y decrease.
Conclusion Remarks
In this paper, we have studied the numerical solution of QVI. We obtain the KKT system of a QVI and present a semismooth Newton method to solve the equations. We also establish its global convergence. Numerical results show that the performance of the proposed algorithm is promising.
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The work was supported by the Natural Science Foundation of China (grant No. 11601188, 11761037), by the Educational Commission of Guangdong Province, China (grant No. 2019KTSCX172), by the Natural Science Foundation of Jiangxi Province (grant No. 20181BAB201009).
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Xie, SL., Sun, Z. & Xu, HR. A new semismooth Newton method for solving finitedimensional quasivariational inequalities. J Inequal Appl 2021, 132 (2021). https://doi.org/10.1186/s13660021026712
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DOI: https://doi.org/10.1186/s13660021026712