# The Second-Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints

Authors: Li Wang; Xingxu Chen; Juhe Sun
Journal: Complexity
 (2021)

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## Abstract

In this paper, the variational inequality with constraints can be viewed as an optimization problem. Using Lagrange function and projection operator, the equivalent operator equations for the variational inequality with constraints under the certain conditions are obtained. Then, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints. We prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to the variational inequality with constraints. In the end, one example with three kinds of different cases by using this differential equation system is solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.


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## Body

## 1. Introduction

We consider the variational inequality with constraints, denoted by VIPK,F: find x∗∈K such that(1)Fx∗,y−x∗≥0,∀y∈K,where K=x∈Ω|gx≤0, F:ℜn⟶ℜn is a monotone mapping, g:ℜn⟶ℜm is a convex and differentiable mapping, and Ω⊆ℜn is a nonempty closed convex set.Variational inequality problems arise in physics, mechanics, economics, optimization, control, equilibrium model in transportation, and so forth. The finite dimensional variational inequality is an active field with abundant intension. In the past 30 years, great progress has been made in the study of numerical algorithms for solving the variational inequality problems and many numerical methods have emerged in large numbers, such as nonsmooth equation methods, smoothing methods, projection iteration methods, interior methods, multisplitting methods, homotopy approaches, Tikhonov regularization, and proximal point methods. The book by Facchinei and Pang [1] is the most famous reference to give a detailed numerical treatment of variational inequality problems and complementarity problems in the area of mathematical programming. But this book did not include the differential equation approaches for solving variational inequality problems.The differential equation method is used to solve constrained nonlinear optimization problems, which was firstly proposed by Arrow and Hurwicz [2]; and the equity-constrained optimization problem was considered. Fiacco and McCormick [3] studied the constraint qualification by using differential equation method. Evtushenko [4] studied the problem of equality constraint earlier. Yamashita [5], Evtushenko et al. [6–10], and Pan [11] developed and improved the differential equation methods. In particular, Evtushenko and Zhadan have carried out a great deal of research on differential equation methods for nonlinear programming problems and constraint problems on general closed sets by using the stability theory of differential equations since 1973, which enriches the differential equation method of nonlinear programming problem. There are other scholars who have done a lot of research work on differential equation methods, and they have established a variety of differential equation systems to solve the optimization problems. Brown et al. [12, 13] proposed the differential equation system based on the penalty function. Zhang and Constantinides [14] presented a differential equation system based on Lagrange function to solve the equality constraint problem. Since the nonlinear Lagrange function can be used to construct a dual algorithm for solving nonlinear optimization problems and the dual algorithm has no limitation on the feasibility of the original variables, many differential equation systems involved dual variables, such as those in the works of Zhou et al. [15] and Jin et al. [16]. Recently, Jin et al. [17] formulated the differential systems based on the first derivatives and the second derivatives of the approximate augmented Lagrangian. Under suitable conditions, the asymptotic stability of the differential systems and local convergence properties of their Euler discrete schemes are analyzed. Zhang [18] studied the differential equation method for quadratic programming.At the same time, there are a series of neural network methods for solving variational inequality and complementarity problems. In fact, the model of neural network is often described by the system of differential equations. Zhou et al. [19] transformed the nonlinear complementarity problem into an equivalent optimization problem, and a system of differential equations is constructed to solve the optimization problem. On the basis of projection operator, Gao et al. [20] presented a system of differential equations for solving variational inequality problems with linear and nonlinear constraints. He and Yang [21] studied a new neural network model for nonsymmetric linear variational inequality. The classical neural network model for solving linear variational inequality based on projection operator and compression method is extended. Liao et al. [22] studied the differential equation system of nonlinear complementarity problem. Nazemi and Sabeghi [23] applied a gradient neural network model to efficiently solve the convex second-order cone constrained variational inequality problem. Nazemi and Sabeghi [24] considered a new neural network model to simply solve the convex second-order cone constrained variational inequality problem.Different from the above results, Antipin [25] considered the first-order and second-order differential equation systems for solving the minimization function on a simple set. The asymptotic and exponential stability of such processes was proved without using Lyapunov function. Furthermore, Antipin [26] studied the differential equation system and the controlled proximal differential equation system for solving the minimization function. The convergence theorems were obtained. Antipin and Antipin [27, 28] established the differential equation system and the controlled proximal differential equation system for solving the saddle problem and proved that the trajectory of the system converges monotonically in norm to one of the equilibrium points. Antipin [29] considered the differential equation system with the controlled process for solving the fixed point of the extremal mapping and proved that the trajectory of the system converges to the solution with exponential rate. Antipin [30] established the differential equation systems and the differential equation system with the controlled process for optimization problem, saddle problem, and the extremal mapping, respectively. The corresponding convergence theorems were also proved. The variational inequality problem with coupled constraints and the fixed point of the extremal mapping with coupled constraints were studied by Antipin and Antipin [31, 32], where symmetric functions were introduced and the differential equation systems with the controlled process for global convergence were proposed. Wang and Wang [33] established the first-order differential equation system to solve the variational inequality with constraints problem (1).We think that the research of differential equation method for solving variational inequality is not much. In the past, the problem is firstly transformed into a smooth unconstrained optimization problem; then the differential equation method is constructed by using negative gradient for the unconstrained optimization. Inspired by the work of Antipin, our approach is different, which directly starts from the nonsmooth system of equations to establish the differential equation method, which requires relatively weak conditions. At the same time, due to the fact that the differential equation system has mature numerical solution methods, such as Runge-Kutta method, using these pieces of numerical software to solve the problem of variational inequality may achieve good results in practical calculation. It is hoped that our research on the differential equation method for the variational inequality will contribute to the development of the neural network method for solving the variational inequality problem.In this paper, we will establish the second-order differential equation system with the controlled process for solving the variational inequality with constraints problem (1). The remainder of this paper is organized as follows. In the next section, based on Lagrange function and projection operator, we prove a lemma that gives the equivalent operator equation under certain conditions for problem (1). In Section 3, we establish the second-order differential equation system with the controlled process for the variational inequality with constraints (1) and prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to problem (1). One example with three kinds of different cases is solved by the second-order differential equation system with the controlled process in Section 4, and the transient behaviors of the trajectories of this kind of differential equation system for every case are illustrated.


## 2. Preliminaries

The projection operator to a convex set is quite useful in reformulating the variational inequality with constraints (1) as an equation.LetC be a convex closed set; for every x∈ℜn, there is a unique x^ in C such that(2)x−x^=minx−y|y∈C.The pointx^ is the projection of x onto C, denoted by ΠCx. The projection operator ΠC:ℜn⟶C is well defined over ℜn and it is a nonexpansive mapping.Lemma 1.
(see [34]). Let H be a real Hilbert space and let C⊂H be a closed convex set. For a given z∈H, u∈C satisfies the inequality(3)u−z,v−u≥0,∀v∈C,if and only if u−ΠCz=0.In order to obtain the second-order differential equation system, we need to transform the variational inequality with constraints problem (1), which can be viewed as the following optimization problem:(4)minfys.t.gy≤0,y∈Ω,where fy=Fx∗,y−x∗ and fy≥0.Define the Lagrange function of the minimization problem (4):(5)ℒx∗,y,u=Fx∗,y−x∗+u,gy,∀y∈Ω,u∈ℜ+m,where y and u are the primal and dual variables, respectively.Sincex∗ is the minimum of fy, the pair x∗,u∗ (under certain regularity conditions) is a saddle point of ℒx∗,y,u. That is,(6)Fx∗,x∗−x∗+u,gx∗≤Fx∗,x∗−x∗+u∗,gx∗≤Fx∗,y−x∗+u∗,gy,∀y∈Ω and u∈ℜ+m.System (6) can be represented in an equivalent manner as(7)x∗∈argminFx∗,y−x∗+u∗,gy|y∈Ω,u∗∈argmaxu,gx∗|u∈ℜ+m.Lemma 2.
Suppose thatg:ℜn⟶ℜm is a differentiable mapping and Ω⊂ℜn is a nonempty closed convex set. Then the pair x∗,u∗ is the saddle point of the Lagrange function ℒx∗,y,u if and only if x∗,u∗ satisfies the following operator equation:(8)x∗=ΠΩx∗−αFx∗+Jgx∗Tu∗,u∗=Π+u∗+αgx∗,where Π+⋅ and ΠΩ⋅ are the operators that project a vector onto the positive octant R+m and the set Ω, respectively, and α>0 is a parameter.Proof.
System (4) or system (5) implies that x∗,u∗ satisfies the following inequalities:(9)Fx∗,y−x∗+u∗,gy−gx∗≥0,∀y∈Ω,−gx∗,u−u∗≥0,∀u∈ℜ+m.
If the functiong is differentiable, we have(10)Fx∗+Jgx∗Tu∗,y−x∗≥0,∀y∈Ω,−gx∗,u−u∗≥0,∀u∈ℜ+m,where Jgx∗ is the Jacobian of the mapping g at x∗.
The variational inequality system (10) can be computed by(11)x∗−x∗+αFx∗+Jgx∗Tu∗,y−x∗≥0,∀y∈Ω,u∗−u∗−αgx∗,u−u∗≥0,∀u∈ℜ+m,where α>0 is a parameter.
With the help of projection operators and Lemma1, the above system of variational inequalities is represented in the form of operator equation as(12)x∗=ΠΩx∗−αFx∗+Jgx∗Tu∗,u∗=Π+u∗+αgx∗,where Π+⋅ and ΠΩ⋅ are the operators that project a vector onto the positive octant R+m and the set Ω, respectively. This completes the proof.Furthermore, ifg:ℜn⟶ℜm is convex, we have for the second term in the first inequality of the variational inequality (10) that(13)Jgx∗Tu∗,y−x∗=u∗,Jgx∗y−x∗≤u∗,gy−gx∗,for all y∈Ω. Thus, we can rewrite the variational inequality (10) as follows:(14)Fx∗,y−x∗+u∗,gy−gx∗≥0,∀y∈Ω,−gx∗,u−u∗≥0,∀u∈ℜ+m.Remark 1.
When the functiong is convex and differentiable, the relations in (6)–(14) are equivalent to each other.


## 3. Second-Order Differential Equation System

In this section, we consider the second-order differential equation system for solving the variational inequality with constraints problem (1).The second-order differential equation system is constructed based on equation (8) as follows:(15)μ1d2xdt2+β1dxdt+x=ΠΩx−αFx+JgxTu,xt0=x0,x˙t0=x˙0,μ2d2udt2+β2dudt+u=Π+u+αgx,ut0=u0,u˙t0=u˙0,where μ1>0,β1>0,μ2>0,β2>0, and α>0 are parameters.In order to provide the convergence of the trajectory of the second-order differential equation system to a saddle pointx∗,u∗ of the Lagrange function ℒx∗,y,u, we introduce the following second-order differential equation system with the controlled process inspired by Antipin et al. [25–32] and Wang and Wang [33].(16)μ1d2xdt2+β1dxdt+x=ΠΩx−αFx¯+Jgx¯Tu¯,xt0=x0,x˙t0=x˙0,μ2d2udt2+β2dudt+u=Π+u+αgx¯,ut0=u0,u˙t0=u˙0,where(17)x¯=ΠΩx−αFx+JgxTu¯,u¯=Π+u+αgx.Ifμ1=μ2=0 and β1=β2=1, then system (16) coincides with (8) in the work of Wang and Wang [33].For simplicity, we denotex¨=d2x/dt2 and u¨=d2u/dt2. Similarly, the second-order differential equation system with the controlled process (16) can be represented in the form of variational inequalities as follows:(18)μ1x¨+β1x˙+αFx¯+Jgx¯Tu¯,y−x−μ1x¨−β1x˙≥0,∀y∈Ω,(19)μ2u¨+β2u˙−αgx¯,v−u−μ2u¨−β2u˙≥0,∀v∈ℜ+m,(20)x¯−x+αFx+JgxTu¯,y−x¯≥0,∀y∈Ω,(21)u¯−u−αgx,v−u¯≥0,∀v∈ℜ+m.Subsequently, we assume thatg,F, and JgT are Lipschitz continuous with the constants g,F, and J, respectively. In addition, we suppose that u¯≤C0 and C0 is a constant.With regard to these conditions and (16), it is easy to show the following inequalities since the projection operator is a nonexpansive mapping.(22)u+μ2u¨+β2u˙−u¯=Π+u+αgx¯−Π+u+αgx≤u+αgx¯−u−αgx≤αgx¯−x,(23)x+μ1x¨+β1x˙−x¯=‖ΠΩx−αFx¯+Jgx¯Tu¯−ΠΩx−αFx+JgxTu¯‖≤x−αFx¯+Jgx¯Tu¯−x+αFx+JgxTu¯=αFx−Fx¯+JgxTu¯−Jgx¯Tu¯≤αFx−Fx¯+αJgxTu¯−Jgx¯Tu¯≤αF+C0Jx−x¯=αCx−x¯,where C=F+C0J.Now we discuss the convergence of the trajectoriesxt, ut, x˙t, and u˙t of the second-order differential equation system (16).Theorem 1.
Suppose that the set of solutionsΩ∗ of problem (1) is nonempty, F is monotone and Lipschitz continuous with the constant F, and g is convex, differentiable, and Lipschitz continuous with the constant g. Suppose that ut≤C0 for all t⟶∞, u˙t is bounded for all t⟶∞, JgT is Lipschitz continuous with the constant J, and Ω⊆ℜn is a convex closed set. Then the accumulation point of the trajectory xt of the second-order differential equation system (16) with 1<β1<μ1<1/2β12, 0<μ2<1/2β22, and α2F+C0J2+g2<1/2 is the solution to the variational inequality with constraints problem (1).Proof.
Settingy=x∗∈Ω∗ in the variational inequality (18), we have(24)μ1x¨+β1x˙+αFx¯+Jgx¯Tu¯,x∗−x−μ1x¨−β1x˙≥0.
Lety=x+μ1x¨+β1x˙ in the variational inequality (20); we obtain(25)x¯−x+αFx+JgxTu¯,x+μ1x¨+β1x˙−x¯≥0,which implies that(26)x¯−x,x+μ1x¨+β1x˙−x¯+αFx¯,x+μ1x¨+β1x˙−x¯−αFx¯−Fx,x+μ1x¨+β1x˙−x¯+αJgx¯Tu¯,x+μ1x¨+β1x˙−x¯−αJgxTu¯−JgxTu¯,x+μ1x¨+β1x˙−x¯≥0.
Summing inequalities (24) and (26), we infer that(27)μ1x¨+β1x˙,x∗−x−μ1x¨−β1x˙+αFx¯,x∗−x¯+αJgx¯Tu¯,x∗−x¯+x¯−x,x+μ1x¨+β1x˙−x¯+αFx−Fx¯,x+μ1x¨+β1x˙−x¯+αJgxTu¯−Jgx¯Tu¯,x+μ1x¨+β1x˙−x¯≥0.
By using inequality (23), we compute the above inequality as follows:(28)μ1x¨,x∗−x−μ1x¨−β1x˙+β1x˙,x∗−x−μ1x¨−β1x˙+x¯−x,x+μ1x¨+β1x˙−x¯+αFx¯,x∗−x¯+αJgx¯Tu¯,x∗−x¯+αC2x¯−x2≥0,where C=F+C0J.
Sinceg is convex, we transform the above inequality by using inequality (14) as follows:(29)μ1x¨,x∗−x−μ1x¨−β1x˙+β1x˙,x∗−x−μ1x¨−β1x˙+x¯−x,x+μ1x¨+β1x˙−x¯+αFx¯,x∗−x¯+αu¯,gx∗−gx¯+αC2x¯−x2≥0.
Lettingy=x¯ in the first inequality of (14), we have(30)αFx∗,x¯−x∗+αu∗,gx¯−gx∗≥0.
Adding the above inequality into (29), we have(31)μ1x¨,x∗−x−μ1x¨−β1x˙+β1x˙,x∗−x−μ1x¨−β1x˙+x¯−x,x+μ1x¨+β1x˙−x¯+αFx¯−Fx∗,x∗−x¯+αu¯−u∗,gx∗−gx¯+αC2x¯−x2≥0
Settingv=u∗ in the variational inequality (19) and v=u+μ2u¨+β2u˙ in the variational inequality (21), we deduce that(32)μ2u¨+β2u˙,u∗−u−μ2u¨−β2u˙−αgx¯,u∗−u−μ2u¨−β2u˙≥0,u¯−u,u+μ2u¨+β2u˙−u¯+αgx¯−gx,u+μ2u¨+β2u˙−u¯−αgx¯,u+μ2u¨+β2u˙−u¯≥0.
Summing the above two inequalities and taking into account (22), we obtain(33)μ2u¨+β2u˙,u∗−u−μ2u¨−β2u˙+u¯−u,u+μ2u¨+β2u˙−u¯−αgx¯,u∗−u¯+αg2x¯−x2≥0.
In view ofu¯,gx∗≤0 and u∗,gx∗=0, we conclude that(34)μ2u¨+β2u˙,u∗−u−μ2u¨−β2u˙+u¯−u,u+μ2u¨+β2u˙−u¯+αgx∗−gx¯,u∗−u¯+αg2x¯−x2≥0.
SinceF is monotone, we sum (31) and (34) to obtain(35)μ1x¨,x∗−x−μ1x¨−β1x˙+β1x˙,x∗−x−μ1x¨−β1x˙+x¯−x,x+μ1x¨+β1x˙−x¯+μ2u¨,u∗−u−μ2u¨−β2u˙+β2u˙,u∗−u−μ2u¨−β2u˙+u¯−u,μ2u¨+β2u˙+u−u¯+α2C2+g2x¯−x2≥0.
The above inequality yields that(36)μ1x¨,x−x∗+μ12x¨2+μ1β1x¨,x˙+β1x˙,x−x∗+μ1β1x˙,x¨+β12x˙2−x¯−x,x+μ1x¨+β1x˙−x¯+μ22u¨2+μ2u¨,u−u∗+μ2β2u¨,u˙+β2u˙,u−u∗+μ2β2u˙,u¨+β22u˙2−u¯−u,u+μ2u¨+β2u˙−u¯−α2C2+g2x¯−x2≤0.
Using the relations(37)x¯−x,x+μ1x¨+β1x˙−x¯=12μ1x¨+β1x˙2−12x¯−x2−12x+μ1x¨+β1x˙−x¯2,u¯−u,u+μ2u¨+β2u˙−u¯=12μ2u¨+β2u˙2−12u¯−u2−12u+μ2u¨+β2u˙−u¯2,12μ1x¨+β1x˙2=12μ12x¨2+μ1β1x¨,x˙+12β12x˙2,12μ2u¨+β2u˙2=12μ22u¨2+μ2β2u¨,u˙+12β22u˙2,inequality (35) can be transformed into the following:(38)μ1x¨,x−x∗+β122x˙2+β1x˙,x−x∗+μ122x¨2+μ1β1x˙,x¨+μ2u¨,u−u∗+β222u˙2+β2u˙,u−u∗+μ2β2u˙,u¨+μ222u¨2+12u¯−u2+12−α2C2+g2x¯−x2≤0.
According to the relations(39)12d2dt2x−x∗2=x˙2+x−x∗,x¨,12ddtx˙2=x˙,x¨,12ddtx−x∗2=x˙,x−x∗,12d2dt2u−u∗2=u˙2+u−u∗,u¨,12ddtu˙2=u˙,u¨,12ddtu−u∗2=u˙,u−u∗,inequality (38) means that(40)μ12d2dt2x−x∗2+β122−μ1x˙2+β12ddtx−x∗2+μ122x¨2+μ1β12ddtx˙2+μ22d2dt2u−u∗2+β222−μ2u˙2+β22ddtu−u∗2+μ222u¨2+μ2β22ddtu˙2+12u−u¯2+12−α2C2+g2x¯−x2≤0.
Letϕx=1/2x−x∗2 and ψu=1/2u−u∗2; inequality (40) yields that(41)μ1d2dt2ϕx+β1ddtϕx+β122−μ1x˙2+μ122x¨2+μ1β12ddtx˙2+μ2d2dt2ψu+β2ddtψu+β222−μ2u˙2+μ222u¨2+μ2β22ddtu˙2+12u−u¯2+12−α2C2+g2x¯−x2≤0.
Sinceμ1<1/2β12, μ2<1/2β22, and α2C2+g2<1/2, inequality (41) can be integrated from t0 to t:(42)μ1ddtϕx+β1ϕx+β122−μ1∫t0tx˙2dτ+μ122∫t0tx¨2dτ+μ1β12x˙2+μ2ddtψu+β2ψu+β222−μ2∫t0tu˙2dτ+μ222∫t0tu¨2dτ+μ2β22u˙2+12∫t0tu¯−u2dτ+12−α2C2+g2∫t0tx¯−x2dτ≤B,where B=μ1d/dtϕx0+β1ϕx0+μ1β1/2x˙02+μ2d/dtψu0+β2ψu0+μ2β2/2u˙02.
Sinceut and u˙t are all bounded for all t⟶∞, we deduce that d/dtψu has the lower bound because(43)ddtψu=u˙,u−u∗≤u˙u−u∗.
Thus, there exists a constantB1 such that(44)μ1ddtϕx+β1ϕx≤B1,which implies that(45)μ1exp−β1μ1tddtexpβ1μ1tϕx≤B1.
That is,(46)ddtexpβ1μ1tϕx≤B11μ1expβ1μ1t.
By integrating (45), we have(47)expβ1μ1tϕx≤B1β1expβ1μ1t+C1,where C1 is a constant. We conclude that(48)ϕx≤B1β1+C1exp−β1μ1t,which means that ϕx is bounded for all t⟶∞.
The functionϕx is strongly convex, and it is well known that each of its Lebesgue sets is bounded. Thus, the trajectory xt is bounded. That is, there exists a constant B2 such that(49)xt−x∗≤B2.
Now we claim that∫t0tx˙2dτ<∞,∫t0tx¨2dτ<∞,∫t0tu˙2dτ<∞,∫t0tu¨2dτ<∞,∫t0tu¯−u2dτ<∞ and ∫t0tx¯−x2dτ<∞.
We firstly show thatx˙ is bounded. It follows from inequality (42) that(50)ddtϕx+β1μ1ϕx+β12x˙2≤B3,where B3 is a constant. From ϕx=1/2x−x∗2, the above inequality means that(51)x˙,x−x∗+β12μ1x−x∗2+β12x˙2≤B3.
In the light ofx˙,x−x∗=−1/2x˙2−1/2x−x∗2+1/2x˙+x−x∗2, the above inequality infers that(52)12β1μ1−1x−x∗2+12β1−1x˙2≤B3.
Noting that1<β1<μ1, we conclude that x˙2 is bounded in the following:(53)12β1−1x˙2≤B3+121−β1μ1x−x∗2≤B3+121−β1μ1B2,which means that x˙2 is bounded. It follows, from(54)ddtϕx=x˙,x−x∗≤x˙x−x∗,that d/dtϕx also has lower bound. Thus, there exists a constant B5 such that(55)β122−μ1∫t0tx˙2dτ+μ122∫t0tx¨2dτ+β222−μ2∫t0tu˙2dτ+μ222∫t0tu¨2dτ+12∫t0tu¯−u2dτ+12−α2C2+g2∫t0tx¯−x2dτ≤B5,which yields that the integrals,(56)∫t0tx˙2dτ<∞,∫t0tx¨2dτ<∞,∫t0tu˙2dτ<∞,∫t0tu¨2dτ<∞,∫t0tu¯−u2dτ<∞,∫t0tx¯−x2dτ<∞,converge as t⟶∞.
Assuming that there existsε>0 such that x¨t≥ε,u¨t≥ε, x˙t≥ε, u˙t≥ε, u¯t−ut≥ε, and x¯t−xt≥ε for all t≥t0, we obtain a contradiction to the convergence of integrals. Hence, there exists a subsequence of time moments ti⟶∞ such that x¨ti⟶0,x˙ti⟶0,u¨ti⟶0, x˙ti⟶0, u¯ti−uti⟶0, and x¯ti−xti⟶0. Since xt and ut are bounded, we know that xti and uti are bounded. We choose the subsequences xtij and utij of xt and ut. Then there exist x′ and u′ such that xtij⟶x′,utij⟶u′ , x¨tij⟶0, x˙tij⟶0, u¨tij⟶0, u˙tij⟶0, u¯tij−utij⟶0, and x¯tij−xtij⟶0 as j⟶∞.
Let us consider inequality (16) for all tij and take the limit as j⟶∞; we have(57)x′=ΠΩx′−αFx′+Jgx′Tu′,u′=Π+u′+αgx′,which means that x′∈Ω∗ is a solution of variational inequality with constraints (1) by using Lemma 2. This completes the proof. □


## 4. Numerical Results

In this section, we test the example with three kinds of different cases by our differential equation system (16). The transient behaviors of the proposed differential equation systems are demonstrated in each case. The numerical implementation is coded by Matlab R2019a running on a PC with Intel i5 9400F of 2.9 GHz CPU and the ordinary differential equation solver adopted is ode45, which uses a Runge-Kutta (4, 5) formula.Example 1.
Consider the variational inequality with constraints problem withF:ℜ5⟶ℜ5 given by Fx=Mx+ρCx+q and K=x∈ℜ+5|gx=10−∑i=15xi≤0, where M is an asymmetric positive definite matrix and Cix=arctanxi−2,i=1,2,…,5. The parameter ρ is used to vary the degree of asymmetry and nonlinearity, and the data of the example are illustrated as follows:(58)M=0.726−0.9490.266−1.193−0.5041.6450.6780.333−0.217−1.443−1.016−0.2250.7690.9341.0071.0630.567−1.1440.550−0.548−0.2591.453−1.0730.5091.026,and q=5.308,0.008,−0.938,1.024,−1.312T, which have been considered in the work of Dang et al. [35]. Its solution is x∗=2,2,2,2,2T.
Since the parameterρ is used to vary the degree of asymmetry and nonlinearity, we apply the second-order differential equation system (16) to solve the variational inequalities in three cases of ρ<1, ρ=1, and ρ>1.Case 1.
Whenρ<1, we take ρ=0.1, which means that the degree of asymmetry of Fx is higher than that of the nonlinearity of Fx. In this experiment, we set μ1=μ2=2.02, β1=β2=2.01, and α=0.5 in the second-order differential equation system with the controlled process (16). Figure 1 describes the convergence behaviors of xt, ut, x˙t, and x˙t of the second-order differential equation system with the controlled process (16) from the given initial points for the example with ρ=0.1.Figure 1
Transient behavior ofxt and ut of the second-order differential equation system (16) in the example with ρ=0.1.Case 2.
We takeρ=1, which means that the degree of asymmetry of Fx is the same as the degree of the nonlinearity of Fx. In this experiment, we let μ1=μ2=2.02, β1=β2=2.01, and α=0.4 in the second-order differential equation system with the controlled process (16). The solution trajectories xt, ut, x˙t, and x˙t of the second-order differential equation system with the controlled process (16) from the given initial points for the example with ρ=1 are shown in Figure 2.Figure 2
Transient behavior ofxt and ut of the second-order differential equation system (16) in the example with ρ=1.Case 3.
Whenρ>1, we take ρ=10, which means that the degree of asymmetry of Fx is lower than that of the nonlinearity of Fx. In this experiment, we set μ1=μ2=2.02, β1=β2=2.01, and α=0.06 in the second-order differential equation system with the controlled process (16). Figure 3 describes the convergence behaviors of xt, ut, x˙t, and u⋅t of the second-order differential equation system with the controlled process (16) from the given initial points for the example with ρ=10.Figure 3
Transient behavior ofxt and ut of the second-order differential equation system (16) in the example with ρ=10.Remark 2.
In Theorem1, the conditions of the parameters μ1,β1,μ2,β2, and α with 1<β1<μ1<1/2β12, 0<μ2<1/2β22, and α2F+C0J2+g2<1/2 are sufficient but not necessary for the convergence of the trajectory of the second-order differential equation system with the controlled process (16) for solving the variational inequality with constraints problem (1). The following lists are given to illustrate the influence on the rate of convergence of trajectory of the second-order differential equation system with the controlled process (16) for solving the variational inequality with constraints problem in the above example. From the condition with 1<β1<μ1<1/2β12, we have that 2<β1<μ1. Hence, we give a discussion in the following and take the first case as an example; that is, ρ=0.1 in our example.
Fixingμ2=2.02,β2=2.01, and α=0.5, the convergence time will be changed as follows when μ1 and β1 are changed (Table 1). In this experiment, when μ2,β2, and α were fixed, we found that the smaller μ1 and β1 are, the shorter the convergence time is. Furthermore, we also found that the change of β1 had a great influence on the rate of the convergence when μ1 is fixed; meanwhile the change of μ1 has a little effect on the rate of the convergence when β1 is fixed.
Fixingμ1=2.02,β1=2.01, and α=0.5, the convergence time will be changed as follows when μ2 and β2 are changed (Table 2). In this experiment, when μ1,β1, and α were fixed, we found that the smaller μ2 and β2 are, the shorter the convergence time is. Furthermore, we also found that the change of β2 had a great influence on the rate of the convergence when μ2 is fixed; meanwhile the change of μ2 has a little effect on the rate of the convergence when β2 is fixed.
Fixingμ1=2.02,β1=2.01,μ2=2.02, and β2=2.01, the convergence time will be changed as follows when α is changed (Table 3). In this experiment, when μ1,β1,μ2, and β2 were fixed, we found that the convergence time is the shortest at about α=0.5.Table 1
Convergence time ofxt by the second-order differential equation system (16).
μ1β1Timemsxt2.022.0118.36912.0000,2.0000,2.0000,2.0000,2.0000T2.122.1122.05022.0000,2.0000,2.0000,2.0000,2.0000T2.222.2122.55892.0000,2.0000,2.0000,2.0000,2.0000T2.322.3123.22632.0000,2.0000,2.0000,2.0000,2.0000T2.422.4123.56762.0000,2.0000,2.0000,2.0000,2.0000TTable 2
Convergence time ofxt by the second-order differential equation system (16).
μ2β2Timemsxt2.022.0118.36912.0000,2.0000,2.0000,2.0000,2.0000T2.122.1118.44302.0000,2.0000,2.0000,2.0000,2.0000T2.222.2121.74732.0000,2.0000,2.0000,2.0000,2.0000T2.322.3121.75032.0000,2.0000,2.0000,2.0000,2.0000T2.422.4121.93882.0000,2.0000,2.0000,2.0000,2.0000TTable 3
Convergence time ofxt by the second-order differential equation system (16).
αTimemsxt0.274.44802.0000,2.0000,2.0000,2.0000,2.0000T0.340.58882.0000,2.0000,2.0000,2.0000,2.0000T0.423.48822.0000,2.0000,2.0000,2.0000,2.0000T0.518.36912.0000,2.0000,2.0000,2.0000,2.0000T0.5539.58632.0000,2.0000,2.0000,2.0000,2.0000T


## 5. Conclusions

In this paper, based on the Lagrange function and projection operator, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints problem (1). The convergence of the trajectories for this kind of differential equation system is proved. Furthermore, we test one example with three kinds of different cases by the second-order differential equation system with the controlled process and illustrate the transient behaviors of the trajectories for this differential equation system in each case, which verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.It is an interesting issue that the second-order differential equation systems are used to solve the optimization problems.



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