A Dynamical System for Strongly Pseudo-monotone Equilibrium Problems

In this paper, we consider a dynamical system for solving equilibrium problems in the framework of Hilbert spaces. First, we prove that under strong pseudo-monotonicity and Lipschitz-type continuity assumptions, the dynamical system has a unique equilibrium solution, which is also globally exponentially stable. Then, we derive the linear rate of convergence of a discrete version of the proposed dynamical system to the unique solution of the problem. Global error bounds are also provided to estimate the distance between any trajectory and this unique solution. Some numerical experiments are reported to confirm the theoretical results.

games, the fixed point problem, and others; see, for instance, [1][2][3][4][5][6][7] and references quoted therein. The interest of this problem is that it unifies all these particular problems in a convenient way. In recent years, a large number of applications have been described successfully via the concept of equilibrium solution and therefore many researchers devoted their efforts to study EPs. For an excellent survey on existence of equilibrium points and solution methods for finding them, we refer the readers to the new monograph [8].
Since EP is a general model, the methods for solving particular problems can be often extended to EPs; see for example, [8][9][10][11][12][13][14][15][16][17][18][19]. Among them, fixed point-type methods play an important role, because they are simple in form and useful in practice. The idea of these methods comes from the proximal point and the projection methods for solving a variational inequality (VI for short) [1,2,13,20]. It is proved in [19] that fixed point methods can efficiently solve the class of strongly monotone EPs and that the convergence rate of these methods is linear [17].
In recent years, dynamical systems have been widely investigated for solving fixed point problems, variational inequalities and monotone inclusions [21][22][23][24][25][26][27][28][29]. As a natural extension, it is interesting to study EPs from a continuous time perspective. This is the aim of this paper. To do so, we first look at the EP as a fixed point problem of a suitable operator. Then, the solutions set of the EP is approached by considering a dynamical system associated with the fixed point map. Under standard assumptions, namely strong pseudo-monotonicity and Lipschitz continuity, we prove that the proposed dynamical system has a unique equilibrium point. Moreover, we also prove that this equilibrium solution is globally exponentially stable. A discrete version of the proposed dynamical system is also considered, allowing one to prove the linear convergence of the corresponding algorithms. Finally, we provide global error bounds in order to estimate the distance between any arbitrary trajectory and the unique solution under the given assumptions.
The remaining part of the paper is organized as follows. In Sect. 2, we recall some preliminary results. Section 3 describes the proposed dynamical system and its global stability. Global error bounds are discussed in Sect. 4. Finally, some numerical experiments are reported in Sect. 5 to illustrate the obtained theoretical results.
When f (x, y) = F(x), y −x for all x, y ∈ H , the equilibrium problem EP( f , C) reduces to the classical variational inequality V I (F, C), which consists in finding a point x * ∈ C such that where F : H → H is a continuous mapping. The solution set of this problem is abbreviated to Sol(F, C).
Another important particular case of equilibrium problem is the saddle point problem: Given two sets C 1 ⊂ H 1 and C 2 ⊂ H 2 , where H 1 and H 2 are two real Hilbert spaces, a saddle point of a function G : holds for any y = (y 1 , y 2 ) ∈ C 1 × C 2 . Finding a saddle point of G amounts to solving Indeed, a saddle point of G is a Nash equilibrium in a two-person zero-sum game, that is a noncooperative game where the cost function of the first player is G and the cost function of the second player is −G (see, e.g., [2,8].) We recall some well-known definitions useful in the sequel. A mapping f : C × C → R is said to be (a) strongly monotone with modulus γ > 0 on C, if for all x, y ∈ C.
We provide in the following a class of functions f , which is strongly pseudomonotone but not strongly monotone. The example is a generalization of examples given in [30][31][32].

Remark 2.2
When EP reduces to VI, i.e., f (x, y) = F(x), y − x for all x, y ∈ H , the (generalized) monotonicity of function f defined above corresponds to the well known (generalized) monotonicity of mapping F (see [33]).
On the other hand, a mapping f : C × C → R is said to satisfy a Lipschitz-type condition on C if there exists a constant L > 0 such that

Remark 2.3
The Lipschitz-type condition (2) was introduced by Quoc and Muu [10]. We note that (2) is weaker than the Lipschitz-type condition introduced by Antipin [2], which can be written as Indeed, taking u = z in (3) we obtain (2). On the other hand, (2) implies the Lipschitztype condition in the sense of Mastroeni [19] f (x, y) where c 1 , c 2 are two given positive constants.
, EP collapses to a VI problem, then f satisfies (2) if F is Lipschitz continuous with a Lipschitz constant L > 0. Indeed, since F is Lipschitz continuous, using the Cauchy-Schwarz inequality, we have To establish the main results of the paper, we need to recall the stability concepts of an equilibrium point of the general dynamical systeṁ where T is a continuous mapping from H to H and x : Definition 2.1 [24] (a) A point x * ∈ H is an equilibrium point for (4) denotes the open ball with center x * and radius r ; (d) An equilibrium point x * of (4) is exponentially stable if there exist δ > 0 and constants μ > 0 and η > 0 such that, for every solution x(t) of (4) with x(0) ∈ B(x * , δ), one has Furthermore, x * is globally exponentially stable if (5) holds true for all solutions x(t) of (4).

A Fixed Point-Type Dynamical System
For every λ > 0, we define a function ψ λ : C → C by Since f (x, ·) is convex, for every x ∈ C and λ > 0, the problem is a strongly convex problem, hence it has a unique solution. Therefore, the function ψ λ is well defined and has single values on C. Moreover, if x * is a fixed point of function ψ λ then x * ∈ Sol( f , C). Indeed, we have Proof See e.g., [11].
where P C denotes the projection operator onto C. In this case, Theorem 3.1 reduces to the well-known characterization of the solution of V I (F, C): , see e.g., [20].

Remark 3.2
The explicit discretization of dynamical system (8) with respect to the time variable t, with step size h n > 0 and initial point x 0 ∈ C, yields the following iterative scheme: or equivalently which is the relaxed projection-type method for solving EP( f , C) with relaxation parameter ρh n . The convergence of this method without relaxation, i.e., ρh n = 1 for all n, was investigated in [17,19] when the function f is strongly monotone.
It is known that, if the function f is strongly pseudo-monotone and continuous, then the equilibrium problem EP( f , C) has a unique solution x * , see [32]. It follows from Theorem 3.1 that x * is a fixed point of ψ λ , i.e., x * is the unique equilibrium point of dynamical system (8). Therefore, as a consequence, we have the following existence result.

Corollary 3.1 Suppose that the function f is continuous and strongly pseudo-monotone
with modulus γ > 0 on a nonempty closed convex set C. Then, the dynamical system (8) has a unique equilibrium point, which is the unique solution of EP( f , C).
We are now in a position to establish the globally stability of dynamical system (8).
For that purpose, we first give an estimation which plays the main role in the study of stability of the dynamical system (8).

Proposition 3.1 Let C be a nonempty closed convex subset of a real Hilbert space
H . Let f be a γ -strongly pseudo-monotone function from C × C into R satisfying the Lipschitz-type condition with modulus L > 0. Let x * be the unique solution of EP( f , C) and λ > 0. Then, Proof Since f satisfies the Lipschitz-type condition (2), using the Cauchy-Schwarz inequality, we have for all x, y, z ∈ C that Hence, (10) can be deduced from [34,Proposition 4.2].
We establish now the main result of this section.

Theorem 3.2
Let all the assumptions in Proposition 3.1 be satisfied and let λ < 2γ L 2 . Then, the unique equilibrium solution x * of the dynamical system (8) is globally exponentially stable.

Proof Consider the Lyapunov function
From (8), time derivative of V can be expressed aṡ Therefore, it follows from Cauchy-Schwarz inequality and (10) thaṫ Therefore, for all t > 0, we have This means that the equilibrium solution x * of the dynamical system (8) is globally exponentially stable.

Remark 3.4
According to Theorem 3.2, the unique equilibrium point x * is globally exponentially stable if parameter λ is small enough. Therefore, it is interesting to study the value of λ which maximizes the convergence rate of the trajectories. Considering α in (11) as a function of λ ∈ 0, 2γ /L 2 , we can see that the maximum convergence rate of the trajectory x(t) is which is attained at λ = λ * = γ /L 2 .
Since any strongly monotone function f is strongly pseudo-monotone, we have directly the following consequence:

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H . Let f be a γ -strongly monotone function from C × C into R satisfying the Lipschitztype condition with modulus L > 0. Let x * be the unique solution of EP( f , C) and
λ ∈ 0, 2γ L 2 . Then, x * is globally exponentially stable.
In the following, we prove the linear convergence of a discrete version of the dynamical system (8), i.e., the scheme (9).

Theorem 3.3
Let all the assumptions in Proposition 3.1 be satisfied and let λ < 2γ L 2 . Let {ρ n } ⊂ [ρ, 1] for all n ≥ 0 where ρ > 0 and let x 0 ∈ C. Then, the sequence {x n } generated by converges linearly to the unique solution x * of EP( f , C).

Remark 3.5
In the case ρ n = 1 for all n ≥ 0, (12) reduces to the proximal point algorithm studied by Antipin in [2], where the linear convergence was obtained under strong convexity assumption. The linear convergence of proximal point algorithm for solving strongly pseudo-monotone EPs is established in [34,Theorem 4.4]. A threestep proximal point algorithm can be found in [35,Theorem 3], where the speed of convergence is slower than linearity.
It is known that the over-relaxation may sometimes improve the convergence performance. We give an over-relaxation result in the following Theorem. Proof From (13), we have Hence, (0, 1). Therefore, {x n } converges linearly to x * .
As a consequence of Theorems 3.3 and 3.4, we have that the sequence {x n } converges linearly to the unique solution of EP( f , C) when ρ n ⊂ [ρ, ρ].
When f (x, y) = F(x), y − x for all x, y ∈ H , we obtain the relaxed version of the projected gradient method for solving variational inequalities considered in [30].

Global Error Bounds
In this section, our aim is to establish a global bound for the distance between any arbitrary trajectory x and the solution set of strongly pseudo-monotone EPs. Specially, given an arbitrary trajectory x ∈ C, we wish to measure how close x is to the unique solution x * of the problem EP( f , C) and to obtain a bound on such measures in terms of some easily computable quantities depending solely on x and the data of EP( f , C).

Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H . Let
f be a γ -strongly pseudo-monotone function from C×C into R satisfying the Lipschitztype condition with modulus L > 0. Let x * be the unique solution of EP( f , C) and λ > 0. Then, for every arbitrary vector x ∈ C, it holds that Proof For every x ∈ C, denote by z := ψ λ (x) ∈ C, the unique solution of the strongly convex minimization problem (7). The optimality condition associated with (7) implies that there exists s ∈ ∂ 2 f (x, z) such that where N C (z) denotes the normal cone to C at z. Hence, by definition of this cone, we obtain that On the other hand, since s ∈ ∂ 2 f (x, z), we have Combining (14) and (15), we obtain Substituting y := x * ∈ C into (16), we obtain Since x * is the unique solution of EP( f , C) and z ∈ C, we have f (x * , z) ≥ 0 and by the strong pseudo-monotonicity of f Combining (17) and (18), we obtain On the other hand, the Lipschitz-type continuity of f gives It follows from (19), (20) and the Cauchy-Schwarz inequality that which implies Using again the Cauchy-Schwarz inequality, we deduce which completes the proof.

Remark 4.1
Since the error bound holds globally, it can be used as a stopping condition in the algorithm for solving EP( f , C), i.e., one can stop the algorithm when x n − ψ λ (x n ) is small enough.
When f (x, y) = F(x), y − x for all x, y ∈ C, we recover the following global error bound for variational inequality as established in [31]. Then, for every arbitrary vector x ∈ C, it holds that The following result states that the distance from an arbitrary vector x to the solution set of EP( f , C) can also be lower bounded by the quantity x − ψ λ (x) . Theorem 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H . Let f be a γ -strongly pseudo-monotone function from C×C into R satisfying the Lipschitztype condition with modulus L > 0. Let x * be the unique solution of EP( f , C) and λ > 0. Then, for every arbitrary vector x ∈ C, it holds that Proof Setting z := ψ λ (x) ∈ C, it follows from (21) that Hence, where we have used the Cauchy-Schwarz inequality in the last estimation. Therefore, If λ ≤ 4γ L 2 then (22) follows from the last inequality. Otherwise, it is trivial.
When f (x, y) = F(x), y − x for all x, y ∈ C, we obtain a new result of global error bound for variational inequality. Then, for every arbitrary vector x ∈ C, it holds that

Numerical Illustrations
In this section, we consider some numerical results to illustrate the global exponential stability of the unique equilibrium point of dynamical system (8). The codes are implemented in MATLAB. The STOP condition is x(t) − ψ λ (t) ≤ for all test problems, where = 10 −5 .

Problem 1
The bifunction f of the equilibrium problem comes from the Cournot-Nash equilibrium model considered in [10]. It is defined for each x, y ∈ R 5 , by f (x, y) = Px + Qy + r , y − x where r ∈ R 5 , and P and Q are two square matrices of order 5 such that P − Q is symmetric positive definite. It was proved in [10] that the function f is strongly monotone with modulus γ = λ min (P − Q), the smallest eigenvalue of P − Q and f satisfies the Lipschitz-type condition with modulus L = P − Q . As in [10], in our test, the vector r and the matrices P and Q are chosen as follows:  The constraint set of this problem is defined by For this problem, γ = λ min (P − Q) = 0.7192 and L = P − Q = 2.905, and we choose λ = 1.9 * γ /L 2 . Figure 1 displays the trajectories generated by the dynamical system (8). It is clear that x(t) converges exponentially to the unique equilibrium point x * . Figure 2 compares the behavior of the non-relaxed algorithm (ρ = 1) with the inner-relaxed algorithm (ρ = 0.8) and the over-relaxed algorithm (ρ = 1.2) when x 0 = (2, 1, 4, −1, −2) T . We can see that the over-relaxed version takes advantage in this example.
We know that f is strongly pseudo-monotone with modulus γ = a − b > 0 and not (strongly) monotone. Moreover, f satisfies the Lipschitztype condition with L = a + 2b and x * = 0 is the unique solution of EP( f , C). Indeed, re-writing g(x, y) = ξ(x)x, y − x , it follows from [30] that F(x) := ξ(x)x is Lipschitz continuous with L = a + 2b. Hence, the conclusion follows from (4). In Fig. 3, we present a simulation for Problem 2 where H = R 5 , a = 10, b = 6 and λ = γ /L 2 . Figure 4 compares the behavior of the non-relaxed algorithm (ρ = 1) with the inner-relaxed algorithm (ρ = 0.8) and the over-relaxed algorithm (ρ = 1.2) with a random initial x 0 . Again, the over-relaxed version takes advantage in this example.

Conclusions
We have proposed and studied a dynamical system for solving equilibrium problems in Hilbert spaces. Under mild conditions, we have established the solution existence and uniqueness as well as the stability of the solution of the equilibrium problem. We have also provided the linear convergence analysis of a discrete version of the proposed dynamical system. Relaxing the strong (pseudo)-monotonicity assumption could be an interesting topic for further research.   . 4 Comparison of the non-relaxed algorithm (ρ n = 1) with the lower-relaxed (ρ n = ρ = 0.8) and over-relaxed (ρ n = 1.2) algorithms with a random initial x 0 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.