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Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery
Advances in Difference Equations volume 2021, Article number: 410 (2021)
Abstract
In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms.
Introduction
Let H be a real Hilbert space with the inner product \(\langle \cdot,\cdot \rangle \) and induced norm \(\\cdot \\) and let F be an open convex subset of H. In 1992, Muu and Oettli [25] introduced the equilibrium problem associated with ψ, which is finding \(s\in C\) such that
where C is a nonempty closed and convex subset of F and \(\psi: F\times F\rightarrow \mathbb {R}\) is a bifunction with \(\psi (s, s) = 0\) for all \(s \in C\). The set of solutions of the problem (1.1) is denoted by \(EP(\psi, C)\). For the Minty equilibrium problem (MEP), it was introduced by Castellani and Giuli [9] in 2013. This problem is associated with the equilibrium problem (1.1), which is to find \(s \in C\) such that
The solution set of the Minty equilibrium problem is represented as SM.
The equilibrium problem has been widely applied to study real world applications, which were unified by including as particular cases in applied mathematics like variational inequality, Nash equilibria, fixed point problem, optimization problem, saddlepoint problems, complementarity problem; see, for instance, [1–3, 5, 15, 16]. When solving some problems in applications of engineering, economics, management science, and other areas, one needs to formulate them in equilibrium form; see, for example, [4, 6–8, 10, 12, 13, 18, 21, 22, 24, 25, 28].
In 2003, Dinh and Kim [11] introduced the projection algorithm with line search of a bifunction which is not required to be pseudomonotone to solve the equilibrium problem. A weak convergence theorem was proved under continuity and convexity assumptions on the bifunction ψ, which is not required to have any monotonicity property, and assuming the solution set of Minty equilibrium problem (1.2) is nonempty.
In 1964, the inertial extrapolation technique was introduced by Polyak [27] to speed up the convergence of the algorithm. After that, many mathematicians have improved it in many ways, see [19, 23, 26]. In 2018, Iyiola et al. [14] motivated the inertialtype algorithms with the algorithm of Dinh and Kim [11], they obtained convergence theorems and presented the following inertialtype iterative method with Armijo line search stepsize which is faster and more efficient than the algorithm by Dinh and Kim [11].
Algorithm 1.1
Step 1: Choose a sequence \({\{\epsilon _{n}\}}^{\infty }_{n=1} \in l_{1}\) and take \(\sigma \in (0,1),\rho > 0\). Select arbitrary points \(s_{0} \in C_{0}, s_{1} \in C_{1}; C_{0} =C_{1}=C\), and \(\theta \in [0,1)\). Set \(n:=1\).
Step 2: Given the iterates \(s_{n1}\) and \(s_{n}, n\geq 1\), choose \(\theta _{n}\) such that \(0 \leq \theta _{n}\leq \bar{\theta }_{n}\), where
Step 3: Compute
Step 4: Compute
if \(u_{n}=s_{n}\), then stop. Otherwise go to Step 5.
Step 5: Find \(m_{n}\) as the smallest nonnegative integer m satisfying
Set \(\sigma _{n}:=\sigma ^{m_{n}},v_{n}=v_{m,n},w_{n}=w_{n,m_{n}}\).
Step 6: Compute
where \(C_{n+1} = C_{n} \cap H_{n}, H_{n} = \{{x \in H: h_{n}(x) \leq 0} \}\), and
Step 7: Set \(n \leftarrow n+1\) and go to Step 2.
In this work, we focus on the common equilibrium problem (CEP), which is to find \(s \in C\) such that
where \(\psi _{i}: F \times F \rightarrow \mathbb {R}\) is a bifunction with \(\psi _{i}(s, s) = 0\) for all \(i = 1,2,\dots,N\). Denote the solution set of the common Minty equilibrium problem (1.6) by \(CS_{M}\).
Very recently, the parallel method was used to solve common problems in many improved algorithms. One of such is a parallel viscositytype subgradient extragradient algorithm (PVTSE) introduced by Suantai et al. [29] for solving common variational inequalities. In this work, PVTSE algorithm was applied for solving image recovery problems with common types of blur effects. Note the similarity with the modified parallel hybrid subgradient extragradient (MHPSE), which is the algorithm that was used to solve common variational inequalities, constructed by Kitisak et al. [17].
Inspired and encouraged by the previous works, in this paper we proposed an inertialtype parallel monotone hybrid algorithm with Armijo line search for solving common nonmonotone equilibrium problems. A weak convergence theorem is established under some suitable conditions imposed on the bifunction \(\psi _{i}\). In the last section, we apply our algorithms for solving unconstrained image recovery problems and compare our main algorithms with PVTSE [29] and MHPSE [17] algorithms. It is remarkable that our method has a better convergence rate.
Preliminaries
This section contains some definitions and basic results that will be used in our subsequent analyses. We next recall some properties of the projection, see [4] for more details. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let \(\{x_{n}\}\) be a sequence in H, we denote the weak convergence (strong convergence) of \(\{x_{n}\}\) to a point \(x\in H\) by \(x_{n} \rightharpoonup x\) (\(x_{n}\rightarrow x\)), respectively.
Lemma 2.1
Let \(h:H\rightarrow \mathbb{R}\) be a realvalued function and K be a subset of H defined by \(K:= \{{u \in H: h(u) \leq 0}\}\). If K is nonempty and h is Lipschitz continuous on C with modulus \(\theta > 0 \), then
Definition 2.2
Let C be a nonempty closed and convex subset of a Hilbert space H. A function \(\psi: C \rightarrow H\) is called Lipschitz continuous if there exists a real constant \(K \geq 0\) such that
Definition 2.3
A bifunction \(\psi: C \times C \rightarrow \mathbb {R}\) is called jointly weakly continuous on \(C \times C \) if for all \(s, t \in C \) and \(\{s_{n}\}, \{t_{n}\} \) being two sequences in C converging weakly to s and t, respectively, \(\psi (s_{n}, t_{n})\) converges to \(\psi (s,t)\).
We now state the following assumptions which will be required in the sequel:
(A1) \(\psi (u,\cdot )\) is convex on H for every \(s \in H \);
(A2) ψ is jointly weakly continuous on \(H \times H\).
For each \(s,t \in H \), by \(\partial _{2}\psi (s,t)\) we denote the subdifferential of the convex function \(\psi (s,\cdot )\) at t, i.e.,
In particular,
In our main theorem, the following lemmas will be used in the convergence analysis.
Lemma 2.4
([30])
Let \(\psi: H \times H \longrightarrow \mathbb {R}\) be a function satisfying conditions (A1) and (A2). Let \(\overline{s}, \overline{t} \in H\) and \(\{ s_{n}\},\{ t_{n}\}\) be two sequences in H converging weakly to \(\overline{s},\overline{t}\), respectively. Then, for any \(\epsilon > 0\), there exist \(\eta > 0\) and \(n_{\epsilon } \in \mathbb {N}\) such that
for every \(n \geq n_{\epsilon }\), where B denotes the closed unit ball in H.
Lemma 2.5
([11])
Suppose the bifunction ψ satisfies the assumptions (A1) and (A2). If \(\{s_{n}\} \subset H\) is a sequence which converges strongly to s̅ and a sequence \(\{u_{n}\}\), with \(u_{n} \in \partial _{2}\psi (s_{n},s_{n})\), converges weakly to u̅, then \(\overline{u} \in \partial _{2}\psi (\overline{s},\overline{s})\).
Lemma 2.6
([11])
Suppose the bifunction ψ satisfies the assumptions (A1) and (A2). If \(\{s_{n}\} \subset C\) is bounded, \(\rho > 0\), and \(\{t_{n}\}\) is a sequence such that
then \(\{t_{n}\}\) is bounded.
Lemma 2.7
([20])
Assume \(\phi _{n} \in [ 0, \infty )\) and \(\delta _{n} \in \phi _{n} \in [ 0, \infty )\) satisfy:

1.
\(\phi _{n+1}  \phi _{n} \leq \theta _{n}(\phi _{n}  \phi _{n1}) + \delta _{n} \),

2.
\(\sum_{n=1}^{\infty }\delta _{n} < \infty \),

3.
\(\{\theta _{n}\} \subset [0,\theta ]\), where \(\theta \in (0,1)\).
Then the sequence \(\{\phi _{n}\}\) is convergent with \(\sum_{n=1}^{\infty }[\phi _{n+1}  \phi _{n}]_{+} < \infty \), where \([t]_{+}:= \max \{t,0\}\) (for any \(t \in \mathbb {R}\)).
Main results
In this section, we introduce a modified parallel method with line search rule for solving the common equilibrium problem (1.6) and provide some comments regarding the iteration parameters.
Algorithm 3.1
Step 1: Choose \(\sigma \in (0,1),\rho > 0\). Select arbitrary points \(s_{0} \in C_{0}, s_{1} \in C_{1} ; C_{0} =C_{1}=C\) and \(\{\theta _{n}\}\subset [0,\theta ]\) for some \(\theta \in [0,1)\). Set \(n:=1\).
Step 2: Compute
Step 3: For each \(i = 1,2,\dots,N\), compute
if \(u_{n}^{i}=t_{n}\), \(\forall i = 1,2,\dots,N\), then stop. Otherwise go to Step 4.
Step 4: Find \(m_{n}^{i}\) as the smallest nonnegative integer \(m^{i}\) satisfying
Set \(\sigma _{n}^{i}=\sigma ^{m_{n}^{i}}\), \(v_{n}^{i}=v_{n,m}^{i}\), \(w_{n}^{i}=w_{n,m_{n}}^{i}\).
Step 5: Compute
where \(C^{i}_{n+1} = C^{i}_{n} \cap H^{i}_{n}, H^{i}_{n} = \{{x \in H: f^{i}_{n}(x) \leq 0} \}\), and
Step 6: Compute
Step 7: Set \(n \leftarrow n+1\) and go to Step 2.
Remark 3.2
(1) It is clear that if \(u^{i}_{n} = t_{n}\) for all \(i = 1,2,\dots,N\), then \(t_{n}\) is a common solution of equilibrium problem (1.6).
(2) If \(N=1\), then Algorithm 3.1 reduces to Algorithm 1.2 of Iyiola et al. [14].
Lemma 3.3
Suppose the solution set \(CS_{M}\) of the Minty equilibrium problem is nonempty. Then, the following hold:

(i)
There exists an integer number \(m_{i}>0\) satisfying the following inequality:
$$\begin{aligned} \bigl\langle w^{i}_{n,m},t_{n}u^{i}_{n} \bigr\rangle \geq \frac{\rho }{2} \bigl\Vert t_{n}u^{i}_{n} \bigr\Vert ^{2} \quad\textit{for all } w^{i}_{n,m} \in \partial _{2}\psi _{i}\bigl(v^{i}_{n,m},v^{i}_{n,m} \bigr); \end{aligned}$$ 
(ii)
The sequence \(\{s_{n}\}\) generated by Algorithm 3.1is well defined and belong to \(C_{n}^{i}\) for all \(i=1,\dots, N\).
Proof
(i) We assume by contradiction that there exists \(i\in \{1,\dots, N\}\) and for every positive integer \(m_{i}\) and \(v^{i}_{n,m} = (1\sigma ^{m_{i}})t_{n} + \sigma ^{m_{i}}u^{i}_{n}\), there exists \(w^{i}_{n,m} \in \partial _{2}\psi _{i}(v^{i}_{n,m}, v^{i}_{n,m})\) such that
Observe that \(v^{i}_{n,m}\) → \(t_{n}\) as m → ∞ and therefore, by Lemma 2.5, the sequence \({\{w^{i}_{n,m}\}}^{\infty }_{m=1}\) is bounded. Thus, we suppose that \(w^{i}_{n,m}\) converges weakly to \(\bar{w} \in C\). Taking the limit as \(m \rightarrow \infty \) (noting that \(v^{i}_{n,m}\) → \(t_{n}\) and \(w^{i}_{n,m}\) ⇀ w̄) and using Lemma 2.6, we get \(\bar{w} \in \partial _{2}\psi _{i}(t_{n}, t_{n})\) and
Moreover, since \(\bar{w} \in \partial _{2}\psi _{i}(t_{n}, t_{n})\), we have
Combining with (3.5) yields
which contradicts the fact that
Thus, the line search is well defined.
(ii) We first show that \(C^{i}_{n}\) is nonempty. Indeed, by the assumption \(CS_{M} \neq \emptyset \), for each \(x^{*} \in CS_{M}\), we get \(\psi _{i}(y,x^{*}) \leq 0,\forall y \in C, \forall i = 1,\dots,N \). So, \(\psi _{i}(v_{n}^{i}, x^{*}) \leq 0, \forall n\). From the convexity of \(\psi _{i}(v_{n}^{i}, \cdot )\), we have
Therefore,
This implies that for each \(i=1,2,\dots,N\), there exists \(x_{n}^{i}\in C^{i}_{n+1}\). This means that \(\{s_{n}\}\) is well defined. □
Theorem 3.4
Suppose \(CS_{M}\neq \emptyset \) and let Assumptions (A1), (A2) hold. If
then the sequence {\(s_{n}\)} generated by Algorithm 1.1converges weakly to z \(\in EP(C,\psi _{i})\) for all \(i = 1,\dots,N\).
Proof
We split our proof into four steps below for the sake of clarity.
Step 1. We first show that \(\{s_{n}\}\) is bounded and there exists a weak cluster point of \(\{s_{n}\}\). Let \(x^{*}\in CS_{M}\). Then from Lemma 3.3, we have that \(x^{*} \in C^{i}_{n}\),
Also,
Observe that
Thus, from (3.8) and (3.9), we have
By the definition of \(\{s_{n}\}\), we have
Since \({\sum }^{\infty }_{n=1} \theta _{n} \s_{n}  s_{n1}\^{2} < \infty \), letting \(\delta _{n} = 2\theta _{n}\s_{n}  x\^{2}\) and \(\psi _{n} = \s_{n}  x^{*}\^{2}\), we deduce from Lemma 2.7 that the sequence \(\{\s_{n}  x^{*}\^{2}\}\) is convergent. Thus, \(\{s_{n}\}\) is bounded and \({\sum }^{\infty }_{n=1}[\s_{n+1}  x^{*}\^{2}  \s_{n}  x^{*}\^{2}] < \infty \). Furthermore, since \(\{s_{n}\}\) is bounded, there exists a subsequence \(\{s_{n_{k}}\}\) of \(\{s_{n}\}\) such that \(s_{n_{k}}\rightharpoonup p\in H\).
Step 2. We now show that for each \(i=1,\dots,N\), any weak accumulation point p of the sequence {\(s_{n}\)} belongs to \(C^{i}_{n}\) for all n. Suppose that \(\{s_{n_{j}}\} \subset \{s_{n}\}, s_{n_{j}} \rightharpoonup p\) as \(j \rightarrow \infty \), and there exists \(n_{0}\) such that \(p \notin C^{i}_{n_{0}}\). Then by the closedness and convexity of \(C^{i}_{n_{0}},C^{i}_{n_{0}}\) is also weakly closed. Hence, there exists \(n_{j_{0}} > n_{0}\) such that \(s_{n_{j}} \notin C^{i}_{n_{0}}\) for all \(n_{j} \geq n_{j_{0}}\), in particular \(s_{n_{j_{0}}} \notin C^{i}_{n_{0}}\). This contradicts the fact that \(s_{n_{j_{0}}} \in C^{i}_{n_{j_{0}}}1 \subset \cdots \subset C^{i}_{n_{0}+1} \subset C^{i}_{n_{0}}\). Therefore, \(p \in C^{i}_{n}, \forall n\) or \(p \in \bigcap^{\infty }_{n=0} C^{i}_{n}\). Since \(C^{i}_{n} \subset B^{i}_{n}, \forall n\), this implies that \(p \in \bigcap^{\infty }_{n=0}B^{i}_{n}\).
Step 3. Show that \(p \in EP(C,\psi _{i})\) for all \(i = 1,\dots,N \).
Using Algorithm 3.1, we have
Hence,
This implies that
From (3.10) and (3.14), we have
Clearly, from \(P_{C}s \in C\) and \(\langle s  P_{C}s, P_{C}s  y \rangle \geq 0, \forall y \in C\), we get \(\langle P_{C^{i}_{n+1}}t_{n}  t_{n}, P_{C^{i}_{n+1}}t_{n}  x^{*} \rangle \leq 0\), and \(\lim_{n\to \infty }\theta _{n } \s_{n}  s_{n1} \^{2}=0\) from the assumption \(\sum_{n=1}^{\infty }\theta _{n}\s_{n}s_{n1}\^{2}<\infty \). Thus from (3.15), we conclude that
From \(t_{n} = s_{n} + \theta _{n}(s_{n}  s_{n1})\), we get
and hence
Since \(s_{n_{j}}\) ⇀ p, it follows from (3.17) that \(t_{n_{j}}\)⇀ p. Observe that \(h^{i}_{n}\) is Lipschitz continuous with modulus \(M_{i} > 0\) for all \(i=1,\dots,N\). It follows from Lemmas 2.1 and 3.3 that
From (3.17) and (3.18), we obtain that
It follows from (3.16) that \(\sigma _{n}^{i} \ t_{n}u_{n}^{i} \^{2}\rightarrow 0\) as \(n\rightarrow \infty \) for all \(i=1,\dots,N\).
Case I. Suppose that for each \(i = 1,\dots,N\),
Then
which implies that
Thus,
for all \(i=1,\dots,N\). From (3.17) and (3.21), we get
Since \(s_{n_{j}}\) ⇀p and due to (3.22), it follows that \(u_{n_{j}}^{i} \rightharpoonup p\) as \(j \rightarrow \infty \) for all \(i = 1,\dots,N\). By the definition of \(u^{i}_{n_{j}}\), we have
So, there exist
This implies that
Combining with
we have
Taking \(j \rightarrow \infty \), by the jointly weak continuity of \(\psi _{i}\), (3.17) and (3.21), we obtain that
Hence
which implies that \(p \in {EP} (C,\psi _{i})\) for all \(i=1,\dots,N\).
Case II. Suppose \(\lim_{n \rightarrow \infty } \sigma _{n}^{i}=0\) for all \(i = 1,2,\dots,N\).
Then from the boundedness of \(\{u_{n}^{i}\}\), there exists \(\{u_{n_{k}}^{i}\} \subset \{u_{n}^{i}\}\) such that \(u_{n_{k}}^{i} \rightharpoonup \overline{u}^{i}\) as \(k \rightarrow \infty \). Replacing y by \(t_{n_{k}}\) in (3.23), we have
Moreover, by the Armijo line search rule (3.1), for \(m^{i}_{n_{k}1}\), there exists \(w^{i}_{n_{k},m^{i}_{n_{k}1}} \in \partial _{2}\psi _{i} (v^{i}_{n_{k},m^{i}_{n_{k}1}} , v^{i}_{n_{k},m^{i}_{n_{k}1}})\) such that
By the convexity of \(\psi _{i}(v^{i}_{n_{k},m^{i}_{n_{k}1}}, \cdot )\) and due to (3.25), for
we have
From (3.24) and (3.26), we obtain
By (3.1), since \(v^{i}_{n_{k},m^{i}_{n_{k}1}} = (1  \sigma ^{m^{i}_{n_{k}1}})t_{n_{k}} + \sigma ^{m^{i}_{n_{k}1}}u^{i}_{n_{k}}, \sigma _{m^{i}_{n_{k}1}} \rightarrow 0\), and \(t_{n_{k}}\) converges weakly to \(p, u^{i}_{n_{k}}\) converges weakly to \(\overline{u}^{i}\) for all \(i = 1,2,\dots,N\), this implies that \(v^{i}_{n_{k},m^{i}_{n_{k}1}} \rightharpoonup p\) as \(k \rightarrow \infty \). Since \(\{\ u^{i}_{n_{k}}  t_{n_{k}} \^{2}\}\) is bounded, without loss of generality, we may assume that \(\lim_{k \to \infty } \ u^{i}_{n_{k}}  t_{n_{k}}\^{2}\) exists for all \(i=1,\dots,N\). Hence, we get in the limit (3.27) that
Therefore, \(\psi _{i}(p, \overline{u}^{i}) = 0\), \(\overline{u}^{i} = p, \forall i = 1,\dots,N\) and \(\lim_{k \rightarrow \infty }\ u^{i}_{n_{k}}  t_{n_{k}} \ ^{2} = 0\). By Case I, it is immediate that \(p\in EP (C, \psi _{i})\) for all \(i = 1,\dots,N\).
Step 4. We show that \(\{s_{n}\}\) converges weakly to a point \(p\in EP (C, \psi _{i})\). Now, let \(x^{\ast }\) and p be two accumulation points of \(\{s_{n}\}\). Then there exists \(\{s_{n_{j}}\} \subset \{s_{n}\}\) such that \(s_{n_{j}}\rightharpoonup p\) and \(\{s_{n_{k}}\} \subset \{s_{n}\}\) such that \(s_{n_{k}}\rightharpoonup x^{\ast }\). Using similar arguments as in Step 2 above, we can show that \(x^{\ast }\), \(p\in \bigcap^{\infty }_{n=0}C^{i}_{n}\). Let \(\lim_{n \rightarrow \infty }\ s_{n}  x^{\ast } \^{2} = \alpha \). Then
Therefore, \(\ p  x^{\ast } \= 0\), and so \(\{s_{n}\}\) converges weakly to p. This completes the proof. □
Application to image restoration problems
The image restoration problem can be modeled by the linear equation system which is equivalent to a matrix equation of the form
where \(s\in \mathbb{R}^{n\times 1}\) is the original image, \(b\in \mathbb{R}^{n\times 1}\) is the observed image, \(\upsilon \in \mathbb{R}^{n\times 1}\) is additive noise, and \(A\in \mathbb{R}^{n \times n}\) is the blurring operation. In order to solve problem (4.1), we aim to approximate the original image, vector s, by minimizing the additive noise, which is known as the following least squares (LS) problem:
where \(\\cdot \\) is the \(\ell _{2}\)norm defined by \(\s\_{2} = \sqrt{\sum_{i=1}^{n}s_{i}^{2}}\). The solution of (4.2) can be estimated by many wellknown iteration methods.
Blur is an unsharp image area caused by camera or subject movement, inaccurate focusing, or the use of an aperture that gives a shallow depth of field. The blur effects are filters that smooth transitions and decrease contrast by averaging the pixels next to hard edges of defined lines and areas where there is significant color transition. In a digital image there are many types of blur effects, i.e., Gaussian blur, out of focus blur, and motion blur. In image restoration problems, the goal is to deblur an image without knowing the blurring operator. So, we define this goal to the following problem:
where s is the original true image, \(A_{i}\) is the blurring matrix, \(b_{i}\) is the blurred image obtained by using the blurring matrix \(A_{i}\) for all \(i = 1, 2, \dots, N\). The set of common solutions of the problem (4.3) is denoted by Ω, which is nonempty. We can apply the proposed algorithm (Algorithm 3.1) to solve the problem (4.3) by setting \(\psi _{i}(s,t)=\langle A^{T}_{i}(A_{i}sb_{i}),ts\rangle \) for all \(s,t\in \mathbb{R}^{n\times 1}\). Both theoretical and experimental results demonstrate the convergence properties of the proposed algorithm in this section. However, for showing the effectiveness of the proposed algorithm, the PVTSE [29] and MHPSE [17] algorithms are also applied to compare.
The Cauchy error of PVTSE, MHPSE, and the proposed algorithm is defined as \(\s_{n}s_{n1}\_{\infty } < 10^{8}\). The performance of the compared algorithms at \(s_{n}\) on the image restoring process is measured quantitatively by the means of the peak signaltonoise ratio (PSNR), which is defined by
where \(\operatorname{MSE} = \s_{n}s\_{2}^{2}\).
Next, we will present the restoration of images that have been corrupted by the following blur types:
 Type I.:

Gaussian blur of filter size \(9\times 9\) with standard deviation \(\sigma = 4\) (the original image has been degraded by the blurring matrix \(A_{1}\)).
 Type II.:

Out of focus blur (disk) with radius \(r = 6\) (the original image has been degraded by the blurring matrix \(A_{2} \)).
 Type III.:

Motion blur specified with motion length of 21 pixels (len =21) and motion orientation 11^{∘} (\(\theta = 11\)) (the original image has been degraded by the blurring matrix \(A_{3} \)).
The RGB format for the color image shown in Fig. 1 is used to demonstrate the effectiveness and practicality of our algorithm compared with PVTSE and MHPSE algorithms.
The three different types of the original RGB image degraded by the blurring matrices \(A_{1}, A_{2}\), and \(A_{3}\) are shown in Fig. 2.
We choose \(\eta =0.1\) and \(\rho =0.5\), and \(\theta _{n}\) can be chosen such that \(0\leq \theta _{n}\leq \bar{\theta }_{n}\) and
After that, we apply PVTSE, MHPSE, and the proposed algorithms to get the solution of the deblurring problem with one of the three blurring matrices \(A_{1}, A_{2}\), and \(A_{3}\). The results of the PVTSE, MHPSE, and the proposed algorithms are demonstrated on the following cases:
 Case I.:

Inputting \(A_{1}\) to PVTSE, MHPSE, and the proposed algorithms.
 Case II.:

Inputting \(A_{2}\) to PVTSE, MHPSE, and the proposed algorithms.
 Case III.:

Inputting \(A_{3}\) to PVTSE, MHPSE, and the proposed algorithms.
 Case IV.:

Inputting \(A_{1}\) and \(A_{2}\) to PVTSE, MHPSE, and the proposed algorithms.
 Case V.:

Inputting \(A_{1}\) and \(A_{3}\) to PVTSE, MHPSE, and the proposed algorithms.
 Case VI.:

Inputting \(A_{2}\) and \(A_{3}\) to PVTSE, MHPSE, and the proposed algorithms.
 Case VII.:

Inputting \(A_{1}\), \(A_{2}\), and \(A_{3}\) to the PVTSE, MHPSE, and the proposed algorithms.
Next, the common solutions of the deblurring problem for all cases under the three blurring matrices \(A_{1}, A_{2}\), and \(A_{3}\) by using PVTSE, MHPSE, and the proposed algorithms are presented. The restored images using these three algorithms after 500 iterations for all seven cases are shown in Figs. 3–5.
From Figs. 3–5, we see that the common solution of the deblurring problem with \((N > 1)\) improves the quality of the considered image. And when all blurring matrices are used in finding the common solution of the deblurring problem, we get the best quality of the recovered RGB image. Moreover, it has been found that the recovered RGB image obtained by the proposed algorithm has the highest PSNR compared with the PVTSE and MHPSE algorithms.
Next, the behavior of Cauchy error, the peak signaltonoise ratio (PSNR), and the number of line search steps per each iteration for recovering processes of the degraded RGB image by using the PVTSE, MHPSE, and the proposed algorithms with \(20{,}000\) iterations are demonstrated.
The quality improvements of the reconstructed RGB images based on PSNR being used are also illustrated for these three algorithms in Fig. 6. Their PSNR are also increased as the number of iterations is increased. The proposed method always gives a maximum value of PSNR when more than one blurring matrix is used in finding the common solution of the deblurring problem compared with PVTSE and MHPSE methods.
The Cauchy error plots show the validity and confirm the convergence of PVTSE, MHPSE, and the proposed methods. It is remarkable that the Cauchy error plot of MHPSE is decreased as the number of iterations is increased. There was an oscillation on the Cauchy error plot throughout the iterations of the PVTSE algorithm. And a gentle oscillation has occurred at the beginning of the iteration for the proposed algorithm. After that the Cauchy error plot of the proposed algorithm is also decreased as the number of iterations is increased. Moreover, it can be seen that the proposed algorithms always give the smallest number of line search steps on each iteration compared with PVTSE and MHPSE algorithms. Through these results, it is shown that the proposed algorithm produces excellent efficiency compared with PVTSE and MHPSE algorithms.
Conclusion
In this work, we use a parallel method combining inertial hybrid algorithm with Armijo line search for solving common nonmonotone equilibrium problems. A weak convergence theorem is established under some suitable conditions imposed on the bifunction \(\psi _{i}\). Moreover, we apply our algorithms for solving unconstrained image recovery problems and show superior efficiency of our proposed algorithm when the number of subproblems are increased; see Fig. 5. Finally, we compare our main algorithms with PVTSE [29] and MHPSE [17] algorithms. It is remarkable that our proposed algorithm has a better convergence rate; see Figs. 6–7.
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Acknowledgements
S. Suantai would like to thank Chiang Mai University, Thailand. W. Cholamjiak would like to thank Thailand Science Research and Innovation under the project IRN62W0007 and University of Phayao, Thailand. D. Yambangwai would like to thank the Thailand Science Research and Innovation Fund and the University of Phayao (Grant No. FF64UoE002).
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Chiang Mai University, Thailand.
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The authors equally conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
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Suantai, S., Yambangwai, D. & Cholamjiak, W. Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery. Adv Differ Equ 2021, 410 (2021). https://doi.org/10.1186/s13662021035659
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DOI: https://doi.org/10.1186/s13662021035659
MSC
 47H04
 54H25
 47H10
Keywords
 Armijo line search
 Nonmonotone equilibrium problems
 Bifunctions
 Hilbert space
 Inertial technique