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A new fixedtime stability of neural network to solve split convex feasibility problems
Journal of Inequalities and Applications volumeÂ 2023, ArticleÂ number:Â 138 (2023)
Abstract
In this paper, we propose a novel neural network that achieves stability within the fixed time (NFxNN) based on projection to solve the split convex feasibility problems. Under the bounded linear regularity assumption, the NFxNN admits a solution of the split convex feasibility problem. We introduce the relationships between NFxNN and the corresponding neural networks. Additionally, we also prove the fixedtime stability of the NFxNN. The convergence time of the NFxNN is independent of the initial states. The effectiveness and superiority of the NFxNN are also demonstrated by numerical experiments compared with the other methods.
1 Introduction
Since the split convex feasibility problem (SCFP) was introduced in [1] in 1994, the SCFP was widely used in retrieval problems [1], image restoration [2], signal processing [3], IMRT [4], systems biology [5]. Many numerical algorithms have been proposed to solve the split convex feasibility problem (see, e.g., [5â€“11]). The SCFP is as follows:
where \(C \subset \mathbb{R}^{n}\) and \(Q \subset \mathbb{R}^{m}\) are two nonempty, closed and convex sets, and \(A:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\) is a linear mapping. Among them, some classical and semialternating CQ algorithms were proposed to solve the split convex feasible problems in [3, 7]. In fact, the CQ algorithm is a projected gradient method, and its constrained minimization problem is as follows:
As we know, projection method only is related to the orthogonal projection that is easy to compute, and it does not require matrix inversion. The CQ algorithm is an efficient and popular method to solve SCFP. The split convex feasibility problem can be converted into the CQ algorithm with the following iterative format:
where \(P_{C}\), \(P_{Q}\) are the orthogonal projections onto C, Q, respectively, Î» is suitable positive constant, \(A^{T}\) is the transpose of A, and I is the identical operator,
In addition, a wide variety of continuoustime algorithms attracted increasing attention in areas such as optimization, artificial intelligence, image restoration, and optimal control (see, e.g., [12â€“15]). The dynamic system method was first proposed in [12] to solve the linear programming problem of the electronic analog computer. Subsequently, a recurrent neural network was presented to deal with the nonlinear projection formulation in [13]. To further solve the optimization problem, a new continuous dynamical system of a gradientprojection type was proposed and studied in [14]. Some neural network models were introduced to solve projection formulation in [15]. The continuation of the split convex feasibility problem has recently been studied. AÂ projection dynamical system was proposed to solve the split equality problem in [16]. Continuous CQ algorithms were proposed, and an exponential convergence was obtained under the bounded linear regularity property in [17]. Although neural networks and their variants have been widely adopted, discrete algorithms generally focus on the analysis of convergence and complexity, and continuous algorithms mainly focus on the convergence rate, and theoretical research mainly deals with the asymptotic or exponential stability (see, e.g., [17â€“21]). Among the existing works on the CQ algorithm and variations for the SCFP, the convergence and convergence rate of algorithms have received great attention (see, e.g., [5, 22]), but a few papers deal with convergence time.
In practical production, more attention is paid to the convergence time of the method. As discussed in [23], the design of finitetime convergence based on asymptotic or exponential stability was of more practical significance than the classical asymptotic and exponential stability. Nevertheless, the finitetime terminating neural network method is highly dependent on the initial conditions. To overcome this drawback, the notion of convergence within a fixed time was introduced in [24]. Recently, some neural networks, which achieve convergence within a finite time or fixed time, were proposed to solve some mathematical models, such as \(\ell _{1}\)minization problem, absolute value equation, and mixed variational inequalities (see, e.g., [25â€“31]). First, a novel proximal dynamical system, which achieves the fixedtime stability, was presented to deal with mixed variational inequality problems in [26]. For different problems, a new proximal neurodynamic network that is fixedtime converging (FXPNN) was proposed to solve equilibrium problems (EPs) in [28]. Recently, a dynamical system was released to deal with the split feasibility problem within the fixed time in [30]. To further decrease the convergence time of the dynamical system, a novel dynamical system that can update automatically, dynamical stepsize was established to deal with pseudomonotone mixed variational inequalities in [31]. As far as we know, few papers exist for solving split convex feasible problems in fixedtime stability. We want to combine continuous algorithms with convergence time to make the algorithm more intuitive and efficient.
Motivated by the above works, we investigate a novel fixedtime stability neural network (NFxNN) to deal with the SCFP. Further, the fixedtime stability of the NFxNN is obtained under the bounded linear regularity assumption. Under mild conditions, we prove that the equilibrium point of the NFxNN globally converges to a solution of the SCFP. Besides, the convergence time of the NFxNN is uniformly bounded and is independent of any initial condition. Numerical examples present the efficiency and competitiveness of the NFxNN compared with the neural network systems.
Compared with the existing related work, the contributions of this paper can be summarized as follows:

(i)
To solve the SCFP, we propose a new fixedtime neural network model. The stability of the neural network is proved. It is proved that the NFxNN converges to the solution within a fixed time. The convergence time is uniformly bounded and is independent of the initial condition.

(ii)
The convergence results are compared with the relative methods (such as [17, 30]). We provide an upper bound formula for the convergence time independent of the initial condition. Compared with other methods, the NFxNN is faster to solve the SCFP. Examples in the paper show that the convergence time of the NFxNN is slightly faster than the fixedtime system in [30] and is faster than the neural network in [17].

(iii)
The relations between the parameters of the NFxNN and the fixedtime stability are discussed through the combination of theory and simulation. In this paper, the influence of parameters Î³, \(\alpha _{1} \), \(\alpha _{2} \) on the NFxNN is presented. This provides a reference for us to choose the appropriate parameters when solving the SCFP with this model.
The rest of this article is structured as follows: Sect.Â 2 reviews useful notions and definitions. AÂ new fixedtime stability neural network model and the consistent discretization of the NFxNN are presented, and its global convergence is proved in Sect.Â 3. Numerical examples also confirm the effectiveness of the method in Sect.Â 4. SectionÂ 5 gives a conclusion.
2 Preliminaries
In this paper, let \(\mathbb{R}^{n}\) be the ndimensional Euclidean space with the inner products \(\langle \cdot ,\cdot \rangle \) and the Euclidean norm \(\Vert \cdot \Vert \). Î˜ denotes the solution set of the split convex feasibility problem. For the sets \(U\subseteq \mathbb{R}^{n}\) and \(V \subseteq \mathbb{R}^{m} \), the family of ktimes continuously differentiable functions from U to V is denoted by \(C^{k}(U,V)\), i.e., \(C^{k}(U,V):=\{f : f: U\to V \text{ is }k\text{times continuously differentiable mapping}\} \). The supremum of a set S, denoted as \(\sup (S)\), is the smallest real number M such that M is greater than or equal to every element inÂ S.
2.1 Definitions and notions
We first review some common definitions and facts.
Definition 1
[32] Let \(F:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n} \) be a mapping. F is called

(i)
Î¼Lipschitz continuous on \(\mathbb{R}^{n} \) if there is a constant \(\mu >0 \) that satisfies
$$\begin{aligned} \mu \Vert xy \Vert \geq \bigl\Vert F(x)F(y) \bigr\Vert ,\quad \forall x,y\in \mathbb{R}^{n}, \end{aligned}$$and F is said to be nonexpansive if it is 1Lipschitz continuous.

(ii)
Î² strongly monotone on \(\mathbb{R}^{n} \) if there is a constant \(\beta >0 \) to satisfy
$$\begin{aligned} \bigl\langle F(x)F(y), xy\bigr\rangle \geq \beta \bigl\Vert F(x)F(y) \bigr\Vert ^{2},\quad \forall x,y\in \mathbb{R}^{n} . \end{aligned}$$F is said to be firmly nonexpansive if it is 1inverse strongly monotone.

(iii)
Ï±averaged mapping if it is reformulated into
$$\begin{aligned} F(x)=(1\varrho )I+\varrho T, \end{aligned}$$where \(\varrho \in (0,1)\), I is the identity mapping, and \(T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n} \) denotes a nonexpansive mapping.

(iv)
Firmly nonexpansive iff
$$\begin{aligned} \Vert xy \Vert ^{2} \bigl\Vert (IF)x(IF)y \bigr\Vert ^{2}\geq \bigl\Vert F(x)F(y) \bigr\Vert ^{2},\quad \forall x,y\in \mathbb{R}^{n} . \end{aligned}$$
Property 1
We give the nonexpansive projection mapping:

(i)
Both \(P_{C}\) and \(IP_{C}\) are firmly nonexpansive, iff
$$\begin{aligned} \bigl\langle xP_{C}(x),yP_{C}(x)\bigr\rangle \leq 0, \quad \forall x\in \mathbb{R}^{n}, y\in C, \end{aligned}$$and
$$\begin{aligned} \bigl\langle P_{C}(x)P_{C}(y),xy\bigr\rangle \geq \bigl\Vert P_{C}(x)P_{C}(y) \bigr\Vert ^{2}, \quad \forall x,y\in \mathbb{R}^{n} . \end{aligned}$$ 
(ii)
A mapping is said to be nonexpansive if it is averaged or firmly nonexpansive.
Definition 2
[17] Suppose that the solution set Î˜ of the SCFP is nonempty, the SCFP is called to satisfy the bounded linear regularity property if, for any \(r > 0\), such that \(\overline{\mathbb{B}(0,r)}\cap \Theta \neq \varnothing \), there is a \(\varpi _{r}>0\) to satisfy
where Î˜ is the solution set of the SCFP, and \(\operatorname{dis}(\cdot ,\cdot )\) is the distance function, \(\mathbb{B}(x,r)\) and \(\overline{\mathbb{B}(x,r)}\) denote the open and closed ball with center at x and radius r, respectively.
We consider the following differential equation:
where \(f:\mathbb{R}^{n}\to \mathbb{R}^{n} \) is a vector valued function, and \(f(\mathbf{0})=\mathbf{0} \). Suppose that the solution of differential equation (5) exists. Without loss of generality, we suppose that the origin is the unique equilibrium point of equation (5).
Definition 3
[33] The origin is called to be a finitetime stability equilibrium of equation (5) if equation (5) is Lyapunov stable and finitetime convergent, i.e., for any \(y(0)\in \mathcal{O}\setminus \{\mathbf{0}\}\), \(\lim_{t \rightarrow T} y(t)=\mathbf{0}\), where \(\mathcal{O}\) is some open neighborhood of the origin and \(T=T(y(0)) < \infty \). The origin is called to be a globally finitetime stability equilibrium of equation (5) whenever \(\mathcal{O}=\mathbb{R}^{n}\).
Definition 4
[24] The origin is called to be a fixedtime stability equilibrium of equation (5) if it is globally finitetime stability, and the setting time \(T(y(0)) \) is uniformly bounded, i.e., there exists \(\bar{T}<\infty \) such that \(\sup_{x(0)\in \mathbb{R}^{n}}T(y(0))\leqslant \bar{T} \).
Lemma 1
[33] Suppose that there is a positive definite function \(V \in C^{1}(D,\mathbb{R}) \) to satisfy
where \(D\subseteq \mathbb{R} ^{n} \) is a neighborhood of the origin, an open neighborhood \(\mathcal{V}\subseteq D \) of the origin, \(c>0 \) and \(\alpha \in (0,1)\) are constants. Then, the origin is a finitetime stability equilibrium point of equation (5) with the setting time \(T \leqslant \frac{V(y(0))^{1\alpha}}{c (1\alpha )} \).
Lemma 2
[24] Suppose that there exists a positive definite function \(V\in C^{1}(\mathbb{R}^{n},\mathbb{R})\) to satisfy
where \(m,n>0\), \(0< a<1\) and \(b >1 \). Then, equation (5) is a fixedtime stability with settling time \(T\leqslant \frac{1}{m(1a)}+\frac{1}{n(b1)}\).
2.2 Neural network approach
In this subsection, we present a projectionbased neural network approach to solve the SCFP (1). We give some decision methods for solving the SCFP (1).
It follows that \(x\in C\) is a solution of the SCFP (1) iff it is a solution of the following:
Based on CQ algorithm (3), the paper [17] proposed a continuous neural network (NN) to deal with the SCFP (1). For simplicity, let us use the following auxiliary functions:
where \(x\in \mathbb{R}^{n}\) and \(\gamma \in (0,\frac{2}{\Vert A\Vert ^{2}})\). The neural network (NN) method is as follows:
where the function U is defined by (8) for \(\gamma \in (0,\frac{2}{\Vert A\Vert ^{2}})\) and \(\lambda >0 \). As a convenience, we let \(\Psi (x)=xP_{C}\circ Ux\).
Lemma 3
[34] Assume that the solution set Î˜ of the SCFP is nonempty. The following statements are the same:

(i)
\(x^{*} \) solves the SCFP.

(ii)
\(P_{C}\circ Ux^{*}=x^{*}\), i.e., \(\Theta =\{x\vert \Psi (x)=0\}\).

(iii)
\(x^{*}\in C\) and \(Ux^{*}=x^{*}\).

(iv)
\(x^{*}\in C\) and \(Ax^{*}=P_{Q}Ax^{*}\).
2.3 Blanket assumptions
Assumption 1
The SCFP is satisfied with bound linear condition with a constantÂ Ï–.
Assumption 2
The parameter \(\gamma \in (0,\frac{2}{L})\) with \(L=\Vert A\Vert ^{2}\).
Assumption 3
If \(\varpi ^{2}L<1\), we take \(\gamma \in (0,\frac{1\sqrt{1\varpi ^{2}L}}{L})\) or \(\gamma \in (\frac{1+\sqrt{1\varpi ^{2}L}}{L},\frac{2}{L})\).
Assumption 4
There exists \(x\in \mathbb{R}^{n}\), such that \(\Psi (x)=0\).
Remark 1
AssumptionÂ 2 guarantees that the NFxNN (10) has a unique solution. Assumptions 1 and 3 guarantee that the NFxNN (10) achieves stability within the fixed time. AssumptionÂ 4 guarantees a consistent discretization of the NFxNN (10).
Lemma 4
[3] If AssumptionÂ 2holds, then \(U:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is an averaged mapping.
Lemma 5
AssumptionÂ 2holds. Then,
Proof
It is proved in [17, TheoremÂ 1, p.Â 2993].â€ƒâ–¡
3 Fixedtime stability of the NFxNN and its convergence time estimation
A novel neural network is introduced to deal with the split convex feasibility problem. The fixedtime stability of the proposed neural network is obtained, and the upper bound forum of its convergence time is uniformly bounded. Moreover, it is irrelevant to the initial states.
3.1 A fixedtime converging neural network to deal with the SCFP
Inspired by [17, 31], we now present a new fixedtime converging neural network (NFxNN) to solve the SCFP (1):
where \(\lambda >0\), \(\gamma \in (0,\frac{2}{\Vert A\Vert ^{2}})\), U is defined by (8), and
Where \(\rho _{1},\rho _{2},\rho _{3}>0\), \(\alpha _{1}\in (0,1) \), \(\alpha _{2}>1\) are tunable parameters, \(\Psi (x)=xP_{C}\circ Ux\), and \(\text{E}(\Psi )=\{x \in \mathbb{R}^{n} \vert \Psi (x)=\mathbf{0}\} \) is the equilibrium points set ofÂ Î¨.
Carefully observe NFxNN (10), which can also be expressed as:
where \(\omega (x)=\frac{\Psi (x)}{\Vert \Psi (x)\Vert }= \frac{xP_{C}\circ Ux}{\Vert \Psi (x)\Vert }\). From the perspective of algorithm design, \(\omega (x)\) can be deemed as a search direction, and \(\alpha _{t}:=\rho _{1} (\Vert \Psi (x)\Vert ^{\alpha _{1}}(1+ \frac{\rho _{2}}{\rho _{1}}\Vert \Psi (x)\Vert ^{\alpha _{2}\alpha _{1}})+ \frac{\rho _{3}}{\rho _{1}} )\) can be deemed as stepsize at the current iterate \(x = x(t)\) in NFxNN (10). The stepsize \(\alpha _{t}\) can be selfupdate with respect to t whenever \(\rho _{1}\), \(\rho _{2}\), \(\rho _{3}\) and \(\alpha _{1}\), \(\alpha _{2}\) are given. If \(\rho _{3}=1\), the NFxNN (10) is reduced to
If \(\rho _{1}=\rho _{2}=0\) in the NFxNN (10), the NFxNN (10) is reduced to neural network (NN) in [17]
If \(\rho _{2}=\rho _{3}=0\), the NFxNN (10) is reduced to the following finitetime converging neural network (FiNN):
If \(\rho _{3}=0\) in, specially the tunable parameter \(\alpha =1\) in [30], the NFxNN (10) degenerates into the fixedtime dynamical system (FxDS) in [30].
Besides, the stepsize \(\alpha _{t}\) in NFxNN (10) when \(\rho _{3}>0\) is larger than that when \(\rho _{3}=0\). Generally, the larger stepsize can speed up the convergence rate of the numerical algorithms. It will be showed in Sect.Â 5. So, the NFxNN (10) is distinct of the fixedtime dynamical system in [30].
3.2 Convergence analysis of the NFxNN with the fixed time
We establish the globally convergence of the NFxNN. Futhermore, its convergence time estimation is obtained under the bounded linear regularity assumption and the Lipschitz continuity assumption. For proving the convergence of the NFxNN (10), we give the following auxiliary lemmas.
Lemma 6
\(x^{*}\in \Theta \) is a solution of the NN (9) iff it is a solution of the NFxNN (10).
Proof
If \(x^{*}\in \Theta \) is a solution of the NFxNN (10), then
it means that \(x^{*}\) is a solution of the equation (9).
On the other hand, if \(x^{*}\) is a solution of the equation (9), \(\Psi ( x^{*})=0 \), we get
which gets that \(x^{*}\) is a solution of the NFxNN (10). So far, we have completed the proof.â€ƒâ–¡
Remark 2
The significant idea of the projection neural approach is to establish the equivalence between optimization problems and Brouwerâ€™s fixedpoint problems. The SCFP (1) is equivalent to the solution of equation (9). Due to LemmaÂ 6, the NFxNN (10) can solve the SCFP (1).
Lemma 7
Let \(x(t)\) be the trajectory of system (10). Where \(x(t_{0})=x_{0}\in C\) and \(x^{*}\in \Theta \), which satisfies \(x(t)\rightarrow x^{*}\). Let \(x^{*}\in \Theta \) be a solution of the SCFP. Let Assumptions 1, 2, and 3hold. Then, \(\Vert P_{C}\circ Ux P_{\Theta }x\Vert \leq \tau \Vert xP_{\Theta }x\Vert \), where \(0<\tau =\sqrt{1\frac{\gamma (2\gamma L)}{\varpi ^{2}}}<1 \), and U is defined by (8).
Proof
Due to nonexpansive property of projection mapping, we get
Recalling the definition of U in (8), we have
We now estimate the last two terms of the above formula. Using (iii) of LemmaÂ 3 and the L definition of AssumptionÂ 2, we have that
From \((i)\) of PropertyÂ 1 and \((\mathit{iv})\) of LemmaÂ 3, we get
Substitute the above two inequalities into (15), we get
Sine \(x(t)\in C\) for all \(t\geq t_{0}\) was proved in [17, LemmaÂ 3.4, p.Â 2998], and the bounded linear regularity property holds, there is a \(\varpi >0\) and \(\bar{t}\geq t_{0}\) such that
So,
If \(\varpi ^{2}L\geq 1\), then \(0<\frac{\gamma (2\gamma L)}{\varpi ^{2}}\leq 1\). If \(\varpi ^{2}L<1\), we take \(\gamma \in (0,\frac{1\sqrt{1\varpi ^{2}L}}{L})\) or \(\gamma \in (\frac{1+\sqrt{1\varpi ^{2}L}}{L},\frac{2}{L})\), then \(0<\frac{\gamma (2\gamma L)}{\varpi ^{2}}<1\). We can get that
By LemmaÂ 3, we have \(P_{C}\circ U(P_{\Theta }x)=P_{\Theta }x\). It follows from (15) and (16) that
with \(\tau =\sqrt{1\frac{\gamma (2\gamma L)}{\varpi ^{2}}}\in (0,1)\). It completes the proof.â€ƒâ–¡
Theorem 1
Let \(x(t)\) be the trajectory of system (10). Where \(x(t_{0})=x_{0}\in C\) and \(x^{*}\in \Theta \) satisfy \(x(t)\rightarrow x^{*}\). Let \(x^{*}\in \Theta \) be a solution of the SCFP. Assumptions 1, 2, and 3hold. Then the solution of the SCFP is a globally fixedtime equilibrium point of the NFxNN, and the convergence time is given as
where \(x_{0}\) is an initial point of the NFxNN (10), and \(p_{1}=\frac{1+\alpha _{1}}{2}\), \(p_{2}=\frac{1+\alpha _{2}}{2}\), \(N_{1}=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1\tau}{(1+\tau )^{1\alpha _{1}}}\), \(N_{2}=2^{\frac{1+\alpha _{2}}{2}}(1\tau )^{\alpha _{2}}(\rho _{2}+ \rho _{3}\frac{1}{\Vert \Psi (x)\Vert ^{\alpha _{2}}})\) for \(x\in \mathbb{R}^{n}\) and \(\tau =\sqrt{1\frac{\gamma (2\gamma L)}{\varpi ^{2}}}\in (0,1)\).
Proof
It follows from the global version of the PicardLindelÃ¶f theorem in [35, TheoremÂ 2.2, p.Â 38] that the NFxNN (10) can achieve a unique solution under the Assumptions 1, 2, and 3. Since \(x^{*} \) is a solution of the NFxNN (10), we deduce by the global version of the PicardLindelÃ¶f theorem and LemmaÂ 6 that \(x^{*}\in \mathbb{R}^{n}\) is a unique solution of the NFxNN (10). To simplify the proof, we derive from LemmaÂ 7 that
where the inequality can be obtained by LemmaÂ 7, i.e.,
The solution of the NFxNN (10) exists uniquely for any initial state according to [26, PropositionÂ 2]. Now, consider a candidate Lyapunov function:
The timederivation of the candidate Lyapunov function V along the solution of (10), we get
where the first inequality is established by inequality (17), and the second by LemmaÂ 7. Then,
where \(M_{1}=\rho _{1}\frac{1\tau}{(1+\tau )^{1\alpha _{1}}}\) and \(M_{2}=(1\tau )^{\alpha _{2}} (\rho _{2}+\rho _{3}\Vert \Psi (x)\Vert ^{ \alpha _{2}} )\).
It can be seen from (18) and (21) that
where \(p_{i}=\frac{1+\alpha _{i}}{2}\), \(i=(1,2)\), \(N_{i}=M_{i}2^{p_{i}}\), \(i=(1,2)\). And \(N_{1}=M_{1}2^{p_{1}}>0\), \(p_{1}=\frac{1+\alpha _{1}}{2}\in (0,0.5)\) since \(\alpha _{1}\in (0,1)\), \(N_{2}=M_{2}^{p_{2}}>0\), \(p_{2}=\frac{1+\alpha _{2}}{2}\in (1,\infty )\) since \(\alpha _{2}\in (1,\infty )\). It then follows from LemmaÂ 2, (ii) and (iv) of LemmaÂ 3, and the intersection of two convex sets is still convex, that the equilibrium point \(x^{*}\) of the NFxNN (10) is globally fixedtime stability, and the settling time satisfies
where \(x(0)\) is an initial condition of the NFxNN (10) and \(N_{1}=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1\tau}{(1+\tau )^{1\alpha _{1}}}\) and \(N_{2}=2^{\frac{1+\alpha _{2}}{2}}(1\tau )^{\alpha _{2}}(\rho _{2}+ \rho _{3}\frac{1}{\Vert \Psi (x)\Vert ^{\alpha _{2}}})\). Here, \(\Vert \Psi (x)\Vert ^{\alpha _{2}}>0\), for \(x\in \mathbb{R}^{n}\), and since \(\rho _{3}>0\), \(\rho _{3}\frac{1}{\Vert \Psi (x)\Vert ^{\alpha _{2}}}>0\). We can find that the NFxNN (10) reduces the upper bound of the convergence time, and the method is also independent of the initial state. The proof is completed.â€ƒâ–¡
Theorem 2
Let \(x(t)\) be the trajectory of the FiNN (13). Where \(x(t_{0})=x_{0}\in C\), and \(x^{*}\in \Theta \) satisfies \(x(t)\rightarrow x^{*}\). Assumptions 1, 2, and 3hold. For each \(\alpha _{1}\in (0,1) \), \(\rho _{1}>0\), the FiNN (13) is finitetime stability with the convergence time
where \(c=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1\tau}{(1+\tau )^{1\alpha _{1}}}>0\), \(\beta =\frac{1+\alpha _{1}}{2}\in (0.5,1)\), \(0\leq \tau =\sqrt{1\frac{\gamma (2\gamma L)}{\varpi ^{2}}}<1 \) and \(V(x(t))=\frac{1}{2}\operatorname{dist}^{2}(x(t),\Theta )\).
Proof
Follow the proof of TheoremÂ 1. Then, the timederivation of the candidate Lyapunov function V along the solution of (13), one has
where the first inequality is established by the CauchySchwarz inequality, and the second holds by LemmaÂ 7, \(\Psi (x)=P_{C}\circ Uxx\) and \(\psi (x)=P_{C}\circ Ux\). Using LemmaÂ 7, there exists \(0\leq \tau =\sqrt{1\frac{\gamma (2\gamma L)}{\varpi ^{2}}}<1 \) such that
Combining (23) with (24), we have
where \(c=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1\tau}{(1+\tau )^{1\alpha _{1}}}>0\), \(\beta =\frac{1+\alpha _{1}}{2}\in (0.5,1)\). Therefore, it follows from LemmaÂ 1 that for each \(\alpha _{1}\in (0,1)\), the FiNN (13) is globally finitetime stability with the convergence time
which means that \(x^{*} \) is a finitetime stability solution of the FiNN (13).â€ƒâ–¡
Remark 3
(i) TheoremÂ 1 presents the upper bound of the convergence time of the NFxNN (10), which is less, and the accuracy is higher. Besides, the the upper bound of the convergence time of the NFxNN (10) is less than that in [30].
(ii) We can see from Fig.Â 1 that the convergence time of the NFxNN (10) is about 0.01 second, and the convergence time of the FiNN (13) and the FxDS (14) is about 0.025 second. The convergence time of the NN (12) is about 0.5 second. It can thus be seen that the convergence time of the NFxNN (10) is less than that of the NN (12), the FiNN model (13), and the FxDS model (14), respectively. The convergence responses of the errors are different in the NN (12), FiNN (13), FxDS (14), and NFxNN (10). The NFxNN (10) can reach it quickly under the equal error condition, and NN (12) is much slower than them.
3.3 Consistent discretization of the NFxNN
Continuoustime neural networks provide an effective way to accelerate the design of schemes for solving the split convex feasibility problems. Typically, discretetime implementations use iterative methods to solve problems. We note that the fixedtime convergence of continuoustime neural networks may not hold in discrete cases. Uniform discretization refers to the discretization scheme that still has the convergence result of the continuoustime neural network under discrete conditions. In this section, we give the uniform discretization of the NFxNN (10). We next give the result of the forwardEuler discretization of the NFxNN (10):
where \(\eta >0\) is the timestep and \(\rho (\cdot )\) is given by (10), with \(\rho _{1},\rho _{2},\rho _{3}>0\), and \(x_{k}:=x(t)\) with \(t=\eta n\) and \(\dot{x}=\frac{dx}{dt}\).
Theorem 3
Let \(x^{*}\) be a trajectory of the SCFP. All assumptions of TheoremÂ 1hold. Then, for every \(x_{0}\in \mathbb{R}^{n}\), \(\varepsilon >0\) and \(\gamma \in (0,\frac{2}{L})\), satisfying Assumptions 2, 3, and 4, there exist \(\delta \in (0.5,1)\), \(N_{1}\), \(N_{2}\) and \(\eta ^{*}>0\) satisfied for any \(\eta \in (0,\eta ^{*}]\), the sequence \(x_{n}\) generated by (25) holds:
where \(n^{*}=\lceil \frac{\pi}{2\sqrt{N_{1}N_{2}}\eta \delta}\rceil \), \(\lceil x\rceil =[x]+1\). \([x]\) denotes the largest integer not exceeding x, and \(x_{n}\) is a solution of (25) staring from the point \(x_{0}\), and \(x^{*}\in \mathbb{R}^{n}\) is the solution of (10). \(\alpha _{1}\), \(\alpha _{2}\), \(N_{1}\), \(N_{2}\) is given by TheoremÂ 1.
Proof
Since \(P_{C}\circ Ux\) is Lipschitz continuous, \(\Psi (x)\) is continuous. The remainder of proof is like TheoremÂ 4 in [26]. So, it is omitted.â€ƒâ–¡
4 Numerical experiments
In this section, we aim to illustrate the analytical results with some simulations. We demonstrate the validity of the proposed approaches to deal with the split convex feasibility problem by several numerical examples. All programs are edited by Matlab 2018. All programs are executed on a personal computer. Unless otherwise stated, let \(\alpha =0.2\), \(\alpha _{1} =0.2\), \(\alpha _{2} =1.2\), \(\rho _{1} = \rho _{2} =100 \) and \(\gamma = \frac{1}{L} \).
Example 1
[6, 17] Consider the SCFP (1), where
All requirements in TheoremÂ 1 can be met. Numerical simulations of the NN system (12) in [17] achieve the equilibrium point \(x^{\ast}=[ 0.2488, 0.0831, 0.2969]^{T}\in \Theta \). Numerical simulations of the FiNN system (13) obtain the solution \(x^{\ast}=[ 0.2437, 0.0810, 0.2954]^{T}\in \Theta \). Numerical simulations of the FxDS system (14) converge to the point \(x^{\ast}=[ 0.2129, 0.0703, 0.2914]^{T}\in \Theta \). Numerical simulations of the NFxNN system (10) achieve the solution \(x^{\ast}=[ 0.2237, 0.0758, 0.2905]^{T}\in \Theta \). FigureÂ 1 presents the convergence trajectories of the NN (12) in [17] , the FiNN model (13), the NFxNN (10) for initial conditions \(x_{0}=[1,1,1]^{T}\). We can see from Fig.Â 1 that the convergence time of the NFxNN (10) is about 0.01 second, and the convergence time of the FiNN (13) and the FxDS (14) is about 0.025 second. The convergence time of the NN (12) is about 0.5 second. It can thus be seen that the convergence time of the NFxNN (10) is less than that of the NN (12), the FiNN model (13), and the FxDS model (14), respectively. The convergence responses of the errors are different in the NN (12), FiNN (13), FxDS (14), and NFxNN (10). The NFxNN (10) can reach it quickly under the equal error condition, and NN (12) is much slower than them.
Example 2
[17] We consider SCFP (1): the matrix \(A=(a_{ij})_{2\times 2}\) and \(a_{ij}\in (0,1)\) are generated randomly, and \(C=\{u\in \mathbb{R}^{2}\vert c(u)\leq 0\}\) and \(Q=\{v\in \mathbb{R}^{2}\vert q(v)\leq 0\}\) are the nonempty closed convex set with
for all \(u=(u_{1},u_{2})^{T}\in \mathbb{R}^{2}\) and \(v=(v_{1},v_{2})^{T}\in \mathbb{R}^{2}\). We take
generated randomly. All conditions of TheoremÂ 1 are satisfied. The trajectory \(x(t)\) of the NN in [17] achieves the point \(x^{\ast}=[ 0.0003956 , 0.0005248]^{T}\in \Theta \). Numerical simulations of the FiNN system (13) converge to the equilibrium point \(x^{\ast}=[ 0.00001986 , 0.00003241]^{T}\in \Theta \). Numerical simulations of the FxDS (14) obtain the solution \(x^{\ast}=[ 0.00001994 , 0.00002881]^{T}\in \Theta \). Numerical simulations of the NFxNN (10) converge to the point \(x^{\ast}=[ 0.00001994 , 0.00002881]^{T}\in \Theta \).
FigureÂ 2 demonstrates the convergence trajectories of the NN (12) in [17], the FiNN (13), the FxDS (14), and the NFxNN (10) for initial conditions \(x_{0} \). We can observe from Fig.Â 2 that the convergence time of the NFxNN (10) is about 0.05 second. The NFxNN (10) takes about 0.06 second, and the convergence time of the NN (12) is much more than 0.5 second. The FiNN (13) takes about 0.09 second. We can observe the difference from the convergence responses of the errors in Fig.Â 2 (d). It also presents that the convergence rate of the NFxNN (10) is faster than that of the NN (12) and slightly faster than that of the FiNN (13) and FxDS (14). So, the convergence time of the NFxNN (10) is less than that of the NN (12), FiNN (13), and FxDS (14). From the error image, we can also see that the NN (12) achieves less accuracy than the FiNN (13), FxDS (14), and NFxNN (10).
Example 3
[18] Consider linear equation problem (LEP), find \(x\in \mathbb{R}^{n}\) to satisfy
When \(C=\mathbb{R}^{n}\) and \(Q=\{b\}\), the LEP can transform into a split feasible problem. In this case, the algorithm degrades into the following form:
for \(x_{0}\) arbitrary. So, system (9) degrades into the following dynamical system:
We consider LEP with
By means of algebraic calculation, it is easy to verify that \(x^{*}=[9,1,6]^{T}\) is a unique solution. All numerical simulations achieve the unique solution \(x^{\ast}=[9,1,6]^{T}\). FigureÂ 3 displays that the trajectories of dynamical system (12), (13), (14), and (10) with initial point \(x_{0}=[3,5,2]^{T}\). And all globally converge to \(x^{*}\).
FigureÂ 3 presents the convergence trajectories of the NFxNN (10), FxDS (14), FiNN (13), and NN (12) in [17] for solving the split convex feasibility problem.
We can observe from Fig.Â 3 (a), (b), (c) the convergence times of the NFxNN (10), FxDS (14), FiNN (13), and NN (12). The NFxNN (10) takes about 0.1 second. The FxDS (14) and FiNN (13) are about 0.2 second, and the NN (12) is much more than 2 second. So, the convergence speed of the NFxNN (10) is faster than that of the FxDS (14), FiNN (13), and NN (12). The difference on the convergence error of the NFxNN (10), FxDS (14), FiNN (13), and NN (12) is also shown from the convergence errors in Fig.Â 3 (d).
Example 4
We use the continuoustime method to solve the problem of recovering noisy sparse signals from a limited number of samples. The problem was considered in [8]. Let \(x\in \mathbb{R}^{n}\) be a \(K\mathit{sparse}\) signal and \(K\ll n\). The sampling matrix \(B\in \mathbb{R}^{m\times n}\) is stimulated by the standard Gaussian distribution and vector \(b=Bx+e\) with e being an additional noisy. The main work is to recover the signal x from the dataÂ b. The recovering noisy sparse signals can be converted into the LASSO problem:
where \(t>0\) is a given constant. We take \(C=\{x:\Vert x\Vert _{1}\leq t\}\) and \(Q=\{b\}\), the above LASSO problem is converted into SCFP (1). So, we can deal with the problem by dynamical system (12), (13), (14), and (10). In the process of using the system to solve (such as (12), (13), (14) and (10)), we take the following set of parameters: the initial point \(x_{0}\in \mathbb{R}^{n}\) of system (12), (13), (14), (10) and the matrix \(B\in \mathbb{R}^{m\times n}\) are generated randomly. The true signal \(x\in \mathbb{R}^{n}\) with K nonzero elements is generated from uniform distribution in the interval [âˆ’2,2] and vector \(b=Ax\). Let \(m=n=100\), \(t=K=10\), \(\gamma =\frac{1}{\Vert A\Vert ^{2}}\) and \(\lambda =5\).
We compare the NFxNN (10) with NN (12) in [17], the FiNN (13), and FxDS (14). The convergence trajectories of the NFxNN (10), the NN (12) in [17], the FiNN (13), the FxDS (14) are shown in Fig.Â 4. We can observe from Fig.Â 4 (a), (b), (c), (d) the convergence time of the NFxNN (10), FxDS (14), FiNN (12), and NN (9). The NFxNN (10) is about 0.02 second. The FxDS (14) takes about 0.025 second. The FiNN (12) is about 0.09 second. The NN (9) is much more than 0.45 second. So, the convergence rate of the NFxNN (10) is quicker than that of the FxDS (14), FiNN (13), and NN (12). The difference in the convergence rates of the NFxNN (10), FxDS (14), FiNN (13), and NN (12) is also shown from the convergence responses of the errors in Fig.Â 3 (d). FigureÂ 4 means that the convergence rate of the NFxNN (10) is faster than that of NN (12) in [17] and the FxDS (14).
Taking ExampleÂ 4 as an example, we explore the influence of parameters on the NFxNN method. FigureÂ 5 illustrates the influence of parameters Î³, \(\alpha _{1} \), and \(\alpha _{2} \) on the NFxNN (10), i.e., it means that the convergence time of the NFxNN (10) is obtained much less as Î³ increase. When \(\alpha _{1}\) tends to 0, the NFxNN (10) can achieve the solution faster. The NFxNN (10) can converge to the solution faster while \(\alpha _{2}\) tends toÂ 1.
5 Conclusions
The NFxNN for solving the split convex feasibility problem is proposed by using the projection method. Under the bound linear regularity assumption, the NFxNN obtains a solution of the SCFP. At the same time, we derive the relationship between the method and the corresponding neural network. We obtain the results of the global fixedtime stability of the NFxNN, i.e., the convergence time of the NFxNN is independent of the initial conditions, and the convergence time is uniformly bounded. The results obtained are different from the relevant neural network methods (such as [17, 30]). Numerical examples also present the superiority and availability of the method.
Availability of data and materials
All data generated or analysed during this study are included in this published article.
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Acknowledgements
This research was partially supported by the Natural Science Foundation of Chongqing (No. cstc2021jcyjmsxmX0925), and the Education Committee Project Research Foundation of Chongqing (No. KJQN202201802). Jinlan Zheng and Rulan Gan are corresponding authors.
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JZ: Investigation, Methodology, WritingÂ  original draft. RG: Conceptualization, Software, review & editing. XJ: Methodology & editing. XO: Methodology. All authors read and approved the final manuscript.
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Zheng, J., Gan, R., Ju, X. et al. A new fixedtime stability of neural network to solve split convex feasibility problems. J Inequal Appl 2023, 138 (2023). https://doi.org/10.1186/s13660023030465
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DOI: https://doi.org/10.1186/s13660023030465