An inertially constructed projection based hybrid algorithm for fixed point and split null point problems

1.   Introduction

Convex optimization problem (COP) is considered very important in the current literature as it covers a diverse range of problems with possible applications in signal processing, image processing and machine learning. As a consequence, the tremendous progress in studying the COP has led the emergence of a theory of convex optimization and a useful interface linking various branches of sciences.

Monotone operator theory is a prominent framework for various nonlinear problems and closely related with the theory of convex optimization. One of the fundamental themes in monotone operator theory is to compute zeros of the (maximal-) monotone operators. The importance of this concept stems from the fact that the sub-differential operator associated with a proper, convex and lower semicontinuous (PCLS) function is not only a maximal monotone operator but also solves the convex minimization problem. It is remarked that most of the practical phenomenon can be reformulated as zero point problem. This formalism includes variational inequalities, evolution equations, complementarity problems and inclusions ^[12].

The class of split feasibility problems (SFP) has an extraordinary utility and broad applicability in medical image reconstruction, signal processing and computerized tomography ^{[15,17,18,21]}. Some interesting and crucial results regarding the SFP with areas of feasible applications are established in ^[16,19,20]. The first prototype strategies for computing the optimal solution of the split common null point problem (SCNPP) can be found in ^[16]. Since then, different variants of these strategies have been proposed and analyzed for SCNPP and other instances of SFP ^[19,20,29].

Another useful formalism for modelling various nonlinear phenomenon is the fixed point problem (FPP) of the operator under consideration. Most of the problems in diverse areas such as mathematical economics, variational inequality theory, control theory and game theory can be reformulated in terms of FPP. It is remarked that various nonlinear fixed point operators play equivalent important role in COP. In 2015, Takahashi et al.^[31] investigated a unified formalism of null point problem and FPP in Hilbert spaces. Since then, FPP associated with different nonlinear operators are jointly investigated with (split common-) null point problem in this domain. It is therefore natural to investigate FPP associated with an infinite family of operators jointly with SCNPP in Hilbert spaces.

A variety of strategies combining iterative optimization algorithms and fixed point algorithms have been introduced and analyzed to construct an optimal solution of the SCNPP and FPP. Each strategy has certain shortcomings in terms of convergence characteristic and/or rate of convergence. The hybrid shrinking projection algorithm is a well-known strategy for the strong convergence characteristic whereas the inertial extrapolation technique, essentially due to ^[27] and see also ^{[1,2,3,4,5,6,7,8,9,10,11,32]}, enhances the rate of convergence of the algorithm under consideration.

Our main contributions in this ongoing fruitful research direction are as follows:

(1) We posit a framework to jointly investigate SCNPP and FPP associated with an infinite family of operators in Hilbert spaces. For the case of an infinite family of fixed point operators, we exploit the construction of an auxiliary operator defined in ^[28,34];

(2) We employ an algorithmic approach combining the hybrid shrinking projection algorithm with the inertial extrapolation technique to construct the common optimal solution of the problems as mentioned in item (1);

(3) We establish the strong convergence analysis of the proposed algorithm by employing the suitable constraints in accordance with the standard techniques and tools in the current literature;

(4) We posit different frameworks, as an application of the framework mentioned in item (1), for various instances of SFP in Hilbert spaces;

(5) Last but not least, we incorporate an appropriate numerical example for the demonstration of the framework as well as the applicability of the proposed algorithm for signal recovery problem.

2.   Preliminaries

Throughout the rest of the sections, the triplet $(\Xi_{1}, < \cdot, \cdot > , \|\cdot\|)$ indicates the real Hilbert space with the conventional notations of the inner product and the norm and $A_{1}\subseteq \Xi_{1}\times \Xi_{1}$ denotes a set-valued operator with the usual definitions of $dom (A_{1})$, $gra(A_{1})$ and $zer(A_{1})$. We use $\rightarrow$ (resp. $\rightharpoonup$) to represent the symbol for strong convergence (resp. weak convergence). The set of reals and natural numbers are symbolized as $\mathbb{R}$ and $\mathbb{N}$, respectively.

Recall from the celebrated monograph ^[12] that the set-valued operator $A_{1}$ is called monotone if $\langle r-s, u-v\rangle \geq 0, \; \forall(r, u), \; (s, v) \in gra(A_{1})$. In addition, $A_{1}$ is coined as maximal monotone operator provided that the graph of $A_{1}$ could not be properly contained in the graph of any other monotone operator. Let $m > 0$, then the resolvent operator of $A_{1}$ is defined as $J^{A_{1}}_{m} = (Id+m A_{1})^{-1},$ where $Id$ denotes the identity operator. In this connection, $J^{A_{1}}_{m}$ is well-defined, single-valued and firmly nonexpansive operator.

Let $T:H\rightarrow H$ be an operator defined on a nonempty subset $H$ of $\Xi_{1}$. We use $Fix(T) = \{p \in H\; |\; p = Tp\}$ to denote the set of fixed points of the operator $T$. The metric projection operator $\Pi_{H}:\Xi_{1} \rightarrow H$ associated with the nonempty closed convex subset $H$ of $\Xi_{1}$ is defined as $\Pi_{H}(u) = argmin_{v \in H}\Vert u-v\Vert$. It is well-known that the operator $\Pi_{H}$ is firmly nonexpansive and satisfies $\langle u-\Pi_{H}u, \Pi_{H}u-v\rangle \geq 0, \forall\; u \in \Xi_{1}, v \in H.$ Recall that the operator $T$ is known as $\Bbbk$-demicontractive ^[24] provided that $\Bbbk \in [0, 1)$ such that

The class of $\Bbbk$-demicontractive operators has been studied extensively in various instances of fixed point problems in Hilbert spaces. However, we are concerned with the fixed point problem of an infinite family of $\Bbbk$-demicontractive operators in Hilbert spaces via the following construction of auxiliary operator $S_{k}$:

where $0\leq \beta_{m}\leq 1$ and $T^{'}_{m} = \alpha x+(1+\alpha)T_{m}x$ for all $x \in H$ with $T_{m}$ being $\Bbbk$-demicontractive operator and $\alpha \in [\Bbbk, 1)$. It is well-known in the context of operator $S_{k}$ that each $T^{'}_{m}$ is nonexpansive and the limit $\lim_{k\rightarrow \infty}Q_{k, m}$ exists. Moreover

This implies that $Fix(S) = \bigcap^{\infty}_{k = 1}Fix(S_{k})$ ^[28,34].

We now finally introduce the formalism of the proposed problem.

Let $A_{1}\subseteq \Xi_{1}\times \Xi_{1}$ and $A_{2}\subseteq \Xi_{2}\times \Xi_{2}$ be maximal monotone operators such that the domain of $A_{1}$ is the subset of $H$ and let $J^{A_{1}}_{m}$ and $J^{A_{2}}_{m}$ be the resolvents of $A_{1}$ and $A_{2}$, respectively, for $m > 0$. Let $h: \Xi_{1} \rightarrow \Xi_{2}$ be a bounded linear operator and let $h^{\ast}$ be the adjoint operator of $h$. Let $S_{k}$ be the $S$-operator such that $\Gamma: = \Omega \cap Fix(S) \neq \emptyset$, where $\Omega : = \{\hat{p} \in A^{-1}_{1}(0):h \hat{p} \in A^{-1}_{2}(0)\}$ indicates the SCNPP ^[16]. We aim to find

The following crucial results are required in the sequel:

Lemma 2.1. ^[14] Let $U:H \rightarrow H$ be an operator defined on a nonempty closed convex subset $H$ of a real Hilbert space $\Xi_{1}$ and let $(p_{k})$ be a sequence in $H$. If $p_{k}\rightharpoonup p$ and if $(Id-U)p_{k}\rightarrow 0$, then $p \in Fix(U)$.

Lemma 2.2. Let $\mu, \nu \in \Xi_{1}$ and $\theta \in \mathbb{R}$ then

(i) $\Vert \mu+\nu\Vert^{2} \leq \Vert \mu \Vert^{2}+2 \langle \nu, \mu+\nu \rangle$;

(ii) $\Vert \mu-\nu\Vert^{2} \leq \Vert \mu \Vert^{2}-\Vert \nu \Vert^{2}-2 \langle \mu-\nu, \nu \rangle$;

(iii) $\|\theta \mu+(1-\theta)\nu\|^{2} = \theta\|\mu\|^{2}+(1-\theta)\|\nu\|^{2}-\theta(1-\theta)\|\mu-\nu\|^{2}.$

Lemma 2.3. ^[34] Let $H$ be a nonempty closed and convex subset of a real Hilbert space $\Xi_{1}$ and let $T: H \rightarrow H$ be a $\Bbbk$-demicontractive operator with $\Bbbk \in [0, 1)$. Let $\alpha \in [\Bbbk, 1)$ and set $T^{'} = (1-\alpha)Id+\alpha T$, then $T^{'}: H \rightarrow H$ is a nonexpansive operator such that $Fix(T^{'}) = Fix(T)$.

Lemma 2.4. ^[26] Let $H$ be a nonempty closed convex subset of a real Hilbert space $\Xi_{1}$. For every $p, q, t\in \Xi_{1}$ and $\gamma \in \mathbb{R}$, the set

is closed and convex.

Lemma 2.5. ^[34] Let $H$ be a nonempty closed and convex subset of a real Hilbert space $\Xi_{1}$ and let $(T^{'}_{m})$ be a sequence of nonexpansive operators such that $\bigcap^{\infty}_{k = 1}Fix(T^{'}_{k}) \neq \emptyset$ and $0\leq \beta_{m}\leq b < 1$. Then for a bounded subset $K$ of $H$, we have

3.   Algorithm and convergence analysis

We start with the architecture of the algorithm for the construction of an optimal solution of the problem (2.1).

Theorem 3.1. The sequence $(p_{k})$ generated by the Algorithm 1, under the following control conditions,

${\rm(C1)}$ $\sum^{\infty}_{k = 1}\theta_{k}\|p_{k}-p_{k-1}\| < \infty$;

${\rm(C2)}$ $0 < a \leq \liminf_{k \rightarrow \infty} \alpha_{k} \leq \limsup_{k \rightarrow \infty} \alpha_{k} \leq a^{\ast}$;

${\rm(C3)}$ $\liminf_{k \rightarrow \infty} \beta_{k} > 0$;

${\rm(C4)}$ $\liminf_{k \rightarrow \infty} m_{k} > 0$;

converges strongly to an element $p^{\ast} \in \Gamma$.

Proof. Step 1. The Algorithm 1 is well-defined.

Recall that the set $\Gamma$ is closed and convex whereas the closedness and the convexity of $H_{k+1}$, for each $k\geq 1,$ follows from Lemma 2.4. Let $p^{\ast}\in \Gamma$, then recalling the Algorithm 1, we have

Also from Algorithm 1 and Lemma 2.3, we have

Further, we obtain

Recalling the nonexpansivity of $J^{A_{1}}_{m_{k}}$, we obtain

Denote $\lambda_{k} = 2\delta\langle h \bar{w}_{k}-hp^{\ast}, \; (J^{A_{2}}_{m_{k}}-Id)h \bar{w}_{k} \rangle$ and recalling the firm nonexpansivity of $J^{A_{2}}_{m_{k}}$, we get

The estimate (3.3) implies after recalling the estimates (3.4) and (3.5)

The above estimate (3.6) indicates the inclusion $\Gamma \subset H_{k+1}.$ Summarising the above stated facts, this infers that the Algorithm 1 is well-defined.

Step 2. The limit $\lim_{k\rightarrow \infty }\Vert p_{k}-p_{1}\Vert$ exists.

Note that $\Vert p_{k+1}-p_{1}\Vert \leq \Vert p-p_{1}\Vert,$ for all $p\in H_{k+1}$ by employing the fact that $p_{k+1} = \Pi_{H_{k+1}}p_{1}.$ This infers that $\Vert p_{k+1}-p_{1}\Vert \leq \Vert p^{\ast}-p_{1}\Vert,$ for all $p^{\ast}\in \Gamma \subset H_{k+1}$ and consequently establishes the boundedness of $(\Vert p_{k}-p_{1}\Vert)$. Also from $p_{k} = \Pi_{H_{k}}p_{1}$, we have that

The above approximation infers that the sequence $(\Vert p_{k}-p_{1}\Vert)$ is non-decreasing, therefore, we have

Step 3. Prove that $q\in \Gamma.$

The following crucial estimates are required to prove the claim:

By employing the $\lim \sup$, and recalling the estimate (3.7), the above estimate implies that $\limsup_{k\rightarrow \infty }\left\Vert p_{k+1}-p_{k}\right\Vert ^{2} = 0.$ That is

As an easy inference of the approximates (3.8) and (3.9), we get

Since $p_{k+1}\in H_{k+1}$, we have

Recalling the estimate (3.8) and the condition (C1), the above estimate implies that

Recalling the estimates (3.8), (3.11) and the following triangular inequality:

we get

Consider the estimate (3.6) in the following variation:

Letting $k \rightarrow \infty$ and recalling the conditions (C1)–(C2) and the estimate (3.12), we have

The estimate (3.13) also implies that

Recalling the estimates (3.8), (3.14) and the following triangular inequality:

we get

Now recalling the estimates (3.4), (3.5) and Lemma 2.2, we have

Rearranging the estimate (3.16), we have

Now recalling the conditions (C1), (C3), the estimate (3.9) and $\delta \in (0, \frac{2}{\Vert h\Vert^{2}})$, the estimate (3.17) implies that

Recalling the estimates (3.4) and (3.5), we obtain

Denote $\xi_{k} = J^{A_{1}}_{m_{k}}(\bar{w}_{k}+\delta h^{\ast}(J^{A_{2}}_{m_{k}}-Id)h\bar{w}_{k})$ and recalling the estimate (3.19), it follows that

That is

As a consequence, we have

The estimate (3.22), gives that

Recalling the estimate (3.18) and the condition (C3), we have

Reasoning as above, by recalling the definition of $(e_{k})$, the condition (C1) and the estimate (3.24), we get

Note that the existence of a convergent subsequence $(p_{k_{j}})$ of $(p_{k})$ such that $p_{k_{j}} \rightharpoonup q \in \Xi_{1}$ as $j \rightarrow \infty$ follows from the boundedness of $(p_{k})$. This also infers that $\xi_{k_{j}}\rightharpoonup q$ and $\bar{w}_{k_{j}} \rightharpoonup q$ as $j \rightarrow \infty$. To establish the claim, we first prove that $q \in \Omega$.

Let $(u, v) \in gra(A_{1})$. Since $\xi_{k_{j}} = J^{A_{1}}_{m_{k_{j}}}(\bar{w}_{k_{j}}+\delta h^{\ast}(J^{A_{2}}_{m_{k_{j}}}-Id)h \bar{w}_{k_{j}})$, therefore, we have

This implies that

Recalling the monotonicity of $A_{1}$, we have

The above estimate infers that

Since $\xi_{k_{j}} \rightharpoonup q$, we obtain $\lim_{j \rightarrow \infty}\langle u-\xi_{k_{j}}, v\rangle = \langle u-q, v\rangle.$ Now utilizing the estimates (3.18), (3.24) and (3.26), we have $\langle u-q, v\rangle \geq 0.$ This implies that $0 \in A_{1}q$.

Recalling the facts that (i) $h$ is a bounded linear operator implies that $h\bar{w}_{k_{j}} \rightharpoonup hq$ as $j \rightarrow \infty$, (ii) $J^{A_{2}}_{m_{k}}$ is a nonexpansive operator with $Id - J^{A_{2}}_{m_{k}}$ being demiclosed at the origin (Lemma 2.1), we also obtain that $0 \in A_{2}(h q)$. Hence $q \in \Omega$.

We now show that $q \in Fix(S)$. Observe that

Recalling the estimate (3.13) and Lemma 2.5, the above estimate implies that $\lim_{k\rightarrow \infty}\Vert e_{k}-Se_{k}\Vert = 0$. This together with the fact that $e_{k_{j}} \rightharpoonup q$ implies, with the help of Lemma 2.1, that $q \in Fix(S) = \bigcap^{\infty}_{k = 1}Fix(S_{k})$.

Step 4. Prove that $p_{k}\rightarrow p^{\ast} = \Pi_{\Gamma }p_{1}.$

Let $p^{\ast} = \Pi_{\Gamma }p_{1}$. As $p_{k+1} = \Pi_{H_{k+1}}p_{1}$ and $\Gamma \subset H_{k+1}$, therefore, we have

Also

That is

This implies that $\lim_{k\rightarrow \infty }p_{k} = q = p^{\ast} = \Pi_{\Gamma }p_{1}$ and hence completes the proof.

If we take $A_{2} = 0$ and $\Xi_{1} = \Xi_{2}$, then the problem (2.1) reduces to find a point of the following problem:

Hence the following result is an easy consequence of the Theorem 3.1:

Corollary 3.1. Assume that $\Gamma \neq \emptyset$. Then the sequence $(p_{k})$

generated by (3.27), under the control conditions (C1)–(C4), converges strongly to an element $p^{\ast} = \Pi_{\Gamma}p_{1}$.

4.   Applications

In this section, we posit different frameworks, as an application of the framework established in Section 3.

4.1.   Split feasibility problems

The classical SFP, essentially due to Censor and Elfving ^[18], aims to find $\hat{p} \in \omega : = H \cap h^{-1}(G) = \{\bar{q} \in H: h\bar{q} \in G\}$, where $H\subset \Xi_{1}$ and $G\subset \Xi_{2}$ are nonempty, closed and convex subsets of $\Xi_{1}$ and $\Xi_{2}$, respectively. For the implementation of the Theorem 3.1, we recall the indicator operator of a nonempty, closed and convex subset $H$ of $\Xi_{1}$ as

It has been established that the subdifferential $\partial \Phi_{H}$ satisfies the maximal monotonicity provided that the operator $\Phi_{H}$ is proper, convex and lower semicontinuous. Also $\partial \Phi_{G} = \mathcal{N}(\mu, H)$, where $\mathcal{N}(\mu, H)$ is the normal cone of $H$ at $\mu$. Utilizing this fact, we conclude that the resolvent operator of $\partial \Phi_{H}$ is the metric projection operator of $\Xi_{1}$ onto $H$. We, therefore, arrive at the following variant of the problem (2.1):

Corollary 4.1. Assume that $\Gamma = \omega \cap Fix(S) \neq \emptyset$. Then the sequence $(p_{k})$

generated by (4.1), under the control conditions (C1)–(C4), converges strongly to an element $p^{\ast} = \Pi_{\Gamma}p_{1}$.

4.2.   Split equilibrium problems

The equilibrium problem from ^[13] aims to compute a point $p^{\ast} \in H$ such that

where $f:H \times H \rightarrow \mathbb{R}$ is a bifunction satisfying,

$\begin{array}{ll} (A1) & f(p^{\ast}, p^{\ast}) = 0 \text{ for all } p^{\ast}\in H; \\ (A2) & f \text{ is monotone, i.e., } f(p^{\ast}, q^{\ast})+f(q^{\ast}, p^{\ast})\leq 0 \text{ for all } p^{\ast}, q^{\ast} \in H; \\ (A3) & \text{ for each } p^{\ast}, q^{\ast}, t^{\ast}\in H, \limsup_{x\rightarrow 0}f(xt^{\ast}+(1-x)p^{\ast}, q^{\ast})\leq f(p^{\ast}, q^{\ast}); \\ (A4) & \text{for each } p^{\ast}\in H, q^{\ast}\mapsto h(p^{\ast}, q^{\ast}) \text{ is convex and lower semi-continuous.} \end{array}$

The set EP($f$) denotes the set of all solutions associated with the equilibrium problem (4.2). Recall the following auxiliary results for the equilibrium problem:

Lemma 4.1. ^[25] Let $H$ be a nonempty closed convex subset of a real Hilbert space $\Xi_{1}$ and let $f:H\times H\rightarrow \mathbb{R}$ be a bifunction satisfying (A1)–(A4). For $s > 0$ and $p^{\ast} \in \Xi_{1},$ there exists $t^{\ast} \in H$ such that

Moreover, define an operator $U_{s}^{f}:\Xi_{1}\rightarrow H$ by

Lemma 4.2. ^[30] Let $H$ be a nonempty closed convex subset of $\Xi_{1}$ and let $f: H \times H \rightarrow \mathbb{R}$ be a bifunction satisfying (A1)–(A4). Let $A_{f}:\Xi_{1}\rightarrow 2^{\Xi_{1}}$ be a multivalued operator defined as:

Recall that the operator $A_{f}$ is a maximal monotone operator with domain $D(A_{f}) \subset C$ and $EP(f) = A^{-1}_{f}0$. Moreover, $U^{f}_{s} = (Id+sA_{f})^{-1}$ for $s > 0$, i.e., $U^{f}_{s}$ is the resolvent of $A_{f}$. We, therefore, arrive at the following variant of the problem (2.1):

Corollary 4.2. Assume that $\Gamma = EP(f_{1})\cap h^{-1}(EP(f_{2}))\cap Fix(S) \neq \emptyset$. Then the sequence $(p_{k})$

generated by (4.3), under the control conditions (C1)-(C4), converges strongly to an element $p^{\ast} = \Pi_{\Gamma}p_{1}$.

4.3.   Split optimization problems

Let $g: \Xi_{1} \rightarrow (-\infty, \infty]$ be a PCLS function, then the set of minimizer associated with $g$ is defined as

Recall that the $\partial g$ of PCLS function $g$ is a maximal monotone operator and the corresponding resolvent operator of $\partial g$ is called the proximity operator (see^[22]). Hence argmin $g = (\partial g)^{-1}0$. We, therefore, arrive at the following variant of the problem (2.1).

Corollary 4.3. Assume that $\Gamma = \{x \in \arg\min g_{1}:h x \in \arg\min g_{2}\}\cap Fix(S) \neq \emptyset$. Then the sequence $(p_{k})$

generated by (4.4), under the control conditions (C1)–(C4), converges strongly to an element $p^{\ast} = \Pi_{\Gamma}p_{1}$.

5.   Numerical experiment and results

In this section, we present an example that characterizes the performance of our algorithm.

Example 5.1. Let $\Xi_{1} = \Xi_{2} = (\mathbb{R}, \langle \cdot, \cdot\rangle, |\cdot|)$ where $\langle p, q\rangle = pq$. Consider the operators $h, A_{1}, A_{2} : \mathbb{R}\rightarrow \mathbb{R}$ are defined as $h(p) = 3p$, $A_{1}p = 2p$ and $A_{2}p = 3p$, respectively. It is evident from the definition that $A_{1}$, $A_{2}$ are maximal monotone operators such that $\Omega : = \{p^{\ast} \in A^{-1}_{1}0:h p^{\ast} \in A^{-1}_{2}0\} = 0$ and $h$ is a bounded linear operator on $\mathbb{R}$ with the adjoint operator $h^{\ast}$ such that $\|h\| = \|h^{\ast}\| = 3$. Let the sequence of operators $S_{k}: \mathbb{R} \rightarrow \mathbb{R}$ be defined by

Then $S_{k}$ is an infinite family of $\frac{1-k^{2}}{(1+k)^{2}}$-demicontractive operators with $\bigcap^{\infty}_{k = 1}Fix(S_{k}) = \{0\}$. Hence $\Gamma = \Omega \cap \bigcap^{\infty}_{k = 1}Fix(S_{k}) = 0$. We use the following initialization parameters for the computation of the Algorithm 1: $\theta = 0.5$, $\alpha_{k} = \frac{1}{100k+1}$, $\beta_{k} = \frac{k}{100k+1}$, $\delta = \frac{1}{8}$, $L = 3$ and $m$ = 0.02. Also observe that

Let $Error = E_{k} = \Vert p_{k} - p_{k-1}\Vert < 10^{-5}$ denote the stopping criteria. The performance of the Algorithm 1 (i.e., Algo.1, $\theta_{k} \neq 0$) is compared with the non-inertial variant of the Algorithm 1 (i.e., Algo.1, $\theta_{k} = 0$) and Algo. 3.1^[16]. For different choices of the initial inputs $p_{0}$ and $p_{1}$, the values of Algo.1 are summarized in the following table:

Choice A. Choose $x_{0} = (5)$, $x_{1} = (2)$,

Choice B. Choose $x_{0} = (4.2)$, $x_{1} = (1.5)$,

Choice C. Choose $x_{0} = (-7)$, $x_{1} = (4)$.

The error plotting $E_{k}$ against Algorithm 1 and Algorithm 3.1^[16], for each choices in Table 1, has shown in Figure 1.

It is evident from Figure 1 that Algorithm 1 outperforms its noninertial variant and Algorithm 3.1 ^[16] with respect to the computation of error, CPU time and the number of iterations.

5.1.   Signal processing

The mathematical model of the signal recovery problem in an under-determined linear equation system is expressed as follows:

where $\nu \in \mathbb{R}^{D}$ denotes the original unknown signal to be recovered, $\zeta \in \mathbb{R}^{P}$ denotes the observed signal distort via the bounded linear matrix operator $h:\mathbb{R}^{D} \rightarrow \mathbb{R}^{P}, (P < D)$ and the noise $\rho$. One can define a natural convex constrained optimization-theoretic formulation of (5.1) via the following well known LASSO problem ^[33]:

The set of solutions of the $\ell_{1}$-minimization problem (5.1) is equivalent to the set of solutions of (5.2) under certain control conditions on the matrix $h$ ^[15]. The $\ell_{1}$-norm based regularization problems are widely applicable in signal and image processing.

Set $\Gamma = H\cap h^{-1}(G) \neq \emptyset$ with $H = \{\nu\; \vert \; \Vert \nu\Vert_{1} \leq c\}$ and $G = \{\zeta\}$. The experiment is conducted under the matrix $h^{D \times P}$ whose elements are generated from independently distributed normal distributions having 0 as mean and 1 as variance. The sparse vector $\nu$, having $t = spikes$ nonzero elements, is generated via uniform distribution in $[-2, 2]$. The following iterative regularization method, often known as the Richardson method (or the Landweber method) ^[23], is generally employed to solve the problem (5.2):

where $\eta$, the step size, is assumed to be constant. The algorithm (5.3) converges for $0 < \eta < \frac{2}{\epsilon^{2}_{\max}}$, where $\epsilon_{\max}$ is the maximum singular value of $h$. The initial points $\nu_{0}, \nu_{1}$ are chosen randomly. We use the mean squared error indicator to examine the performance of the algorithm for image restoration, i.e., $E_{k} = \frac{1}{N}\Vert \nu_{k}-\nu^{\ast} \Vert < 10^{-4},$ where $\nu^{\ast}$ is the approximation of the signal $\nu$. The computation of the observed signal $\zeta$ is carried out by employing the Gaussian noise associated with the signal-to-noise ratio (SNR = 40). Also set $m_{k} = \frac{1.85}{\Vert h\Vert^{2}}$, $\alpha_{k} = \frac{k}{100k+1}$, $\beta_{k} = \frac{1}{15k+1}$, $\delta = 0.04$, $c = t-0.002$, $\mu = 0$ and $\theta_{k}$ = 0.5.

Performance Test 1: Fix $D = 512$, $P = 256$ and $spikes = 15$.

Performance Test 2: Fix $D = 1024$, $P = 512$ and $spikes = 35$.

It is clear from the Figures 2 and 3 that the Algorithm 1 outperforms its variants and Algorithm 3.1 ^[16] for the signal recovery problem as well as exhibits fast convergence characteristic with regards to the error and number iterations.

6.   Conclusions

In this paper, we have posited a framework for the investigation of the SCNPP and the FPP associated with an infinite family of $\Bbbk$-demicontractive operators in Hilbert spaces. The common optimal solution of the problem is then constructed via an inertial hybrid projection algorithm under the suitable set of constraints. We have incorporated an appropriate numerical example for the demonstration of the framework as well as for the applicability of our algorithm. We found that our algorithm outperforms its variants and Algorithm 3.1^[16]. We have also discussed various instances of the proposed formalism and can pave a way for an important future research direction in the theories of SCNPP and FPP.

Acknowledgments

The authors Y. Arfat and P. Kumam acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005. The author Y. Arfat was supported by the Petchra Pra Jom Klao Ph.D Research Scholarship from King Mongkut's University of Technology Thonburi, Thailand (Grant No.16/2562). Finally, the corresponding author Supak Phiangsungnoen acknowledge the financial support provided by institute of Research and Development, Rajamangala University of Technology Rattanakosin (Fundamental Fund Grant No.FRB6620/25).

Availability of data and material

Data sharing not applicable to this article as no data-sets were generated or analysed during the current study.

Conflict of interest

The authors declare no conflict of interest.