MSC. 47H06, 47H09, 47J05, 47J25.

Keywords. Nonexpansive mapping, minimization problem, inertial method, forward-backward splitting method, fixed point problem.

1 Introduction

Let \(H\) be a real Hilbert space with inner product \(\langle \cdot ,\cdot \rangle \), the induced norm \(\| \cdot \| \) and \(f,g:H \rightarrow \mathbb {R} \cup \{ +\infty \} \) be two proper, lower semi-continuous and convex functions in which \(f\) is Fréchet differentiable on an open set containing the domain of \(g.\) The Convex Minimization Problem (CMP) is formulated as follows:

We denote by \(\Upsilon \) the solution set of 1. The CMP 1 is a general form of the classical minimization problem which is given as:

The minimization problems 1, 2 and their other modifications are known to have notable applications in optimal control, signal processing, system identification, machine learning, and image analysis; see, e.g., [ 5 , 3 , 2 , 26 ] . It is well known that CMP 1 relates to the following fixed point equation:

where \(\beta \) is a positive real number and \(\operatorname {prox}_{g}\) is the proximal operator of \(g\) the Moreau-Yosida resolvent of \(g\) in Hilbert space is defined as follows:

where \(\operatorname {argmin}~ g:= \Big\{ \overline{x} \in H: g(\overline{x}) \leq g(x) ~ for~ all~ x \in H \Big\} \) .

In 2012, Lin and Takahashi [ 28 ] introduced the following forward-backward algorithm:

where \(F:H \rightarrow H\) is a contraction, \(\{ \alpha _n\} \subset (0,1), \{ \beta _n\} \subset (0, +\infty )\), \(\nabla f\) is Lipschitz continuous and g is convex and lower-semicontinuous function. They obtained a strong convergence result of algorithm 5 under the following mild conditions:

where \(L\) is the Lipschitz constant of \(\nabla f\). Also, Wang and Wang [ 39 ] proposed the following forward-backward splitting method: find \(x_0 \in H\) such that

where \(\{ \alpha _n\} \subset (0,1), \{ \mu _n\} \subset (0,2), \{ \gamma _n\} \subset (-2,1)\) and \(\alpha _n + \gamma _n + \mu _n=1\) and \(F:H \rightarrow H\) is a contraction. They proved that the sequence 6 converges strongly to a solution of \(\Upsilon .\)
Bello and Nghia [ 10 ] in 2016 investigated the forward-backward method using linesearch that eliminates the undesired Lipschitz assumption on the gradient of \(f.\) They proposed the following algorithm and established its weak convergence:

Very recently, Kunrada and Cholamjiak [ 27 ] proposed the forward-backward algorithm involving the viscosity approximation method and stepsize that does not require the Lipschitz continuity condition on the gradient as follows:

They proved the strong convergence theorem for algorithm 2 under some weakened assumptions on the stepsize.

We observe that the choice of stepsizes in algorithm 1 and algorithm 2 heavily depend on the linesearches which are known to slow down the rate of convergence in iterative algorithms (see [ 25 , 34 ] ).

The Split Feasibility Problem (SFP) was first introduced in [ 16 ] by Censor and Elfving. Let \(C\) and \(Q\) be nonempty closed convex subsets of real Hilbert spaces \(H_1\) and \(H_2\) respectively and \(A:H_1\to H_2\) be a bounded linear operator. The SFP is defined as follows:

The SFP arises in many fields in the real world, such as signal processing, medical image reconstruction, intensity modulated radiation therapy, sensor network, antenna design, immaterial science, computerized tomography, data denoising and data compression [ 7 , 12 , 11 , 15 , 17 ] . Several SFP variant for different optimization problems have been extensively studied [...]. Let \(C\) and \(Q\) be nonempty closed and convex subsets of real Hilbert spaces \(H_1\) and \(H_2,\) \(g:H_1\to \mathbb {R}\cup \{ +\infty \} \) and \(f:H_2\to \mathbb {R}\cup \{ +\infty \} \) be two proper and lower semi-continuous convex functions. Let \(A:H_1\to H_2\) be a bounded linear operator, then the Split Minimization Problem (SMP) is to find

and such that

Many researchers have employed different types of iterative algorithms to study SMP 7 and 8 in Hilbert and Banach spaces. For instance, Abass et al. [ 5 ] proposed a proximal type algorithm to solve SMP 7 and 8 in Hilbert spaces. They established the sequence generated from the their proposed algorithm strongly converges to the solution set of the SMP. Very recently, Abass et al. [ 3 ] introduced another proximal type algorithm to approximate solutions of systems of SMP and fixed point problems of nonexpansive mappings in Hilbert spaces. They showed that their algorithm converges to a common solution of the SMP and fixed points of the nonlinear mappings.

Constructing iterative schemes with a faster rate of convergence are usually of great interest. The inertial-type algorithm which was originated from the equation for an oscillator with damping and conservative restoring force has been an important tool employed in improving the performance of algorithms and has some nice convergence characteristics. In general, the main feature of the inertial-type algorithms is that we can use the previous iterates to construct the next one. Since the introduction of inertial-like algorithm, many authors have combined the inertial term \([\theta _n(x_n - x_{n-1})]\) together with different kinds of iterative algorithms being Mann, Kranoselski, Halpern, Viscosity, to mention few to approximate solutions of fixed point problems and optimization problems. Most authors were able to prove weak convergence results while few proved strong convergence results.

Polyak [ 33 ] was the first author to propose the heavy ball method, Alvarez and Attouch [ 6 ] employed this to the setting of a general maximal monotone operator using the Proximal Point Algorithm (PPA), which is called the inertial PPA, and is of the form:

They proved that if \(\{ r_n\} \) is non-decreasing and \(\{ \theta _n\} \subset [0,1)\) with

then the Algorithm 9 converges weakly to a zero of \(B\). More precisely, condition 12 is true for \(\theta _n {\lt} \frac{1}{3}\). Here \(\theta _n\) is an extrapolation factor. Also see [ 1 , 4 , 6 , 18 , 20 , 24 , 28 ] for results on inertial method.

We highlight our contributions in this paper as follows:

• Unlike the result of [ 10 ] which proved weak convergence, we proved a strong convergence theorem for the sequence generated by our algorithm. Note that in solving optimization problems, strong convergence algorithms are more desirable than the weak convergence counterparts.

• The stepsize used in our algorithm is chosen self-adaptively and not restricted by any Lipschitz constant. This improves the corresponding results of [ 5 , 10 , 24 ] .

• The method of proof in our convergence analysis is simpler and different from the method of proof used by many other authors such as [ 2 , 10 , 38 , 30 ] .

• The CMP considered in our article generalizes the one considered in [ 3 ] when \(f\) is identically zero.

• We would like to emphasize that the main advantage of our algorithm is that it does not require the information of the Lipschitz constant of the gradient of functions which makes it more practical for computing.

Inspired by the works aforementioned and the ongoing works in this direction, we develop an inertial-Halpern forward-backward splitting method for approximating a common solution of a finite family of SMP associated with two proper, lower semicontinuous and convex functions; and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we establish that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The selection of the stepsizes in our algorithm do not require the Lipschitz continuity condition on the gradient and does not need the prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. Our result extends and complements many related results in the literature.

2 Preliminaries

We state some known and useful results which will be needed in the proof of our main theorem. In the sequel, we denote strong and weak convergence by "\(\rightarrow \)" and "\(\rightharpoonup ,\)" respectively.

and

We briefly recall that the proximal operator \(\operatorname {prox}_{g}:H \rightarrow \operatorname {dom} (g)\) with \(\operatorname {prox}_{g}(z)=(I+ \partial g)^{-1}(z),~ z \in H,\) where \(I\) is the identity operator. It is well known that the proximal operator is single-valued with full domain. it is also known that

3 Main results

Throughout this section, we assume that

1. \(H_1\) and \(H_2\) are real Hilbert spaces, \(A:H_1 \rightarrow H_2\) is a bounded linear operator with \(A\neq \emptyset .\) Let \(f,g:H_1 \rightarrow \mathbb {R} \cup \{ +\infty \} \) are two proper, lower semi-continuous and convex functions with \(\operatorname {dom} g \subseteq \operatorname {dom} f\). The function \(f\) is Fréchet differentiable on an open set containing \(\operatorname {dom} g\). The gradient \(\nabla f\) is uniformly continuous on any bounded subset of \(\operatorname {dom} g\) and maps any bounded subset of \(\operatorname {dom} g\) to a bounded subset in \(H_1\).

2. For each \(j=1,2, \cdots ,m,\) let \(h_j:H_2 \rightarrow \mathbb {R} \cup \{ +\infty \} \) be proper, lower semi-continuous and convex function. Suppose \(S:H_2 \rightarrow H_2\) be a nonexpansive mapping, then we define \(\Gamma :=\Big\{ \min \limits _{z \in H_1}\{ f+g\} ~ \text{and}~ Az \in \operatorname {Fix}(S): Az\in \bigcap \limits _{j=1}^{m}\operatorname {argmin}_{y\in H_2}h_j(y)\Big\} \neq \emptyset .\)

where

On combining 21 and 22, we obtain that

Using 17, we have that

Hence, from 11, we have

Hence, by applying the condition \(\frac{\theta _n}{\beta _n}\| x_n-x_{n-1}\| \rightarrow 0\), there exists a constant \(M_1 {\gt} 0\) such that

and so,

By applying 13 and 11, we observe that

From the convexity of g, we obtain

Also the convexity of \(f\) implies

On combining 28 and 29 with any \(x \in \operatorname {dom}(g) \subseteq \operatorname {dom} (f)\) and \(y=y_n \in \operatorname {dom} (g),\) we obtain

where the last inequality follows from step 3 of 11. It then follows that

Replacing \(x=y_n\) and \(y=w_n\) in 27, we obtain that

This, together with 31 yields

On using

we obtain from 32 that

Since \(z\in \Gamma \), then we obtain from 33, we obtain that

From 11, 26 and 34, we get

Hence, the sequence \(\{ x_n\} \) is bounded. Consequently, it follows that 21-34 that the sequence \(\{ u_n\} , ~ \{ y_n\} \) and \(\{ w_n\} \) are bounded.

Thus, we conclude that

for some \(M_2 {\gt} 0,\) where \(M_2=2\| x_n-x^*\| + \theta _n\| x_n-x_{n-1}\| \). Now from 11 23, 34, 25 and 26, we obtain that

We conclude from algorithm 11 and 27, we have that

Putting \(d_n=\bigg[\frac{\theta _n}{\beta _n}\| x_n-x_{n-1}\| (1-\alpha _n)M_2 + 2\langle u-z, x_{n+1}-z\rangle \bigg]\), in view of lemma 10, we need to proof that \(\limsup \limits _{k\rightarrow \infty }d_{n_k}\leq 0\) for every \(\{ \| x_{n_k}- x^*\| \} \) of \(\{ \| x_n- x^*\| \} \) satisfying the condition

To show this, suppose that \(\{ \| x_{n_k}- x^*\| \} \) is a subsequence of \(\{ \| x_n- x^*\| \} \) such that 30 holds. Then

From 28 and 31, we get

Thus,

which implies that

Hence,

Also, using following the same approach as in 32, we obtain that

Using 28, we get

Thus, we obtain that

From 11, we have

From 34, 31 and 32, we obtain that

More so, applying 32 and 33, we get

Using 11, we obtain that

Considering 35, we achieve

Since \(\{ x_{n_k}\} \) is bounded, there exists a subsequence \(\{ x_{n_{k_j}}\} \) of \(\{ x_{n_k}\} \) which converges weakly to \(x^*\). Also, from 32, 33 and 34, there exist subsequence \(\{ u_{n_{k_j}}\} \) of \(\{ u_{n_k}\} , \{ y_{n_{k_j}}\} \) of \(\{ y_{n_k}\} \) and \(\{ w_{n_{k_j}}\} \) of \(\{ w_{n_k}\} \) which converge weakly to \(x^*\). Using 33, lemma 8, lemma 9 and the fact that \(A\) is a bounded linear operator, \(Ax^* \in F(S\Pi _{j=1}^{m}\operatorname {prox}_{\lambda _{n_k} h_j})\) which implies that \(Ax^* \in F(S) \bigcap F(\operatorname {prox}_{\lambda _{n_k} h_j}), j=1,2,\cdots m \). Hence, we show that \(z \in \Upsilon \).

From the statements in 11, we get that

Since \(w_{n_{k_j}}=\operatorname {prox}_{\lambda _{n_{k_j}}}g(y_{n_{k_j}}-\lambda _{n_{k_j}}\nabla f(y_{n_{k_j}}))\), it follows from 13 that

which implies that

Passing \(j \rightarrow \infty \) and by applying lemma 8 and 31, we obtain that \(x^* \in \Upsilon \). Hence, we conclude that \(x^* \in \Gamma \). Also, we show that

Let \(z=P_{\Gamma }f(z)\), then we have from 36 that

Hence, we obtain that

On substituting 40 into 29 and applying lemma 10, we conclude that \(\{ x_n\} \) converges strongly to \(z\).

4 Numerical example

In this section we give a numerical example in a m-dimensional space of real numbers to support our main result.

Example 16

Let \(H_1=H_2=\mathbb {R}^m\) with the Euclidean norm. For each \(x \in H_1,\) define \(f,g: H_1 \to \mathbb {R} \cup \{ +\infty \} \) by

\[ f(x)=\tfrac {1}{2}\| Ax-b\| ^2, \qquad g(y)=\tfrac {1}{2}\| By-c\| ^2, \]

where \(A, B \in \mathbb {R}^{m \times m}\) and \(b, c \in \mathbb {R}^m.\) It is easy to see that \(f\) and \(g\) are proper lower semicontinuous. Also, we know by [ 21 ] that

\begin{align*} \operatorname {prox}_{\lambda f}(x)& =\operatorname {argmin}_{y \in H}\left[f(y)+\tfrac {1}{2\lambda }\| y-x\| ^2 \right]\nonumber \\ & =(I+A^T A)^{-1}(x+A^T b). \end{align*}

Now, let \(A: H_1 \to H_2\) be defined by

\[ A(x)=\left(\tfrac {x}{1.5} \right), ~ ~ \forall ~ x=\{ x_i\} _{i=1}^{m} \]

then

\[ A^{T}(x)=\left(\tfrac {y}{1.5} \right),~ ~ \forall ~ y=\{ y_i\} _{i=1}^{m}. \]

For each \(j=1,2,\) \(x \in H_2,\) let \(h_j : H_2 \to \mathbb {R} \cup \{ +\infty \} \) be given by

\begin{equation*} h_j(x)=\tfrac {1}{2}\| P_j x-q_j\| ^2, \end{equation*}

where \(P_1, P_2 \in \mathbb {R}^{m \times m}\) and \(q_1, q_2 \in \mathbb {R}^m.\) As before,

\begin{equation*} \operatorname {prox}_{\lambda h_j}(x)=(I+P_{j}^{T}P_j)^{-1}(x+P_jq_j). \end{equation*}

Let the mapping \(S: H_2 \to H_2\) be defined by \(S(x)=\frac{x}{2}.\) For this example, choose \(\beta _n=\frac{1}{2n+3},\) \(u=\frac{1}{2},\) \(\delta =\frac{1}{4},\) \(\mu =\frac{1}{8},\) \(\sigma =\frac{3}{2000}\) and \(\epsilon _n =\frac{1}{n^{1.2}}.\) We choose the initial points \(x_0, x_1 \in \mathbb {R}^m\) randomly in (0,1). By using \(\| x_{n+1}-x_n\| ^2 \leq 10^{-4}\) as our stopping criterion, we conduct this example for various values of \(m.\)

Case I: \(m=10;\)
Case II: \(m=15;\)
Case III: \(m=20;\)
Case IV: \(m=50.\)

The results of this experiment are reported in figure 1.

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