A Fixed-Point Subgradient Splitting Method for Solving Constrained Convex Optimization Problems

Many applications in science and engineering have shown a huge interest in solving an inverse problem of finding $x\in {\mathbb{R}}^n$ satisfying where $b\in {\mathbb{R}}^r$ is the observed data and $B^{r\times n}$ is the corresponding nonzero matrix. Actually, the inverse problem is typically facing the ill-condition of the matrix B so that it may have no solution. Then, an approach for finding an approximate solution by minimizing the squared norm of the residual term has been considered:

Observe that the problem (2) has several optimal solutions; in this situation, it is not clear which of these solutions should be considered. One strategy for pursuing the best optimal solution among these many solutions is to add a regularization term to the objective function. The classical technique is to consider the celebrated Tikhonov regularization [1] of the form where $\lambda >0$ is a regularization parameter. In this setting, the uniqueness of the solution to (3) is acquired. However, note that from a practical point of view, the shortcoming of this strategy is that the unique solution to the regularization problem (3) may probably not optimal in the original sense as in (2), see [2,3] for further discussions.

To over come this, we should consider the strategy of selecting a specific solution among optimal solutions to (2) by minimizing an additional prior function over these optimal solutions. This brings the framework of the following bilevel optimization problem, where $f:{\mathbb{R}}^n\to \mathbb{R}$ and $h:{\mathbb{R}}^m\to \mathbb{R}$ are convex functions, and $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a nonzero linear transformation. It is very important to point out that many problems can be formulated into this form. For instance, if $m:=n-1$, $f:={\parallel ·\parallel }_1$, $h:={\parallel ·\parallel }_1$, and the problem (4) becomes the fused lasso [4] solution to the problem (2). This situation also occurs in image denoising problems ($r=n$ and B is the identity matrix), and in image inpainting problems ($r=n$ and B is a symmetric diagonal matrix), where the term ${\parallel Ax\parallel }_1$ is known as an 1D total variation [5]. When $m=n$, $f={\parallel ·\parallel }^2$, $h={\parallel ·\parallel }_1$, and A is the identity matrix, the problem (4) becomes the elastic net [6] solution to the problem (2). Moreover, in wavelet-based image restoration problems, the matrix A is given by an inverse wavelet transform [7].

Let us consider the constrained set of (4), it is known that introducing the Landweber operator $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ of the form yields T is firmly nonexpansive and the set of all fixed points of T is nothing else that the set ${argmin}_{x\in {\mathbb{R}}^n}{\parallel Bu-b\parallel }^2$, see [8] for more details. Motivated by this observation, the bilevel problem (4) can be considered in the general setting as where $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a nonlinear operator with $FixT:=\{x\in {\mathbb{R}}^n:Tx=x\}\ne \varnothing$. Note that problem (5) encompasses not only the problem (4), but also many problems in the literature, for instance, the minimization over the intersection of a finite number of sublevel sets of convex nonsmooth functions (see Section 5.2), the minimization over the intersection of many convex sets in which the metric projection on such intersection can not be computed explicitly, see [9,10,11] for more details.

There are some existing methods for solving convex optimization problems over fixed-point set in the form of (5), but the celebrated one is due to the hybrid steepest descent method, which was firstly investigated in [12]. Note that the algorithm proposed by Yamada [12] goes on with the hypotheses that the objective functions are strongly convex and smooth and the operator T is nonexpansive. Several variants and generalizations of this well-known method are, for instance, Yamada and Ogura [11] considered the same scheme for solving the problem (5) when T belongs to a class of so called quasi-shrinking operator. Cegielski [10] proposed a generalized hybrid steepest descent method by using the sequence of quasi-nonexpansive operators. Iiduka [13,14] considered a nonsmooth convex optimization problem (5) with fixed-point constraints of certain quasi-nonexpansive operators.

On the other hand, in the recent decade, the split common fixed point problem [15,16] turns out to be one of the attractions among several nonlinear problems due to its widely applications in many image and signal processing problems. Actually, for given a nonzero linear transformation $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$, and two nonlinear operators $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $S:{\mathbb{R}}^m\to {\mathbb{R}}^m$ with $\mathrm{Fix}\left(T\right)\ne \varnothing$, and $\mathcal{R}\left(A\right)\cap \mathrm{Fix}\left(S\right)\ne \varnothing$, the split common fixed point problem is to find a point $x^{*}\in {\mathbb{R}}^n$ in which The key idea of this problem is to find a point in the fixed point of a nonlinear operator in a primal space, and subsequently its image under an appropriate linear transformation forms a fixed point of another nonlinear operator in another space. This situation appears, for instance, in dynamic emission tomographic image reconstruction [17] and in the intensity-modulated radiation therapy treatment planning, see [18] for more details. Of course, there are many authors that have investigated the study of iterative algorithms for split common fixed point problems and proposed their generalizations in several aspects, see, for example, [9,19,20,21,22] and references therein.

The aim of this paper is to present a nonsmooth and non-strongly convex version of the hybrid steepest descent method for minimizing the sum of two convex functions over the fixed-point constraints of the form: where $f:{\mathbb{R}}^n\to \mathbb{R}$ and $h:{\mathbb{R}}^m\to \mathbb{R}$ are convex nonsmooth functions, $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a nonzero linear transformation, $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $S:{\mathbb{R}}^m\to {\mathbb{R}}^m$ are certain quasi-nonexpansive operators with $\mathrm{Fix}\left(T\right)\ne \varnothing$, and $\mathcal{R}\left(A\right)\cap \mathrm{Fix}\left(S\right)\ne \varnothing$, and $X\subset {\mathbb{R}}^n$ is a simple closed convex bounded set. We prove the convergence of function value to the minimum value where some control conditions on a stepsize sequence and a parameter are imposed.

The paper is organized as follows. After recalling and introducing some useful notions and tools in Section 2, we present our algorithm and discuss the convergence analysis in Section 3. Furthermore, in Section 4, we discuss an important implication of our problem and algorithm to the minimizing sum of convex functions over coupling constraints. In Section 5, we discuss in detail some remarkably practical applications, and Section 6 describes the results of numerical experiments on fused lasso like problem. Finally, the conclusions are given in Section 7.

We summarize some useful notations, definitions, and properties, which we will utilize later.

For further details, the reader can consult the well-known books, for instance, in [8,23,24,25].

Let ${\mathbb{R}}^n$ be an n-dimensional Euclidean space with an inner product $〈·,·〉$ and the corresponding norm $\parallel ·\parallel$.

Let $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be an operator. We denote the set of all fixed points of T by $FixT$, that is, We say that T is $\rho$-strongly quasi-nonexpansive ($\rho$-SQNE), where $\rho \ge 0$, if $FixT\ne \varnothing$ and for all $x\in {\mathbb{R}}^n$ and $z\in FixT$. If $\rho >0$, then T is called strongly quasi-nonexpansive (SQNE). If $\rho =0$, then T is called quasi-nonexpansive (QNE), that is, for all $x\in {\mathbb{R}}^n$ and $z\in FixT$. Clearly, if T is SQNE, then it is QNE. We say that T is cutter if $FixT\ne \varnothing$ and for all $x\in {\mathbb{R}}^n$ and all $z\in FixT$. We say that T is firmly nonexpansive (FNE) if for all $x,y\in {\mathbb{R}}^n$.

The following properties will be applied in the next sections.

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a function and $x\in {\mathbb{R}}^n$. The subdifferential of f at x is the set If $\partial f\left(x\right)\ne \varnothing$, then an element $f^{\prime }\left(x\right)\in \partial f\left(x\right)$ is called a subgradient of f at x.

As we work on the n-dimensional Euclidean space, we will use the notion of matrix instead of the notion of linear transformation throughout this work. Denote ${\mathbb{R}}^{m\times n}$ by the set of all real-valued $m\times n$ matrices. Let $A\in {\mathbb{R}}^{m\times n}$ be given. We denote by $\mathcal{R}\left(A\right):=\{y\in {\mathbb{R}}^m:y=Ax\mathrm{for}\mathrm{some}x\in \mathbb{R}{}^n\}$ its range, and $A^{\top }$ its (conjugate) transpose. We denote the induced norm of A by $\parallel A\parallel$, which is given by $\parallel A\parallel =\sqrt{{\lambda }_{max}\left(A^{\top }A\right)}$, where ${\lambda }_{max}\left(A^{\top }A\right)$ is the maximum eigenvalue of the matrix $A^{\top }A$.

Now, we formulate the composite nonsmooth convex minimization problem over the intersections of fixed-point sets which we aim to investigate throughout this paper.

Throughout this work, we denote the solution set of Problem 1 by $\Gamma$ and assume that it is nonempty.

Problem 1 can be viewed as a bilevel problem in which data given from two sources in a system. Actually, let us consider the system of two users in different sources (they can have differently a number of factors n and m) in which they can communicate to each other via the transformation A. The first user aims to find the best solutions with respect to criterion f among many feasible points represented in the form of fixed point set of an appropriate operator T. Similarly, the second user has its own objective in the same fashion of finding the best solutions among feasible points in $FixS$ with priori criterion h. Now, to find the best solutions of this system, we consider the fixed-point subgradient splitting method (in short, FSSM) as follows, see Algorithm 1.

To study the convergence properties of a function values of a sequence generated by Algorithm 1, we start with the following technical result.

The following lemma is very useful for the convergence result.

For the sake of simplicity, we let and assume that ${(f+h\circ A)}^{*}>-\infty$.

We consider a convergence property in objective values with diminishing stepsize as the following theorem.

The aim of this section is to show that Algorithm 1 and their convergence properties can be employed when solving a convex minimization involving sum of a finite number of composite functions.

Let us take a look the composite convex minimization problem: where, we assume further that, for all $i=1,\dots ,l$, there hold

In this section, we assume that the solution set of (13) is denoted by $\Omega$ and asuume that it is nonempty.

Denote the product of spaces equipped with the addition the scalar multiplication with the inner product defined by and the norm by for all $\mathbf{x}=(x_1,x_2,\dots ,x_l)$, $\mathbf{y}=(y_1,y_2,\dots ,y_l)\in {\mathbb{R}}^m$, is again a Euclidean space (see [24], Example 2.1). Define a matrix $\mathbf{A}\in {\mathbb{R}}^{n\times m}$ by and an operator $\mathbf{S}:{\mathbb{R}}^m\to {\mathbb{R}}^m$ by for all $\mathbf{y}=(y_1,y_2,\dots ,y_l)\in {\mathbb{R}}^m$. Note that the operator $\mathbf{S}$ is cutter with Furthermore, defining a function $\mathbf{h}:{\mathbb{R}}^m\to \mathbb{R}$ by for all $\mathbf{x}=(x_1,x_2,\dots ,x_l)\in {\mathbb{R}}^m$, we also have that the function $\mathbf{h}$ is a convex function (see [24], Proposition 8.25). By hte above setting, we can rewrite the problem (13) as which is nothing else than Problem 1.

Here, to investigate the solving of the problem (13), we state the Algorithm 2 as follow.

As an above consequence, we note that for all $\mathbf{x}=(x_1,x_2,\dots ,x_l)\in {\mathbb{R}}^m$. Furthermore, we know that see ([25], Corollary 2.4.5). Putting ${\mathbf{d}}_k:=(d_{k,1},\dots ,d_{k,l})$ where $d_{k,i}\in \partial h_i\left(S_i{A}_i{x}_k\right),i=1,\dots ,l$, for all $k\ge 1$, we obtain that for all $k\ge 1$. Notice that for all $k\ge 1.$ Thus, Algorithm 2 can be rewrite as for all $k\ge 1$. Since ${\parallel \mathbf{A}\parallel }^2\le {\sum }_{i=1}^l{\parallel A_i\parallel }^2$, the convergence result therefore follows from Theorem 1 and can be stated as the following corollary.

The aim of this section is to show that Algorithm 1 and their convergence results can be employed when solving some well-known problems. Furthermore, we also present some noticeable applications relate to Algorithm 1 and its convergence results. For simplicity, we assume here that $S=I$.