# S-Subgradient Projection Methods with S-Subdifferential Functions for Nonconvex Split Feasibility Problems

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## Abstract

**:**

## 1. Introduction

- The boundedness of subset Q;
- The full column rank of matrix A.

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2**

**Definition**

**3**

**Lemma**

**1**

**Definition**

**4**

**Lemma**

**2**

## 3. Nonconvex Split Feasibility Problem

- (1)
- continuous, but not necessarily convex functions $c:{\mathbb{R}}^{n}\to \mathbb{R}$ and $q:{\mathbb{R}}^{m}\to \mathbb{R}$ are the S-subdifferential, and c and q are locally Lipschitzian in addition.
- (2)
- the lower level sets of c and q at height $\xi \in \mathbb{R},\xi >0$ are defined by ${C}_{\xi}=\left\{u\in {\mathbb{R}}^{n}:c(u)\le \xi \right\}$ and ${Q}_{\xi}=\left\{v\in {\mathbb{R}}^{m}:q(v)\le \xi \right\}.$
- (3)
- the set of solutions to $SFP$ is nonempty, that is there exists at least one element $\tilde{u}\in {C}_{\xi}$ such that $A\tilde{u}\in {Q}_{\xi}$, where A is an $m\times n$ matrix.
- (4)
- $U\subseteq {\mathbb{R}}^{n}$ and $V\subseteq {\mathbb{R}}^{m}$ are closed convex subsets such that ${C}_{\xi}\subseteq U$ and ${Q}_{\xi}\subseteq V$.
- (5)
- c and q are the S-subdifferential on ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$ with respect to U and V, respectively.
- (6)
- ${\partial}_{U,{r}_{c}}c(u)$ and ${\partial}_{V,{r}_{q}}q(v)$ are the S-subdifferential of c and q with respect to U and V, respectively.
- (7)
- both ${\partial}_{U,{r}_{c}}c(u)$ and ${\partial}_{V,{r}_{q}}q(v)$ are not empty; let ${s}_{c}(u)\in {\partial}_{U,{r}_{c}}c(u)$ and ${s}_{q}(v)\in {\partial}_{V,{r}_{q}}q(v)$.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

- 1,
- Can the result presented in Theorem 1 hold in infinity spaces?
- 2,
- Since we only obtain weak convergence of the proposed algorithm in this paper, how do we modify the algorithm so that the strong convergence is guaranteed?

**Remark**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Chen, J.; Postolache, M.; Yao, Y.
*S*-Subgradient Projection Methods with *S*-Subdifferential Functions for Nonconvex Split Feasibility Problems. *Symmetry* **2019**, *11*, 1517.
https://doi.org/10.3390/sym11121517

**AMA Style**

Chen J, Postolache M, Yao Y.
*S*-Subgradient Projection Methods with *S*-Subdifferential Functions for Nonconvex Split Feasibility Problems. *Symmetry*. 2019; 11(12):1517.
https://doi.org/10.3390/sym11121517

**Chicago/Turabian Style**

Chen, Jinzuo, Mihai Postolache, and Yonghong Yao.
2019. "*S*-Subgradient Projection Methods with *S*-Subdifferential Functions for Nonconvex Split Feasibility Problems" *Symmetry* 11, no. 12: 1517.
https://doi.org/10.3390/sym11121517