# Partially Projective Algorithm for the Split Feasibility Problem with Visualization of the Solution Set

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Algorithm**

**1.**

**Definition**

**1**

- T is said to be nonexpansive, if$$\u2225Tx-Ty\u2225\le \u2225x-y\u2225,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x,y\in H.$$
- T is said to be an averaged operator if $T=(1-\alpha )I+\alpha N$, where $\alpha \in (0,1)$, I is the identity map and $N:H\to H$ is a nonexpansive mapping.
- T is called monotone if$$\u2329Tx-Ty,x-y\u232a\ge 0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x,y\in H.$$
- Assume $\nu >0$. Then, T is called ν-inverse strongly monotone (ν-ism) if$$\u2329Tx-Ty,x-y\u232a\ge \nu {\u2225Tx-Ty\u2225}^{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x,y\in H.$$
- Any 1-ism T is also known as being firmly nonexpansive; that is,$$\u2329Tx-Ty,x-y\u232a\ge {\u2225Tx-Ty\u2225}^{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x,y\in H.$$

**Lemma**

**1**

- (i)
- Each firmly nonexpansive mapping is averaged and each averaged operator is nonexpansive.
- (ii)
- T is a firmly nonexpansive mapping if and only if its complement $I-T$ is firmly nonexpansive.
- (iii)
- The composition of a finite number of averaged operators is averaged.
- (iv)
- An operator N is nonexpansive if and only if its complement $I-N$ is a $\frac{1}{2}$-ism.
- (v)
- An operator T is averaged if and only if its complement is a ν-ism, for some $\nu >\frac{1}{2}$. Moreover, if $T=(1-\alpha )I+\alpha N$, then $I-T$ is a $\frac{1}{2\alpha}$-ism.
- (vi)
- If T is a ν-ism and $\gamma >0$, then $\gamma T$ is $\frac{\nu}{\gamma}$-ism.

**Lemma**

**2.**

- (i)
- $\u2329x-{P}_{C}(x),{P}_{C}(x)-c\u232a\ge 0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall c\in C$.
- (ii)
- ${P}_{C}$ is a firmly nonexpansive operator, hence also averaged and nonexpansive.

**Lemma**

**3**

**Lemma**

**4**

**Lemma**

**5**

## 3. Main Results

**Algorithm**

**2**

- (1)
- (2)
- the iteration function inside Algorithm 2, namely $T={P}_{C}[I-\gamma {A}^{*}(I-{P}_{Q})A]$ is nonexpansive, for properly chosen $\gamma $ (see Lemma 3.1 in [12]).

**Algorithm**

**3.**

**Lemma**

**6.**

**Proof.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Theorem**

**6.**

## 4. Numerical Simulation

**Example**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Bejenaru, A.; Postolache, M.
Partially Projective Algorithm for the Split Feasibility Problem with Visualization of the Solution Set. *Symmetry* **2020**, *12*, 608.
https://doi.org/10.3390/sym12040608

**AMA Style**

Bejenaru A, Postolache M.
Partially Projective Algorithm for the Split Feasibility Problem with Visualization of the Solution Set. *Symmetry*. 2020; 12(4):608.
https://doi.org/10.3390/sym12040608

**Chicago/Turabian Style**

Bejenaru, Andreea, and Mihai Postolache.
2020. "Partially Projective Algorithm for the Split Feasibility Problem with Visualization of the Solution Set" *Symmetry* 12, no. 4: 608.
https://doi.org/10.3390/sym12040608