# Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Augmented Weak Sharpness of Solution Sets

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (1)
- $K=\left\{k\right|{x}^{k}\notin \overline{S}\}$ is a finite set;
- (2)
- When $K=\left\{k\right|{x}^{k}\notin \overline{S}\}$ is an infinite sequence, there exist a set-valued mapping $H:\overline{S}\to {2}^{{R}^{n}}$ such that
- (a)
- there is a constant $\alpha >0,$ satisfying$$\alpha B\subset H\left(z\right)+{\left[{T}_{S}\left(z\right)\bigcap {\widehat{N}}_{\overline{S}}\left(z\right)\right]}^{\circ},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall z\in \overline{S}.$$
- (b)
- for $\forall {z}^{k}\in {P}_{\overline{S}}\left({x}^{k}\right)$ and $\forall {v}^{k}\in H\left({z}^{k}\right)$, there is$$\phantom{\rule{3.33333pt}{0ex}}lim{\mathrm{sup}}_{k\in K,k\to \infty}{\psi}_{k}=\frac{1}{\parallel {x}^{k}-{z}^{k}\parallel}\u2329F\left({x}^{k}\right)-{v}^{k},{x}^{k}-{z}^{k}\u232a\ge 0.$$

**Example**

**1.**

- (i)
- $\left\{{x}^{k}\right\}\subset \left\{x\in {R}^{2}|{x}_{1}\le \frac{\pi}{4}\right\}\cup \left\{x\in {R}^{2}|{x}_{1}\ge \frac{\pi}{4},{x}_{1}+{x}_{2}\ge \frac{\pi}{2}\right\};$
- (ii)
- ${lim}_{k\to \infty}dist({x}^{k},\overline{S})=0.$

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

## 3. Finite Convergence

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

**Corollary**

**1.**

**Remark**

**2.**

**Corollary**

**2.**

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Rockafellar, R.T. Monotone operators and the proximal point algorithm. SIAM J. Control. Optim.
**1976**, 14, 877–898. [Google Scholar] [CrossRef] - Polyak, B.T. Introduction to Optimization; Optimization Software: New York, NY, USA, 1987. [Google Scholar]
- Ferris, M.C. Weak Sharp Minima and Penalty Functions in Mathematical Programming. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 1988. [Google Scholar]
- Ferris, M.C. Finite termination of the proximal point algorithm. Math. Program.
**1991**, 50, 359–366. [Google Scholar] [CrossRef] - Matsushita, S.Y.; Xu, L. Finite termination of the proximal point algorithm in Banach spaces. J. Math. Anal. Appl.
**2012**, 387, 765–769. [Google Scholar] [CrossRef] - Xiu, N.H.; Zhang, J.Z. On finite convergence of proximal point algorithms for variational inequalities. J. Math. Anal. Appl.
**2005**, 312, 148–158. [Google Scholar] [CrossRef] - Al-Khayyal, F.; Kyparisis, J. Finite convergence of algorithms for nonlinear programs and variational inequalities. J. Optim. Theory Appl.
**1991**, 70, 319–332. [Google Scholar] [CrossRef] - Burke, J.V.; Moré, J.J. On the identification of active constraints. SIAM J. Numer. Anal.
**1988**, 25, 1197–1211. [Google Scholar] [CrossRef] - Fischer, A.; Kanzow, C. On finite termination of an iterative method for linear complementarity problem. Math. Program.
**1996**, 74, 279–292. [Google Scholar] [CrossRef] - Solodov, M.V.; Tseng, P. Some methods based on the D-gap function for solving monotone variational inequalities. Comput. Optim. Appl.
**2000**, 17, 255–271. [Google Scholar] [CrossRef] - Wang, C.Y.; Zhao, W.L.; Zhou, J.H.; Lian, S.J. Global convergence and finite termination of a class of smooth penalty function algorithms. Optim. Methods Softw.
**2013**, 28, 1–25. [Google Scholar] [CrossRef] - Xiu, N.H.; Zhang, J.Z. Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math.
**2003**, 152, 559–585. [Google Scholar] [CrossRef] - Marcotte, P.; Zhu, D. Weak sharp solutions of variational inequalities. SIAM J. Optim.
**1998**, 9, 179–189. [Google Scholar] [CrossRef] - Khan, A.A.; Tammer, C.; Zalinescu, C. Set-Valued Optimization; Springer: New York, NY, USA, 2014; pp. 110–112. [Google Scholar]
- Rockafellar, R.T.; Wets, R.J. Variational Analysis; Springer: New York, NY, USA, 1998. [Google Scholar]
- Wang, C.Y.; Zhang, J.Z.; Zhao, W.L. Two error bounds for constrained optimization problem and their application. Appl. Math. Optim.
**2008**, 57, 307–328. [Google Scholar] [CrossRef] - Zhou, J.C.; Wang, C.Y. New characterizations of weak sharp minima. Optim. Lett.
**2012**, 6, 1773–1785. [Google Scholar] [CrossRef]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, W.; Wang, R.; Zhang, H.
Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems. *Math. Comput. Appl.* **2017**, *22*, 36.
https://doi.org/10.3390/mca22020036

**AMA Style**

Zhao W, Wang R, Zhang H.
Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems. *Mathematical and Computational Applications*. 2017; 22(2):36.
https://doi.org/10.3390/mca22020036

**Chicago/Turabian Style**

Zhao, Wenling, Ruyu Wang, and Hongxiang Zhang.
2017. "Finite Convergence for Feasible Solution Sequence of Variational Inequality Problems" *Mathematical and Computational Applications* 22, no. 2: 36.
https://doi.org/10.3390/mca22020036