# Local Sharp Vector Variational Type Inequality and Optimization Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- (i)
- approximate$\eta $-pseudoconvex type-I at${x}_{0}\in X$if for any$\tau >0,$there exists$\delta >0,$such that, whenever$x,y\in B({x}_{0},\delta )\cap X$and$$\langle {x}^{*},\eta (y,x)\rangle \ge 0,\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}some\phantom{\rule{3.33333pt}{0ex}}{x}^{*}\in \partial \phi \left(x\right),$$$$\phi \left(y\right)-\phi \left(x\right)\ge -\tau \parallel y-x\parallel ;$$
- (ii)
- approximate$\eta $-pseudoconvex type-$II$(strictly approximate$\eta $-pseudoconvex type-$II$) at${x}_{0}\in X$if for any$\tau >0,$there exists$\delta >0,$such that, whenever$x,y\in B({x}_{0},\delta )\cap X$and$$\langle {x}^{*},\eta (y,x)\rangle +\tau \parallel y-x\parallel \ge 0,\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}some\phantom{\rule{3.33333pt}{0ex}}{x}^{*}\in \partial \phi \left(x\right),$$$$\phi \left(y\right)\ge (>)\phi \left(x\right);$$
- (iii)
- approximate$\eta $-quasiconvex type-I at${x}_{0}\in X$if for any$\tau >0,$there exists$\delta >0,$such that, whenever$x,y\in B({x}_{0},\delta )\cap X$and$$\phi \left(y\right)\le \phi \left(x\right),$$$$\langle {x}^{*},\eta (y,x)\rangle -\tau \parallel y-x\parallel \le 0,\phantom{\rule{3.33333pt}{0ex}}\forall {x}^{*}\in \partial \phi \left(x\right);$$
- (iv)
- approximate$\eta $-quasiconvex type-$II$(strictly approximate$\eta $-quasiconvex type-$II$) at${x}_{0}\in X$if for any$\tau >0,$there exists$\delta >0,$such that, whenever$x,y\in B({x}_{0},\delta )\cap X$and$$\phi \left(y\right)\le (<)\phi \left(x\right)+\tau \parallel y-x\parallel ,$$$$\langle {x}^{*},\eta (y,x)\rangle \le 0,\phantom{\rule{3.33333pt}{0ex}}\forall {x}^{*}\in \partial \phi \left(x\right).$$

**(VOP):**A vector optimization problem (VOP) is formulated as follows:

**Definition**

**3.**

**.**

- (i)
- A vector${x}_{0}\in X$is said to be a local sharp efficient solution of (VOP), if for any$\tau >0$there exists a$\delta $-neighborhood of${x}_{0}$, such that for all$x\in B({x}_{0},\delta )\cap X$,$$\underset{1\le i\le p}{max}\left(\right)open="\{"\; close="\}">{f}_{i}\left(x\right)-{f}_{i}\left({x}_{0}\right)$$
- (ii)
- A vector${x}_{0}\in X$is said to be a weak local sharp efficient solution of (VOP), if for any$\tau >0$, there exists a$\delta $-neighborhood of${x}_{0}$, such that for all$x\in B({x}_{0},\delta )\cap X$,$$\underset{1\le i\le p}{max}\left(\right)open="\{"\; close="\}">{f}_{i}\left(x\right)-{f}_{i}\left({x}_{0}\right)$$$$\overline{X}:=\left(\right)open="\{"\; close="\}">x\in X\mid f\left(x\right)=f\left({x}_{0}\right)$$

## 3. Local Sharp Vector Variational Type Inequalities

**(LSVVTI):**For finding ${x}_{0}\in X,$ there exists a $\delta $-neighborhood of ${x}_{0}$ and for any $\tau >0$, such that $x\in B({x}_{0},\delta )\cap X$ and$$\underset{1\le i\le p}{max}\underset{{x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right)}{max}\langle {x}_{{0}_{i}}^{*},\eta (x,{x}_{0})\rangle \ge \tau \parallel x-{x}_{0}\parallel ,\phantom{\rule{3.33333pt}{0ex}}\forall {x}_{{0}_{i}}^{*}\in \phi {f}_{i}\left({x}_{0}\right).$$**(WLSVVTI):**For finding ${x}_{0}\in X,$ there exists a $\delta $-neighborhood of ${x}_{0}$ and for any $\tau >0,$ such that $x\in B({x}_{0},\delta )\cap X$ and$$\underset{1\le i\le p}{max}\underset{{x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right)}{max}\langle {x}_{{0}_{i}}^{*},\eta (x,{x}_{0})\rangle \ge \tau d(x,\overline{X}),\forall {x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right),$$$$\overline{X}=\left(\right)open="\{"\; close="\}">x\in X\mid f\left(x\right)=f\left({x}_{0}\right)$$

**Special Cases:**Assume that, if $\eta (x,{x}_{0})=x-{x}_{0}$. Then,

- (1) reduces to local sharp vector variational inequalities (LSVVI): for finding ${x}_{0}\in X,$ there exists a $\delta $-neighborhood of ${x}_{0}$ and for any $\tau >0$, such that $x\in B({x}_{0},\delta )\cap X$ and$$\underset{1\le i\le p}{max}\underset{{x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right)}{max}\langle {x}_{{0}_{i}}^{*},x-{x}_{0}\rangle \ge \tau \parallel x-{x}_{0}\parallel ,\forall {x}_{{0}_{i}}^{*}\in \phi {f}_{i}\left({x}_{0}\right).$$
- In addition, (2) reduces to weak local sharp vector variational inequalities (WLSVVI) for finding ${x}_{0}\in X,$ there exists a $\delta $-neighborhood of ${x}_{0}$ and for any $\tau >0,$ such that $x\in B({x}_{0},\delta )\cap X$ and$$\underset{1\le i\le p}{max}\underset{{x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right)}{max}\langle {x}_{{0}_{i}}^{*},x-{x}_{0}\rangle \ge \tau d(x,\overline{X}),\phantom{\rule{3.33333pt}{0ex}}\forall {x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right),$$$$\overline{X}=\left(\right)open="\{"\; close="\}">x\in X\mid f\left(x\right)=f\left({x}_{0}\right)$$
- Again, we note that if $\eta (x,{x}_{0})=x-{x}_{0}$, then the solution of (LSVVI) is also a solution of (AVVI)${}_{1}$ (defined by [17]), but the converse need not be true:
**Example:**consider the function$$f\left(x\right)=({f}_{1}\left(x\right),{f}_{2}\left(x\right)),x\in \mathbb{R},$$If we take ${x}_{0}=0,$ then for any $\tau >0,$ there does not exist any $\delta >0$ such that$$\langle {x}_{{0}_{i}}^{*},x-{x}_{0}\rangle \le \tau \parallel x-{x}_{0}\parallel ,\forall i\in \{1,2\},{x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right),x\in B({x}_{0},\delta )\cap \mathbb{R},$$$$\underset{1\le i\le p}{max}\underset{{x}_{{0}_{i}}^{*}\in \partial {f}_{i}\left({x}_{0}\right)}{max}\langle {x}_{{0}_{i}}^{*},x-{x}_{0}\rangle \ge \tau \parallel x-{x}_{0}\parallel ,$$

**(C)**- For the bi-function $\eta :X\times X\to X$ and the mappings ${f}_{i}:X\to \mathbb{R},\phantom{\rule{3.33333pt}{0ex}}i=1,\cdots ,p,$$$\langle {f}_{i}\left(x\right),\eta (x,x)\rangle =0$$

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 4. Minty Local Sharp Vector Variational Type Inequalities

**(MLSVVTI):**Finding ${x}_{0}\in X,$ there exists a $\delta $-neighborhood of ${x}_{0}$ and any $\tau >0,$ such that $x\in B({x}_{0},\delta )\cap X$ and$$\underset{1\le i\le p}{max}\underset{{x}_{i}^{*}\in \partial {f}_{i}\left(x\right)}{max}\langle {x}_{i}^{*},\eta (x,{x}_{0})\rangle \ge \tau \parallel x-{x}_{0}\parallel ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {x}_{i}^{*}\in \partial {f}_{i}\left(x\right).$$**(MWLSVVTI):**For finding ${x}_{0}\in X,$ there exists a $\delta $-neighborhood of ${x}_{0}$ and any $\tau >0,$ such that $x\in B({x}_{0},\delta )\cap X$ and$$\underset{1\le i\le p}{max}\underset{{x}_{i}^{*}\in \partial {f}_{i}\left(x\right)}{max}\langle {x}_{i}^{*},\eta (x,{x}_{0})\rangle \ge \tau d(x,\overline{X}),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall {x}_{i}^{*}\in \partial {f}_{i}\left(x\right),$$

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Anh, P.N.; Thach, H.T.C.; Kim, J.K. Proximal-like subgradient methods for solving multi-valued variational inequalities. Nonlinear Funct. Anal. Appl.
**2020**, 25, 437–451. [Google Scholar] - Ansari, Q.H.; Ahmad, R.; Salahuddin. A perturbed Ishikawa iterative algorithm for general mixed multivalued mildly nonlinear variational inequalities. Adv. Nonlinear Var. Inequal.
**2020**, 3, 53–64. [Google Scholar] - Fan, K. A generalization of Tychonoff’s fixed-point theprem. Math. Ann.
**1961**, 142, 305–310. [Google Scholar] [CrossRef] - Giannessi, F. Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems; Cottle, R.W., Giannessi, F., Lions, J.-L., Eds.; Wiley: New York, NY, USA, 1980; pp. 151–186. [Google Scholar]
- Huang, H. Strict vector variational inequalities and strict Pareto efficiency in nonconvex vector optimization. Positivity
**2015**, 19, 95–109. [Google Scholar] [CrossRef] - Kim, J.K.; Salahuddin. The existence of deterministic random generalized vector equilibrium problems. Nonlinear Funct. Anal. Appl.
**2015**, 20, 453–464. [Google Scholar] - Kim, J.K.; Salahuddin. Extragradient methods for generalized mixed equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Funct. Anal. Appl.
**2017**, 22, 693–709. [Google Scholar] - Lee, B.S.; Khan, M.F.; Salahuddin. Generalized vector variational-type inequalities. Comput. Math. Appl.
**2008**, 55, 1164–1169. [Google Scholar] [CrossRef] [Green Version] - Lee, B.S.; Salahuddin. Minty lemma for inverted vector variational inequalities. Optimization
**2017**, 66, 351–359. [Google Scholar] [CrossRef] - Liu, Y.; Cui, Y. A modified gradient projection algorithm for variational inequalities and relatively nonexpansive mappings in Banach spaces. Nonlinear Funct. Anal. Appl.
**2017**, 22, 433–448. [Google Scholar] - Salahuddin. Convergence analysis for hierarchical optimizations. Nonlinear Anal. Forum
**2015**, 20, 229–239. [Google Scholar] - Salahuddin. A coordinate wise variational method with tolerance functions. J. Appl. Nonlinear Dyna
**2020**, 9, 541–549. [Google Scholar] [CrossRef] - Loridan, P. ϵ-solutions in vector minimization problems. J. Optim. Theory Appl.
**1984**, 43, 265–276. [Google Scholar] [CrossRef] - White, D.J. Epsilon efficiency. J. Optim. Theory Appl.
**1986**, 49, 319–337. [Google Scholar] [CrossRef] - Burke, J.V.; Ferris, M.C. Weak sharp minima in mathematical programming. SIAM J. Control Optim.
**1993**, 31, 1340–1359. [Google Scholar] [CrossRef] [Green Version] - Zhu, S.K. Weak sharp efficiency in multiobjective optimization. Optim. Lett.
**2016**, 10, 1287–1301. [Google Scholar] [CrossRef] - Mishra, S.K.; Laha, V. On approximately star-shaped functions and approximate vector variational inequalities. J. Optim. Theory Appl.
**2013**, 156, 278–293. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Kim, J.K.; Salahuddin.
Local Sharp Vector Variational Type Inequality and Optimization Problems. *Mathematics* **2020**, *8*, 1844.
https://doi.org/10.3390/math8101844

**AMA Style**

Kim JK, Salahuddin.
Local Sharp Vector Variational Type Inequality and Optimization Problems. *Mathematics*. 2020; 8(10):1844.
https://doi.org/10.3390/math8101844

**Chicago/Turabian Style**

Kim, Jong Kyu, and Salahuddin.
2020. "Local Sharp Vector Variational Type Inequality and Optimization Problems" *Mathematics* 8, no. 10: 1844.
https://doi.org/10.3390/math8101844