# Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

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## Abstract

**:**

## 1. Introduction

- The weak minimum point of a multiobjective function $f=({f}_{1},\dots ,{f}_{p})$ over a closed set $S\subseteq {\mathbb{R}}^{p}$ is any $\overline{x}\in S$ such that for any $y\in S,\phantom{\rule{0.222222em}{0ex}}\exists i$ such that ${f}_{i}\left(y\right)-{f}_{i}\left(\overline{x}\right)\ge 0$. Finding a weak minimum point can be reduced to solving an equilibrium problem by virtue of setting$$F(x,y)=\underset{i=1,\dots ,p}{max}[{f}_{i}\left(y\right)-{f}_{i}\left(x\right)].$$
- The Stampacchia variational inequality problem demands finding $\overline{x}\in S$ such that$$<G\left(\overline{x}\right),y-\overline{x}>\ge 0,\phantom{\rule{0.222222em}{0ex}}\forall y\in S$$$$F(x,y)=<G\left(x\right),y-x>.$$
- Nash equilibrium problems in a non-cooperative game with p players where each player i has a set of possible strategies ${K}_{i}\subseteq {\mathbb{R}}^{{n}_{i}}$ aim to minimize a loss function ${f}_{i}:K\to \mathbb{R}$ with $K={K}_{1}\times \dots \times {K}_{p}$. Thus, a Nash equilibrium point is any $\overline{x}\in K$ such that no player can reduce its loss by unilaterally changing their strategy, i.e., any $\overline{x}\in K$ such that$${f}_{i}\left(\overline{x}\right)\le {f}_{i}\left(\overline{x}\left({y}_{i}\right)\right)$$$$F(x,y)={\displaystyle \sum _{i=1}^{p}}[{f}_{i}\left(x\left({y}_{i}\right)\right)-{f}_{i}\left(x\right)].$$

## 2. Preliminaries

**Definition**

**1.**

**Example**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Notation**

**1.**

**Definition**

**5.**

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Step 1.**We have to prove that $(0,0)\notin W$. By reduction ad absurdum, if $(0,0)\in W\phantom{\rule{0.222222em}{0ex}}\Rightarrow \exists {x}_{0}\in {S}_{1}$, such that

**Step 2.**We will prove that there exists a multiplier $v\in {\mathbb{R}}_{+}^{p}$. As W is an open set and the separation theorem holds (see Theorem 2.13 and Remark 2.14 in [28]) or [3]), there exists $(v,u)\ne (0,0)\in {\mathbb{R}}^{p}\times {\mathbb{R}}^{p}$ such that

**Step 3.**We will prove that $v\ne 0$, thus is, $v\in {\mathbb{R}}_{+}^{p}\backslash \left\{0\right\}$. By reduction ad absurdum, if $v=0$, from Equation (11) we get

**Step 4.**We will prove the first KKT condition.

**Step 5.**We will prove the second KKT condition. As

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

## 4. Application

**Corollary**

**1.**

**Corollary**

**2.**

**Proof.**

**Remark**

**3.**

**Example**

**2.**

## 5. Conclusions

- The need for an extension of the concept of convex set to that of totally convex.
- The use of an adequate definition of differential functions in similar terms to those of directional derivatives in Euclidean space using an exponential Riemannian map.
- Generalizing the invexity definition by extending its classical definition given by Hanson [14] in order to obtain sufficient optimality conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ruiz-Garzón, G.; Osuna-Gómez, R.; Ruiz-Zapatero, J.
Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds. *Symmetry* **2019**, *11*, 1037.
https://doi.org/10.3390/sym11081037

**AMA Style**

Ruiz-Garzón G, Osuna-Gómez R, Ruiz-Zapatero J.
Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds. *Symmetry*. 2019; 11(8):1037.
https://doi.org/10.3390/sym11081037

**Chicago/Turabian Style**

Ruiz-Garzón, Gabriel, Rafaela Osuna-Gómez, and Jaime Ruiz-Zapatero.
2019. "Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds" *Symmetry* 11, no. 8: 1037.
https://doi.org/10.3390/sym11081037