# Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

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## Abstract

**:**

## 1. Introduction

- (a)
- Nonconvex constrained problems in ${\mathbb{R}}^{n}$ are transformed into convex ones in the Hadamard manifolds (see [1]).
- (b)
- Moreover, for example, the set $X=\left\{\right(cost,sint):\phantom{\rule{0.222222em}{0ex}}t\in [\pi /4,3\pi /4\left]\right\}$ is not convex in the usual sense with $X\subset {\mathbb{R}}^{2}$, but X is a geodesic convex on the Poincaré upper-plane model $({\mathbb{H}}^{2},{g}_{\mathbb{H}})$, as it is the image of a geodesic segment (see [2]).

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

**Definition**

**5.**

**Definition**

**6.**

- (a)
- When $\theta $ is a locally Lipschitz convex function, we have ${\theta}^{0}(x;v)={\theta}^{\prime}(x;v)$ for all $x\in M$. For a convex function $\theta :M\to \mathbb{R}$, the directional derivative of $\theta $ at the point $x\in M$ in the direction $v\in {T}_{y}M$ is defined by$${\theta}^{\prime}(x,v)=\underset{t\to {0}^{+}}{lim}\frac{\theta \left({\mathrm{exp}}_{x}\left(tv\right)\right)-\theta \left(x\right)}{t}$$$$\partial \theta \left(x\right)=\{A\in {T}_{x}M|\phantom{\rule{0.222222em}{0ex}}{\theta}^{\prime}(x;v)\ge <A,v>,\phantom{\rule{0.222222em}{0ex}}\forall v\in {T}_{x}M\}$$
- (b)
- If $\theta $ is differentiable at $x\in M$, we define the gradient of $\theta $ as the unique vector $\mathrm{grad}\theta \left(x\right)\in {T}_{x}M$ that satisfies$$d{\theta}_{x}\left(v\right)=<grad\phantom{\rule{0.222222em}{0ex}}\theta \left(x\right),v>\phantom{\rule{1.em}{0ex}}\forall v\in {T}_{x}M$$

**Definition**

**7.**

- (a)
- Invex (IX) at $\overline{x}$ with respect to η on X if, $\forall x\in X\subseteq M$, there exist some $\eta (x,\overline{x})\in {T}_{\overline{x}}M$, $A\in \partial f\left(\overline{x}\right)$ such that$$f\left(x\right)-f\left(\overline{x}\right)-A\eta (x,\overline{x})\in {\mathbb{R}}_{+}^{p}.$$
- (b)
- Pseudoinvex (PIX) at $\overline{x}$ with respect to η on X if, $\forall x\in X\subseteq M$, there exist some $\eta (x,\overline{x})\in {T}_{\overline{x}}M$, $A\in \partial f\left(\overline{x}\right)$ such that$$f\left(x\right)-f\left(\overline{x}\right)\in -\mathrm{int}{\mathbb{R}}_{+}^{p}\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{0.222222em}{0ex}}A\eta (x,\overline{x})\in -\mathrm{int}{\mathbb{R}}_{+}^{p}.$$
- (c)
- Strong pseudoinvex (SGPIX) at $\overline{x}$ with respect to η on X if, $\forall x\in X\subseteq M$, there exist some $\eta (x,\overline{x})\in {T}_{\overline{x}}M$, $A\in \partial f\left(\overline{x}\right)$ such that$$f\left(x\right)-f\left(\overline{x}\right)\in -{\mathbb{R}}_{+}^{p}\backslash \left\{0\right\}\phantom{\rule{3.33333pt}{0ex}}\Rightarrow A\eta (x,\overline{x})\in -\mathrm{int}{\mathbb{R}}_{+}^{p}.$$The function f is said to be invex (resp. pseudoinvex, strong pseudoinvex) with respect to η on X if, for every $x\in X$, f is invex (resp. pseudoinvex, strong pseudoinvex) at x with respect to η on X.

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

- (a)
- If ${p}_{1}<0$ or ${p}_{1}>1$, then$${f}^{0}(p,v)=\langle \left(\begin{array}{cc}{p}_{2}^{2}& 0\\ 0& {p}_{2}^{2}\end{array}\right)\left(\begin{array}{cc}1& 0\\ -1& 0\end{array}\right),\phantom{\rule{0.222222em}{0ex}}\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)\rangle =\left(\begin{array}{cc}{p}_{2}^{2}& 0\\ -{p}_{2}^{2}& 0\end{array}\right)\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)=\left(\begin{array}{c}{p}_{2}^{2}{v}_{1}\\ -{p}_{2}^{2}{v}_{1}\end{array}\right)$$
- (b)
- If ${p}_{1}=0$ or ${p}_{1}=1$ and $-1\le a\le 0$, then$${f}^{0}(p,v)=\langle \left(\begin{array}{cc}{p}_{2}^{2}& 0\\ 0& {p}_{2}^{2}\end{array}\right)\left(\begin{array}{cc}1& 0\\ a& 0\end{array}\right),\phantom{\rule{0.222222em}{0ex}}\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)\rangle =\left(\begin{array}{cc}{p}_{2}^{2}& 0\\ {p}_{2}^{2}a& 0\end{array}\right)\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)=\left(\begin{array}{c}{p}_{2}^{2}{v}_{1}\\ {p}_{2}^{2}a{v}_{1}\end{array}\right)$$
- (c)
- If $0<{p}_{1}<1$, then$${f}^{0}(p,v)=\langle \left(\begin{array}{cc}{p}_{2}^{2}& 0\\ 0& {p}_{2}^{2}\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),\phantom{\rule{0.222222em}{0ex}}\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)\rangle =\left(\begin{array}{cc}{p}_{2}^{2}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)=\left(\begin{array}{c}{p}_{2}^{2}{v}_{1}\\ 0\end{array}\right)$$

## 3. Relations between Solutions of Vector Optimization Problems and Vector Critical Points on Hadamard Manifolds

**Definition**

**8.**

- (a)
- An efficient solution for (VOP) if there does not exist another feasible point x such that$$f\left(x\right)-f\left(\overline{x}\right)\in -{\mathbb{R}}_{+}^{p}\backslash \left\{0\right\}.$$
- (b)
- A weakly efficient solution for (VOP) if there does not exist another feasible point x such that$$f\left(x\right)-f\left(\overline{x}\right)\in -\mathrm{int}{\mathbb{R}}_{+}^{p}.$$

**Definition**

**9.**

**Theorem**

**1.**

**Proof.**

- (a)
- We consider two points $x,\overline{x}\in X$ and assume that $f\left(x\right)-f\left(\overline{x}\right)\in -\mathrm{int}{\mathbb{R}}_{+}^{p}$. Then, $\overline{x}$ is not a weakly efficient solution of (VOP). By the hypothesis, we derive that $\overline{x}$ is not a VCP with respect to $\eta $, i.e., there do not exist some $x\in X\subseteq M$ with $\eta (x,\overline{x})\in {T}_{\overline{x}}M$ not identically zero and $\lambda \in {\mathbb{R}}_{+}^{p}\backslash \left\{0\right\}$ such that ${\lambda}^{T}A\eta (x,\overline{x})=0$ for some $A\in \partial f\left(\overline{x}\right)$. It follows from ([29], Theorem 5.1) that $A\eta (x,\overline{x})\in -\mathrm{int}{\mathbb{R}}_{+}^{p}$ and f is PIX.
- (b)
- For any points $x,\overline{x}\in X$ such that $f\left(x\right)-f\left(\overline{x}\right)\notin -\mathrm{int}{\mathbb{R}}_{+}^{p}$, we define $\eta (x,\overline{x})=0$, and therefore f is PIX with respect to $\eta $ on X.

**Corollary**

**1.**

**Example**

**5.**

**Corollary**

**2.**

- (a)
- θ is invex (IX) with respect to η on X.
- (b)
- Every critical point (CP) of θ with respect to η on X is a global minimum of θ on X.
- (c)
- θ is PIX with respect to η on X.

**Proof.**

- Firstly, if$$\theta \left(x\right)-\theta \left(\overline{x}\right)<0$$$$\theta \left(x\right)-\theta \left(\overline{x}\right)\ge 0,\phantom{\rule{0.222222em}{0ex}}\forall x\in X$$Therefore, $A\eta (x,\overline{x})<0$. Then, as ${\theta}^{0}(x,\xb7)$ is positively homogeneous, it follows that $\theta \left(x\right)-\theta \left(\overline{x}\right)\ge A\eta (x,\overline{x})$, and thus $\theta $ is IX with respect to $\eta (x,\overline{x})=tv$, where t is an arbitrary positive real number.
- Secondly, if$$\theta \left(x\right)-\theta \left(\overline{x}\right)\ge 0$$

## 4. Relations between Solutions of the Constrained VOP and KKT VCPs on Hadamard Manifolds

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Example**

**6.**

## 5. Application: Relations between Nash Equilibrium Points and Nash Critical Points

**Definition**

**13.**

**Definition**

**14.**

**Theorem**

**3.**

**Proof.**

**Example**

**7.**

## 6. Conclusions

- There is a need to extend the different concepts of invexity to Hadamard manifolds and clarify the relationships between them.
- It is important to use an adequate definition of VCPs or KKT-VCPs.
- There are applications of invexity in the search for equilibrium points, which are so desirable in economics.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A.
Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions. *Symmetry* **2020**, *12*, 804.
https://doi.org/10.3390/sym12050804

**AMA Style**

Ruiz-Garzón G, Osuna-Gómez R, Rufián-Lizana A.
Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions. *Symmetry*. 2020; 12(5):804.
https://doi.org/10.3390/sym12050804

**Chicago/Turabian Style**

Ruiz-Garzón, Gabriel, Rafaela Osuna-Gómez, and Antonio Rufián-Lizana.
2020. "Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions" *Symmetry* 12, no. 5: 804.
https://doi.org/10.3390/sym12050804