# ON THE CODERIVATIVE OF THE PROJECTION OPERATOR
ONTO THE POSITIVE CONE IN HILBERT SPACES

Le Van Hien111Faculty of Pedagogy, HaTinh University, Hatinh, Vietnam; email:
<EMAIL_ADDRESS>and Nguyen Viet Quan222Faculty of Pedagogy, HaTinh
University, Hatinh, Vietnam; email<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the generalized differentiability of the metric
projection operator onto the positive cone in Hilbert spaces. We first
establish the formula for exactly computing the regular coderivative and the
Mordukhovich coderivative of the metric projection operator onto the positive
cone in Euclidean spaces. Then, these results are also established for the
projection operator onto the positive cone in the real Hilbert space $l_{2}$.

Key words. metric projection, positive cone, regular coderivative,
Mordukhovich coderivative, Hilbert spaces

2010 AMS subject classification. 49J53, 90C31, 90C46

##  1 Introduction

Let $H$ be a Hilbert space with norm $\|.\|$ and let $C$ be a nonempty closed
convex subset of $H$. The metric projection operator $P_{C}$ is defined by

$P_{C}(x)=\\{y\in C:\inf\limits_{z\in C}\|x-z\|=\|x-y\|\\},\ \ \mbox{for all}\
\ x\in H.$

The metric projection operator $P_{C}$ is a well-defined single-valued mapping
that has many useful properties, such as continuity, monotonicity, and non-
expansiveness. Researching the metric projection operator in Hilbert space is
an interesting topic and receives the attention of many people, because of its
applications in optimization, theory of equation, control theory and many
other fields; see [3, 4, 8, 9, 11, 12, 13, 15, 14, 16, 17, 18, 19, 23, 27, 28]
and the references therein.

In addition to the above properties, the smoothness of the projection operator
is also of special interest. Recently, in their articles, Li has studied the
Gâteaux directional differentiability of the projection operator in Bochner
space [11] and the projection operator in the real uniformly convex and
uniformly smooth Banach space [13], in this paper, Li gave the exact
representations of the directional derivatives of $P_{C}$ with $C$ is a closed
ball or a closed and convex cone (including proper closed subspaces); studied
the strict Fréchet differentiability of the projection operator in Hilbert
spaces [14], the Fréchet differentiability of the projection operator in
uniformly convex and uniformly smooth Banach spaces [12]. In that direction of
research, Hien [7] also studied the strict Fréchet differentiability of the
projection operator onto closed balls with centerat arbitrarily given point in
Hilbert spaces and onto the second-order cones in Euclidean spaces.

Generalized differentiation lies at the heart of variational analysis andits
applications. In [21], Mordukhovich introduced the concept of generalized
differentiation and provide many useful properties, of which the coderivative
is an important tool because of its applications in optimization and
algorithms. Therefore, research on the coderivative is one of the interesting
topics, attracting the attention of many people; see [2, 5, 10, 20, 21, 22,
24, 25, 29] and the references therein. Recently, calculating the coderivative
of the metric projection operator has also obtained some interesting results.
In [24], Outrata and Sun have calculated the coderivative of the metric
projection onto the second-order cone in $\mathbb{R}^{n}$, then, this result
was used to obtain a sufficientcondition for the Aubin property of the
solution map of a parameterized secondorder cone complementarity problem and
to derive necessary optimality conditionsfor a mathematical program with a
second-order cone complementarity problemamong the constraints. Li [15, 16,
17] investigated the properties of the regular coderivative of the metric
projection operator onto some closed balls, closed and convex cylinders and
positive cones in Hilbert spaces, in uniformly convex and uniformly smooth
Banach spaces, and in some general Banach spaces. In [6], Hien gave the
complete calculation formula for the regular coderivative of the projection
operator onto closed balls with center at arbitrarily given point in Hilbert
spaces.

The aim of this paper is to establish a formula for calculating the
coderivative of the metric projection operator onto the positive cone in
Hilebert spaces. This problem has been studied by Li [15], however the results
on the regular coderivative of the metric operator projected onto the positive
cones in [15] is only given in some special cases and has an inaccurate
conclusion. More precisely, we first establish aformula for exactly computing
the regular coderivative and Mordukhovich coderivative of the metric
projection operator onto the positive cone in Euclidean space
$\mathbb{R}^{n}$. Then, these formulas are also established for the projection
operator onto the positive cone in the real Hilbert space $l_{2}$.

The structure of the paper is as follows. In section 2, we recall some
preliminary materials. In section 3, we present results on the regular
coderivative and Mordukhovich coderivative of the metric projection operator
onto the positive cone in $\mathbb{R}^{n}$. Then, in section 4, we give the
exact calculation formula for the regular coderivative and Mordukhovich
coderivative of the metric projection operator onto the positive cone in
$l_{2}$. Finally, we conclude the paper in section 5 where we discuss some
perspectives of the obtained results and future works.

##  2 Preliminaries

We first give the following notations that will be used throughout the paper.
Let $X$ be Hilbert space with scalar product $\langle.,.\rangle$ and norm
$\|.\|$. Given a set $C\subset X,$ we denote by $C^{o}$ its interior. Denote
$t\downarrow 0$ means that $t\to 0$ with $t>0$. The symbol $\mathbb{N}$ is the
set of natural numbers.

Below are basic notions and facts from variational analysis, which are
frequently used in the following; see [1, 14, 21, 25, 26] for more details.

First, We recall the concepts of Gâteaux directional differentiability and the
Fréchet differentiability of the metric projection operator
$P_{C}:H\rightarrow C.$ For $\bar{x}\in X$ and $w\in X\backslash\\{\theta\\}$,
if the following limit exists,

$P_{C}^{{}^{\prime}}(\bar{x})(w)=\lim\limits_{t\downarrow
0}\dfrac{P_{C}(\bar{x}+tw)-P_{C}(\bar{x})}{t},$

then, $P_{C}$ is said to be Gâteaux directionally differentiable at point
$\bar{x}$ along direction $w$ and $P_{C}^{{}^{\prime}}(\bar{x})(w)$ is called
the Gâteaux directional derivative of $P_{C}$ at point $\bar{x}$ along
direction $w$.

$P_{C}$ is called Fréchet differentiable at $\bar{x}\in X$ if there is a
linear continuous mapping $\nabla P_{C}(\bar{x}):X\to X$ such that

$\lim\limits_{h\to 0}\dfrac{\|P_{C}(\bar{x}+h)-P_{C}(\bar{x})-\nabla
P_{C}(\bar{x})(h)\|}{\|h\|}=0.$

Then, $\nabla P_{C}(\bar{x})$ is said to be the Fréchet derivative of $P_{C}$
at $\bar{x}$.

In particular, the mapping $P_{C}$ is called strictly Fréchet differentiable
at $\bar{x}\in X$ if

$\lim\limits_{(u,v)\to(\bar{x},\bar{x})}\dfrac{\|P_{C}(u)-P_{C}(v)-\nabla
P_{C}(\bar{x})(u-v)\|}{\|u-v\|}=0.$

The mapping $P_{C}$ is called Gâteaux directionally differentiable (Fréchet
differentiable, strictly Fréchet differentiable) on $\Omega\subset X$ if
$P_{C}$ is Gâteaux directionally differentiable (Fréchet differentiable,
strictly Fréchet differentiable, respectively) at every point $x\in\Omega$.

######  Definition 2.1.

(see [21, 25]) Let $\Omega$ be a nonempty subset of a Hilbert space $H$ and
$\bar{x}\in\Omega$, we denote by $\widehat{N}_{\Omega}(\bar{x})$ the Fréchet
(regular) normal cone to $\Omega$ at $\bar{x}$, defined by

$\widehat{N}_{\Omega}(\bar{x}):=\Big{\\{}z\in
H|\limsup\limits_{u\xrightarrow{\Omega}\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle}{\|u-\bar{x}\|}\leq 0\Big{\\}}.$

The limiting (Mordukhovich) normal cone to $\Omega$ at $\bar{x}$, denoted
$N_{\Omega}(\bar{x})$, is defined by

$N_{\Omega}(\bar{x}):=\mathop{{\rm Lim}\,{\rm
sup}}\limits_{u\xrightarrow{\Omega}\bar{x}}\widehat{N}_{\Omega}(x)$

where "Lim sup" is the Painlevé-Kuratowski outer limit of sets (see [25])

If $\bar{x}\not\in\Omega$, one puts
$\widehat{N}_{\Omega}(\bar{x})=N_{\Omega}(\bar{x})=\emptyset$ by convention.
When the set $\Omega$ is convex,
$\widehat{N}_{\Omega}(\bar{x})=N_{\Omega}(\bar{x})$ amounts to the classic
normal cone in the sense of convex analysis.

######  Definition 2.2.

(see [21, 25]) Let $H$ be Hilbert space and $\Phi:H\rightrightarrows H$ be a
set-valued mapping with its graph gph$\Phi:=\\{(x,y)|y\in\Phi(x)\\}$ and its
domain Dom$\Phi:=\\{x|\Phi(x)\not=\emptyset\\}$. The multifunction
$\widehat{D}^{\ast}\Phi(\bar{x},\bar{y}):H\rightrightarrows H$, defined by

$\widehat{D}^{\ast}\Phi(\bar{x},\bar{y})(w):=\big{\\{}v\in
H|(v,-w)\in\widehat{N}_{\mbox{\rm gph}\Phi}(\bar{x},\bar{y})\\}\ \mbox{for
all}\ w\in H$

is called regular coderivative of $\Phi$ at $(\bar{x},\bar{y}).$

Analogously, the multifunction
$D^{\ast}\Phi(\bar{x},\bar{y}):H\rightrightarrows H$, defined by

$D^{\ast}\Phi(\bar{x},\bar{y})(w):=\big{\\{}v\in H|(v,-w)\in N_{\mbox{\rm
gph}\Phi}(\bar{x},\bar{y})\\}\ \mbox{for all}\ w\in H$

is called limiting (Mordukhovich) coderivative of $\Phi$ at
$(\bar{x},\bar{y}).$

In the case $\Phi(\bar{x})=\\{\bar{y}\\},$ we usually write
$\widehat{D}^{\ast}\Phi(\bar{x})(D^{\ast}\Phi(\bar{x})).$ If
$\Phi:H\rightarrow H$ is strictly Fréchet differentiable at $\bar{x}$, then
$D^{\ast}\Phi(\bar{x})(y)=\widehat{D}^{\ast}\Phi(\bar{x})(y)=\\{\nabla\Phi(\bar{x})(y)\\},\
\mbox{for all}\ y\in H.$

Note that gph$\Phi\subset H\times H$, where $H\times H$ be the orthogonal
product of $H$ equipped with the inner product $\langle.,.\rangle_{H\times
H}$, which is abbreviated as $\langle.,.\rangle$. It is defined by,

$\langle(x,y),(u,v)\rangle=\langle x,v\rangle+\langle y,v\rangle,\ \ \mbox{for
any}\ (x,y),(u,v)\in H\times H.$

Let $\|.\|_{H\times H}$ be the norm in $H\times H$ induced by the inner
product $\langle.,.\rangle$. Then, we have

$\|(x,y)\|_{H\times
H}=\sqrt{\langle(x,y),(x,y)\rangle}=\sqrt{\|x\|^{2}+\|y\|^{2}}\ \ \mbox{for
all}\ x,y\in H.$

The following result is given in [6], which provides a useful formula to
calculate the Fréchet coderivatives of single-valued mappings on Hilbert
spaces.

######  Lemma 2.3.

(see [6]) Let $H$ be a Hilbert space, $f:H\rightarrow H$ be a Lipschitz
continuous mapping on $H$. Then, the regular coderivatives of $f$ at a point
$\bar{x}\in H$ satisfies that, for any $y\in H$,

$\widehat{D}^{\ast}f(\bar{x})(y)=\left\\{z\in
H\big{|}\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,f(u)-f(\bar{x})\rangle}{\|u-\bar{x}\|}\leq
0\right\\}.$

##  3 Coderivative of the metric projection onto the positive cone in
$\mathbb{R}^{n}$

Let $\mathbb{R}^{n}$ denote the n-dimensional Euclidean space with origin
$\theta$. We recall some notations and concepts from [14], which will be used
in this section. For any $x=(x_{1},...,x_{n})\in\mathbb{R}^{n}$, we define
three subsets of the set $\\{1,2,...,n\\}$ with respect to the given $x$ by

$x^{+}:=\\{i\in\\{1,...,n\\}:x_{i}>0\\};$
$x^{-}:=\\{i\in\\{1,...,n\\}:x_{i}<0\\};$
$x^{\bullet}:=\\{i\in\\{1,...,n\\}:x_{i}=0\\}.$

Let K denote the positive cone of $\mathbb{R}^{n}$, which is defined by

$K:=\\{x\in\mathbb{R}^{n}:x^{-}=\emptyset\\}.$

Then, $K$ is a pointed closed and convex cone in $\mathbb{R}^{n}$ with the
interior given by

$K^{o}=\\{x\in\mathbb{R}^{n}:x^{-}=x^{\bullet}=\emptyset\\}.$

Define

$\widehat{K}:=\\{x\in\mathbb{R}^{n}:x^{+}\not=\emptyset,x^{-}\not=\emptyset,x^{\bullet}=\emptyset\\},$

and

$\Delta\mathbb{R}^{n}=\\{x\in\mathbb{R}^{n}:x^{\bullet}\not=\emptyset\\}.$

For any given fixed $x\in\widehat{K}$, we define a mapping
$b(x;.):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, by

$b(x;w)_{i}:=\begin{cases}w_{i}\ \ \ \ \mbox{if}\ i\in x^{+},\\\ 0\ \ \ \ \ \
\mbox{if}\ i\not\in x^{+},\end{cases}\ \ \mbox{for all}\ w\in\mathbb{R}^{n}.$

It follows from [14] that the metric projection operator
$P_{K}:\mathbb{R}^{n}\rightarrow K$ has the following properties:

(i)

$P_{K}(x)_{i}:=\begin{cases}x_{i}\ \ \ \ \mbox{if}\ i\in x^{+},\\\ 0\ \ \ \ \
\mbox{if}\ i\not\in x^{+},\end{cases}\ \ \mbox{for all}\ x\in\mathbb{R}^{n}.$

(ii) $P_{K}$ is a Lipschitz continuous mapping on $\mathbb{R}^{n}$ with
Lipschitz constant $L=1$, i.e

$\|P_{K}(x)-P_{K}(y)\|\leq\|x-y\|,\ \ \mbox{for all}\ x,y\in\mathbb{R}^{n}.$

(iii) $P_{K}$ is strictly Fréchet differentiable on $K^{o},(-K)^{o}$ and
$\widehat{K}$; $P_{K}$ is Gâteaux directionally differentiable but not Fréchet
differentiable on $\Delta\mathbb{R}^{n}$.

The following result provides the formula for calculating the regular
coderivatives of the metric projection onto the positive cone in
$\mathbb{R}^{n}$.

######  Theorem 3.1.

Let $K$ be the positive cone of $\mathbb{R}^{n}$. Then, the regular
coderivative of the projection operator $P_{K}$ at $\bar{x}\in\mathbb{R}^{n}$
is given by

$\widehat{D}^{\ast}P_{K}(\bar{x})(y)=\left\\{z\in\mathbb{R}^{n}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq
z_{i}\leq y_{i},i\in\bar{x}^{\bullet}\right\\},\ \ \mbox{for all}\
y\in\mathbb{R}^{n}.$ (3.1)

Proof.

Let $y\in\mathbb{R}^{n}$, thanks to the Lipschitz continuity of $P_{K}$, by
Lemma 2.3, we have

$z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)\iff\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\leq 0.$ (3.2)

Suppose that $z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)$. We take a directional
line segment in the limit in (3.2), $u=(u_{i})\in\mathbb{R}^{n}$ with
$u_{i}=\bar{x}_{i}+t(z_{i}-y_{i}),\ \mbox{for}\ i\in\bar{x}^{+};\
u_{i}=\bar{x}_{i}+tz_{i},\ \mbox{for}\ i\in\bar{x}^{-};\ u_{i}=0,\ \mbox{for}\
i\in\bar{x}^{\bullet}$, where $t\downarrow 0$. It follows that

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\limsup\limits_{t\downarrow
0}\dfrac{t\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})^{2}+t\sum\limits_{i\in\bar{x}^{-}}z_{i}^{2}}{\sqrt{t^{2}\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})^{2}+t^{2}\sum\limits_{i\in\bar{x}^{-}}z_{i}^{2}}}\\\
&=\sqrt{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})^{2}+\sum\limits_{i\in\bar{x}^{-}}z_{i}^{2}}.\end{array}$

This implies that

$z_{i}=y_{i},\ \mbox{for all}\ i\in\bar{x}^{+}\ \ \mbox{and}\ \ z_{i}=0,\
\mbox{for all}\ i\in\bar{x}^{-}.$ (3.3)

Assume, by the way of contradiction, that that there exists
$i_{0}\in\bar{x}^{\bullet}$ such that $z_{i_{0}}>y_{i_{0}}$. Choosing
$u=(u_{i})\in\mathbb{R}^{n}$ with $u_{i}=\bar{x}_{i},\ \mbox{for }\
i\not=i_{0};\ u_{i_{0}}=t$ for $t\downarrow 0$. We have

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\limsup\limits_{t\downarrow
0}\dfrac{t\left(z_{i_{0}}-y_{i_{0}}\right)}{t}\\\
&=z_{i_{0}}-y_{i_{0}}.\end{array}$

This means $z_{i_{0}}\leq y_{i_{0}}$. This contradiction proves that

$z_{i}\leq y_{i}\ \ \ \ \ \mbox{for all}\ i\in\bar{x}^{\bullet}.$ (3.4)

Similarly, suppose there exists $i_{0}\in\bar{x}^{\bullet}$ such that
$z_{i_{0}}<0$. By choosing $u=(u_{i})\in\mathbb{R}^{n}$ with
$u_{i}=\bar{x}_{i},\ \mbox{for }\ i\not=i_{0};\ u_{i_{0}}=-t$ for $t\downarrow
0$. We have

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\ast}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\limsup\limits_{t\downarrow 0}\dfrac{-tz_{i_{0}}}{t}\\\
&=-z_{i_{0}}.\end{array}$

This shows that

$z_{i}\geq 0,\ \ \ \mbox{for all}\ i\in\bar{x}^{\bullet}.$ (3.5)

By (3.3), (3.4) and (3.5) together, we have

$z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)\Longrightarrow\begin{cases}z_{i}=y_{i}\
\ \ \ \ \ \ \ \mbox{if}\ \ i\in\bar{x}^{+}\\\ z_{i}=0\ \ \ \ \ \ \ \ \
\mbox{if}\ i\in\bar{x}^{-}\\\ 0\leq z_{i}\leq y_{i}\ \ \ \mbox{if}\
i\in\bar{x}^{\bullet}\end{cases}$ (3.6)

Next, we prove the converse, assuming $z=(z_{i})\in\mathbb{R}^{n}$ with
$z_{i}=y_{i},\ i\in\bar{x}^{+};$$z_{i}=0,\ i\in\bar{x}^{-}$ and $0\leq
z_{i}\leq y_{i},\ i\in\bar{x}^{\bullet}$. We need to show that

$\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\leq 0.$

Indeed, we have

$\begin{array}[]{rl}&\dfrac{\langle z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&=\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\dfrac{\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\dfrac{\sum\limits_{i\in\bar{x}^{\bullet}\cap
u^{+}}(z_{i}-y_{i})u_{i}+\sum\limits_{i\in\bar{x}^{\bullet}\backslash
u^{+}}z_{i}u_{i}}{\|u-\bar{x}\|}\\\ &\leq 0\ \ \ \ \mbox{for all}\
u\in\mathbb{R}^{n}\backslash\\{\bar{x}\\}.\end{array}$

So, $\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\leq 0$. This means

$\begin{cases}z_{i}=y_{i}\ \ \ \ \ \ \ \ \mbox{if}\ \ i\in\bar{x}^{+}\\\
z_{i}=0\ \ \ \ \ \ \ \ \ \mbox{if}\ i\in\bar{x}^{-}\\\ 0\leq z_{i}\leq y_{i}\
\ \ \mbox{if}\ i\in\bar{x}^{\bullet}\end{cases}\Longrightarrow
z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y).$ (3.7)

Then, by (3.6) and (3.7), (3.1) is proved. $\square$

Using Theorem 3.1, we obtain the following result, part of which was given by
Li in [15].

######  Corollary 3.2.

(see [15, Theorem 4.1])  Let $K$ be the positive cone of $\mathbb{R}^{n}$ with
negative cone $-K$. Then, the regular coderivative of the projection operator
$P_{K}$ have the following representations.

(i) For any $y\in\mathbb{R}^{n}$, we have

$\widehat{D}^{\ast}P_{K}(\bar{x})(y)=\begin{cases}\\{y\\}\ \ \ \ \ \ \ \ \
\mbox{if}\ \bar{x}\in K^{o}\\\ \\{\theta\\}\ \ \ \ \ \ \ \ \ \mbox{if}\
\bar{x}\in(-K)^{o}\\\ \\{b(\bar{x};y)\\}\ \ \ \mbox{if}\
\bar{x}\in\widehat{K}\\\ \end{cases}$

(ii) For any $\bar{x}\in\Delta\mathbb{R}^{n}$, we have

(a) $\widehat{D}^{\ast}P_{K}(\bar{x})(\theta)=\\{\theta\\};$

(b) For any $y\in\mathbb{R}^{n}$,

$y^{-}\cap\bar{x}^{\bullet}\not=\emptyset\Longrightarrow\lambda
y\not\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)\ \mbox{for all}\ \lambda<1.$

In particular,

$y^{-}\cap\bar{x}^{\bullet}\not=\emptyset\Longrightarrow\theta\not\in\widehat{D}^{\ast}P_{K}(\bar{x})(y).$

(c) For any $y\in\mathbb{R}^{n}$,
$y^{-}\cap\bar{x}^{\bullet}\not=\emptyset\Longrightarrow\widehat{D}^{\ast}P_{K}(\bar{x})(y)=\emptyset.$

(d) $\widehat{D}^{\ast}P_{K}(\bar{x})(\bar{x})=\\{P_{K}(\bar{x})\\}$.

(e)
$\widehat{D}^{\ast}P_{K}(\theta)(y)=\displaystyle\prod_{i=1}^{n}[0,y_{i}].$

Proof. (i) For any $\bar{x}\in K^{o}$, we have
$\bar{x}^{-}=\bar{x}^{\bullet}=\emptyset$. For any $y\in\mathbb{R}^{n}$, using
(3.1), we get

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{K}(\bar{x})(y)&=\left\\{z\in\mathbb{R}^{n}:z_{i}=y_{i},i\in\bar{x}^{+}\right\\}.\\\
&=\\{z\in\mathbb{R}^{n}:z_{i}=y_{i},\forall i=1,2,...,n\\}\\\
&=\\{y\\}.\end{array}$

For any $\bar{x}\in(-K)^{o}$, we have
$\bar{x}^{+}=\bar{x}^{\bullet}=\emptyset$. For any $y\in\mathbb{R}^{n}$, using
(3.1), we get

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{K}(\bar{x})(y)&=\left\\{z\in\mathbb{R}^{n}:z_{i}=0,i\in\bar{x}^{-}\right\\}.\\\
&=\\{z\in\mathbb{R}^{n}:z_{i}=0,\forall i=1,2,...,n\\}\\\
&=\\{\theta\\}.\end{array}$

For any $\bar{x}\in\widehat{K}$, we have $\bar{x}^{\bullet}=\emptyset$. For
any $y\in\mathbb{R}^{n}$, using (3.1), we get

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{K}(\bar{x})(y)&=\left\\{z\in\mathbb{R}^{n}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-}\right\\}.\\\
&=\left\\{z\in\mathbb{R}^{n}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\not\in\bar{x}^{+}\right\\}.\\\
&=\\{b(\bar{x};y)\\}.\end{array}$

(ii.a) Using (3.1) with $y=0$, we have

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{K}(\bar{x})(\theta)&=\left\\{z\in\mathbb{R}^{n}:z_{i}=0,i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq
z_{i}\leq 0,i\in\bar{x}^{\bullet}\right\\}\\\
&=\\{z\in\mathbb{R}^{n}:z_{i}=0,\forall i=1,2,...,n\\}\\\
&=\\{\theta\\}.\end{array}$

(ii.b) If $y^{-}\cap\bar{x}^{\bullet}\not=\emptyset$, then there exists
$i_{0}\in\bar{x}^{\bullet}$ such that $y_{i_{0}}<0$. We have $\lambda
y_{i_{0}}>y_{i_{0}}$ for all $\lambda<1$. By (3.1), $\lambda
y\not\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)\ \mbox{for all}\ \lambda<1.$ In
particular, for $\lambda=0$, we get
$\theta\not\in\widehat{D}^{\ast}P_{K}(\bar{x})(y).$

(ii.c) Since $y^{-}\cap\bar{x}^{\bullet}\not=\emptyset$, there exists
$i_{0}\in\bar{x}^{\bullet}$ such that $y_{i_{0}}<0$. This means that there
does not exist $z=(z_{i})\in\mathbb{R}^{n}$ such that $0\leq z_{i_{0}}\leq
y_{i_{0}}$, and $\widehat{D}^{\ast}P_{K}(\bar{x})(y)=\emptyset.$

(ii.d) Using (3.1) with $y=\bar{x}$, we have

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{K}(\bar{x})(\bar{x})&=\left\\{z\in\mathbb{R}^{n}:z_{i}=\bar{x}_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq
z_{i}\leq 0,i\in\bar{x}^{\bullet}\right\\}\\\
&=\left\\{z\in\mathbb{R}^{n}:z_{i}=\bar{x}_{i},i\in\bar{x}^{+};z_{i}=0,i\not\in\bar{x}^{+}\right\\}\\\
&=\\{P_{K}(\bar{x})\\}.\end{array}$

(ii.e) Using (3.1) with $\bar{x}=\theta$, we have
$\bar{x}^{+}=\bar{x}^{-}=\emptyset,\ \bar{x}^{\bullet}=\\{1,2,...,n\\}$ and

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{K}(\theta)(y)&=\left\\{z\in\mathbb{R}^{n}:0\leq
z_{i}\leq y_{i},i\in\bar{x}^{\bullet}\right\\}.\\\
&=\left\\{z\in\mathbb{R}^{n}:0\leq z_{i}\leq y_{i},\forall
i=1,2,...,n\right\\}\\\ &=\displaystyle\prod_{i=1}^{n}[0,y_{i}].\end{array}$

$\square$

######  Remark 3.3.

By Corollary 3.2 (ii.d), the statements of [15, Theorem 4.1 (iv.c)]

"Let $\bar{x}\in\Delta\mathbb{R}^{n}\backslash\\{\theta\\}$. For any
$z\in\mathbb{R}^{n}$, we have,

$z\not=\theta\Longrightarrow
z\not\in\widehat{D}^{\ast}P_{K}(\bar{x})(\bar{x}),$ (3.8) $\mbox{and}\ \
\bar{x}^{+}\not=\emptyset\Longrightarrow\widehat{D}^{\ast}P_{K}(\bar{x})(\bar{x})=\emptyset.$
(3.9)

"

are incorrect.

Next, we compute the limiting (Mordukhovich) coderivative
$D^{\ast}P_{K}(\bar{x})$. Since $P_{K}(.)$ is continuous, the graph of $P_{K}$
is closed and, by [25, Equation 8(18)], we know that

$D^{\ast}P_{K}(\bar{x})(y)=\mathop{{\rm Lim}\,{\rm
sup}}\limits_{x\rightarrow\bar{x},y^{\prime}\rightarrow
y}\widehat{D}^{\ast}P_{K}(x)(y^{\prime})$ (3.10)

This, together with Theorem 3.1, allows us to provide a complete
characterization of $D^{\ast}P_{K}(\bar{x})$.

######  Theorem 3.4.

Let $K$ be the positive cone of $\mathbb{R}^{n}$. Then, the Mordukhovich
coderivative of the projection operator $P_{K}$ at $\bar{x}\in\mathbb{R}^{n}$
is given by

$\begin{array}[]{rl}D^{\ast}P_{K}(\bar{x})(y)&=\widehat{D}^{\ast}P_{K}(\bar{x})(y)\\\
&=\left\\{z\in\mathbb{R}^{n}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq
z_{i}\leq y_{i},i\in\bar{x}^{\bullet}\right\\},\end{array}$ (3.11)

$\mbox{for all}\ y\in\mathbb{R}^{n}.$

Proof. Since $\widehat{D}^{\ast}P_{K}(\bar{x})(y)\subset
D^{\ast}P_{K}(\bar{x})(y)$, it suffices to prove

$D^{\ast}P_{K}(\bar{x})(y)\subset\widehat{D}^{\ast}P_{K}(\bar{x})(y).$

Take $z\in D^{\ast}P_{K}(\bar{x})(y)$. Then by definition, there exist
$y^{k}\rightarrow y,x^{k}\rightarrow\bar{x}$ and $z^{k}\rightarrow z$ such
that $z^{k}\in\widehat{D}^{\ast}P_{K}(x^{k})(y^{k})$, for all
$k\in\mathbb{N}.$ By (3.1), we have

$\begin{cases}z^{k}_{i}=y^{k}_{i}\ \ \ \ \ \ \ \ \mbox{if}\ \
i\in(x^{k})^{+}\\\ z^{k}_{i}=0\ \ \ \ \ \ \ \ \ \ \mbox{if}\
i\in(x^{k})^{-}\\\ 0\leq z^{k}_{i}\leq y^{k}_{i}\ \ \ \mbox{if}\
i\in(x^{k})^{\bullet}\end{cases}\ \ \ \ \mbox{for all}\ k\in\mathbb{N}.$

Since $x^{k}\rightarrow\bar{x}$, we have
$\bar{x}^{+}\subset(x^{k})^{+},\bar{x}^{-}\subset(x^{k})^{-}$ and
$(x^{k})^{\bullet}\subset\bar{x}^{\bullet}$. So, for all
$i\in\bar{x}^{+}\subset(x^{k})^{+}$, we have

$z_{i}=\lim\limits_{k\rightarrow\infty}z^{k}_{i}=\lim\limits_{k\rightarrow\infty}y^{k}_{i}=y_{i}.$
(3.12)

For all $i\in\bar{x}^{-}\subset(x^{k})^{-}$,

$z_{i}=\lim\limits_{k\rightarrow\infty}z^{k}_{i}=\lim\limits_{k\rightarrow\infty}0=0.$
(3.13)

For $i\in\bar{x}^{\bullet}$, using a subsequence if necessary, we consider the
following cases.

Case 1. $i\in(x^{k})^{+},$ for all $k\in\mathbb{N}$. We have
$z_{i}=\lim\limits_{k\rightarrow\infty}z^{k}_{i}=\lim\limits_{k\rightarrow\infty}y^{k}_{i}=y_{i}.$

Case 2. $i\in(x^{k})^{-},$ for all $k\in\mathbb{N}$. We have
$z_{i}=\lim\limits_{k\rightarrow\infty}z^{k}_{i}=\lim\limits_{k\rightarrow\infty}0=0.$

Case 3. $i\in(x^{k})^{\bullet},$ for all $k\in\mathbb{N}$. We have $0\leq
z^{k}_{i}\leq y^{k}_{i}$ for all $k\in\mathbb{N}.$ Taking the limits as
$k\rightarrow\infty$, we get $0\leq z_{i}\leq y_{i}$.

Thus, for any $i\in\bar{x}^{\bullet}$,

$0\leq z_{i}\leq y_{i}$ (3.14)

Then, by (3.12), (3.13), (3.14) and (3.1),
$z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y).$ The proof is complete. $\square$

To end this section, we give some examples to demonstrate the results of
Theorems 3.1 and 3.4.

######  Example 3.5.

Let $\bar{x}=(\bar{x}_{1},\bar{x}_{2})\in\mathbb{R}^{2}$. For all
$y=(y_{1},y_{2})\in\mathbb{R}^{2},$ we have

$\widehat{D}^{\ast}P_{K}(\bar{x})(y)=D^{\ast}P_{K}(\bar{x})(y)=\begin{cases}\\{y\\}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ \bar{x}_{1},\bar{x}_{2}>0,\\\
\left\\{(0,0)\right\\}\ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \
\bar{x}_{1},\bar{x}_{2}<0,\\\ \left\\{(\bar{x}_{1},0)\right\\}\ \ \ \ \ \ \ \
\ \ \ \ \mbox{if}\ \ \bar{x}_{1}>0,\bar{x}_{2}<0,\\\
\left\\{(0,\bar{x}_{2})\right\\}\ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \
\bar{x}_{1}<0,\bar{x}_{2}>0,\\\ \left\\{y_{1}\right\\}\times[0,y_{2}]\ \ \ \ \
\ \ \mbox{if}\ \ \bar{x}_{1}>0,\bar{x}_{2}=0,\\\
[0,y_{1}]\times\left\\{y_{2}\right\\}\ \ \ \ \ \ \ \mbox{if}\ \
\bar{x}_{1}=0,\bar{x}_{2}>0,\\\ \left\\{0\right\\}\times[0,y_{2}]\ \ \ \ \ \ \
\ \ \mbox{if}\ \ \bar{x}_{1}<0,\bar{x}_{2}=0,\\\
[0,y_{1}]\times\left\\{0\right\\}\ \ \ \ \ \ \ \ \ \mbox{if}\ \
\bar{x}_{1}=0,\bar{x}_{2}<0,\\\ [0,y_{1}]\times[0,y_{2}]\ \ \ \ \ \ \
\mbox{if}\ \ \bar{x}_{1}=\bar{x}_{2}=0.\end{cases}$

######  Example 3.6.

Let $a,b,c>0.$ Then, for any $y=(y_{1},y_{2},y_{3})\in\mathbb{R}^{3}$ we have

$\widehat{D}^{\ast}P_{K}(a,b,c)(y)=D^{\ast}P_{K}(a,b,c)(y)=\\{y\\}.$
$\widehat{D}^{\ast}P_{K}(-a,-b,-c)(y)=D^{\ast}P_{K}(-a,-b,-c)(y)=\\{(0,0,0)\\}.$
$\widehat{D}^{\ast}P_{K}(a,b,0)(y)=D^{\ast}P_{K}(a,b,0)(y)=\\{y_{1}\\}\times\\{y_{2}\\}\times[0,y_{3}].$
$\widehat{D}^{\ast}P_{K}(a,b,-c)(y)=D^{\ast}P_{K}(a,b,-c)(y)=\\{(y_{1},y_{2},0)\\}.$
$\widehat{D}^{\ast}P_{K}(a,-b,-c)(y)=D^{\ast}P_{K}(a,-b,-c)(y)=\\{(y_{1},0,0)\\}.$
$\widehat{D}^{\ast}P_{K}(0,-b,-c)(y)=D^{\ast}P_{K}(0,-b,-c)(y)=[0,y_{1}]\times\\{0\\}\times\\{0\\}.$
$\widehat{D}^{\ast}P_{K}(a,0,-c)(y)=D^{\ast}P_{K}(a,0,-c)(y)=\\{y_{1}\\}\times[0,y_{2}]\times\\{0\\}.$
$\widehat{D}^{\ast}P_{K}(a,0,0)(y)=D^{\ast}P_{K}(a,0,0)(y)=\\{y_{1}\\}\times[0,y_{2}]\times[0,y_{3}].$
$\widehat{D}^{\ast}P_{K}(0,0,-c)(y)=D^{\ast}P_{K}(0,0,-c)(y)=[0,y_{1}]\times[0,y_{2}]\times\\{0\\}.$
$\widehat{D}^{\ast}P_{K}(0,0,0)(y)=D^{\ast}P_{K}(0,0,0)(y)=[0,y_{1}]\times[0,y_{2}]\times[0,y_{3}].$

##  4 Coderivative of the metric projection onto the positive cone in real
Hilbert space $l_{2}$

In this section, we consider the real Hilbert space $l_{2}$ with norm $\|.\|$
and inner product $\langle.,.\rangle$. Let $\mathbb{K}$ denote the positive
cone of $l_{2}$ defined by

$\mathbb{K}:=\left\\{x=(x_{1},x_{2},...)\in l_{2}:x_{i}\geq 0,\ \mbox{for
all}\ i\in\mathbb{N}\right\\}.$

Let $P_{\mathbb{K}}:l_{2}\rightarrow\mathbb{K}$ be the metric projection
operator from $l_{2}$ onto $\mathbb{K}$. It follows from [14] that for any
$x\in l_{2}$, $P_{\mathbb{K}}(x)$ is represented as follows

$P_{\mathbb{K}}(x)_{i}=\begin{cases}x_{i}\ \ \ \ \ \mbox{if}\ x_{i}>0;\\\ 0\ \
\ \ \ \ \mbox{if}\ x_{i}\leq 0;\end{cases}\mbox{for}\ i\in\mathbb{N}.$

Furthermore, $P_{\mathbb{K}}$ is a Lipschitz continuous mapping on $l_{2}$
with Lipschitz constant $L=1$, i.e

$\|P_{\mathbb{K}}(x)-P_{\mathbb{K}}(y)\|\leq\|x-y\|,\ \mbox{for all }\ x,y\in
l_{2}.$

And $P_{\mathbb{K}}$ is positive homogeneous, i.e

$P_{\mathbb{K}}(\lambda x)=\lambda P_{\mathbb{K}}(x),\ \ \mbox{for any}\ \
x\in l_{2}\ \mbox{and}\ \lambda>0.$

By [14, Theorem 5.3], $P_{\mathbb{K}}$ is Gâteaux directionally differentiable
but not Fréchet differentiable on sets $\mathbb{K}^{+},\mathbb{K}^{-}$ and
$\widehat{\mathbb{K}}$. Where,

$\mathbb{K}^{+}:=\left\\{x=(x_{1},x_{2},...)\in l_{2}:x_{i}>0,\ \mbox{for
all}\ i\in\mathbb{N}\right\\};$
$\mathbb{K}^{-}:=\left\\{x=(x_{1},x_{2},...)\in l_{2}:x_{i}<0,\ \mbox{for
all}\ i\in\mathbb{N}\right\\};$
$\begin{array}[]{rl}\widehat{\mathbb{K}}:=\\{&x=(x_{1},x_{2},...)\in
l_{2}:|x_{i}|>0,\ \mbox{for all}\ i\in\mathbb{N}\ \mbox{and}\\\ &\mbox{there
are at least one pair}\ j,k\in\mathbb{N}\ \mbox{with}\
x_{j}x_{k}<0\\}.\end{array}$

For any $x=(x_{1},x_{2},...)\in l_{2}$, define

$x^{+}:=\\{i\in\mathbb{N}:x_{i}>0\\};$ $x^{-}:=\\{i\in\mathbb{N}:x_{i}<0\\};$
$x^{\bullet}:=\\{i\in\mathbb{N}:x_{i}<0\\}.$

The following result provides the formula for calculating the regular
coderivatives of the metric projection onto the positive cone in $l_{2}$.

######  Theorem 4.1.

Let $\mathbb{K}$ be the positive cone of $l_{2}$. Then, the regular
coderivative of the projection operator $P_{\mathbb{K}}$ at $\bar{x}\in l_{2}$
is given by

$\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)=\left\\{z\in
l_{2}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq z_{i}\leq
y_{i},i\in\bar{x}^{\bullet}\right\\},\ \ \mbox{for all}\ y\in l_{2}.$ (4.1)

Proof. Similar to the proof of Theorem 3.1. Take $\bar{x}\in l_{2}$.

Let $y\in l_{2}$, thanks to the Lipschitz continuity of $P_{K}$, by Lemma 2.3,
we have

$z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)\iff\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\leq 0.$ (4.2)

Suppose that $z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)$, we prove

$\begin{cases}z_{i}=y_{i}\ \ \ \ \ \ \ \ \mbox{if}\ \ i\in\bar{x}^{+},\\\
z_{i}=0\ \ \ \ \ \ \ \ \ \mbox{if}\ i\in\bar{x}^{-},\\\ 0\leq z_{i}\leq y_{i}\
\ \ \mbox{if}\ i\in\bar{x}^{\bullet}.\end{cases}$

Indeed, suppose there exists $j\in\bar{x}^{+}$ such that $z_{j}\not=y_{j},$ by
choosing $u\in l_{2}$ with $u_{i}=\bar{x}_{i},$ for $i\not=j$ and
$u_{j}=x_{j}+t(z_{j}-y_{j}),$ for $t\downarrow 0.$ We have
$u\rightarrow\bar{x}$ as $t\downarrow 0$ and

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\limsup\limits_{t\downarrow 0}\dfrac{t(z_{j}-y_{j})^{2}}{t|z_{j}-y_{j}|}\\\
&=|z_{j}-y_{j}|.\end{array}$

This implies that

$z_{i}=y_{i},\ \mbox{for all}\ i\in\bar{x}^{+}.$ (4.3)

Suppose there exists $j\in\bar{x}^{-}$ such that $z_{j}\not=0,$ by choosing
$u\in l_{2}$ with $u_{i}=\bar{x}_{i},$ for $i\not=j$ and $u_{j}=x_{j}+tz_{j},$
for $t\downarrow 0.$ We have $u\rightarrow\bar{x}$ as $t\downarrow 0$ and

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\limsup\limits_{t\downarrow 0}\dfrac{tz_{j}^{2}}{t|z_{j}|}\\\
&=|z_{j}|.\end{array}$

This implies that

$z_{i}=0,\ \mbox{for all}\ i\in\bar{x}^{-}.$ (4.4)

Assume, by the way of contradiction, that that there exists
$j\in\bar{x}^{\bullet}$ such that $z_{j}>y_{j}$. Choosing $u=(u_{i})\in l_{2}$
with $u_{i}=\bar{x}_{i}$, for $i\not=j$ and $u_{j}=t$, for $t\downarrow 0$. We
have $u\rightarrow\bar{x}$ and

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\ast}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{(z_{j}u_{j}-y_{j}(P_{K}(u))_{j})}{t}\\\ &=\limsup\limits_{t\downarrow
0}\dfrac{t\left(z_{j}-y_{j}\right)}{t}\\\ &=z_{j}-y_{j}.\end{array}$

This means $z_{j}\leq y_{j}$. This contradiction proves that

$z_{i}\leq y_{i},\ \ \ \ \ \mbox{for all}\ i\in\bar{x}^{\bullet}.$ (4.5)

Similarly, suppose there exists $j\in\bar{x}^{\bullet}$ such that $z_{j}<0$.
By choosing $u=(u_{i})\in l_{2}$ with $u_{i}=\bar{x}_{i},$ for $j\not=j$ and
$u_{j}=-t$, for $t\downarrow 0.$ We get $u\rightarrow\bar{x}$ and

$\begin{array}[]{rl}0&\geq\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\ast}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&\geq\limsup\limits_{t\downarrow
0}\dfrac{(z_{j}u_{j}-y_{j}(P_{K}(u))_{j})}{t}\\\ &=\limsup\limits_{t\downarrow
0}\dfrac{-tz_{j}}{t}\\\ &=-z_{j}.\end{array}$

This shows that

$z_{i}\geq 0,\ \ \ \mbox{for all}\ i\in\bar{x}^{\bullet}.$ (4.6)

By (4.3), (4.4), (4.5) and (4.6) together, we have

$z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y)\Longrightarrow\begin{cases}z_{i}=y_{i}\
\ \ \ \ \ \ \ \mbox{if}\ \ i\in\bar{x}^{+}\\\ z_{i}=0\ \ \ \ \ \ \ \ \
\mbox{if}\ i\in\bar{x}^{-}\\\ 0\leq z_{i}\leq y_{i}\ \ \ \mbox{if}\
i\in\bar{x}^{\bullet}\end{cases}$ (4.7)

Next, we prove the converse, assuming $z\in l_{2}$ with $z_{i}=y_{i}$, for
$i\in\bar{x}^{+},$ $z_{i}=0,$ for $i\in\bar{x}^{-}$ and $0\leq z_{i}\leq
y_{i},$ for $i\in~{}\bar{x}^{\bullet}$. We need to show that

$\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\leq 0.$

Indeed, we have

$\begin{array}[]{rl}&\dfrac{\langle z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\\\
&=\dfrac{\sum\limits_{i\in\bar{x}^{+}}(z_{i}-y_{i})(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{-}}z_{i}(u_{i}-\bar{x}_{i})+\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\dfrac{\sum\limits_{i\in\bar{x}^{\bullet}}(z_{i}u_{i}-y_{i}(P_{K}(u))_{i})}{\|u-\bar{x}\|}\\\
&=\dfrac{\sum\limits_{i\in\bar{x}^{\bullet}\cap
u^{+}}(z_{i}-y_{i})u_{i}+\sum\limits_{i\in\bar{x}^{\bullet}\backslash
u^{+}}z_{i}u_{i}}{\|u-\bar{x}\|}\\\ &\leq 0\ \ \ \ \mbox{for all}\ u\in
l_{2}\backslash\\{\bar{x}\\}.\end{array}$

So, $\limsup\limits_{u\rightarrow\bar{x}}\dfrac{\langle
z,u-\bar{x}\rangle-\langle
y,P_{K}(u)-P_{K}(\bar{x})\rangle}{\|u-\bar{x}\|}\leq 0$. This means

$\begin{cases}z_{i}=y_{i}\ \ \ \ \ \ \ \ \mbox{if}\ \ i\in\bar{x}^{+}\\\
z_{i}=0\ \ \ \ \ \ \ \ \ \mbox{if}\ i\in\bar{x}^{-}\\\ 0\leq z_{i}\leq y_{i}\
\ \ \mbox{if}\ i\in\bar{x}^{\bullet}\end{cases}\Longrightarrow
z\in\widehat{D}^{\ast}P_{K}(\bar{x})(y).$ (4.8)

Then, by (4.7) and (4.8), (4.1) is proved. $\square$

Let $N$ be a nonempty subset of $\mathbb{N}$ with complementary $\bar{N}$. We
define some subsets and an ordering relation in $l_{2}$ with respect to $N$.

$\mathbb{R}^{N}=\left\\{x=(x_{i})\in l_{2}:x_{i}=0,\ \mbox{for}\
i\in\bar{N}\right\\};$ $\mathbb{K}_{N}=\left\\{x=(x_{i})\in l_{2}:x_{i}\geq
0,\ \mbox{for}\ i\in N\right\\};$
$\partial\mathbb{K}_{N}=\left\\{x=(x_{i})\in\mathbb{K}_{N}:x_{i}=0,\
\mbox{for}\ i\in N\right\\};$ $\mathbb{Z}_{N}=\left\\{x=(x_{i})\in
l_{2}:x_{i}>0,\ \mbox{for all}\ i\in N\ \mbox{and}\ x_{i}=0,\ \mbox{for all}\
i\in\bar{N}\right\\};$ $z\preceq_{N}y\Leftrightarrow z_{i}\leq y_{i},\
\mbox{for all}\ i\in N\ \mbox{and}\ z_{i}=y_{i},\ \mbox{for all}\
i\in\bar{N}.$

Using Theorem 4.1, we obtain the following result, which was given by Li in
[15].

######  Corollary 4.2.

(see [15, Theorem 5.1])  Let $M$ be a nonempty finite subset of $\mathbb{N}$
with complementary $\bar{M}$. The regular coderivative of $P_{\mathbb{K}}$
have the following representations.

(i) For any $\bar{x}\in l_{2}$, we have

$\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(\theta)=\\{\theta\\};$

(ii) Let $\bar{x}\in\mathbb{Z}_{M}.$ For any $y\in l_{2}$,

$y\in\mathbb{K}_{\bar{M}}\Longleftrightarrow
y\in\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y);$ (4.9)

(iii) Let $\bar{x}\in\mathbb{Z}_{M}.$ For any $y\in\mathbb{K}_{\bar{M}}$, we
have

$\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)=\left\\{z\in\mathbb{K}_{\bar{M}}:z\preceq_{M}y\right\\}.$
(4.10)

In particular,

$y\in\partial\mathbb{K}_{\bar{M}}\Longrightarrow\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)=\\{y\\}.$

Proof.

(i) Using (4.1) with $y=\theta$, we have

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(\theta)&=\left\\{z\in
l_{2}:z_{i}=0,i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq z_{i}\leq
0,i\in\bar{x}^{\bullet}\right\\}\\\ &=\\{z\in l_{2}:z_{i}=0,\forall
i\in\mathbb{N}\\}\\\ &=\\{\theta\\}.\end{array}$

(ii, iii) Since $\bar{x}\in\mathbb{Z}_{M}$,
$\bar{x}^{+}=M,\bar{x}^{-}=\emptyset$ and $\bar{x}^{\bullet}=\bar{M}.$ By
(4.1), for any $y\in l_{2}$, we have

$\begin{array}[]{rl}\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)&=\left\\{z\in
l_{2}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq z_{i}\leq
y_{i},i\in\bar{x}^{\bullet}\right\\}\\\ &=\left\\{z\in l_{2}:z_{i}=y_{i},i\in
M;0\leq z_{i}\leq y_{i},i\in\bar{M}\right\\}\\\ &=\\{z\in
l_{2}:z\preceq_{\bar{M}}y,\ z_{i}\geq 0,i\in\bar{M}\\}\\\
&=\left\\{z\in\mathbb{K}_{\bar{M}}:z\preceq_{\bar{M}}y\right\\}.\end{array}$

So,

$\begin{array}[]{rl}y\in\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)&\Longleftrightarrow
y\in\left\\{z\in\mathbb{K}_{\bar{M}}:z\preceq_{\bar{M}}y\right\\}\\\
&\Longleftrightarrow y\in\mathbb{K}_{\bar{M}}.\end{array}$

In particular,

$\begin{array}[]{rl}y\in\partial\mathbb{K}_{\bar{M}}&\Longrightarrow y_{i}=0,\
\mbox{for all}\ i\in\bar{M}\\\
&\Longrightarrow\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)=\left\\{z\in\mathbb{K}_{\bar{M}}:z\preceq_{\bar{M}}y\right\\}\\\
&\Longrightarrow\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)=\\{y\\}.\end{array}$

$\square$

Similar to Theorem 3.4, the following result is deduced from Theorem 4.1 and
the definition of the Mordukhovich coderivative.

######  Theorem 4.3.

Let $\mathbb{K}$ be the positive cone of $l_{2}$. Then, the Mordukhovich
coderivative of the projection operator $P_{\mathbb{K}}$ at $\bar{x}\in l_{2}$
is given by

$\begin{array}[]{rl}D^{\ast}P_{\mathbb{K}}(\bar{x})(y)&=\widehat{D}^{\ast}P_{\mathbb{K}}(\bar{x})(y)\\\
&=\left\\{z\in l_{2}:z_{i}=y_{i},i\in\bar{x}^{+};z_{i}=0,i\in\bar{x}^{-};0\leq
z_{i}\leq y_{i},i\in\bar{x}^{\ast}\right\\},\end{array}$ (4.11)

$\mbox{for all}\ y\in l_{2}.$

Proof. Proof of this Theorem is similar to the proof of Theorem 3.4.
Therefore, it is omitted here.

##  5 Concluding remarks

The main results of this paper exhibits formulas for computing the regular
coderivative and Mordukhovich coderivative of the metric projection operator
onto positive cones in Euclidean spaces and in real Hilbert space $l_{2}$. In
the near future, we will continue to research other generalized differential
structures of metric projection operators in Hilbert spaces and general Banach
spaces.

.

## Acknowledgments

The first author would like to thank Vietnam Institute for Advanced Study in
Mathematics for hospitality during his post-doctoral fellowship of the
Institute in 2022–2023.

## References

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f((0,0,1))=4,$ $\displaystyle f((1,1,0))=3,\quad f((1,0,1))=5,\quad
f((0,1,1))=4,\quad f((1,1,1))=5.$

We also have the following deformation retractions:

  1. (1)

$r_{1}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{0\\}}\right)=S(0,1,2,3)$,

$r_{1}(5)=r_{1}(3)=3,~{}r_{1}(4)=r_{1}(2)=2,~{}r_{1}(1)=1,~{}r_{1}(0)=0;$

  2. (2)

$r_{2}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{0\\}\boxdot
I}\right)=S(0,1,4,5)$,

$r_{2}(5)=5,~{}r_{2}(4)=4,~{}r_{2}(3)=r_{2}(1)=1,~{}r_{2}(2)=r_{2}(0)=0;$

  3. (3)

$r_{3}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{0\\}\boxdot
I^{\boxdot 2}}\right)=K_{3}(0,2,4)$,

$r_{3}(5)=r_{3}(4)=4,~{}r_{3}(3)=r_{3}(2)=2,~{}r_{3}(1)=r_{3}(0)=0;$

  4. (4)

$r_{4}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{1\\}}\right)=\\{4\rightarrow 5\\}$,

$r_{4}(5)=r_{4}(3)=r_{4}(1)=5,~{}r_{4}(4)=r_{4}(2)=r_{4}(0)=4;$

  5. (5)

$r_{5}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{1\\}\boxdot
I}\right)=S(2,3,4,5)$,

$r_{5}(5)=5,~{}r_{5}(4)=4,~{}r_{5}(3)=r_{5}(1)=3,~{}r_{5}(2)=r_{5}(0)=2;$

  6. (6)

$r_{6}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{1\\}\boxdot
I^{\boxdot 2}}\right)=K_{3}(1,3,5)$,

$r_{6}(5)=r_{6}(4)=5,~{}r_{6}(3)=r_{6}(2)=3,~{}r_{6}(1)=r_{6}(0)=1.$

######  Example A.5 ($\operatorname{NE}(P)=4$, $\operatorname{NF}(P)=4$).

The admissible pair $(P,\operatorname{Supp}(P))$ with
$\operatorname{NE}(P)=4$, $\operatorname{NF}(P)=4$ is given by

$P=e_{0135}-e_{0235}+e_{0245}-e_{0145},$

$\operatorname{Supp}(P)=$
---
$0$
---
$1$
---
$3$
---
$5$
---
$2$
---
$4$
---

The singular cubical map (non-unique) $f:I^{\boxdot
3}\rightarrow\operatorname{Supp}(P)$ could be given by

$\displaystyle f((0,0,0))=0,\quad f((1,0,0))=1,\quad f((0,1,0))=2,\quad
f((0,0,1))=2,$ $\displaystyle f((1,1,0))=3,\quad f((1,0,1))=4,\quad
f((0,1,1))=4,\quad f((1,1,1))=5.$

We also have the following deformation retractions:

  1. (1)

$r_{1}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{0\\}}\right)=S(0,1,2,3)$,

$r_{1}(5)=3,~{}r_{1}(4)=2,~{}r_{1}(3)=3,~{}r_{1}(2)=2,~{}r_{1}(1)=1,~{}r_{1}(0)=0;$

  2. (2)

$r_{2}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{0\\}\boxdot
I}\right)=S(0,1,2,4)$,

$r_{2}(5)=4,~{}r_{2}(4)=4,~{}r_{2}(3)=1,~{}r_{2}(2)=2,~{}r_{2}(1)=1,~{}r_{2}(0)=0;$

  3. (3)

$r_{3}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{0\\}\boxdot
I^{\boxdot 2}}\right)=\\{0\rightarrow 2\rightarrow 4\\}$,

$r_{3}(5)=4,~{}r_{3}(4)=4,~{}r_{3}(3)=2,~{}r_{3}(2)=2,~{}r_{3}(1)=2,~{}r_{3}(0)=0;$

  4. (4)

$r_{4}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{1\\}}\right)=\\{2\rightarrow 4\rightarrow 5\\}$,

$r_{4}(5)=5,~{}r_{4}(4)=4,~{}r_{4}(3)=4,~{}r_{4}(2)=2,~{}r_{4}(1)=2,~{}r_{4}(0)=2;$

  5. (5)

$r_{5}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{1\\}\boxdot
I}\right)=S(2,3,4,5)$,

$r_{5}(5)=5,~{}r_{5}(4)=4,~{}r_{5}(3)=3,~{}r_{5}(2)=2,~{}r_{5}(1)=2,~{}r_{5}(0)=2;$

  6. (6)

$r_{6}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{1\\}\boxdot
I^{\boxdot 2}}\right)=S(1,3,4,5)$,

$r_{6}(5)=5,~{}r_{6}(4)=4,~{}r_{6}(3)=3,~{}r_{6}(2)=1,~{}r_{6}(1)=1,~{}r_{6}(0)=1.$

The (3-step) homotopy $F$ between $i\circ r_{1}$ (similarly, $i\circ r_{2}$,
$i\circ r_{5}$, $i\circ r_{6}$) and
$\operatorname{Id}_{\operatorname{Supp}(P)}$ can be found in Subsection 3.4 of
[17]. Alternatively, if we relax the condition
$r\big{|}_{H}=\operatorname{Id}_{H}$ in the definition of retraction, we can
also construct the following convenient digraph maps:

  1. (1)

$s_{1}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{0\\}}\right)=S(0,1,2,3)$,

$s_{1}(5)=3,~{}s_{1}(4)=2,~{}s_{1}(3)=1,~{}s_{1}(2)=0,~{}s_{1}(1)=0,~{}s_{1}(0)=0;$

  2. (2)

$s_{2}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{0\\}\boxdot
I}\right)=S(0,1,2,4)$,

$s_{2}(5)=4,~{}s_{2}(4)=2,~{}s_{2}(3)=1,~{}s_{2}(2)=0,~{}s_{2}(1)=0,~{}s_{2}(0)=0;$

  3. (3)

$s_{3}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{0\\}\boxdot
I^{\boxdot 2}}\right)=\\{0\rightarrow 2\rightarrow 4\\}$,

$s_{3}(5)=4,~{}s_{3}(4)=2,~{}s_{3}(3)=2,~{}s_{3}(2)=2,~{}s_{3}(1)=0,~{}s_{3}(0)=0;$

  4. (4)

$s_{4}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{1\\}}\right)=\\{2\rightarrow 4\rightarrow 5\\}$,

$s_{4}(5)=5,~{}s_{4}(4)=4,~{}s_{4}(3)=5,~{}s_{4}(2)=4,~{}s_{4}(1)=4,~{}s_{4}(0)=2;$

  5. (5)

$s_{5}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{1\\}\boxdot
I}\right)=S(2,3,4,5)$,

$s_{5}(5)=5,~{}s_{5}(4)=5,~{}s_{5}(3)=5,~{}s_{5}(2)=3,~{}s_{5}(1)=4,~{}s_{5}(0)=2;$

  6. (6)

$s_{6}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{1\\}\boxdot
I^{\boxdot 2}}\right)=S(1,3,4,5)$,

$s_{6}(5)=5,~{}s_{6}(4)=5,~{}s_{6}(3)=5,~{}s_{6}(2)=4,~{}s_{6}(1)=3,~{}s_{6}(0)=1.$

These maps could play the key step in the homotopy to prove that
$r_{S}:\operatorname{Supp}(P)\rightarrow\\{0\\}$ and
$r_{E}:\operatorname{Supp}(P)\rightarrow\\{5\\}$ are deformation retractions.

######  Example A.6 ($\operatorname{NE}(P)=4$, $\operatorname{NF}(P)=6$, i.e.
Example 3.14).

The admissible pair $(P,\operatorname{Supp}(P))$ with
$\operatorname{NE}(P)=4$, $\operatorname{NF}(P)=6$ is given by

$P=e_{0136}-e_{0236}+e_{0256}-e_{0146},$

$3$
---

$0$
---

$1$
---
$2$
---
$4$
---
$5$
---
$6$
---
$\operatorname{Supp}(P)=$
---

The singular cubical map (non-unique) $f:I^{\boxdot
3}\rightarrow\operatorname{Supp}(P)$ could be given by

$\displaystyle f((0,0,0))=0,\quad f((1,0,0))=1,\quad f((0,1,0))=0,\quad
f((0,0,1))=2,$ $\displaystyle f((1,1,0))=4,\quad f((1,0,1))=3,\quad
f((0,1,1))=5,\quad f((1,1,1))=6.$

We also have the following deformation retractions:

  1. (1)

$r_{1}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{0\\}}\right)=K_{3}(0,1,4)$,

$r_{1}(6)=4,~{}r_{1}(5)=0,~{}r_{1}(4)=4,~{}r_{1}(3)=1,~{}r_{1}(2)=0,~{}r_{1}(1)=1,~{}r_{1}(0)=0;$

  2. (2)

$r_{2}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{0\\}\boxdot
I}\right)=S(0,1,2,3)$,

$r_{2}(6)=3,~{}r_{2}(5)=2,~{}r_{2}(4)=1,~{}r_{2}(3)=3,~{}r_{2}(2)=2,~{}r_{2}(1)=1,~{}r_{2}(0)=0;$

  3. (3)

$r_{3}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{0\\}\boxdot
I^{\boxdot 2}}\right)=K_{3}(0,2,5)$,

$r_{3}(6)=5,~{}r_{3}(5)=5,~{}r_{3}(4)=0,~{}r_{3}(3)=2,~{}r_{3}(2)=2,~{}r_{3}(1)=0,~{}r_{3}(0)=0;$

  4. (4)

$r_{4}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{1\\}}\right)=S(2,3,5,6)$,

$r_{4}(6)=6,~{}r_{4}(5)=5,~{}r_{4}(4)=3,~{}r_{4}(3)=3,~{}r_{4}(2)=2,~{}r_{4}(1)=3,~{}r_{4}(0)=2;$

  5. (5)

$r_{5}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{1\\}\boxdot
I}\right)=S(0,4,5,6)$,

$r_{5}(6)=6,~{}r_{5}(5)=5,~{}r_{5}(4)=4,~{}r_{5}(3)=6,~{}r_{5}(2)=5,~{}r_{5}(1)=4,~{}r_{5}(0)=0;$

  6. (6)

$r_{6}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{1\\}\boxdot
I^{\boxdot 2}}\right)=S(1,3,4,6)$,

$r_{6}(6)=6,~{}r_{6}(5)=3,~{}r_{6}(4)=4,~{}r_{6}(3)=3,~{}r_{6}(2)=3,~{}r_{6}(1)=1,~{}r_{6}(0)=1.$

The (3-step) homotopy $F$ between $i\circ r_{4}$ (similarly, $i\circ r_{6}$)
and $\operatorname{Id}_{\operatorname{Supp}(P)}$ can be found in Subsection
3.4 of [17].

######  Example A.7 ($\operatorname{NE}(P)=5$, $\operatorname{NF}(P)=6$, i.e.
Example 3.12).

The admissible pair $(P,\operatorname{Supp}(P))$ with
$\operatorname{NE}(P)=5$, $\operatorname{NF}(P)=6$ is given by

$P=e_{0136}-e_{0156}+e_{0456}+e_{0246}-e_{0236},$

$0$
---
$1$
---
$2$
---
$3$
---
$4$
---
$5$
---
$\operatorname{Supp}(P)=$
---
$6$
---

The singular cubical map (non-unique) $f:I^{\boxdot
3}\rightarrow\operatorname{Supp}(P)$ could be given by

$\displaystyle f((0,0,0))=0,\quad f((1,0,0))=1,\quad f((0,1,0))=2,\quad
f((0,0,1))=4,$ $\displaystyle f((1,1,0))=3,\quad f((1,0,1))=5,\quad
f((0,1,1))=4,\quad f((1,1,1))=6.$

We also have the following obvious deformation retractions:

  1. (1)

$r_{2}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{0\\}\boxdot
I}\right)=S(0,1,4,5)$,

$r_{2}(6)=5,~{}r_{2}(5)=5,~{}r_{2}(4)=4,~{}r_{2}(3)=1,~{}r_{2}(2)=0,~{}r_{2}(1)=1,~{}r_{2}(0)=0;$

  2. (2)

$r_{3}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{\\{0\\}\boxdot
I^{\boxdot 2}}\right)=K_{3}(0,2,4)$,

$r_{3}(6)=4,~{}r_{3}(5)=4,~{}r_{3}(4)=4,~{}r_{3}(3)=2,~{}r_{3}(2)=2,~{}r_{3}(1)=0,~{}r_{3}(0)=0;$

  3. (3)

$r_{4}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I^{\boxdot
2}\boxdot\\{1\\}}\right)=K_{3}(4,5,6)$,

$r_{4}(6)=6,~{}r_{4}(5)=5,~{}r_{4}(4)=4,~{}r_{4}(3)=6,~{}r_{4}(2)=4,~{}r_{4}(1)=5,~{}r_{4}(0)=4;$

  4. (4)

$r_{5}:\operatorname{Supp}(P)\rightarrow\operatorname{im}\left(f\big{|}_{I\boxdot\\{1\\}\boxdot
I}\right)=S(2,3,4,6)$,

$r_{5}(6)=6,~{}r_{5}(5)=6,~{}r_{5}(4)=4,~{}r_{5}(3)=3,~{}r_{5}(2)=2,~{}r_{5}(1)=3,~{}r_{5}(0)=2;$

######  Remark A.8.

There is another minimal $3$-paths with $5$ elementary path components up to
the global orientation of edges and isomorphism of digraphs. That is,

$P=e_{0147}-e_{0247}+e_{0257}-e_{0357}+e_{0367}.$

Its supporting digraph is given by the following one555This digraph is
isomorphic to $\operatorname{Supp}(P)$ in Subsection 3.4 of [17], where
$P=e_{0137}-e_{0237}+e_{0267}-e_{0157}+e_{0457}.$ .

$\operatorname{Supp}(P)=$
---
$0$
---
$1$
---
$2$
---
$4$
---
$5$
---
$7$
---
$3$
---
$6$
---

One can see that $\operatorname{Supp}(P)$ has $7$ face components, thus
$P\notin\mathcal{P}_{\operatorname{adm},3}$.

######  Example A.9 ($\operatorname{NE}(P)=6$, $\operatorname{NF}(P)=6$).

The admissible pair $(P,\operatorname{Supp}(P))$ with
$\operatorname{NE}(P)=6$, $\operatorname{NF}(P)=6$ is given by

$P=e_{0147}-e_{0157}+e_{0357}-e_{0367}+e_{0267}-e_{0167},$

and the supporting digraph $\operatorname{Supp}(P)$ is nothing but the 3-cube
$I^{\boxdot 3}$. The corresponding six retraction digraph maps are the
standard projections.

## References

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# Thermal gauge theories with Lagrange multiplier fields

F. T. Brandt<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo,
São Paulo, SP 05508-090, Brazil  J. Frenkel<EMAIL_ADDRESS>Instituto de
Física, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil  S.
Martins-Filho<EMAIL_ADDRESS>Instituto de Física, Universidade de
São Paulo, São Paulo, SP 05508-090, Brazil  D. G. C. McKeon<EMAIL_ADDRESS>Department of Applied Mathematics, The University of Western Ontario, London,
Ontario N6A 5B7, Canada  Department of Mathematics and Computer Science,
Algoma University, Sault Ste. Marie, Ontario P6A 2G4, Canada  G. S. S. Sakoda
<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, São
Paulo, SP 05508-090, Brazil

###### Abstract

We study the Yang-Mills theory and quantum gravity at finite temperature, in
the presence of Lagrange multiplier fields. These restrict the path integrals
to field configurations which obey the classical equations of motion. This has
the effect of doubling the usual one–loop thermal contributions and of
suppressing all radiative corrections at higher loop order. Such theories are
renormalizable at all temperatures. Some consequences of this result in
quantum gravity are briefly examined.

finite temperature field theory, gauge theories, gravity

###### pacs:

11.10.Wx,11.15.-q,04.60.-m

##  I Introduction

The standard theory of quantum gravity, which is based upon the Einstein-
Hilbert (EH) Lagrangian, is non-renormalizable tHooft:1974bx ; Goroff:1986th ;
vandeVen:1991gw . It requires an infinite number of counterterms which involve
higher-order terms in the curvature tensor, to cancel all the ultraviolet
divergences Gomis:1995jp ; PhysRevD.100.026018 . Many studies have been
devoted to obtain a quantum field theory that is renormalizable and unitary
while having general relativity as a classical limit. These entail, for
example, the introduction of additional terms and fields into the action, like
higher derivatives models Mannheim:2011ds  and supergravity Freedman:2012zz ,
or the appeal to non-perturbative properties of renormalization group
functions Nagy:2012ef . These attempts have met with varying degrees of
success. Moreover, it has been argued that quantum gravity may be an effective
field theory obtained in the low energy limit of a string theory
green_schwarz_witten_2012 ; polchinski_1998 .

Recently, it has been proposed that there may be an alternative way of
quantizing the EH action that removes such divergences in a simpler manner,
preserves unitarity and produces the expected tree level effects McKeon:1992rq
; PhysRevD.100.125014 ; Brandt:2018lbe ; McKeon:2020pkm ; Brandt:2021ycu .
This involves the introduction of a Lagrangian multiplier (LM) field that
restricts the path integral used to quantize the theory to paths that satisfy
the classical Euler-Lagrange equations. One finds that this procedure yields,
at zero temperature, twice the usual one–loop contributions and that all
higher order radiative corrections vanish.

We are thus motivated to extend this analysis to quantum gravity with LM
fields at finite temperature. However, since such calculations are rather
involved, we will follow in this work Feynman’ s approach Klauder:1972lsv  :
“The algebraic complexity of the gravitational field equations is so great
that it is very difficult to investigate it…The Yang-Mills theory is also a
non-linear theory which might show the same kind of problems, and yet be
easier to handle algebraically. This proved to be the case, and thereafter all
the work was done first with the Yang–Mills theory and then the corresponding
expressions for gravitation were worked out.” Thus, in section 2, we study the
Yang-Mills theory with a Lagrange multiplier field at finite temperature
Braaten:1990mz ; Frenkel:1989br ; lebellac:book96 ; kapusta . To this end, we
introduce the method of forward scattering amplitudes Frenkel:1989br ;
Barton:1990fk  which greatly simplifies the computations. With these insights,
we extend in section 3 the analysis to thermal quantum gravity with Lagrange
multiplier fields. We find that, up to one–loop order, the thermal effects are
twice those obtained in the usual theories, while all contributions beyond
one–loop order are eliminated. In section 4, we present a short discussion of
the results and their possible implications in quantum gravity.

##  II Thermal Yang-Mills theory with LM fields

The usual Yang-Mills (YM) theory is described by the Lagrangian

${\cal
L}_{YM}=-\frac{1}{4}f^{a}_{\mu\nu}f^{a\mu\nu};\;\;\;f_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c},$
(1)

where $A_{\mu}^{a}$ is the gauge field. The Lagrange multiplier (LM)
$\lambda_{\mu}^{a}$ field is introduced as

$\lambda_{\mu}^{a}\frac{\partial{{\cal L}_{YM}}}{\partial
A^{a}_{\mu}}=-\lambda_{\mu}^{a}D^{ab}_{\nu}f^{b\mu\nu},$ (2)

where $D^{ab}_{\nu}=\partial_{\nu}\delta^{ab}+gf^{acb}A^{c}_{\nu}$ is the
covariant derivative. The gauge fixing terms in the Feynman gauge, together
with the corresponding ghost terms, have the form

$-\frac{1}{2}(\partial\cdot A)^{2}-(\partial\cdot
A^{a})(\partial\cdot\lambda^{a})-{\bar{c}}^{a}\partial\cdot
D^{ab}(A)d^{b}-{\bar{d}}^{a}\partial\cdot
D^{ab}(A)c^{b}-{\bar{c}}^{a}\partial\cdot D^{ab}(A+\lambda)c^{b},$ (3)

(see Eqs. (13) and (17) of McKeon:1992rq ) where $c$, $d$ and $\bar{c}$,
$\bar{d}$ are respectively the ghost and anti-ghost fields. It may be verified
that the complete Lagrangian is invariant under BRST transformations.

The quadratic part of the Lagrangian is

${\cal
L}^{(2)}=\frac{1}{2}\begin{pmatrix}A^{a\mu},&\lambda^{a\mu}\end{pmatrix}\begin{pmatrix}\partial^{2}&\partial^{2}\\\
\partial^{2}&0\end{pmatrix}\begin{pmatrix}A^{a}_{\mu}\\\
\lambda^{a}_{\mu}\end{pmatrix}-\begin{pmatrix}{\bar{c}}^{a},&{\bar{d}}^{a}\end{pmatrix}\begin{pmatrix}\partial^{2}&\partial^{2}\\\
\partial^{2}&0\end{pmatrix}\begin{pmatrix}{c}^{a}\\\ d^{a}\end{pmatrix}$ (4)

which leads to the following matrix propagator

$\Delta=\begin{pmatrix}\partial^{2}&\partial^{2}\\\
\partial^{2}&0\end{pmatrix}^{-1}=\begin{pmatrix}0&1/\partial^{2}\\\
1/\partial^{2}&-1/\partial^{2}\end{pmatrix}.$ (5)

We see that there is no $\langle AA\rangle$ propagator. There are, however,
mixed $\langle A\lambda\rangle=\langle\lambda A\rangle$ propagators as well as
a propagator $\langle\lambda\lambda\rangle$ for the LM field. Moreover, all
vertices have at most a single LM field. As a result, one cannot draw a
Feynman diagram with more than one loop. Consequently, no higher loop diagrams
can contribute.

The starting point of the thermal field theory is the new definition of an
observable $O$ for a system in contact with a thermal bath at temperature $T$

$\langle O\rangle_{T}=Z^{-1}{\rm Tr}\left(e^{-H/T}\,O\right),$ (6)

where $Z$ is the partition functionlebellac:book96 . The Boltzmann factor
$\exp(-H/T)$ weights the occupation number of states that are accessible to
the system. In the imaginary time formalism, in momentum space, a thermal
quantum field theory in $3+1$ dimensions reduces to a $3$-dimensional
Euclidean theory with an infinite summation over the Matsubara frequencies
$Q^{0}_{n}=i2\pi nT$ ($n=0,\pm 1,\pm 2,\dots$ or $n=\pm 1/2,\pm 3/2,\pm 5/2$
for Bosons or Fermions respectively).

The thermal field theory is renormalizable, provided the zero-temperature
theory is so. This is intuitively obvious as the thermal perturbative
corrections come with a Bose-Einstein factor

$N_{B}(\omega)=\frac{1}{e^{\omega/T}-1}$ (7)

that cuts off any ultraviolet divergence.

###  II.1 Forward scattering amplitudes

In the imaginary time formalism, the Bosonic Matsubara summations can be done
using the relation kapusta

$T\sum_{n=-\infty}^{\infty}f(Q^{0}_{n})=\int_{-i\infty}^{i\infty}\frac{dQ^{0}}{2\pi
i}\frac{f(Q^{0})+f(-Q^{0})}{2}+\int_{-i\infty+\delta}^{i\infty+\delta}\frac{dQ^{0}}{2\pi
i}\frac{f(Q^{0})+f(-Q^{0})}{e^{Q^{0}/T}-1}$ (8)

which allows us to separate the $T=0$ part (first term) from the finite-
temperature contribution (second term).

Figure 1: A generic one–loop diagram.

Using Eq. (8), the thermal part of a generic one-loop diagram containing a
number $L$ of internal lines, as shown in Fig. 1, is given by

$\displaystyle\int\,\frac{{\rm
d}^{3}Q}{(2\pi)^{3}}\int_{-i\infty+\delta}^{i\infty+\delta}\frac{{\rm
d}Q^{0}}{2\pi
i}\,\frac{f(Q^{0},\vec{Q},k_{1},k_{2},\dots,k_{L})+f(-Q^{0},\vec{Q},k_{1},k_{2},\dots,k_{L})}{e^{Q^{0}/T}-1}.$
(9) $\displaystyle=$ $\displaystyle\int\,\frac{{\rm
d}^{3}Q}{(2\pi)^{3}}\int_{-i\infty+\delta}^{i\infty+\delta}\frac{{\rm
d}Q^{0}}{2\pi
i}\,\frac{1}{e^{Q^{0}/T}-1}\left(f(Q^{0},\vec{Q},k_{1},k_{2},\dots,k_{L})+Q\leftrightarrow-Q\right).$

From Eq. (5) we know that $f(Q^{0},\vec{Q},k_{1},k_{2},\dots,k_{L})$ has the
following structure

$f(Q^{0},\vec{Q},k_{1},k_{2},\dots,k_{L})=\frac{1}{{Q^{0}}^{2}-\vec{Q}^{2}}\frac{1}{(Q^{0}+k_{1}^{0})^{2}-(\vec{Q}+\vec{k}_{1})^{2}}\cdots\frac{t(Q,k_{1},k_{2}\cdots
k_{L})}{(Q^{0}-k_{L}^{0})^{2}-(\vec{Q}-\vec{k}_{L})^{2}};\;\;\;(k_{1}+k_{2}+\cdots+k_{L}=0).$
(10)

where $t(Q,k_{1},k_{2}\cdots k_{L})$ is a tensor (or a scalar in the case of a
scalar field theory) which is determined by the interaction vertices of the
theory. Using partial fraction decomposition like

$\frac{1}{(Q^{0}+{k}^{0})^{2}-(\vec{Q}+\vec{k})^{2}}=\frac{1}{2\,|\vec{Q}+\vec{k}|}\left[\frac{1}{Q^{0}+k^{0}-|\vec{Q}+\vec{k}|}-\frac{1}{Q^{0}+k^{0}+|\vec{Q}+\vec{k}|}\right]$
(11)

($(k^{0},\vec{k})$ represents any of the following possibilities:
$(0,\vec{0})$, $(k^{0}_{1},\vec{k}_{1})$,
$(k^{0}_{1}+k^{0}_{2},\vec{k}_{1}+\vec{k}_{2})$, $\dots$,
$(-k^{0}_{l},-\vec{k}_{l})$) the $Q^{0}$ integral can be done upon closing the
integration contour on the right hand side of the complex plane and using
Cauchy’s integral formula. Then, performing shifts in the momentum $\vec{Q}$
and using the property $\exp(x+i2\pi m)=\exp(x)$; $m=0,\pm 1,\pm 2,\dots$, one
can show that

$\left[\begin{array}[]{l}\includegraphics[scale={0.5}]{one_loop.eps}\end{array}\right]_{\rm
thermal}=-\int\,\frac{{\rm
d}^{3}Q}{(2\pi)^{3}}\frac{1}{2|\vec{Q}|}\frac{1}{e^{|\vec{Q}|/T}-1}{\cal
A}(Q,k_{1},k_{2},\cdots,k_{L}),$ (12)

(the minus sign arises from the clockwise contour on the right hand complex
plane) where

$\displaystyle{\cal A}(Q,k_{1},k_{2},\cdots,k_{L})$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{c}\includegraphics[scale={0.7}]{Nforward}\end{array}+Q\leftrightarrow-Q\right)_{Q^{0}=|\vec{Q}|}$
(14) $\displaystyle+$ $\displaystyle\mbox{ cyclic permutations of
}V_{1},V_{2},\dots,V_{L}.$ (15)

The tree amplitude ${\cal A}(Q,k_{1},k_{2},\cdots,k_{L})$ is a forward
scattering amplitude which describes the scattering of a on-shell particle of
momentum $Q$ by external particles of momentum $k_{1}$, $k_{2}$, $\dots$,
$k_{L}$ Barton:1990fk . It is a rational function of $k^{0}$ which can be
analytically continued to all continuous values of the external energy.

One can now evaluate the leading high-temperature contributions, which come
from the hard thermal region where $|\vec{Q}|\gg|\vec{k}_{i}|,|k^{0}_{i}|$. To
this end, we expand the denominators of ${\cal A}(Q,k_{1},k_{2},\cdots,k_{L})$
as

$\frac{1}{k^{2}+2k\cdot Q}=\frac{1}{2k\cdot Q}-\frac{k^{2}}{(2k\cdot
Q)^{2}}+\dots.$ (16)

($k$ represents any of the following possibilities: $k_{1}$, $k_{1}+k_{2}$,
$\dots$, $-k_{L}$) After combining with the contributions from the numerator
of ${\cal A}(Q,k_{1},k_{2},\cdots,k_{L})$, the expansions (16) produce terms
of different degrees in $|\vec{Q}|$; the highest degree yields the leading
high temperature behaviour, which can be found using the formula

$\int_{0}^{\infty}d|\vec{Q}|\frac{|\vec{Q}|^{n-1}}{e^{|\vec{Q}|/T}-1}=\Gamma(n)\zeta(n)T^{n}$
(17)

where $\zeta(n)$ is the Riemann zeta function.

Figure 2: One-loop contribution to the thermal self-energy of the gauge Boson.
Springy lines represent the LM field; full lines denote the $d$ ghosts and
traced lines represent the $c$ ghosts.

###  II.2 The gluon self-energy

Let us consider the contributions from the gluon self-energy diagrams shown in
Fig. 2. These are twice those found in the usual YM theory, since in the later
case there is a combinatorial factor of $1/2$ associated with purely gluonic
graphs and only one type of ghost. Using Eqs. (14) and (12), we can write the
thermal contribution from Fig. (2a) as

${\Pi^{(1)}}^{ab}_{\mu\nu}(k)=-\int\frac{d^{3}Q}{(2\pi)^{3}}\frac{N_{B}(|\vec{Q}|)}{2|\vec{Q}|}{{\cal
A}^{(1)}}^{ab}_{\mu\nu}(Q,k),$ (18)

where the forward scattering amplitude ${{\cal A}^{(1)}}^{ab}_{\mu\nu}(Q,k)$
is shown in Fig. (3a) (added with the permutation
$(\mu\leftrightarrow\nu,k\leftrightarrow-k)$ and $Q\leftrightarrow-Q$).

Figure 3: The forward scattering amplitudes corresponding to Fig. 2. Diagrams
(a), (c) and (d) have to be added with
$(\mu\leftrightarrow\nu,k\leftrightarrow-k)$ as well as $Q\leftrightarrow-Q$
as indicated in the Eq. (14).

Next, let us consider the thermal contribution from the tadpole graph shown in
Fig. (2b). Proceeding as before, we can express it in terms of the forward
scattering amplitude as indicated in Fig. (3 b). Finally, we must consider the
thermal contributions associated with the ghost loops shown in Figs. (2 c) and
(2 d). These may also be expressed in terms of the forward scattering
amplitudes shown in Figs. (3 c) and (3 d) together with the graphs obtained by
making $(\mu\leftrightarrow\nu,k\leftrightarrow-k,Q\leftrightarrow-Q)$.

Denoting by ${\cal A}^{ab}_{\mu\nu}(Q,k)$ the total forward scattering
amplitude, we can express the complete thermal contribution as

$\Pi^{ab}_{\mu\nu}(k)=-\int\frac{d^{3}Q}{(2\pi)^{3}}\frac{N_{B}(|\vec{Q}|)}{2|\vec{Q}|}{\cal
A}^{ab}_{\mu\nu}(Q,k).$ (19)

Using the expansion of Eq. (16) and the formula (17), we find for the leading
thermal contribution from the gluon self-energy, the expression

$\Pi^{ab}_{\mu\nu}(k)=\frac{1}{3}g^{2}C_{A}T^{2}\delta^{ab}\int\frac{d\Omega}{4\pi}\left(\frac{k_{\mu}\hat{Q}_{\nu}+k_{\nu}\hat{Q}_{\mu}}{k\cdot\hat{Q}}-\frac{k^{2}}{(k\cdot\hat{Q})^{2}}\hat{Q}_{\mu}\hat{Q}_{\nu}-\eta_{\mu\nu}\right),$
(20)

where ${\hat{Q}}=(1,{\vec{Q}}/{|\vec{Q}|})$ and $C_{A}$ is the group factor.
Eq. (20) satisfies the transversality condition
$k^{\mu}\Pi^{ab}_{\mu\nu}(k)=0$.

Finally, upon using identities for the integral over the directions
$\vec{Q}/|\vec{Q}|$ Eq. (20) can be further simplified, yielding Brandt:1993mj

$\Pi^{ab}_{\mu\nu}(k)=\frac{2}{3}C_{A}g^{2}T^{2}\delta^{ab}\int\frac{d\Omega}{4\pi}\left(k_{0}\frac{{\hat{Q}}_{\mu}{\hat{Q}}_{\nu}}{k\cdot{\hat{Q}}}-\eta_{\mu
0}\eta_{\nu 0}\right).$ (21)

This result leads to a screening thermal mass squared given by

$m^{2}=\frac{2}{3}C_{A}g^{2}T^{2},$ (22)

which is twice that obtained in the usual YM theory lebellac:book96 , as
expected.

##  III Thermal Quantum Gravity with LM fields

Next, let us consider the Einstein-Hilbert (EH) Lagrangian

${\cal L}_{EH}=\frac{1}{\kappa^{2}}\sqrt{-g}g^{\mu\nu}R_{\mu\nu}(\Gamma),$
(23)

where $\kappa$ is related to Newton’s constant $G$ as $\kappa=\sqrt{16\pi G}$
and the Christoffel symbol is

$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\sigma}\left(g_{\mu\sigma,\nu}+g_{\nu\sigma,\mu}-g_{\mu\nu,\sigma}\right).$
(24)

The tensor $R_{\mu\nu}$ is given by

$R_{\mu\nu}(\Gamma)=\Gamma^{\rho}_{\mu\rho,\nu}-\Gamma^{\rho}_{\mu\nu,\rho}+\Gamma^{\rho}_{\mu\sigma}\Gamma^{\sigma}_{\nu\rho}-\Gamma^{\sigma}_{\mu\nu}\Gamma^{\rho}_{\sigma\rho}.$
(25)

For our purpose, it is convenient to expand the metric tensor $g^{\mu\nu}$ in
terms of the deviation from the Minkowski metric $\eta^{\mu\nu}$ as follows

$\sqrt{-g}g^{\mu\nu}=\eta^{\mu\nu}+\kappa h^{\mu\nu}.$ (26)

This allows to evaluate perturbatively the thermal Green’ s functions by
expanding the EH Lagrangian in powers of $\kappa$.

The Lagrange multiplier field $\Lambda^{\mu\nu}$ is introduced as

$\Lambda^{\mu\nu}\frac{\partial{\cal L}_{EH}}{\partial
h^{\mu\nu}}=\frac{2}{\kappa}\Lambda^{\mu\nu}R^{l}_{\mu\nu}+{\cal O}(\kappa),$
(27)

where $R^{l}_{\mu\nu}$ is the linearized Ricci tensor.

It is simpler to work in the de Donder gauge by choosing the gauge fixing
Lagrangian

${\cal L}_{\rm
fix}=(\partial_{\mu}h^{\mu\nu})^{2}-\Lambda^{\mu\nu}\left(\partial_{\mu}\partial_{\sigma}h^{\sigma}_{\;\nu}+\partial_{\nu}\partial_{\sigma}h^{\sigma}_{\;\mu}\right),$
(28)

which leads to a contribution of the gravitational ghosts given by

$\displaystyle{\cal L}_{\rm ghost}$ $\displaystyle=$
$\displaystyle{\bar{c}}_{\nu}\partial^{2}\eta^{\mu\nu}d_{\mu}+\kappa{\bar{c}}_{\nu}\partial_{\mu}\left[h^{\mu\rho}\partial_{\rho}d^{\nu}+h^{\nu\rho}\partial_{\rho}d^{\mu}-\partial_{\rho}(h^{\mu\nu}d^{\rho})\right]+\\{\bar{c}\rightarrow\bar{d},\;\;d\rightarrow
c\\}$ (29) $\displaystyle+$
$\displaystyle{\bar{c}}_{\nu}\partial^{2}\eta^{\mu\nu}c_{\mu}+\kappa{\bar{c}}_{\nu}\partial_{\mu}\left[(h^{\mu\rho}+\Lambda^{\mu\rho})\partial_{\rho}c^{\nu}+(h^{\nu\rho}+\Lambda^{\nu\rho})\partial_{\rho}c^{\mu}-\partial_{\rho}((h^{\mu\nu}+\Lambda^{\mu\nu})c^{\rho})\right].$

Using a similar procedure to that employed in the YM theory, it turns out that
there is no $\langle hh\rangle$ propagator. There are, however, mixed $\langle
h\Lambda\rangle=\langle\Lambda h\rangle$ propagators as well as a propagator
$\langle\Lambda\Lambda\rangle$ for the LM fields, which have the form

$D{}^{h}_{\alpha\beta,}{}^{\Lambda}_{\mu\nu}(Q)=-D{}^{\Lambda}_{\alpha\beta,}{}^{\Lambda}_{\mu\nu}(Q)=\frac{1}{2}\frac{1}{Q^{2}}\left(\eta_{\alpha\mu}\eta_{\beta\nu}+\eta_{\alpha\nu}\eta_{\beta\mu}-\eta_{\mu\nu}\eta_{\alpha\beta}\right).$
(30)

As in the YM theory, all the radiative corrections in quantum gravity with LM
fields vanish beyond the one–loop order. The Feynman diagrams which contribute
to the one–loop graviton self-energy function are shown in Fig. 2. The
associated forward scattering amplitudes of a thermal graviton with on-shell
momenta are indicated in Fig. 3. As we have seen, such amplitudes must be
multiplied by the corresponding Bose-Einstein factor and integrated over the
3-momentum. In this way, we can express the thermal self-energy graviton
function as

$\Pi_{\alpha\beta,\mu\nu}(k)=\frac{1}{4\pi^{2}}\int_{0}^{\infty}dQ|\vec{Q}|N_{B}(|\vec{Q}|)\int\frac{d\Omega}{4\pi}{\cal
A}_{\alpha\beta,\mu\nu}(Q,k).$ (31)

The leading high-temperature limit of the forward scattering amplitude ${\cal
A}_{\alpha\beta,\mu\nu}(Q,k)$ is governed by the contributions quadratic in
$Q$, for large values of $Q$. To obtain these, one needs to expand the Feynman
denominators as shown in Eq. (16). Using the formula (17) we find that

${\Pi}_{\alpha\beta,\mu\nu}(k)=\frac{\pi^{2}T^{4}}{60}\int\frac{d\Omega}{4\pi}{\cal
A}_{\alpha\beta,\mu\nu}(\hat{Q},k),$ (32)

where the forward amplitude ${\cal A}_{\alpha\beta,\mu\nu}(\hat{Q},k)$ is
given by the expression (${\hat{Q}}=(1,{\vec{Q}}/{|\vec{Q}|})$)

$\displaystyle{\cal A}_{\alpha\beta,\mu\nu}(\hat{Q},k)$ $\displaystyle=$
$\displaystyle
2\kappa^{2}\left[\frac{k_{\nu}{\hat{Q}}_{\alpha}{\hat{Q}}_{\beta}{\hat{Q}}_{\mu}}{k\cdot{\hat{Q}}}+\frac{k_{\mu}{\hat{Q}}_{\alpha}{\hat{Q}}_{\beta}{\hat{Q}}_{\nu}}{k\cdot{\hat{Q}}}+\frac{k_{\alpha}{\hat{Q}}_{\beta}{\hat{Q}}_{\mu}{\hat{Q}}_{\nu}}{k\cdot{\hat{Q}}}\right.$
(33) $\displaystyle+$
$\displaystyle\left.\frac{k_{\beta}{\hat{Q}}_{\alpha}{\hat{Q}}_{\mu}{\hat{Q}}_{\nu}}{k\cdot{\hat{Q}}}-\frac{k^{2}{\hat{Q}}_{\alpha}{\hat{Q}}_{\beta}{\hat{Q}}_{\mu}{\hat{Q}}_{\nu}}{(k\cdot{\hat{Q}})^{2}}\right]$

The expression (32) is consistent with the Ward identity relating the
self–energy to the one–particle graviton function $\Gamma_{\mu\nu}$
Brandt:1993dk :

$(2\eta^{\alpha\lambda}k^{\beta}-\eta^{\alpha\beta}k^{\lambda})\Pi_{\alpha\beta,\mu\nu}(k)=-\kappa(k_{\mu}\Gamma_{\nu}^{\;\lambda}+k_{\nu}\Gamma_{\mu}^{\;\lambda})$
(34)

which follows in consequence of the invariance under general coordinate
transformations.

We note that the result (32) yields a screening thermal graviton mass squared
given by

$m_{g}^{2}=2(32\pi G)\frac{\pi^{2}T^{4}}{45},$ (35)

which is twice that obtained in thermal quantum gravity Rebhan:1990yr .

Figure 4:  Feynman diagrams contributing to the thermal three-graviton vertex
function (a), and the associated forward scattering amplitudes (b). Graphs
obtained by cyclic permutations of external gravitons are to be understood.

One can also evaluate the leading $T^{4}$ temperature corrections of the
thermal three-point graviton function. A typical Feynman diagram is shown in
Fig. (4a), while the corresponding forward scattering amplitude is indicated
in Fig. (4b). This calculation is much more involved, but the result for the
total forward scattering amplitude turns out to be twice that obtained in the
usual quantum gravity theory Brandt:1993dk .

##  IV Discussion

We have examined the YM and quantum gravity theories with Lagrange multiplier
fields at finite temperature. We have shown that the one–loop thermal
contributions are twice those obtained in the usual theories, with all
higher–order radiative corrections being suppressed. Such theories are
renormalizable at all temperatures, as well as unitary. In the present theory,
the Lagrange multiplier fields are regarded as being dynamical fields. It can
be shown that this approach leads to a unitary theory with a positive energy
spectrum, where the Lagrange multiplier particles appear asymptotically only
in pairs Brandt:2021ycu . Such a feature is reminiscent of that observed in
the case of the strange particles, which are produced by strong interactions
only pairwise. Since the Lagrange multiplier theory reduces to Einstein’s
theory in the classical limit at zero temperature McKeon:1992rq ;
PhysRevD.100.125014 ; Brandt:2018lbe ; McKeon:2020pkm ; Brandt:2021ycu , it is
compatible with general relativity in the low energy domain.

However, unlike the case at zero temperature, the thermal zeroth order effects
also appear to be twice those obtained in the conventional gauge theories.
This may be seen, for instance, by considering the partition function $Z$ in
the Yang-Mills theory which, in the free-field case, is given by
lebellac:book96

$Z^{(0)}=\det\Delta={\det}^{2}\left(\frac{1}{\partial^{2}}\right)=\exp\left[2{\rm
Tr}\ln\left(\frac{1}{\partial^{2}}\right)\right],$ (36)

where $\Delta$ is the matrix propagator of Eq. (5). This leads to a radiation
pressure

$P=\frac{T}{V}\ln Z^{(0)}=2\frac{\pi^{2}T^{4}}{45},$ (37)

which is twice that found in the usual gauge theories. This fact is due to the
extra degrees of freedom associated with the LM field. This outcome depends
only on the total number of degrees of freedom, so that the same result would
also be obtained in quantum gravity. Thus, one may inquire about some possible
implications of this fact in gravity. For example, in a star, it is the
pressure of the electromagnetic radiation emitted during the carbon cycle that
equilibrates the weight of the matter. In this case, the pressure of the
gravitational radiation is negligible. Moreover, in the present epoch, the
contribution of the pressure to the stress tensor for the matter distribution
on a large scale, can be disregarded.

The LM field interacts with gravitons, but it does not directly couple to
matter Brandt:2021ycu . Hence, such a field exhibits a behaviour resembling
that of dark matter, which has gravitational interactions but does not
otherwise couple to ordinary matter. Thus, the LM field may provide an extra
gravity which could possibly affect the galactic dynamics.

Certainly, the only way of deciding the correctness of any approach to
quantizing gravity is to appeal to experiment. Although such observations are
not yet feasible, there have been several proposals for quantum gravity
phenomenology. These include, for example, possible observations of quantum
gravity effects in the cosmic microwave background Akhmedov2014  and of
quantum decoherence induced by space-time fluctuations Oniga:2017pyq . It may
be worth studying further the predictions of quantum gravity theories for such
phenomena, which could be tested in the foreseeable future.

###### Acknowledgements.

D. G. C. M. acknowledges discussions with Roger Macleod. F. T. B., J. F., S.
M.-F. and G. S. S. S. thank CNPq (Brazil) for financial support.

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