# User Association and Resource Allocation in Large Language Model Based
Mobile Edge Computing System over Wireless Communications

Liangxin Qian, Jun Zhao
School of Computer Science and Engineering
Nanyang Technological University
Singapore
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the rapidly evolving landscape of large language models (LLMs) and mobile
edge computing, the need for efficient service delivery to mobile users with
constrained computational resources has become paramount. Addressing this, our
paper delves into a collaborative framework for model training where user data
and model adapters are shared with servers to optimize performance. Within
this framework, users initially update the first several layers of the
adapters while freezing the other layers of them, leveraging their local
datasets. Once this step is complete, these partially trained parameters are
transmitted to servers. The servers, equipped with more robust computational
capabilities, then update the subsequent layers. After this training, they
send the enhanced parameters back to the users. This collaborative training
approach ensures that mobile users with limited computational capacities can
still benefit from advanced LLM services without being burdened by exhaustive
computations. Central to our methodology is the DASHF algorithm, which
encapsulates the Dinkelbach algorithm, alternating optimization, semidefinite
relaxation (SDR), the Hungarian method, and a pioneering fractional
programming technique from a recent IEEE JSAC paper [1]. The crux of DASHF is
its capability to reformulate an optimization problem as Quadratically
Constrained Quadratic Programming (QCQP) via meticulously crafted
transformations, making it solvable by SDR and the Hungarian algorithm.
Through extensive simulations, we demonstrate the effectiveness of the DASHF
algorithm, offering significant insights for the advancement of collaborative
LLM service deployments.

###### Index Terms:

Large language model, mobile edge computing, wireless communications, resource
allocation.

##  I Introduction

The proliferation of large language models (LLMs) marks a monumental leap in
the realms of artificial intelligence and natural language processing. These
models, with their deep structures and vast parameter sizes, offer
capabilities that redefine the benchmarks of machine-human interactions [2].
However, the very nature of their size and intricacy means they cannot be
effortlessly deployed, especially in constrained environments like mobile
devices [3].

Mobile Edge Computing (MEC) environments, designed to bring computation closer
to the data source, seem like a perfect fit for deploying LLMs. Still, they
present their own set of challenges. Mobile devices are constrained by
computational resources and battery life, making it strenuous to run these
heavyweight LLMs efficiently [4]. Additionally, the unpredictability of
wireless communication, with its fluctuating data rates and potential for high
latency, complicates the seamless integration of LLMs [5].

To meet the growing demand for on-the-fly LLM services, there’s a pressing
need to address these issues. This involves optimizing the LLMs for
constrained devices and innovating on the wireless communication front. A
potential solution lies in a collaborative approach: a synergy where local
computations on mobile devices are harmoniously complemented by offloading
specific, intensive tasks to more capable servers. Such a paradigm can make
the promise of LLMs in MEC environments a tangible reality.

###  I-A Studied problem

In this paper, we explore the LLM-driven MEC system and introduce the novel
concept of the user service experience-cost ratio (ECR), represented as
$\frac{\text{experience score}}{\text{cost}}$. This metric eloquently captures
the balance between user service experience scores and the crucial factors of
delay and energy consumption within mobile computing environments. The user
service experience score amalgamates a user’s wireless and computational
resources as perceived by the server. Cost consumption embodies the cumulative
delay and energy expenditure of both users and servers. Given the
computational challenges posed by LLMs, we hypothesize that users begin by
training the initial layers of the adapters with their local data. Once this
preliminary training is completed, users send these trained parameters to the
servers. Servers, equipped with more computational resources, take over from
this point, training the subsequent layers of the adapters. Specifically,
servers then assign both wireless and computational resources to each user.
This includes bandwidth, user and server transmission power, and the GPU
computing resources of users and servers. Upon completion, these refined
parameters are then relayed back to the users by servers.

###  I-B Main contributions

To the best of our knowledge, our paper is the first to explore user
association and resource allocation in the LLM wireless communication
scenario. Our contributions include a novel joint optimization problem, the
introduction of ECR, and a novel alternating optimization algorithm as
follows:

$\bullet$ Joint Optimization of User-Sercer Adapter Parameter Training Ratio
and User Association: We propose a joint optimization problem that optimizes
user adapter training offloading and user association for tailored LLM service
to users.

$\bullet$ Introduction of the User Service Experience-Cost Ratio: The concept
of the user service experience-cost ratio (ECR) is introduced. ECR quantifies
the balance between user service experience scores and the overall delay and
energy consumption in the entire uplink and downlink communication. It
provides a valuable metric for assessing the trade-off between user experience
and resource efficiency.

$\bullet$ Innovative Alternating Optimization Approach: We propose an
innovative Alternating Optimization (AO) approach called DASHF, which
represents the combination of the Dinkelbach algorithm, alternating
optimization, semidefinite relaxation (SDR), the Hungarian algorithm, and a
novel fractional programming (FP) technique by [1] published in IEEE JSAC
recently. The most challenging part of DASHF is to rewrite an optimization
problem as Quadratically Constrained Quadratic Programming (QCQP) via
carefully constructed transformations, in order to leverage SDR and the
Hungarian algorithm to obtain a solution. Initially, it addresses the
optimization of user connections and adapter parameter training ratios as a
single QCQP problem. Subsequently, it delves into the optimization of
communication-computation resource allocation (for bandwidth, transmit power
of users and servers, and computing frequency of users and servers), providing
an effective solution for the non-convex FP problem.

$\bullet$ The simulation results substantiate the effectiveness of the
proposed DASHF algorithm in achieving the joint optimization of user adapter
parameter offloading, resource allocation, experience-cost ratio, and
communication-computation resource allocation, demonstrating its practical
applicability and benefits.

The rest of this paper is organized as follows. The system model and
optimization problem formulation are presented in Section II. We propose a
novel DASHF algorithm to solve the optimization problem in Section III. The
numerical results are provided in Section IV. We conclude this paper in
Section V.

##  II LLM-Empowered MEC System and Optimization Problem Formulation

In this section, we first introduce the system scenario, then analyze the
delay and energy consumption in the system model, then introduce the concept
of user service experience-cost ratio (ECR), and formulate the optimization
problem.

Figure 1: Optimizing the ECR of an LLM system with $N$ users and $M$ servers
through joint optimization of user association, offloading, and resource
allocation.

###  II-A System scenario

As presented in Fig. 1, the LLM-based mobile edge computing system contains
multiple servers that distribute tailored LLM models or emulators to various
mobile users. Given the computational constraints of some mobile users, a
hybrid approach is adopted: users train the first few layers locally with
their datasets while freezing the other layers. After training, users send
these layer parameters of their adapters to the server. Once the server
receives those layer parameters, it completes the training of the remaining
layers while freezing the layers trained by the users. After training, the
server sends the refined parameters back to the users. This collaborative
mechanism ensures efficient and personalized LLM services, compensating for
individual users’ computational limitations.

###  II-B System model

We consider a system comprising $N$ mobile users and $M$ LLM servers. We use
$n$ and $m$ as indices for a VR user and a LLM server, respectively, where
$n\in\mathcal{N}:=\\{1,2,\cdots,N\\}$ and
$m\in\mathcal{M}:=\\{1,2,\cdots,M\\}$. Each user is connected to one and only
one server; i.e., $\sum_{m\in\mathcal{M}}x_{n,m}=1$. We introduce indicator
variables $x_{n,m}\in\\{0,1\\}$ to characterize the connection between users
and servers; specifically, $x_{n,m}=1$ (resp, $0$) means that the $n$-th user
is connected (resp., not connected) to the $m$-th server. For example, if
$x_{n,m}=1$, it means that the $n$-th user only connects to the $m$-th server
and $x_{n,m^{\prime}}=0$ for $m{{}^{\prime}}\in\mathcal{M}\setminus\\{m\\}$.

####  II-B1 Time consumption

We consider frequency-division multiple access (FDMA) so that communication
among users and servers would not interfere. The transmission rate from user
$n$ to the chosen edge server $m$ is
$r_{n,m}(b_{n,m},p_{n})=b_{n,m}\log_{2}(1+\frac{g_{n,m}p_{n}}{\sigma^{2}b_{n,m}})$,
where $\sigma^{2}$ is the noise power, $b_{n,m}$ is the allocated bandwidth
between user $n$ and server $m$, $p_{n}$ is the transmit power of user $n$,
$g_{n,m}$ is the channel gain and can be further expressed by
$g_{n,m}=h_{n,m}l_{n,m}$, where $h_{n,m}$ is the large-scale slow-fading
component capturing effects of path loss and shadowing and $l_{n,m}$ is the
small-scale Rayleigh fading.

In Parameter-Efficient Fine-Tuning (PEFT) strategies for large language
models, the concept is to introduce ”adapters” - smaller neural network
components. These are placed within the model, enabling task-specific
customization while largely keeping the pre-trained parameters unchanged. By
doing this, there’s no need for the exhaustive retraining of the complete
model. If we consider inserting an adapter between two layers with dimensions
$d_{in}$ and $d_{out}$, the design usually involves: 1. A down-projection from
$d_{in}$ to a reduced dimension $d_{adapt}$. 2. An up-projection from
$d_{adapt}$ back to $d_{out}$. Accounting for weights and biases in these
transformations, the total size of the adapter’s parameters, $d$, can be
captured as: $d=d_{in}\times d_{adapt}+d_{adapt}\times
d_{out}+d_{adapt}+d_{out}$. For any given user, represented by $n$, the
parameter size of the adapter, $d_{n}$, can differ. This could be due to user-
specific requirements or constraints. Hence, $d_{n}$ links the inherent
complexity of the LLM, the architecture of the adapter, and the unique data
requirements of user $n$.

User training and sending adapter parameter phases. Based on the above
discussion, assume the total adapter parameter size at user $n$ is $d_{n}$.
The adapter parameter size trained by user $n$ is $\varphi_{n}d_{n}$,
$\varphi_{n}\in[0,1]$. User $n$ trains the first $\varphi_{n}d_{n}$ layer
parameters with the local datasets. The training time consumed is
$T^{(t_{1})}_{n,m}=\frac{t_{n}\varphi_{n}d_{n}e_{n}}{g_{n}F_{n}}$. $t_{n}$ is
the FLOPs of all tokens for each adapter parameter of the user $n$. $e_{n}$ is
the number of local training epochs. $g_{n}$ is the available GPU number of
user $n$. $F_{n}$ is the available GPU computation speed of user $n$, defined
as floating point operations (FLOPs). After local training and the user-server
connection algorithm (this can be complicated by choosing the nearest neighbor
server sets, then choosing the server with the lowest transmission time, and
finally finishing all user-server connections), user $n$ transmits
$\varphi_{n}d_{n}$ adapter parameters to the server $m$. The transmission time
from the user $n$ to the server $m$ is
$T^{(t_{2})}_{n,m}=\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}$, where
$\omega_{b}$ is the bits number used to represent each parameter. For example,
if we use the “float32” floating-point number, $\omega_{b}$ will be 32.

Server training and returning adapter parameter phases. After receiving the
partial adapter parameters $\varphi_{n}d_{n}$ from the user $n$, the server
$m$ trains the remaining part of the adapter with the shared user datasets and
the training delay is
$T^{(t_{3})}_{n,m}=\frac{t_{n}(1-\varphi_{n})x_{n,m}d_{n}e_{m}}{g_{m}F_{m}}$.
$e_{m}$ is the number of server training epochs. $g_{m}$ is the available GPU
number of server $m$. $F_{m}$ is the available GPU computation speed of server
$m$. Then, Server $m$ transmits the results to the user $n$, and the delay is
$T^{(t_{4})}_{n,m}=\frac{x_{n,m}(1-\varphi_{n})d_{n}\omega_{b}}{r_{m,n}}$,
where $r_{m,n}=b_{n,m}\log_{2}(1+\frac{g_{n,m}p_{m}}{\sigma^{2}b_{n,m}})$. We
assume the path loss and bandwidth between the downlink and uplink are the
same. The time consumed on the server side is

$\textstyle T_{s,n,m}=T^{(t_{2})}_{n,m}+T^{(t_{3})}_{n,m}.$ (1)

The time consumed on the user side is

$\textstyle T_{u,n,m}=T^{(t_{1})}_{n,m}+T^{(t_{4})}_{n}.$ (2)

Therefore, the total delay will be

$\textstyle T_{total}$ $\textstyle=T_{s,n,m}+T_{u,n,m}$
$\textstyle=\frac{t_{n}\varphi_{n}d_{n}e_{n}}{g_{n}F_{n}}\\!\\!+\\!\\!\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}\\!\\!+\\!\\!\frac{t_{n}(1-\varphi_{n})x_{n,m}d_{n}e_{m}}{g_{m}F_{m}}$
$\textstyle+\frac{x_{n,m}(1-\varphi_{n})d_{n}\omega_{b}}{r_{m,n}}.$ (3)

####  II-B2 Energy consumption

Based on the delay discussion, we then compute the energy consumption in this
system. Energy used for user $n$ training the adapter locally can be
calculated by
$E^{(t_{1})}_{n,m}=e_{n}\kappa_{n}\varphi_{n}d_{n}t_{n}F^{2}_{n}$ [6].
$\kappa_{n}$ is the computational efficiency of user $n$’s GPUs, denoting the
power growth rate corresponding to rising computing speeds. Energy used for
transmitting data from the $n$-th user to the server $m$ is given as
$E^{(t_{2})}_{n,m}=p_{n}T^{(t_{2})}_{n,m}=p_{n}\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}$.
Energy for server training $\varphi_{n}d_{n}$ adapter parameters is given as
$E^{(t_{3})}_{n,m}=e_{m}\kappa_{m}x_{n,m}(1-\varphi_{n})d_{n}t_{n}F_{m}^{2}$.
$\kappa_{m}$ is the computational efficiency of server $m$’s GPUs. Energy
caused by $m$-th server transmitting trained adapter parameters to user $n$ is
$E^{(t_{4})}_{n,m}=p_{m}T^{(t_{4})}_{n,m}=p_{m}\frac{x_{n,m}(1-\varphi_{n})d_{n}\omega_{b}}{r_{m,n}}$.
Thus, the total energy consumption can be formulated as follows:

$\textstyle E_{total}$
$\textstyle=\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}(E_{u,n,m}+E_{s,n,m})$
$\textstyle=\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}(E^{(t_{1})}_{n,m}+E^{(t_{2})}_{n,m}+E^{(t_{3})}_{n,m}+E^{(t_{4})}_{n,m})$
$\textstyle=\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}(p_{n}\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}+p_{m}\frac{x_{n,m}(1-\varphi_{n})d_{n}\omega_{b}}{r_{m,n}}$
$\textstyle+\\!e_{n}\kappa_{n}\varphi_{n}d_{n}t_{n}F^{2}_{n}\\!+\\!e_{m}\kappa_{m}x_{n,m}(1\\!\\!-\\!\\!\varphi_{n})d_{n}t_{n}F_{m}^{2}).$
(4)

####  II-B3 User service experience score

We denote the service experience score of user $n$ that is connected to server
$m$ as:

$\textstyle
v_{n,m}=\varpi_{1}\ln[1+\varpi_{2}(\frac{p_{m}}{p_{max}^{(m)}}+\frac{F_{n,m}}{F_{max}^{(m)}}+\frac{b_{n,m}}{b_{max}})],$
(5)

where $\varpi_{1}$ determines the range of function value, $\varpi_{2}$ is
used for normalization of
$(\frac{p_{m}}{p_{max}^{(m)}}+\frac{F_{n,m}}{F_{max}^{(m)}}+\frac{b_{n,m}}{b_{max}})$.
This function is jointly concave of $p_{m}$, $F_{n,m}$, and $b_{n,m}$ [7].
This user service experience score function is effective and sensitive in all
value ranges of
$(\frac{p_{m}}{p_{max}^{(m)}}+\frac{F_{n,m}}{F_{max}^{(m)}}+\frac{b_{n,m}}{b_{max}})$,
which can describe each user’s subjective experience of the communication and
computing resources obtained from the server.

###  II-C Optimization Problem

User connection $\bm{x}=(x_{n,m})|_{n\in\mathcal{N},m\in\mathcal{M}}$,
$\bm{\varphi}=(\varphi_{n})|_{n\in\mathcal{N}}$, bandwidth
$\bm{b}=(b_{n,m})|_{n\in\mathcal{N},m\in\mathcal{M}}$, transmission power
$\bm{p_{u}}=(p_{n})|_{n\in\mathcal{N}}$ and
$\bm{p_{s}}=(p_{m})|_{m\in\mathcal{M}}$, and GPU computation speed
$\bm{f_{u}}=(F_{n})|_{n\in\mathcal{N}}$ and
$\bm{f_{s}}=(F_{m})|_{m\in\mathcal{M}}$. Our goal is to maximize the user
service experience-cost ratio (ECR):

$\textstyle\frac{\mathcal{V}}{\omega_{t}T_{total}+\omega_{e}E_{total}}=$
$\textstyle\frac{\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}x_{n,m}v_{n,m}}{\omega_{t}(T_{s,n,m}+T_{u,n,m})+\omega_{e}(\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}(E^{(t_{1})}_{n,m}+E^{(t_{2})}_{n,m}+E^{(t_{3})}_{n,m}+E^{(t_{4})}_{n,m}))},$
(6)

where $\omega_{t}$ and $\omega_{e}$ represent the weight values of delay and
energy, respectively. In order to linearize the “maximize” term of
$T_{total}$, we add an auxiliary variable $T$, which is constrained to be
greater than or equal to $T_{total}$. Besides, we utilize Dinkelbach’s
Algorithm [8] by adding an additional variable $y$, which is obtained from the
ECR value in the previous iteration. Then, the fractional programming in the
trust-cost ratio is transformed into the following problem:

$\displaystyle\max\limits_{\bm{x},\bm{\varphi},\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T}$
$\displaystyle\big{\\{}\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\![x_{n,m}v_{n,m}-y\omega_{e}(E_{u,n,m}\\!\\!+\\!\\!E_{s,n,m})]\big{\\}}-y\omega_{t}T$
(7) s.t. $\displaystyle x_{n,m}\in\\{0,1\\},\forall n,m$ (7a)
$\displaystyle\sum\limits_{m\in\mathcal{M}}x_{n,m}=1,\forall n$ (7b)
$\displaystyle\varphi_{n}\in[0,1],\forall n$ (7c)
$\displaystyle\sum\limits_{n\in\mathcal{N}}x_{n,m}b_{n,m}\leq b_{max},\forall
m$ (7d) $\displaystyle p_{n}\leq p^{(n)}_{max},\forall n$ (7e)
$\displaystyle\sum\limits_{n\in\mathcal{N}}x_{n,m}p_{n,m}\leq
p^{(m)}_{max},\forall m$ (7f) $\displaystyle F_{n}\leq F^{(n)}_{max},\forall
n$ (7g) $\displaystyle\sum\limits_{n\in\mathcal{N}}x_{n,m}F_{m}\leq
F^{(m)}_{max},\forall m$ (7h) $\displaystyle
b_{n,m}\\!\geq\\!0,p_{n}\\!\geq\\!0,p_{n,m}\\!\geq\\!0,F_{n}\\!\geq\\!0,F_{m}\\!\geq\\!0,\forall
n,m$ (7i) $\displaystyle T_{s,n,m}+T_{u,n,m}\leq T,\forall n,m$ (7j)

Based on Dinkelbach’s Algorithm, we iteratively optimize $y$ and problem (7).
Specifically, at the $i$-th iteration, given $y^{(i-1)}$, we first obtain
$\bm{x}^{(i)},\bm{\varphi}^{(i)},\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)},T^{(i)}$
by solving the optimization problem 7; then we calculate $y^{(i)}$ with the
given
$\bm{x}^{(i)},\bm{\varphi}^{(i)},\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)},T^{(i)}$.
Repeat the above operations until the solutions converge. In the following
section, we consider using the alternating optimization method (AO) to tackle
the complex problem (7).

Roadmap of the whole algorithm. First, we decompose the outer fractional
structure of the original ECR problem using Dinkelbach algorithm and
sequentially optimize $\bm{x},\bm{\varphi}$ and
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$ using the AO method. In
the first step of AO, we fix
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$ and optimize
$\bm{x},\bm{\varphi},T$. We transform the optimization problem in the first
step of AO into a Quadratically Constrained Quadratic Program (QCQP) and solve
it using Semidefinite Relaxation (SDR) and the Hungarian algorithm. In the
second step of AO, we fix $\bm{x},\bm{\varphi}$ and optimize
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$. During the
optimization in the second step of AO, we propose a new fractional programming
method to transform this non-convex problem into a convex one. Finally, we
calculate $y$ based on the obtained solutions and repeat the aforementioned
process until $y$ converges. In this algorithm, since we utilize Dinkelbach’s
algorithm, alternating optimization, semidefinite relaxation, Hungarian
algorithm, and fractional programming, we refer to this algorithm as the DASHF
Algorithm.

##  III Our proposed DASHF Algorithm to solve the optimization problem

Assuming that $y$ is given, we need to optimize
$\bm{x},\bm{\varphi},\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$. In
the outermost loops, we iteratively optimize $y$; In the innermost loops, we
iteratively optimize
$\bm{x},\bm{\varphi},\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$.
However, it is still difficult to optimize them in parallel. Thus, we consider
operating two inner AO steps to solve it. At the $i$-th iteration,

  1. 1.

Optimize $\bm{x}$, $\bm{\varphi}$, $T$, given $\bm{b}$, $\bm{p_{u}}$,
$\bm{p_{s}}$, $\bm{f_{u}}$, $\bm{f_{s}}$. Assuming that $\bm{b}^{(i-1)}$,
$\bm{p_{u}}^{(i-1)},$$\bm{p_{s}}^{(i-1)}$, $\bm{f_{u}}^{(i-1)}$,
$\bm{f_{s}}^{(i-1)}$, $y^{(i-1)}$ are given, we optimize $\bm{x}^{(i)}$,
$\bm{\varphi}^{(i)}$, $T^{(i)}$.

  2. 2.

Optimize $\bm{b}$,$\bm{p_{u}}$,$\bm{p_{s}}$,$\bm{f_{u}}$,$\bm{f_{s}}$,$T$,
given $\bm{x}$, $\bm{\varphi}$. Assuming that $\bm{x}^{(i-1)}$,
$\bm{\varphi}^{(i-1)}$,$y^{(i-1)}$ are given, we optimize
$\bm{b}^{(i)}$,$\bm{p_{u}}^{(i)}$,$\bm{p_{s}}^{(i)}$,$\bm{f_{u}}^{(i)}$,$\bm{f_{s}}^{(i)}$,$T^{(i)}$.

###  III-A AO Part 1: Optimizing $\bm{x},\bm{\varphi},T$, given
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}}$

Given $\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$, we optimize
$\bm{x},\bm{\varphi},T$. The optimization problem will be:

$\displaystyle\\!\\!\\!\max\limits_{\bm{x},\bm{\varphi},T}\\!\\!\\!\\!$
$\displaystyle\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\![x_{n,m}v_{n,m}\\!-\\!y\omega_{e}(E_{u,n,m}\\!+\\!E_{s,n,m})]-y\omega_{t}T$
(8) s.t.
$\displaystyle(\text{\ref{x_constr1}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{T_constr}}).$

$x_{n,m}$ are binary variables and this is a mixed-integer nonlinear
programming problem. We rewrite $x_{n,m}\in\\{0,1\\},\forall n,m$ as
$x_{n,m}(x_{n,m}-1)=0,\forall n,m$. The optimization problem will be rewritten
as:

$\displaystyle\\!\\!\\!\\!\max\limits_{\bm{x},\bm{\varphi},T}\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\!\\!\\![x_{n,m}v_{n,m}\\!-\\!y\omega_{e}(E_{u,n,m}\\!+\\!E_{s,n,m})]-y\omega_{t}T$
(9) $\displaystyle\text{s.t.}\quad x_{n,m}(x_{n,m}-1)=0,\forall n,m$ (9a)
$\displaystyle\quad\quad(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{T_constr}}).$

We substitute the expression of $E_{u,n,m}+E_{s,n,m}$ into problem (9) and
convert the $\max$ problem in problem (9) to a $\min$ problem (10).

$\displaystyle\min\limits_{\bm{x},\bm{\varphi},T}y\omega_{t}T\\!\\!+\\!\\!\\!\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\!\\!\\!\\{y\omega_{e}(\frac{p_{n}d_{n}\omega_{b}}{r_{n,m}}\\!\\!-\\!\\!\frac{p_{m}d_{n}\omega_{b}}{r_{m,n}}\\!\\!-\\!e_{m}\kappa_{m}d_{n}t_{n}F_{m}^{2})$
$\displaystyle
x_{n,m}\varphi_{n}+[y\omega_{e}(\frac{p_{m}d_{n}\omega_{b}}{r_{m,n}}+e_{m}\kappa_{m}d_{n}t_{n}F_{m}^{2})-v_{n,m}]x_{n,m}$
$\displaystyle\quad+y\omega_{e}(e_{n}\kappa_{n}d_{n}t_{n}F_{n}^{2})\varphi_{n}\\}$
(10)
$\displaystyle\text{s.t.}\quad\quad(\text{\ref{x_constr1_new}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{T_constr}}).$

Let
$G_{n,m}\\!=\\!y\omega_{e}(\frac{p_{n}d_{n}\omega_{b}}{r_{n,m}}-\frac{p_{m}d_{n}\omega_{b}}{r_{m,n}}-e_{m}\kappa_{m}d_{n}t_{n}F_{m}^{2})$,
$A_{n,m}=y\omega_{e}(e_{n}\kappa_{n}d_{n}t_{n}F_{n}^{2})$, and
$B_{n,m}=y\omega_{e}(\frac{p_{m}d_{n}\omega_{b}}{r_{m,n}}+e_{m}\kappa_{m}d_{n}t_{n}F_{m}^{2})-v_{n,m}$.
Let $\mathbf{A}=[A_{n}]|_{n\in\mathcal{N}}$,
$\mathbf{B}=[B_{n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$, and
$\mathbf{G}=[G_{n,m}]|_{n\in\mathcal{N},m\in\mathcal{M}}$. The optimization
problem (10) can be rewritten as

$\displaystyle\mathcal{P}_{1}:$
$\displaystyle\min\limits_{\bm{x},\bm{\varphi},T}y\omega_{t}T\\!\\!+\\!\\!\sum\limits_{n\in\mathcal{N}}A_{n}\varphi_{n}+\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!(B_{n,m}x_{n,m}+$
$\displaystyle\quad\quad\quad\quad G_{n,m}x_{n,m}\varphi_{n})$ (11)
$\displaystyle\text{s.t.}\quad(\text{\ref{x_constr1_new}}),(\text{\ref{x_constr2}}),(\text{\ref{varphi_constr}}),(\text{\ref{x_b_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{T_constr}}).$

This is a quadratically constrained quadratic program (QCQP) problem. Then, we
need to get the standard form of the QCQP problem. Let
$\bm{Q}=(\bm{\varphi}^{\intercal},\bm{x_{1}}^{\intercal},\cdots,\bm{x_{M}}^{\intercal})^{\intercal}$,
where $\bm{\varphi}=(\varphi_{1},\cdots,\varphi_{N})^{\intercal}$ and
$\bm{x_{m}}=(x_{1,m},\cdots,x_{n,m})^{\intercal},\forall m\in\mathcal{M}$. Let
$\bm{e}_{i}$ denotes
$(0,\cdots,\underset{\underset{i\text{-th}}{\uparrow}}{1},\cdots,0)^{\intercal}_{NM+N\times
1}$. Let $\bm{e}_{i,j}$ denotes
$(\bm{e}_{i},\cdots,\bm{e}_{j})^{\intercal},i<j$. Then,
$\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}G_{n,m}x_{n,m}\varphi_{n}$ can
be expressed as $\bm{Q}^{\intercal}\text{diag}(\bm{G},\bm{0}_{NM\times
1})(\bm{e}_{N+1,NM+N},\bm{0}_{NM\times NM+N})^{\intercal}\bm{Q}$.Let

$\mathbf{P}_{0}=\text{diag}(\bm{G},\bm{0}_{NM\times
1})(\bm{e}_{N+1,NM+N},\bm{0}_{NM\times NM+N})^{\intercal}.$ (12)

$\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}B_{n,m}x_{n,m}$ can be expressed
as $\mathbf{W}_{0}^{\intercal}\bm{Q}$, where
$\mathbf{W}_{0}^{\intercal}=(\bm{B}_{1}^{\intercal},\cdots,\bm{B}_{M}^{\intercal})\bm{e}_{N+1,NM+N}$
and $\bm{B}_{i}^{\intercal}=(B_{1,i},\cdots,B_{N,i}),\forall i\in\mathcal{M}$.
$\sum_{n\in\mathcal{N}}A_{n}\varphi_{n}$ can be expressed as
$\mathbf{W}_{1}^{\intercal}\bm{Q}$, where
$\mathbf{W}_{1}^{\intercal}=(A_{1},\cdots,A_{N})\bm{e}_{1,N}$. Let
$P^{(\text{T1})}_{n,m}=-\frac{d_{n}\omega_{b}}{r_{m,n}}+\frac{d_{n}\omega_{b}}{r_{n,m}}-\frac{t_{n}d_{n}e_{m}}{g_{m}F_{m}}$,
$P^{(\text{T2})}_{n}=\frac{t_{n}d_{n}e_{n}}{g_{n}F_{n}}$, and
$P^{(\text{T3})}_{n,m}=\frac{t_{n}d_{n}e_{m}}{g_{m}F_{m}}+\frac{d_{n}\omega_{b}}{r_{m,n}}$.
Let
$\mathbf{P^{(\text{T1})}}=[P^{(\text{T1})}]|_{n\in\mathcal{N},m\in\mathcal{M}}$,
$\mathbf{P^{(\text{T2})}}=[P^{(\text{T2})}]|_{n\in\mathcal{N}}$,
$\mathbf{P^{(\text{T3})}}=[P^{(\text{T3})}]|_{n\in\mathcal{N},m\in\mathcal{M}}$.
Let
$\mathbf{P^{(\text{T4})}}=\text{diag}(\bm{P^{(\text{T1})}},\bm{0}_{NM\times
1})(\bm{e}_{N+1,NM+N},\bm{0}_{NM\times NM+N})^{\intercal}$,
$\mathbf{P^{(\text{T5})}}_{1}^{\intercal}=(P^{(\text{T2})}_{1},\cdots,P^{(\text{T2})}_{N})\bm{e}_{1,N}$,
${\mathbf{P^{(\text{T6})}}}^{\intercal}=({\bm{P^{(\text{T3})}}_{1}}^{\intercal},\cdots,{\bm{P^{(\text{T3})}}_{M}}^{\intercal})\bm{e}_{N+1,NM+N}$,
where
${\bm{P^{(\text{T3})}}_{i}}^{\intercal}=(P^{(\text{T3})}_{1,i},\cdots,P^{(\text{T3})}_{N,i}),\forall
i\in\mathcal{M}$. Therefore, the optimization problem can be expressed as

$\displaystyle\mathcal{P}_{2}:$
$\displaystyle\min\limits_{\bm{Q},T}\quad\bm{Q}^{\intercal}\mathbf{P}_{0}\bm{Q}+\mathbf{W}_{0}^{\intercal}\bm{Q}+\mathbf{W}_{1}^{\intercal}\bm{Q}+y\omega_{t}T$
(13) s.t.
$\displaystyle\text{diag}(\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q})(\text{diag}(\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q})-\mathbf{I})=\mathbf{0}$
(13a)
$\displaystyle\text{diag}(\bm{e}_{\overline{1},\overline{M}}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q})=\mathbf{I}$
(13b) $\displaystyle\text{diag}(\bm{e}_{1,N}^{\intercal}\bm{Q})\leq\mathbf{I}$
(13c) $\displaystyle\text{diag}(\bm{e}_{1,N}^{\intercal}\bm{Q})\geq\mathbf{0}$
(13d)
$\displaystyle\bm{B}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q}-B_{max}\leq
0$ (13e)
$\displaystyle\bm{P}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q}-P_{max}^{(m)}\leq
0$ (13f)
$\displaystyle\bm{F}^{\intercal}\bm{e}_{N+1,NM+N}^{\intercal}\bm{Q}-F_{max}^{(m)}\leq
0$ (13g)
$\displaystyle\bm{Q}^{\intercal}\mathbf{P}^{(\text{T4})}\bm{Q}+{\mathbf{P}^{(\text{T5})}}^{\intercal}\bm{Q}+{\mathbf{P}^{(\text{T6})}}^{\intercal}\bm{Q}\leq
T,$ (13h)

where $\bm{e}_{\overline{i},\overline{j}}$ denotes
$(\bm{e}_{\overline{i}},\cdots,\bm{e}_{\overline{j}})^{\intercal},i<j$,
$\bm{B}=(b_{1,1},\cdots,b_{n,m})^{\intercal}$,
$\bm{P}=(P_{1,1},\cdots,p_{n,m})^{\intercal}$, and
$\bm{F}=(F_{1,1},\cdots,F_{n,m})^{\intercal}$. The constraints
$(\text{\ref{x_constr1_new}})$, $(\text{\ref{x_constr2}})$,
$(\text{\ref{varphi_constr}})$, $(\text{\ref{x_b_constr}})$,
$(\text{\ref{x_ps_constr}})$, $(\text{\ref{x_fs_constr}})$,
$(\text{\ref{T_constr}})$ in $\mathcal{P}_{1}$ are transformed into the
constraints $(\text{\ref{x_constr1_qcqp}})$, $(\text{\ref{x_constr2_qcqp}})$,
$\\{(\text{\ref{varphi_constr1_qcqp}})$,
$(\text{\ref{varphi_constr2_qcqp}})\\}$, $(\text{\ref{x_b_constr_qcqp}})$,
$(\text{\ref{x_ps_constr_qcqp}})$, $(\text{\ref{x_fs_constr_qcqp}})$,
$(\text{\ref{T_constr_qcqp}})$ in $\mathcal{P}_{2}$, respectively. Problem
(13) is the standard QCQP form. However, it is still non-convex. Then, we need
to utilize the semidefinite programming (SDP) method to transform this QCQP
problem into a semidefinite relaxation (SDR) problem. Let
$\mathbf{S}=(\bm{Q}^{\intercal},1)^{\intercal}(\bm{Q}^{\intercal},1)$. Let
$\bm{e}_{i\rightarrow j}$ denotes
$(0,\cdots,\underset{\underset{i\text{-th}}{\uparrow}}{1},1,\cdots,\underset{\underset{j\text{-th}}{\uparrow}}{1},0,\cdots,0)^{\intercal},i<j$.
Then we obtain the SDR problem

$\displaystyle\mathcal{P}_{3}:\min\limits_{\bm{S},T}\quad$
$\displaystyle\text{Tr}(\mathbf{P}_{1}\mathbf{S})$ (14) s.t.
$\displaystyle\text{Tr}(\mathbf{P}_{2}\mathbf{S})=0$ (14a)
$\displaystyle\text{Tr}(\mathbf{P}_{3}\mathbf{S})=0$ (14b)
$\displaystyle\text{Tr}(\mathbf{P}_{4}\mathbf{S})\leq 0$ (14c)
$\displaystyle\text{Tr}(\mathbf{P}_{5}\mathbf{S})\leq 0$ (14d)
$\displaystyle\text{Tr}(\mathbf{P}_{6}\mathbf{S})\leq 0$ (14e)
$\displaystyle\text{Tr}(\mathbf{P}_{7}\mathbf{S})\leq 0$ (14f)
$\displaystyle\text{Tr}(\mathbf{P}_{8}\mathbf{S})\leq T$ (14g)
$\displaystyle\mathbf{S}\succeq 0,$ (14h)

where

$\mathbf{P}_{1}=\left(\begin{array}[]{cc}\mathbf{P}_{0}&\frac{1}{2}(\mathbf{W}_{0}+\mathbf{W}_{1})\\\
\frac{1}{2}(\mathbf{W}_{0}+\mathbf{W}_{1})^{\intercal}&y\omega_{t}T\end{array}\right),$
$\mathbf{P}_{2}=\left(\begin{array}[]{cc}\bm{e}_{i}^{\intercal}\bm{e}_{i}&-\frac{1}{2}\bm{e}_{i}\\\
-\frac{1}{2}\bm{e}_{i}^{\intercal}&0\end{array}\right),\forall
i\in\\{1,\cdots,NM\\}$
$\displaystyle\mathbf{P}_{3}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}(\bm{e}_{\overline{i}}\bm{e}_{N+1,NM+N}^{\intercal})\\\
\frac{1}{2}(\bm{e}_{\overline{i}}\bm{e}_{N+1,NM+N}^{\intercal})^{\intercal}&-1\end{array}\right),$
(17) $\displaystyle\forall i\in\\{1,\cdots,N\\}$
$\mathbf{P}_{4}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{e}_{i}\\\
\frac{1}{2}\bm{e}_{i}^{\intercal}&-1\end{array}\right),\forall
i\in\\{1,\cdots,N\\}$
$\mathbf{P}_{5}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{B}\bm{e}_{N+1,NM+N}\\\
\frac{1}{2}(\bm{B}\bm{e}_{N+1,NM+N})^{\intercal}&-B_{max}\end{array}\right),$
$\mathbf{P}_{6}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{P}\bm{e}_{N+1,NM+N}\\\
\frac{1}{2}(\bm{P}\bm{e}_{N+1,NM+N})^{\intercal}&-P_{max}^{(m)}\end{array}\right),$
$\mathbf{P}_{7}=\left(\begin{array}[]{cc}\mathbf{0}_{NM+N\times
NM+N}&\frac{1}{2}\bm{F}\bm{e}_{N+1,NM+N}\\\
\frac{1}{2}(\bm{F}\bm{e}_{N+1,NM+N})^{\intercal}&-F_{max}^{(m)}\end{array}\right),$
$\mathbf{P}_{8}=\left(\begin{array}[]{cc}\mathbf{P}^{(\text{T4})}&\frac{1}{2}(\mathbf{P}^{(\text{T5})}+\mathbf{P}^{(\text{T6})})\\\
\frac{1}{2}(\mathbf{P}^{(\text{T5})}+\mathbf{P}^{(\text{T6})})^{\intercal}&0\end{array}\right).$

The constraints $(\text{\ref{x_constr1_qcqp}})$,
$(\text{\ref{x_constr2_qcqp}})$, $(\text{\ref{varphi_constr1_qcqp}})$,
$(\text{\ref{varphi_constr2_qcqp}})$, $(\text{\ref{x_b_constr_qcqp}})$,
$(\text{\ref{x_ps_constr_qcqp}})$, $(\text{\ref{x_fs_constr_qcqp}})$,
$(\text{\ref{T_constr_qcqp}})$ in $\mathcal{P}_{2}$ are transformed into the
constraints
$(\text{\ref{x_constr1_sdr}}),(\text{\ref{x_constr2_sdr}}),(\text{\ref{varphi_constr_sdr}}),(\text{\ref{S_constr_sdr}}),(\text{\ref{x_b_constr_sdr}}),(\text{\ref{x_ps_constr_sdr}}),(\text{\ref{x_fs_constr_sdr}}),(\text{\ref{T_constr_sdr}})$
in $\mathcal{P}_{3}$, respectively. Drop the constraint
$\text{rank}(\mathbf{S})=1$ and the objective function and the constraints are
all convex. Then this SDR problem will be solved in polynomial time by common
convex solvers. By solving this SDR problem, we can get a continuous solution
of $\bm{Q}$. However, this solution is the lower bound of the optimal solution
and it may not guarantee the constraint $\text{rank}(\mathbf{S})=1$.
Therefore, we need to use rounding techniques to recover the solution. The
latter $NM$ elements in $\bm{Q}$ is $x_{n,m}$, for all
$n\in\mathcal{N},m\in\mathcal{M}$, which means that user $n$ is fractional
connected to server $m$. Then, find all user $n$ that
$\sum\limits_{m\in\mathcal{M}}x_{n,m}>1$. For these users, modify $x_{n,m}$ as
$\frac{x_{n,m}}{|\sum\limits_{m\in\mathcal{M}}x_{n,m}|}$. Use the Hungarian
algorithm [9] with augmented zero vectors to find the best matching with the
maximum weight and denote this matching as a set $\mathcal{X}_{matching}$. For
nodes $n$ and $m$ in $\mathcal{X}_{matching}$, let $x_{n,m}=1$, else
$x_{n,m}=0$, and denote this integer association result as $\bm{x}_{\sharp}$.
Then substitute $\bm{x}_{\sharp}$ into Problem (13) to obtain the optimal
$\bm{\varphi}$.

###  III-B AO Part 2: Optimizing
$\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T$, given
$\bm{x},\bm{\varphi}$

Given $\bm{x}$ and $\bm{\varphi}$, the remaining optimization problem is shown
as (18).

$\displaystyle\max\limits_{\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T}\quad\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}x_{n,m}v_{n,m}-y\bigg{\\{}\omega_{t}T$
$\displaystyle+\omega_{e}\big{\\{}\sum\limits_{n\in\mathcal{N}}e_{n}\kappa_{n}\varphi_{n}d_{n}t_{n}F^{2}_{n}+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\{p_{n}\frac{x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}$
$\displaystyle+\\!e_{m}\kappa_{m}x_{n,m}(1\\!\\!-\\!\\!\varphi_{n})d_{n}t_{n}F_{n,m}^{2}\\!\\!+\\!p_{m}\frac{x_{n,m}(1\\!\\!-\\!\\!\varphi_{n})d_{n}\omega_{b}}{r_{m,n}}\\}\big{\\}}\bigg{\\}}$
(18)
$\displaystyle\text{s.t.}\quad(\text{\ref{x_b_constr}}),(\text{\ref{pu_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{fu_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{nonnegative_constr}}),(\text{\ref{T_constr}}).$

We first Let

$\textstyle\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)=\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!x_{n,m}v_{n,m}-y\omega_{t}T$
$\textstyle-y\omega_{e}\big{\\{}\sum\limits_{n\in\mathcal{N}}e_{n}\kappa_{n}\varphi_{n}d_{n}t_{n}F^{2}_{n}$
$\textstyle+\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!e_{m}\kappa_{m}x_{n,m}(1-\varphi_{n})d_{n}t_{n}F_{n,m}^{2}\big{\\}}.$
(19)

It’s easy to justify that
$\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)$ is concave. Then, the
optimization objective function is

$\textstyle\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)\\!-\\!y\omega_{e}\\!\\!\\!\\!\\!\\!\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}\\!\\!\\!\\!\\!\\!\\!(\frac{p_{n}x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}\\!$
$\textstyle+\\!\frac{p_{m}x_{n,m}(1-\varphi_{n})d_{n}\omega_{b}}{r_{m,n}}),$
(20)

where
$\frac{p_{n}x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{n,m}}+\frac{p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{b}}{r_{m,n}}$
is non-convex or concave. Then, according to the fractional programming
technique introduced in the Section IV in [1], we let
$z_{1,n,m}=\frac{1}{2p_{n}x_{n,m}\varphi_{n}d_{n}\omega_{b}r_{n,m}}$ and
$z_{2,n,m}=\frac{1}{2p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{b}r_{m,n}}$. The
optimization objective function can be expressed as

$\textstyle\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)$
$\textstyle-y\omega_{e}\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}[(p_{n}x_{n,m}\varphi_{n}d_{n})^{2}z_{1,n,m}+$
$\textstyle\frac{1}{4(b_{n,m}\log_{2}(1+\frac{g_{n,m}p_{n}}{\sigma^{2}b_{n,m}}))^{2}z_{1,n,m}}\\!\\!+\\!\\!(p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p})^{2}z_{2,n,m}$
$\textstyle+\frac{1}{4(b_{m,n}\log_{2}(1+\frac{g_{m,n}p_{m}}{\sigma^{2}b_{m,n}}))^{2}z_{2,n,m}}].$
(21)

The complete transformation optimization problem is shown as (22).

$\displaystyle\max\limits_{\bm{b},\bm{p_{u}},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T}\mathcal{F}(\bm{b},\bm{p_{s}},\bm{f_{u}},\bm{f_{s}},T)$
$\displaystyle\quad\quad\quad\quad\quad-y\omega_{e}\sum\limits_{n\in\mathcal{N},m\in\mathcal{M}}[(p_{n}x_{n,m}\varphi_{n}d_{n})^{2}z_{1,n,m}$
$\displaystyle\quad\quad\quad\quad\quad+\frac{1}{4(b_{n,m}\log_{2}(1+\frac{g_{n,m}p_{n}}{\sigma^{2}b_{n,m}}))^{2}z_{1,n,m}}$
$\displaystyle\quad\quad\quad\quad\quad+(p_{m}x_{n,m}\varphi_{n}d_{n}\omega_{p})^{2}z_{2,n,m}$
$\displaystyle\quad\quad\quad\quad\quad+\frac{1}{4(b_{m,n}\log_{2}(1+\frac{g_{m,n}p_{m}}{\sigma^{2}b_{m,n}}))^{2}z_{2,n,m}}]$
(22)
$\displaystyle\quad\text{s.t.}(\text{\ref{x_b_constr}}),(\text{\ref{pu_constr}}),(\text{\ref{x_ps_constr}}),(\text{\ref{fu_constr}}),(\text{\ref{x_fs_constr}}),(\text{\ref{nonnegative_constr}}),(\text{\ref{T_constr}}).$

If $\bm{z}_{1}=(z_{1,1,1},\cdots,z_{1,n,m})^{\intercal}$ and
$\bm{z}_{2}=(z_{2,1,1},\cdots,z_{2,n,m})^{\intercal}$ is given, the objective
function (22) is concave. At the $i$-th iteration, $[y^{(i)}$,
$\bm{z}_{1}^{(i)}$, $\bm{z}_{2}^{(i)}]$ are first calculated with the solution
$[\bm{b}^{(i-1)},\bm{p_{u}}^{(i-1)},\bm{p_{s}}^{(i-1)},\bm{f_{u}}^{(i-1)},\bm{f_{s}}^{(i-1)}]$.
Then,
$[\bm{b}^{(i)},\bm{p_{u}}^{(i)},\bm{p_{s}}^{(i)},\bm{f_{u}}^{(i)},\bm{f_{s}}^{(i)}]$
can be obtained by solving the concave problem (22) with
$[y^{(i)},\bm{z}_{1}^{(i)},\bm{z}_{2}^{(i)}]$. Thus, the optimization problem
is concave and can be solved by common convex solvers. More detailed proofs
can be seen in [1].

(a) Convergence of Algorithm AO-Part 1

(b) Convergence of Algorithm AO-Part 2

(c) Convergence of Algorithm DASHF

Figure 2: Convergence of the proposed Algorithms

(a) Resource consumption and ECR of baselines and proposed DASHF Algorithm

(b) ECR under different total bandwidth

(c) Resources consumption and ECR under different $(\omega_{t},\omega_{e})$

Figure 3: ECR comparisons under different cases

##  IV Simulation Results

In this section, we first introduce the default settings for the numerical
simulations. Subsequently, we verify the convergence of the proposed DASHF
algorithm and compare it with other baselines to validate its effectiveness.
Then, we adjust the available communication and computational resources as
well as the cost weight parameters to analyze their impacts on the ECR.

###  IV-A Default settings

We consider a network topology of 1000 m $\times$ 1000 m with 10 mobile users
and 2 servers. The large-scale fading $h_{n,m}$ between the user $n$ and
server $m$ is modeled as $128.1+37.6\log_{10}d_{n,m}$, where $d_{n,m}$ denotes
the Euclidean distance between the user $n$ and server $m$. The small-scale
fading is the Rayleigh fading. Gaussian noise power $\sigma^{2}$ is $-134$dBm.
The total bandwidth for each server $b_{max}$ is 10 MHz. The maximum transmit
power of mobile users $p_{max}^{(n)}$ is 0.2 W. The maximum transmit power of
servers $p_{max}^{(m)}$ is 10 W. We assume the GPU resource utilization is
$0.55$ for users and servers. The maximum GPU computation speed of mobile
users $F_{max}^{(n)}$ is $19.58$ TFLOPs with four GTX 1080 GPUs and that of
servers $F_{max}^{(m)}$ is $1372.8$ TFLOPs with eight A100 GPUs. The effective
switched capacitance of mobile users and servers ($\kappa_{n}$ and
$\kappa_{m}$) is $10^{-38}$. We refer to the adapter parameter sizes in [3]
and [2]. The adapter parameter sizes of mobile users are randomly selected
from $[1.2,14]$ M. To achieve this, pseudorandom values are generated, which
follow a standard uniform distribution over the open interval $(0,1)$. These
pseudorandom values are then scaled to the range of $[1.2,14]$ M to determine
the specific adapter parameter sizes for each mobile user. The token data
sizes of users are randomly selected from $[10,50]$ M bits. The parameters of
delay and energy consumption ($\omega_{t}$ and $\omega_{e}$) are 0.5 and 0.005
to keep them in the same order. We set $\varpi_{1}$ and $\varpi_{2}$ as
$\frac{10000}{\ln 2}$ and $\frac{1}{3}$ to keep ECR big enough, respectively.
we consider the “float32” method to represent the floating-point number and
$\omega_{b}$ is 32. We consider using the Mosek optimization tool in Matlab to
conduct the simulations.

###  IV-B Convergence of proposed Algorithms

In this section, we evaluate the convergence of the proposed algorithms. We
consider two network topologies with (10 users, 2 servers) and (20 users, 3
servers) and keep other settings as default. Primal objective value means an
estimate for the primal objective value when using Mosek to solve the
optimization problem. When the primal objective value converges, the algorithm
converges to one stationary point. Fig. 2(a) plots the convergence of the
Algorithm AO-Part 1, which converges within 15 iterations. Fig. 2(b) plots the
convergence of the Algorithm AO-Part 2, which converges within 9 iterations.
Fig. 2(c) plots the convergence of the DASHF Algorithm, which converges within
9 iterations. Thus, the proposed DASHF Algorithm is effective in finding one
stationary point of the Problem (7).

###  IV-C Comparison with baselines

In this section, we mainly consider four baselines to carry out the comparison
experiments.

  1. 1.

Random user connection with average resource allocation (RUCAA). In this
algorithm, one server is randomly selected for each user. The server equally
allocates communication and computational resources among the users connected
to it.

  2. 2.

Greedy user connection with average resource allocation (GUCAA). In this
algorithm, each user selects the server with the least number of users
underserving. The server distributes communication and computational resources
evenly to the users connected to it.

  3. 3.

Average resource allocation with user connection optimization (AAUCO). In this
algorithm, the communication and computation resources of each Metaverse
server are equally allocated to each user connected to it. Besides, Algorithm
III-A is leveraged to operate user connection optimization.

  4. 4.

Greedy user connection with resource allocation optimization (GUCRO). In this
algorithm, each user selects the Metaverse server with the least number of
users underserving. Besides, Algorithm III-B is leveraged to operate resource
optimization.

  5. 5.

Proposed DASHF algorithm. Joint optimization of user connection and resource
allocation by utilizing the whole proposed DASHF algorothm.

In Fig. 3(a), we compare the resource consumption and ECR of the proposed
DASHF Algorithms with other baselines. The performances of RUCAA and GUCAA are
worse since no optimization is utilized. GUCRO and AAUCO have better
performances than GUCAA, which confirms the effectiveness of the proposed
Algorithm AO-Part 1 and Part 2. Furthermore, the ECR of AAUCO is higher than
that of GUCRO, which shows that user connection optimization is more effective
than resource optimization in this case. The time consumption of the proposed
DASHF algorithm is the lowest of these five methods and the energy consumption
is also low (just higher than AAUCO), and the ECR is the highest one. This
results from the benefits of joint optimization of user connection and
resource allocation.

###  IV-D ECR versus the total bandwidth

We consider the total bandwidth from 10 MHz to 100 MHz to test the ECR under
different total bandwidths. Other parameters are fixed as default settings.
Fig. 3(b) reveals distinct algorithmic performance trends, with the proposed
DASHF method consistently outperforming GUCRO, AAUCO, RUCAA, and GUCAA in
terms of the ECR. Notably, optimization algorithms (GUCRO and AAUCO)
demonstrate superior or close performance compared to non-optimization
algorithms (RUCAA and GUCAA). AAUCO employs user connection optimization
strategies and performs better than RUCAA, GUCRO, and GUCAA.

###  IV-E Impact of cost weights on ECR

Fig. 3(c) featuring various combinations of ($\omega_{t}$, $\omega_{e}$) that
signifies the trade-off between delay-energy optimization and trust score. As
($\omega_{t}$, $\omega_{e}$) values shift, emphasizing either delay or energy,
distinct performance outcomes are evident. For instance, when prioritizing
energy efficiency (e.g., ($\omega_{t}$, $\omega_{e}$) = (0.1, 0.009)), the
system achieves lower energy consumption but at the expense of higher delay,
resulting in a moderate ECR value. Conversely, balanced settings (e.g.,
($\omega_{t}$, $\omega_{e}$) = (0.5, 0.005)) lead to lower delay and slightly
higher energy consumption, yielding a high ECR value. These findings
underscore the sensitivity of the optimization process to parameter choices
and emphasize the importance of tailoring ($\omega_{t}$, $\omega_{e}$) values
to meet specific application requirements while carefully considering the
trade-offs between delay-energy and trust score.

##  V Conclusion

In this investigation into LLMs and MEC, we’ve delved into the intricacies of
ensuring efficient LLM service delivery amidst the constraints of wireless
communication systems. With their vast linguistic and computational
capabilities, the promise of LLMs is now being actualized in real-world
applications. Our contributions, as presented in this paper, lay the
foundation for seamless collaborative training between mobile users and
servers, addressing the challenges of limited computational and communication
resources. We optimize resource utilization and ensure robust LLM performance
by implementing a framework where initial layers are trained by users and
subsequent layers by servers. In this context, the ECR measures collaboration
efficiency and resource optimization. The DASHF algorithm, central to our
methodology, solidifies these efforts. In conclusion, we anticipate a
landscape where mobile edge computing enables ubiquitous and efficient access
to advanced LLM services, harmonizing computational constraints with the ever-
growing demands of modern applications.

## References

  * [1] J. Zhao, L. Qian, and W. Yu, “Human-centric resource allocation in the Metaverse over wireless communications,” _IEEE Journal on Selected Areas in Communications (JSAC)_ , 2023. [Online]. Available: https://arxiv.org/pdf/2304.00355.pdf
  * [2] P. Gao, J. Han, R. Zhang, Z. Lin, S. Geng, A. Zhou, W. Zhang, P. Lu, C. He, X. Yue _et al._ , “Llama-adapter v2: Parameter-efficient visual instruction model,” _arXiv preprint arXiv:2304.15010_ , 2023.
  * [3] R. Zhang, J. Han, A. Zhou, X. Hu, S. Yan, P. Lu, H. Li, P. Gao, and Y. Qiao, “Llama-adapter: Efficient fine-tuning of language models with zero-init attention,” _arXiv preprint arXiv:2303.16199_ , 2023.
  * [4] L. Dong, F. Jiang, Y. Peng, K. Wang, K. Yang, C. Pan, and R. Schober, “Lambo: Large language model empowered edge intelligence,” _arXiv preprint arXiv:2308.15078_ , 2023.
  * [5] Y. Shen, J. Shao, X. Zhang, Z. Lin, H. Pan, D. Li, J. Zhang, and K. B. Letaief, “Large language models empowered autonomous edge ai for connected intelligence,” _arXiv preprint arXiv:2307.02779_ , 2023.
  * [6] Q. Zeng, Y. Du, K. Huang, and K. K. Leung, “Energy-efficient resource management for federated edge learning with CPU-GPU heterogeneous computing,” _IEEE Transactions on Wireless Communications_ , vol. 20, no. 12, pp. 7947–7962, 2021.
  * [7] D. Yang, G. Xue, X. Fang, and J. Tang, “Incentive mechanisms for crowdsensing: Crowdsourcing with smartphones,” _IEEE/ACM Transactions on Networking_ , vol. 24, no. 3, pp. 1732–1744, 2015.
  * [8] W. Dinkelbach, “On nonlinear fractional programming,” _Management Science_ , vol. 13, no. 7, pp. 492–498, 1967.
  * [9] Y. Dai, D. Xu, S. Maharjan, and Y. Zhang, “Joint computation offloading and user association in multi-task mobile edge computing,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 12, pp. 12 313–12 325, 2018.


# ”Infinite” properties of certain local cohomology modules of determinantal
rings

Peter Schenzel  Martin-Luther-Universität Halle-Wittenberg, Institut für
Informatik, D — 06 099 Halle (Saale), Germany<EMAIL_ADDRESS>halle.de Dedicated to Nguyen Tu Cuong on the occasion of his 70th birthday
and for a long friendship

###### Abstract.

For given integers $m,n\geq 2$ there are examples of ideals $I$ of complete
determinantal local rings $(R,\mathfrak{m}),\dim
R=m+n-1,\operatorname{grade}I=n-1,$ with the canonical module $\omega_{R}$ and
the property that the socle dimensions of $H^{m+n-2}_{I}(\omega_{R})$ and
$H^{m}_{\mathfrak{m}}(H^{n-1}_{I}(\omega_{R}))$ are not finite. In the case of
$m=n$, i.e. a Gorenstein ring, the socle dimensions provide further
information about the $\tau$-numbers as studied in [10]. Moreover, the
endomorphism ring of $H^{n-1}_{I}(\omega_{R})$ is studied and shown to be an
$R$-algebra of finite type but not finitely generated as $R$-module
generalizing an example of [15].

###### Key words and phrases:

local cohomology, determinantal ring, ring of endomorphisms

###### 2010 Mathematics Subject Classification:

Primary: 13D45 ; Secondary: 13H10, 13J10

##  1\. Introduction

Let $I$ denote an ideal of a local ring $(R,\mathfrak{m})$ with
$\Bbbk=R/\mathfrak{m}$ its residue field. Let $M$ be a finitely generated
$R$-module, and let $H^{i}_{I}(M),i\in\mathbb{Z},$ denote the local cohomology
modules of $M$ with respect to $I$ (see [3] or [1] for definitions). By
$\operatorname{grade}(I,M)$ we denote the length of the largest regular
sequence contained in $I$. In the case of $M=R$ , an equicharacteristic
complete regular local ring, the following is known:

  * (a)

The Bass numbers $\dim_{\Bbbk}\operatorname{Ext}_{R}^{i}(\Bbbk,H^{i}_{I}(R))$
are finite (see [8] and [9]).

  * (b)

The natural homomorphism
$R\to\operatorname{Hom}_{R}(H^{c}_{I}(R),H^{c}_{I}(R)),c=\operatorname{grade}I,$
is an isomorphism if and only if $H^{i}_{I}(R)=0$ for $i=d-1,d,$ and
$c<d-1,d=\dim R$ (see [15]).

The socle $\operatorname{Hom}_{R}(\Bbbk,H^{i}_{I}(M))$ is in general not
finite dimensional as has been shown at first by R. Hartshorne (see [5]) by
disproving a question by A. Grothendieck about cofiniteness of local
cohomology, i.e. the finiteness of $\operatorname{Hom}_{R}(R/I,H^{i}_{I}(R))$.
In their paper Huneke and Koh (see [7]) studied cofiniteness of various
ideals. Moreover they showed that for a field $\Bbbk$ of characteristic zero
and a polynomial ring $R=\Bbbk[x_{i,j},1\leq 2,1\leq j\leq 3]$ the local
cohomology $H^{3}_{I}(R)$ is not $I$-cofinite for $I$ the ideal generated by
the maximal minors of the $2\times 3$-matrix $(x_{i,j})$. A large class of
local cohomology modules with infinite socle has been constructed by T. Marley
and C. Vassilev (see [11]).

The main topic of the present note is to discuss and relate the properties (a)
and (b) above in the non-smooth situation of certain determinantal rings. More
precisely:

######  Theorem 1.1.

Let $R$ denote the completion of the coordinate ring of the vanishing ideal of
the $2\times 2$ minors of an $m\times n$-matrix of variables and therefore
$\dim R=m+n-1$. For the ideal $I$ generated by the first $n-1$ columns we have
$\dim R/I=m$ and $\operatorname{grade}I=n-1$. Let $\omega_{R}$ denote the
canonical module of $R$.

  * (a)

$\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(\omega_{R}))$ and
$\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(R))$ are not finite dimensional,
that is $I$ is not cofinite and $H^{i}_{I}(\omega_{R})=0$ for
$i\not=n-1,m+n-2$.

  * (b)

$\operatorname{Hom}_{R}(H^{n-1}_{I}(\omega_{R}),H^{n-1}_{I}(\omega_{R}))$ is
an $R$-algebra of finite type, not finitely generated over $R$.

  * (c)

$\operatorname{Hom}_{R}(\Bbbk,H^{m}_{\mathfrak{m}}(H^{n-1}_{I}(\omega_{R})))$
and
$\operatorname{Hom}_{R}(\Bbbk,H^{0}_{\mathfrak{m}}(H^{m+n-2}_{I}(\omega_{R})))$
are not finite dimensional.

In the case of $m=n$ the ring $R$ is a Gorenstein ring and $\omega_{R}\cong
R$. For $m=n=2$ we recover R. Harsthorne’s example (see [5]) by completely
different arguments. For an ideal $I$ the cohomological dimension is defined
by $\operatorname{cd}I=\sup\\{i\in\mathbb{N}|H^{i}_{I}(R)\not=0\\}$
(introduced by R. Hartshorne (see [4])). Whence, $\operatorname{cd}I=m+n-2$
for the examples above. That is, possible non-cofinite ideals could occur in
the highest non-vanishing cohomological level. In the case of a
$d$-dimensional Gorenstein ring $(R,\mathfrak{m})$ the socle dimension
$\operatorname{Hom}_{R}(\Bbbk,H^{i}_{\mathfrak{m}}(H^{d-j}_{I}(R)))$ are
called $\tau$-numbers of type $(i,j)$ of $I$ (see [10]). In the case of a
regular local ring containing a field they coincide with the Lyubeznik numbers
introduced by Lyubeznik in [9] (see [10, 3.5] for the details). Therefore, the
above results yield further information about these $\tau$-numbers.

In Section 2 we prove the essentials about local cohomology for our purposes.
Theorem 2.2 is the needed technical result for our constructions. Furthermore,
we introduce the basics for the construction of our examples, based about some
results of Segre varieties. In Section 3 we prove the statements of the
examples introduced in Section 2. For the needed results about Commutative
Algebra we refer to Matsumura’s book [12]. For a few homological arguments
resp. some facts about determinantal ideals we refer to [16] resp. to [2].

##  2\. Preliminaries and constructions

At the beginning let us recall a few basics from local duality.

######  Notations and Recalls 2.1.

(A) Let $(R,\mathfrak{m})$ denote a $d$-dimensional Cohen-Macaulay ring with
$E=E_{R}(R/\mathfrak{m})$ the injective hull of its residue field. Suppose
that $R$ possesses a normalized dualizing complex $D$ in the sense of [16,
11.4.6]. Then $R$ admits a canonical module $\omega_{R}$ (see [13]). Moreover,
for an arbitrary $R$-module $X$ there is the following Local Duality Theorem

$H^{i}_{\mathfrak{m}}(X)\cong\operatorname{Hom}_{R}(\operatorname{Ext}_{R}^{d-i}(X,\omega_{R}),E)$

for all $i\geq 0$ (see e.g. [16, 10.4.3]). For further properties of
$\omega_{R}$ we refer to [16] and [13].
(B) With the notation of (A) let $M$ denote a finitely generated $R$-module.
Then $\omega_{R}(M)$, the canonical module of $M$ (in the sense of [13]),
exists and there is an isomorphism
$\operatorname{Ext}_{R}^{c}(M,\omega_{R})\cong\omega_{R}(M)$, where $c=\dim
R-\dim_{R}M$ (see also [13] for more details).
(C) With the notation of (A) it follows that
$\omega_{R}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}D$, a minimal
injective resolution of $\omega_{R}$, is a normalized dualizing complex in the
sense of [16, 11.4.6]. Let $M$ denote a finitely generated $R$-module. Then
there is a natural morphism

$M\to\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(M,D),D)$

that is an isomorphism in cohomology. Let $c=d-\dim_{R}M$, then it induces a
natural homomorphism

$M\to\operatorname{Ext}_{R}^{c}(\operatorname{Ext}_{R}^{c}(M,\omega_{R}),\omega_{R})=\omega_{R}(\omega_{R}(M)).$

(D) It is known that for an $R$-module $M$ the module
$\omega_{R}(\omega_{R}(M))$ is the $S_{2}$-fication of $M$ (see [13]).
Moreover, the natural homomorphism $M\to\omega_{R}(\omega_{R}(M))$ induces a
short exact sequence

$0\to M/u(M)\to\omega_{R}(\omega_{R}(M))\to C\to 0,$

where $u(M)$ denotes the intersection of those primary components of $M$ that
are of highest dimension $\dim_{R}M$ and $\dim_{R}C\leq\dim_{R}M-2$ for $C$
the cokernel. Let $I\subset R$ an ideal of grade $c$ in the Cohen-Macaulay
ring $R$. Then we get an inverse system of homomorphisms
$R/I^{\alpha}\to\operatorname{Ext}_{R}^{c}(\operatorname{Ext}_{R}^{c}(R/I^{\alpha},\omega_{R}),\omega_{R})$.
By passing to the inverse limit there is a homomorphism

$\hat{R}^{I}\to
B:=\varprojlim\operatorname{Ext}_{R}^{c}(\operatorname{Ext}_{R}^{c}(R/I^{\alpha},\omega_{R}),\omega_{R})\cong\operatorname{Ext}_{R}^{c}(H^{c}_{I}(\omega_{R}),\omega_{R}),$

where $\hat{R}^{I}$ denotes the $I$-adic completion of $I$. Note that the
inverse maps are induced by $R/I^{\alpha+1}\to R/I^{\alpha}$. By a slight
modification of the arguments of [15, Section 3] it follows that $B$ is a
commutative ring.
(E) For an arbitrary $R$-module $X$ there is a natural homomorphism
$R\to\operatorname{Hom}_{R}(X,X),r\mapsto\phi(r)$, the multiplication by $r\in
R$ on $X$. Its kernel is $\operatorname{Ann}_{R}X$. In general the
endomorphism ring $\operatorname{Hom}_{R}(X,X)$ is not commutative.

For the property of the ring $B$ in 2.1 (D) we need a few more intrinsic
properties.

######  Theorem 2.2.

Let $(R,\mathfrak{m},\Bbbk)$ be a $d$-dimensional complete local Cohen-
Macaulay ring. Let $\omega_{R}$ denote its canonical module. For an ideal
$I\subset R$ with $c=\operatorname{grade}I$ we have:

  * (a)

There are isomorphisms

$\operatorname{Hom}_{R}(H^{c}_{I}(\omega_{R}),H^{c}_{I}(\omega_{R}))\cong\operatorname{Ext}_{R}^{c}(H^{c}_{I}(\omega_{R}),\omega_{R})\cong\operatorname{Hom}_{R}(H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R})),E).$

  * (b)

The endomorphism ring
$\operatorname{Hom}_{R}(H^{c}_{I}(\omega_{R}),H^{c}_{I}(\omega_{R}))$ is a
finitely generated $R$-module if and only if
$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R})))$
is finite.

  * (c)

The natural ring homomorphism
$\rho:R\to\operatorname{Hom}_{R}(H^{c}_{I}(\omega_{R}),H^{c}_{I}(\omega_{R}))$
is onto if and only if
$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R})))=1$

###### Proof.

(a): In order to prove the first isomorphism we refer to [14, 2.2 (d)]. In
order to proof the second one we have that

$\operatorname{Ext}_{R}^{c}(H^{c}_{I}(\omega_{R}),\omega_{R})\cong\operatorname{Hom}_{R}(H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R})),E)$

as follows by the Local Duality for the complete local Cohen-Macaulay ring $R$
(see 2.1 (A)).
(b): By virtue of (a) we have isomorphisms
$B/\mathfrak{m}^{\alpha}\cong\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(R/\mathfrak{m}^{\alpha},H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R})),E)$
that form an inverse system. By passing to the inverse limit there are
isomorphisms

$\hat{B}^{\mathfrak{m}}=\varprojlim\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(R/\mathfrak{m}^{\alpha},H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R})),E)\cong\operatorname{Hom}_{R}(H^{0}_{\mathfrak{m}}(H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R}))),E)\cong
B$

because of
$\operatorname{Supp}_{R}H^{d-c}_{\mathfrak{m}}(H^{c}_{I}(\omega_{R}))\subseteq
V(\mathfrak{m})$. That is, $B$ is $\mathfrak{m}$-adic complete. Then $B$ is a
finitely generated $R$-module if and only if
$\dim_{\Bbbk}B/\mathfrak{m}B<\infty$ (see e.g. [12, Theorem 8.4]).
(c): Since $H^{c}_{I}(\omega_{R})\not=0$ it follows that the natural
homomorphism

$\rho\otimes
1_{\Bbbk}:R\otimes_{R}\Bbbk\to\operatorname{Hom}_{R}(H^{c}_{I}(\omega_{R}),H^{c}_{I}(\omega_{R}))\otimes_{R}\Bbbk$

is not zero. Then the statement follows by (b). ∎

######  Construction 2.3.

(A) Let $\Bbbk$ denote a field. Let $m,n\geq 2$ denote integers and let

$\mathcal{X}=\begin{pmatrix}x_{11}&\ldots&x_{1n}\\\ \vdots&\ddots&\vdots\\\
x_{m1}&\ldots&x_{mn}\end{pmatrix}$

denote a $m\times n$ matrix of $mn$ variables over $\Bbbk$. Let
$P=\Bbbk[[\mathcal{X}]]$ denote the formal powers series ring in the $mn$
variables of $\mathcal{X}$. We define $R=P/I_{2}$, where
$I_{2}=I_{2}(\mathcal{X})$ denotes the ideal generated by the $2\times 2$
minors of $\mathcal{X}$. Then $R$ is a local Cohen-Macaulay domain of
dimension $m+n-1$ (see [6]).
(B) We put $\underline{x}_{i}=x_{1i},\ldots,x_{mi},i=1,\ldots,n,$ the elements
of the $i$-th column of $\mathcal{X}$ and $\underline{x}=\underline{x}_{n}$
the elements of the last column. We define
$I=(\underline{x}_{1},\ldots,\underline{x}_{n-1})R$ the ideal generated by the
elements of the first $n-1$ columns of $\mathcal{X}$. Then $\dim R/I=m$ and
$0ptI=\operatorname{grade}I=n-1$. In the following we recall the form ring
$G_{I}(R)$ of $I\subset R$. First we define a $m\times n$-matrix

$\mathcal{Y}=\begin{pmatrix}T_{11}&\ldots&T_{1,n-1}&x_{1n}\\\
\vdots&\ddots&\vdots&\vdots\\\ T_{m1}&\ldots&T_{m,n-1}&x_{mn}\end{pmatrix}$

where $T_{ij},1\leq i\leq m,1\leq j\leq n-1,$ are variables over $R$. Let
$J=I_{2}(\mathcal{Y})$ the ideal generated by the $2\times 2$-minors of
$\mathcal{Y}$. Let $\mathcal{T}$ denote the set of all variables
$T_{i,j},1\leq i\leq m,1\leq j\leq n-1$ of degree 1 and let
$\mathcal{G}=(R/I)[\mathcal{T}]$. Then consider the induced homogeneous map

$\Theta:\mathcal{G}/J\to Gr_{I}(R),\;T_{ij}+J\mapsto x_{ij}+I/I^{2},\;1\leq
i\leq m,1\leq j\leq n-1.$

Note that it is well defined and surjective. Clearly, $\mathcal{G}/J$ is a
Cohen-Macaulay determinantal domain with $\dim\mathcal{G}/J=m+n-1$, i.e. $J$
is a prime ideal. Moreover $Gr_{I}(R)$ is a ring with $\dim Gr_{I}(R)=m+n-1$.
Let $\mathcal{J}$ be the preimage of $\ker\Theta$ in $\mathcal{G}$. Let
$\mathcal{P}\subset\operatorname{Spec}\mathcal{G}$ denote the preimage of a
highest dimensional prime ideal in $Gr_{I}(R)$. Then
$J\subseteq\mathcal{J}\subseteq\mathcal{P}$ and
$\dim\mathcal{G}/J=\dim\mathcal{G}/\mathcal{P}$ and therefore $J=\ker\Theta$.
That is, $\Theta$ is an isomorphism and $Gr_{I}(R)$ is a domain.
(C) Next we introduce the ring $S=\Bbbk[[\underline{x},\underline{a}]]$, where
$\underline{x}=x_{1},\ldots,x_{m}$ with $x_{i}=x_{in},i=1,\ldots,m$ and
$\underline{a}=a_{1},\ldots,a_{n-1}$ are sequences of variables over $\Bbbk$.
Then $S$ is a regular local ring with $\underline{a}=a_{1},\ldots,a_{n-1}$ a
regular sequence. Therefore
$Gr_{(\underline{a})}(S)\cong(S/\underline{a})[\underline{T}]\cong\Bbbk[[\underline{x}]][\underline{T}]$
with $\underline{T}=T_{1},\ldots,T_{n-1}$ variables of degree $1$. We define a
ring homomorphism

$\phi:R\to S,\;x_{ij}\mapsto a_{i}x_{j},\;i=1,\ldots,n-1,j=1,\ldots,m.$

Note that $S$ is a domain and $\dim R=\dim S=m+n+1$. Therefore $\phi$ is
injective. The homomorphism $\phi$ extends to a homomorphism

$\psi:Gr_{I}(R)\to Gr_{(\underline{a})}(S),\;T_{ij}+J\mapsto
x_{j}T_{i},\;i=1,\ldots,m,j=1,\ldots,n-1.$

Since both $Gr_{I}(R)$ and $Gr_{(\underline{a})}(S)$ are domains of the same
dimension $\psi$ is injective.

##  3\. Examples and remarks

In the following we denote by $\omega_{R}$ the canonical module of a Cohen-
Macaulay ring $R$. For basic definitions and properties of $\omega_{R}$ we
refer to [2, 3.3], the generalization in [13] and the summary in 2.1. For a
local ring $(R,\mathfrak{m})$ we denote by $E=E_{R}(R/\mathfrak{m})$ the
injective hull of the residue field.

######  Example 3.1.

We fix the notation of 2.3, that is, $R$ is the determinantal Cohen-Macaulay
ring of dimension $m+n-1$ with
$I=(\underline{x}_{1},\ldots,\underline{x}_{n-1})R$ the ideal generated by the
elements of the first $n-1$ columns of $\mathcal{X}$. Then there are the
following results:

  * (a)

$H^{i}_{I}(\omega_{R})\not=0,i=n-1,m+n-2,$ that is,
$\operatorname{cd}(I)=m+n-2$.

  * (b)

$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(\omega_{R}))$ and
$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(R))$ are not finite,
that is $I$ is not cofinite.

  * (c)

$\operatorname{Hom}_{R}(H^{n-1}_{I}(\omega_{R}),H^{n-1}_{I}(\omega_{R}))$ is a
commutative Noetherian ring, not finitely generated over $R$.

  * (d)

$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{m}_{\mathfrak{m}}(H^{n-1}_{I}(\omega_{R})))$
and
$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{0}_{\mathfrak{m}}(H^{m+n-2}_{I}(\omega_{R})))$
are not finite.

###### Proof.

The ring homomorphism $\phi$ of 2.3 (C) induces ring homomorphisms

$\phi^{\alpha}:R/I^{\alpha}\to S/(\underline{a})^{\alpha}S\;\mbox{ for all
}\alpha\geq 1,$

which are well-defined. We claim that $\phi^{\alpha}$ is injective for all
$\alpha\geq 1$. The restriction $\psi^{\alpha}$ of the injection $\psi$ of 2.3
(C) to degree $\alpha$ yields injective homomorphisms

${\psi}^{\alpha}:I^{\alpha}/I^{\alpha+1}\to(\underline{a})^{\alpha}S/(\underline{a})^{\alpha+1}S\;\mbox{
for all }\alpha\geq 0.$

Let $D_{\alpha}:=\operatorname{Coker}\psi^{\alpha}$ and let
$\underline{x}\cdot\underline{a}$ be the sequence of elements
$\\{x_{i}a_{j}:i=1,\ldots,n-1,j=1,\ldots,m\\}$. We define
$C_{\alpha}=\operatorname{Coker}\phi^{\alpha}=S/((\underline{a})^{\alpha}S+S[[\underline{x}\cdot\underline{a}]])$
because of $\operatorname{Im}\phi=S[[\underline{x}\cdot\underline{a}]]$. Now
it follows that

$D_{\alpha}\cong((\underline{a})^{\alpha}S+S[[\underline{x}\cdot\underline{a}]])/((\underline{a})^{\alpha+1}S+S[[\underline{x}\cdot\underline{a}]]).$

as a consequence of the following commutative diagram with exact rows and the
snake lemma

$\begin{array}[]{ccccccccc}0&\to&I^{\alpha}/I^{\alpha+1}&\to&R/I^{\alpha+1}&\to&R/I^{\alpha}&\to&0\\\
&&\downarrow\psi^{\alpha}&&\downarrow\phi^{\alpha+1}&&\downarrow\phi^{\alpha}&&\\\
0&\to&(\underline{a})^{\alpha}S/(\underline{a})^{\alpha+1}S&\to&S/(\underline{a})^{\alpha+1}S&\to&S/(\underline{a})^{\alpha}S&\to&0.\end{array}$

Since $\psi^{\alpha}$ is injective for all $\alpha\geq 0$ as the restriction
of $\phi^{\alpha}$ it follows by induction that $\phi^{\alpha}$ is injective
too. Therefore $D_{\alpha}$ is spanned by
$(\underline{x})^{i}(\underline{a})^{\alpha}$ with $0\leq i<\alpha$ over
$\Bbbk$. As an $R$-module it is finitely generated. For the radical of the
annihilator it follows that
$\operatorname{Rad}_{S}\operatorname{Ann}_{S}D_{\alpha}=(\underline{x},\underline{a})$.
So that $C_{\alpha},\alpha\geq 1,$ is an $R$-module of finite length. There is
the short exact sequence

$0\to R/I^{\alpha}\to S/(\underline{a})^{\alpha}S\to C_{\alpha}\to 0\;\mbox{
for all }\alpha\geq 1.$

Because of $C_{\alpha}\cong\oplus_{\beta=1}^{\alpha-1}D_{\beta}$ it follows
that it is an $R$-module of finite length. Since $S/(\underline{a})^{\alpha}S$
is a finitely generated Cohen-Macaulay $R$-module it implies that
$H^{1}_{\mathfrak{m}}(R/I^{\alpha})\cong C_{\alpha}$ and
$H^{i}_{\mathfrak{m}}(R/I^{\alpha})=0$ for all $i\not=1,m$. The above short
exact sequences form an inverse system of short exact sequences. By passing to
the inverse limit we get a short exact sequence

$0\to R\to S\to\varprojlim H^{1}_{\mathfrak{m}}(R/I^{\alpha})\to 0.$

Note that the first family is given by surjective maps. By virtue of the Local
Duality Theorem for a Cohen-Macaulay ring (see 2.1 (A)) there are isomorphisms

$H^{i}_{\mathfrak{m}}(R/I^{\alpha})\cong\operatorname{Hom}_{R}(\operatorname{Ext}_{R}^{m+n-1-i}(R/I^{\alpha},\omega_{R}),E)$

for all $\alpha\geq 1$ and by passing to the inverse limit $\varprojlim
H^{i}_{\mathfrak{m}}(R/I^{\alpha})\cong\operatorname{Hom}_{R}(H^{m+n-1-i}_{I}(\omega_{R}),E)$
for all $i$. Therefore $H^{i}_{I}(\omega_{R})=0$ for $i\not=n-1,m+n-2$ and

$\varprojlim
H^{1}_{\mathfrak{m}}(R/I^{\alpha})\cong\operatorname{Hom}_{R}(H^{m+n-2}_{I}(\omega_{R}),E)\not=0.$

Because $S$ is not a finitely generated $R$-module
$\operatorname{Hom}_{R}(H^{m+n-2}_{I}(\omega_{R}),E)$ is not finitely
generated too. By the isomorphism

$\Bbbk\otimes_{R}\operatorname{Hom}_{R}(H^{n}_{I}(\omega_{R}),E)\cong\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(\omega_{R})),E)$

and by Matlis Duality
$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(\omega_{R}))$ is not
finite. Moreover, $H^{m+n-2}_{I}(\omega_{R})\cong
H^{m+n-2}_{I}(R)\otimes_{R}\omega_{R}$ so that
$\operatorname{Hom}_{R}(H^{m+n-2}_{I}(\omega_{R}),E)\cong\operatorname{Hom}_{R}(\omega_{R},\operatorname{Hom}_{R}(H^{m+n-2}_{I}(R),E))$.
Because $\omega_{R}$ is a finitely generated $R$-module
$\operatorname{Hom}_{R}(H^{m+n-2}_{I}(R),E)$ is not finitely generated and –
as above – $\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,H^{m+n-2}_{I}(R))$ is not
finite. That is, (a) and (b) of 3.1 are proved.

Now recall that $S/(\underline{a})^{\alpha}S$ is an $m$-dimensional Cohen-
Macaulay ring, finitely generated over $R$ and $\dim_{R}C_{\alpha}=0$.
Therefore,

$S/(\underline{a})^{\alpha}S\cong\omega_{R}(\omega_{R}(R/I^{\alpha}))\cong\operatorname{Ext}_{R}^{n-1}(\operatorname{Ext}_{R}^{n-1}(R/I^{\alpha},\omega_{R}),\omega_{R})$

(by view of 2.1 (D)). By passing to the inverse limit of the corresponding
inverse and by Theorem 2.2 (a) it yields that
$S\cong\operatorname{Hom}_{R}(H^{n-1}_{I}(\omega_{R}),H^{n-1}_{I}(\omega_{R}))$,
whence (c) is shown. The first claim in (d) follows by 2.2 (b) since $S$ is
not finitely generated over $R$. The second one is clear by (b) since
$\operatorname{Supp}_{R}H^{m+n-2}_{I}(R)=V(\mathfrak{m})$. ∎

For the case of $m=n$ in 3.1 we get that $R$ is a Gorenstein ring and
therefore $\omega_{R}\cong R$. For $m=n=2$ we recover Hartshorne’s example in
this different context with additional properties.

######  Question 3.2.

Let $I\subset R$ denote an ideal of a local ring $(R,\mathfrak{m})$ with
$c=\operatorname{grade}I.$ We do not know whether the endomorphism ring
$\operatorname{Hom}_{R}(H^{c}_{I}(R),H^{c}_{I}(R))$ is in general commutative
and Noetherian.

For further results about the endomorphism ring
$\operatorname{Hom}_{R}(H^{c}_{I}(R),H^{c}_{I}(R))$ we refer also to [14].

## References

  * [1] M. Brodmann and R. Sharp, Local cohomology. An algebraic introduction with geometric applications. 2nd ed., Cambridge: Cambridge University Press, 2nd ed. ed., 2012.
  * [2] W. Bruns and J. Herzog, Cohen–Macaulay rings. Rev. ed., Cambridge: Cambridge University Press, rev. ed. ed., 1998.
  * [3] A. Grothendieck, Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne. Lecture Notes in Mathematics. 41. Berlin-Heidelberg-New York: Springer-Verlag. 106 p., 1967.
  * [4] R. Hartshorne, Cohomological dimension of algebraic varieties., Ann. Math. (2), 88 (1968), pp. 403–450.
  * [5] , Affine duality and cofiniteness., Invent. Math., 9 (1970), pp. 145–164.
  * [6] M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay., Adv. Math., 13 (1974), pp. 115–175.
  * [7] C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Camb. Philos. Soc., 110 (1991), pp. 421–429.
  * [8] C. L. Huneke and R. Y. Sharp, Bass numbers of local cohomology modules., Trans. Am. Math. Soc., 339 (1993), pp. 765–779.
  * [9] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of $D$\- modules to commutative algebra)., Invent. Math., 113 (1993), pp. 41–55.
  * [10] W. Mahmood and P. Schenzel, On invariants and endomorphism rings of certain local cohomology modules, J. Algebra, 372 (2012), pp. 56–67.
  * [11] T. Marley and J. C. Vassilev, Local cohomology modules with infinite dimensional socles., Proc. Am. Math. Soc., 132 (2004), pp. 3485–3490.
  * [12] H. Matsumura, Commutative ring theory. Transl. from the Japanese by M. Reid., Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, XIII, 320 p., 1986.
  * [13] P. Schenzel, On the use of local cohomology in algebra and geometry, in Six lectures on commutative algebra. Lectures presented at the summer school, Bellaterra, Spain, July 16–26, 1996, Basel: Birkhäuser, 1998, pp. 241–292.
  * [14] , On endomorphism rings and dimensions of local cohomology modules., Proc. Am. Math. Soc., 137 (2009), pp. 1315–1322.
  * [15] , On the structure of the endomorphism ring of a certain local cohomology module., J. Algebra, 344 (2011), pp. 229–245.
  * [16] P. Schenzel and A.-M. Simon, Completion, Čech and local homology and cohomology. Interactions between them., Cham: Springer, 2018.