# Reciprocity of Algorithms Solving Distributed Consensus-Based Optimization
and Distributed Resource Allocation

Seyyed Shaho Alaviani, Atul Gajanan Kelkar, and Umesh Vaidya Seyyed Shaho
Alaviani is with the Mechanical & Industrial Engineering Department,
University of Minnesota, Duluth, MN 55812 USA e-mail<EMAIL_ADDRESS>Gajanan Kelkar and Umesh Vaidya are with the Department of Mechanical
Engineering, Clemson University, Clemson, SC, 29634 USA e-mails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>preliminary version of this paper has
appeared without proofs in [1]. This work has been done while Seyyed Shaho
Alaviani was at Clemson University, Clemson, South Carolina, USA.

###### Abstract

This paper aims at proposing a procedure to derive distributed algorithms for
distributed consensus-based optimization by using distributed algorithms for
network resource allocation and vice versa over switching networks
with/without synchronous protocol. It is shown that first-order gradient
distributed consensus-based optimization algorithms can be used for finding an
optimal solution of distributed resource allocation with synchronous protocol
under weaker assumptions than those given in the literature for non-switching
(static) networks. It is shown that first-order gradient distributed resource
allocation algorithms can be utilized for finding an optimal solution of
distributed consensus-based optimization. The results presented here can be
applied to time-varying and random directed networks with or without
synchronous protocol with arbitrary initialization. As a result, several
algorithms can now be used to derive distributed algorithms for both
consensus-based optimization and resource allocation, that can overcome
limitations of the existing results. While the focus of this paper is on the
first-order gradient algorithms, it is to be noted that the results also work
with second-order gradient algorithms.

###### Index Terms:

distributed optimization, resource allocation, random networks, asynchronous.

##  I Introduction

Resource Allocation problem appears in a wide spectrum of domains including
higher education [2], economics [3], and health care [4], to name a few.
Resource Allocation problem is to allocate the network resources among a group
of agents while optimizing a certain performance index. Network resource
allocation is also fundamental and an important problem that arises in a
variety of application domains within engineering such as the media access
control in communication networks [5], signal processing [6], the load demand
management [7], and economic dispatch problems in power systems [8] (see the
survey paper [9] for more details).

In centralized allocation schemes [10], the allocation decisions are made
centrally by gathering all network data together and coming up with a decision
to satisfy certain central objective and then sending the decision back to the
agents. However, centralized algorithms may not be effective and practical in
a large-scale network due to heavy communication, computational burden,
private information, or complicated network structure. Therefore, designing
decentralized and/or distributed algorithms for resource allocation is very
important and highly desirable. Several researchers have focused their
attention to develop decentralized and/or distributed algorithms [8],
[11]-[37]. Among these, the authors of [27]-[37] have considered switching
graphs and imposed assumptions such as connectivity at any time,
B-connectivity111There exists a bounded time interval such that the union of
the graphs is strongly connected, and each edge transmits a message at least
once., independent and identically distributed (i.i.d.), or Markov chain on
the underlying graph. In [22] and [32], asynchronous222In a synchronous
protocol, all nodes activate at the same time and perform communication
updates. On the other hand, in an asynchronous protocol, each node has its
concept of time defined by a local timer, which randomly triggers either by
the local timer or by a message from neighboring nodes. The algorithms
guaranteed to work with no a priori bound on the time for updates are called
totally asynchronous, and those that need the knowledge of a priori bound,
known as B-connectivity assumption, are called partially asynchronous (see
[38] and [39, Ch. 6-7]). As the dimension of the network increases,
synchronization becomes an issue.  algorithms have been proposed for solving
distributed resource allocation problem. Hence, only few results exist for
decentralized and/or distributed resource allocation algorithms over switching
graphs or with asynchronous protocol, where the proposed algorithms are first-
order gradient. The authors of [13]-[14] have proposed discrete-time
distributed second-order algorithms over static (non-switching) graphs under
synchronous protocol for resource allocation optimization. The second-order
gradient algorithms provide faster convergence rates.

Another challenge in networked systems is distributed consensus-based
optimization. Distributed consensus-based optimization appears in many
problems such as sensor networks or data regression (see survey papers
[40]-[43] for more applications). The aim of agents in distributed consensus-
based optimization is to get an agreement on some value that is an optimal
solution to the problem. Note that the agents may not necessarily get an
agreement on some value in distributed resource allocation. As seen in surveys
[40]-[43], several distributed algorithms for consensus-based optimization
have been proposed over static, time-varying, or random networks with or
without asynchronous protocols. Recently, the first totally asynchronous (see
footnote 2) first-order gradient algorithm has been proposed in [44] for
distributed consensus-based optimization. To the best of authors’ knowledge, a
totally asynchronous algorithm has not been proposed for distributed resource
allocation.

Upon the review of the literature on distributed consensus-based optimization
and distributed resource allocation and their importance in engineering, the
natural questions that arise are: Question 1) is there a relation between
these two problems?; Question 2) is it possible to use algorithms for
distributed consensus-based optimization for solving distributed resource
allocation problem?; and Question 3) is it possible to utilize algorithms for
distributed resource allocation for solving distributed consensus-based
optimization problem? Although it has been established that there is a
relation between these two problems in one direction (reviewed below), it is
not known if there exist two-way reciprocity for solving these problems. This
paper is addressing this question.

With regard to Question 1, it is known that the Lagrange’s dual problem of
resource allocation is consensus-based optimization, e.g., see [25, Sec.
III.A] for proof; and Fenchel’s dual problem [45] of distributed consensus-
based optimization is resource allocation, e.g., see [46, Sec. III.A] for
proof. It needs to be mentioned that Fenchel’s dual problem is a duality
problem (like Lagrange’s duality) represented by (Fenchel) conjugate functions
(e.g., see [45] for details). Note that solving both Fenchel’s and Lagrange’s
dual problems is a challenging problem since the cost function in dual
problems is not an explicit function of dual variables and, in fact, is
determined by another minimization problem (see Comment 1 is Section IV for
details).

The existing literature presents some answers to Question 2 which are
explained next. In [47], it is mentioned that a distributed consensus-based
optimization algorithm may be used for resource allocation through solving the
dual of resource allocation; however, the authors of [47] did not present any
method to do so. Later, the authors in [24] showed the relation that the
first-order optimality conditions of the two aforementioned problems have when
underlying graph is static, connected, and undirected with synchronous
protocol and when the cost function of each agent is proper, closed, and
convex. This leads to the use of decentralized or distributed consensus-based
optimization first-order gradient algorithms with arbitrary initialization for
solving resource allocation without investigating the Lagrangian dual
relationship between the two problems. Note that the optimality conditions
given in [24] only work for static networks. Recently, it was shown in [25]
how to use distributed consensus-based first-order gradient optimization
algorithms for solving resource allocation over static directed networks with
synchronous protocol when cost function of each agent is strongly convex.
Based on the explanation above, one can apply existing distributed consensus-
based optimization algorithms only over static networks with synchronous
protocols, and we cannot derive algorithms for distributed resource allocation
over switching networks and/or with asynchronous protocol to overcome
restrictions of existing algorithms.

Contribution: In this paper, we present how to utilize distributed consensus-
based optimization algorithms to find an optimal solution of a distributed
resource allocation problem and vice versa over switching networks
with/without synchronous protocol. Therefore, the objective of this paper is
not to propose a distributed algorithm and to analyze its convergence; in
fact, we propose a procedure (i.e., P1-P3) to derive a distributed algorithm
for consensus-based optimization and resource allocation while its convergence
analysis has already been analyzed by imposing suitable assumptions. As such,
several algorithms can be employed, by imposing suitable assumptions, to
derive distributed algorithms for both consensus-based optimization and
resource allocation, that can overcome limitations of the existing results. We
show that first-order gradient algorithms for distributed consensus-based
optimization can also be used to find an optimal solution of distributed
resource allocation over static directed networks with synchronous protocol
under weaker assumptions (than strong convexity in [25]) such as strict
convexity, Lipschitz gradient, or coercivity. We show that distributed
resource allocation first-order gradient algorithms can also be utilized to
find an optimal solution of distributed consensus-based optimization. Thus,
our results show the applicability of distributed resource allocation
algorithms for solving distributed consensus-based optimization, namely to
present an affirmative answer to Question 3. It is important to emphasize that
conditions given here can lead to use distributed algorithms for both
consensus-based optimization and resource allocation over static, time-
varying, or random directed networks with or without asynchronous protocols
under arbitrary initialization. While the technical issue behind is
challenging (see the fifth paragraph in this section) to demonstrate how
significant the improvement is made when fixing the gap in the literature, we
apply the first totally asynchronous (see footnote 2) first-order gradient
algorithm [44] intended for distributed consensus-based optimization to derive
the first totally asynchronous first-order gradient-based algorithm for
distributed resource allocation (which is a missing gap in the literature)
where the algorithm does not require a priori B-connectivity (see footnote 1)
or distribution assumption on switching communication graphs for convergence.
The derivation of first-order distributed consensus-based optimization
algorithms from distributed resource allocation algorithms is similar to the
derivation of distributed resource allocation algorithms from distributed
consensus-based optimization algorithms, namely the P1-P3 are still valid.
While we focus in this paper on first-order gradient algorithms, the results
also work with second-order gradient (see Lemma 1 and Observation 1)
algorithms where the related results are given in [48]. This paper provides
proofs and related details for theorem given in [1] as they were omitted due
to space limitations.

This paper is organized as follows: In Section II, necessary preliminaries on
operators and stochastic convergence are provided. Formulations of the
resource allocation and consensus-based optimization problem are given in
Section III. In Section IV, the procedure for using distributed consensus-
based optimization algorithms to solve distributed resource allocation is
given. Section V presents how to use distributed resource allocation’s
algorithms to solve distributed consensus-based optimization.

Notations: $\Re$ denotes the set of all real numbers. $(.)^{T}$ represents the
transpose of a vector or a matrix. For any vector
$z\in\Re^{n},\|z\|_{2}=\sqrt{z^{T}z},$ and for any matrix $Z\in\Re^{n\times
n},\|Z\|_{2}=\sqrt{\lambda_{max}(Z^{T}Z)}=\sigma_{max}(Z)$ where
$\lambda_{max}$ represents the maximum eigenvalue, and $\sigma_{max}$
represents the largest singular value. Sorted in an increasing order with
respect to real parts, $\lambda_{2}(Z)$ represents the second eigenvalue of a
matrix $Z$. $Re(r)$ represents the real part of the complex number $r$.
$I_{n}$ represents Identity matrix of size $n\times n$ for some
$n\in\mathbb{N}$ where $\mathbb{N}$ denotes the set of all natural numbers.
$\otimes$ represents Kronecker product. $E[x]$ denotes Expectation of the
random variable $x$. $\nabla f(x)$ represents the gradient of the function $f$
at $x$. $\partial f(x)$ represents the sub-differential of the function $f$ at
$x$. $\textbf{0}_{n}$ and $\textbf{1}_{n}$ denote the vector of dimension $n$
whose elements are all zero and one, respectively.

##  II Preliminaries

A vector $v\in\Re^{n},n\in\mathbb{N},$ is said to be a stochastic vector when
its components $v_{i},i=1,2,...,n$, are non-negative and their sum is equal to
1; a square $n\times n$ matrix $V$ is said to be a stochastic matrix when each
row of $V$ is a stochastic vector. A square $n\times n$ matrix $V$ is said to
be doubly stochastic when both $V$ and $V^{T}$ are stochastic matrices.

Let $\mathcal{H}$ be a real Hilbert space with norm $\|.\|$ and inner product
$\langle.,.\rangle$. An operator $A:\mathcal{H}\longrightarrow\mathcal{H}$ is
said to be monotone if $\langle x-y,Ax-Ay\rangle\geq 0$ for all
$x,y\in\mathcal{H}$. $A:\mathcal{H}\longrightarrow\mathcal{H}$ is called
$\rho$-strongly monotone if $\langle x-y,Ax-Ay\rangle\geq\rho\|x-y\|^{2}$ for
all $x,y\in\mathcal{H}$. A function $f(.)$ is $\rho$-strongly convex if
$\langle x-y,\nabla f(x)-\nabla f(y)\rangle\geq\rho\|x-y\|^{2}$ for all
$x,y\in\mathcal{H}$. Therefore, a function is $\rho$-strongly convex if its
gradient is $\rho$-strongly monotone. A function $f(.)$ is strictly convex if
$\forall x,y,x\neq y,\forall\kappa\in(0,1)$, we have

$f(\kappa x+(1-\kappa)y)<\kappa f(x)+(1-\kappa)f(y).$

A mapping $B:\mathcal{H}\longrightarrow\mathcal{H}$ is said to be
$K$-Lipschitz continuous if there exists a $K>0$ such that $\|Bx-By\|\leq
K\|x-y\|$ for all $x,y\in\mathcal{H}$. Let $S$ be a nonempty subset of a
Hilbert space $\mathcal{H}$ and $Q:S\longrightarrow\mathcal{H}$. The point $x$
is called a fixed point of $Q$ if $x=Q(x)$.

Let $(\Omega^{*},\sigma)$ be a measurable space ($\sigma$-sigma algebra) and
$C$ be a nonempty subset of a Hilbert space $\mathcal{H}$. A mapping
$x:\Omega^{*}\longrightarrow\mathcal{H}$ is measurable if $x^{-1}(U)\in\sigma$
for each open subset $U$ of $\mathcal{H}$. The mapping $T:\Omega^{*}\times
C\longrightarrow\mathcal{H}$ is a random map if for each fixed $z\in C$, the
mapping $T(.,z):\Omega^{*}\longrightarrow\mathcal{H}$ is measurable, and it is
continuous if for each $\omega^{*}\in\Omega^{*}$ the mapping
$T(\omega^{*},.):C\longrightarrow\mathcal{H}$ is continuous.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space where
$\Omega$ denotes the sample space, $\mathcal{F}$ denotes a $\sigma$-algebra on
$\Omega$, and $\mathbb{P}$ is a probability measure on $\mathcal{F}$. A
sequence of random variables $x_{t}$ is said to

  * •

converge almost surely to $x$ if there exists a subset $A\subseteq\Omega$ such
that $\mathbb{P}(A)=0$, and for every $\omega\notin A$,
$\lim_{t\longrightarrow\infty}\|x_{t}(\omega)-x(\omega)\|=0.$

  * •

converge in mean square to $x$ if $E[\|x_{t}-x\|^{2}]\longrightarrow 0$ as
$t\longrightarrow\infty.$

Definition 1 [44]: If there exists a point $\hat{x}\in\mathcal{H}$ such that
$\hat{x}=T(\omega^{*},\hat{x})$ for all $\omega^{*}\in\Omega^{*}$, it is
called fixed-value point, and $FVP(T)$ represents the set of all fixed-value
points of $T$.

Definition 2 [45],[49]: The (Fenchel) conjugate of a function
$f:\Re^{n}\longrightarrow\Re$ is defined as

$f^{*}(y):=\sup_{x}(y^{T}x-f(x)).$

Moreover, the conjugate function is a convex function.

Definition 3 [45]: A function $f:\Re^{n}\longrightarrow\Re$ is said to be
$0-$coercive and $1-$coercive when

$\displaystyle\lim_{\|x\|\longrightarrow+\infty}f(x)=+\infty\quad{}\text{and}\quad{}\displaystyle\lim_{\|x\|\longrightarrow+\infty}\frac{f(x)}{\|x\|}=+\infty,$

respectively.

##  III Problem Formulation

We use the switched333Switched systems can be roughly divided into two groups:
those subject to arbitrary, i.e., state-independent switching, and those
subject to state-dependent switching (see [50] for details). We require to
mention that we consider arbitrary switching in this paper. Note that weighted
matrix of the graph, Laplacian, or Adjacency matrix can be used for
information of the network in our formulation. For instance, we use weighted
matrix of the graph in this paper. Laplacian has been utilized in [48] for
second-order gradient algorithms. network topology similar to the one used in
[44] wherein a network of $m\in\mathbb{N}$ nodes labeled by the set
$\mathcal{V}=\\{1,2,...,m\\}$ is considered. The topology of the
interconnections among nodes is not fixed but defined by a set of graphs
$\mathcal{G}(\omega^{*})=(\mathcal{V},\mathcal{E}(\omega^{*}))$ where
$\mathcal{E}(\omega^{*})$ is the ordered edge set
$\mathcal{E}(\omega^{*})\subseteq\mathcal{V}\times\mathcal{V}$ and
$\omega^{*}\in\Omega^{*}$ where $\Omega^{*}$ is the set of all possible
communication graphs, i.e.,
$\Omega^{*}=\\{\mathcal{G}_{1},\mathcal{G}_{2},...,\mathcal{G}_{\bar{N}}\\}$.
We assume that $(\Omega^{*},\sigma)$ is a measurable space where $\sigma$ is
the $\sigma$-algebra on $\Omega^{*}$. We write
$\mathcal{N}_{i}^{in}(\omega^{*})/\mathcal{N}_{i}^{out}(\omega^{*})$ for the
labels of agent $i$’s in/out neighbors at graph $\mathcal{G}(\omega^{*})$ so
that there is an arc in $\mathcal{G}(\omega^{*})$ from vertex $j/i$ to vertex
$i/j$ only if agent $i$ receives/sends information from/to agent $j$. We write
$\mathcal{N}_{i}(\omega^{*})$ when
$\mathcal{N}_{i}^{in}(\omega^{*})=\mathcal{N}_{i}^{out}(\omega^{*})$. We
assume that there are no self-looped arcs in the communication graphs. We also
assume that there is no communication delay or noise in delivering a message
from agent $j$ to agent $i$.

Now, we consider the distributed resource allocation problem with $m$ agents
that share their local resources. For each node $i\in\mathcal{V}$, there is a
private convex cost function $f_{i}:\Re^{n}\longrightarrow\Re$ and a private
finite amount of a local resource, given by an $n$-dimensional vector, owned
by the agent $i$, i.e., $R_{i}\in\Re^{n}$, that are known to node $i$. The
objective of each agent is to collaboratively seek the solution of the
following optimization problem using local information exchange with the
neighbors and possibly switching communication topologies with or without
asynchronous protocol:

$\displaystyle\underset{x_{1},\ldots,x_{m}}{\text{min}}$
$\displaystyle\sum_{i=1}^{m}f_{i}(x_{i})$ (1) subject to
$\displaystyle\sum_{i=1}^{m}x_{i}=\sum_{i=1}^{m}R_{i}$

where $x_{i}\in\Re^{n}$ is the decision variable of agent $i$, and the
constraint represents resource sharing among the agents.

So far, we have introduced the distributed resource allocation optimization
(1). Another problem in multi-agent systems is distributed consensus-based
optimization where the objective of each agent is to collaboratively seek a
solution of the following optimization problem:

$\displaystyle\underset{s}{\text{min}}$ $\displaystyle\sum_{i=1}^{m}h_{i}(s)$
(2)

where $s\in\Re^{n}$, and $h_{i}:\Re^{n}\longrightarrow\Re$ is the private cost
of agent $i$. As a matter of fact, the formulation of distributed consensus-
based optimization by using local variables of agents is:

$\displaystyle\underset{x_{1},\ldots,x_{m}}{\text{min}}$
$\displaystyle\sum_{i=1}^{m}h_{i}(x_{i})$ (3) subject to $\displaystyle
x_{1}=x_{2}=\ldots=x_{m}$

where $x_{i}\in\Re^{n}$ is the decision variable of agent $i$, and the
constraint is achieved by interacting with neighbors and possibly switching
communication topologies with or without asynchronous protocol. Relevantly,
the set

$\mathcal{C}:=\\{x\in\Re^{mn}|x_{i}=x_{j},1\leq i,j\leq m,x_{i}\in\Re^{n}\\}$
(4)

is known as consensus subspace.

In Section IV, we show that first-order gradient algorithms of distributed
consensus-based optimization can also be used to find an optimal solution of
(1) over static directed networks with synchronous protocol under weaker
assumptions (than strong convexity) such as strict convexity, Lipschitz
gradient, or coercivity. In Section V, we show that first-order gradient
algorithms of distributed resource allocation can also be utilized to find an
optimal solution of (3) under certain assumptions.

Remark 1: All assumptions given in this paper can lead to derive distributed
algorithms for both consensus-based optimization and resource allocation over
static, time-varying, or random directed networks with or without asynchronous
protocols under arbitrary initialization.

##  IV Solving (1) by Using Distributed Consensus-Based Optimization
Algorithms

The Lagrangian $\mathcal{L}:\Re^{mn}\times\Re^{n}\longrightarrow\Re$
associated with (1) (see [49]) is defined as

$\mathcal{L}(x,y):=\sum_{i=1}^{m}f_{i}(x_{i})+\sum_{i=1}^{m}y^{T}(R_{i}-x_{i})$
(5)

where $x=[x_{1}^{T},\ldots,x_{m}^{T}]^{T}$, and $y$ is the Lagrange multiplier
associated with the equality constraint in (1). It is known that Lagrange’s
dual problem of (1) is distributed consensus-based optimization (e.g., see
[25, Sec. III.A] for proof), i.e.,

$\displaystyle\underset{y_{1},\ldots,y_{m}}{\text{min}}$ $\displaystyle
G(Y):=\sum_{i=1}^{m}G_{i}(y_{i})$ (6) subject to $\displaystyle
y_{1}=y_{2}=\ldots=y_{m}$

where $Y=[y_{1}^{T},\ldots,y_{m}^{T}]^{T}$, and

$G_{i}(y_{i}):=f^{*}_{i}(y_{i})-y_{i}^{T}R_{i}$ (7)

is the private cost of agent $i$, in which

$f^{*}_{i}(y_{i}):=\sup_{x_{i}}(y^{T}_{i}x_{i}-f_{i}(x_{i}))$ (8)

is the Fenchel conjugate of $f_{i}(x_{i})$ (see Definition 2).

Comment 1: As seen in (7) and (8), the main difficulty in solving (6) is that
the functions $G_{i}(y_{i}),i=1,\ldots,m,$ are not explicit functions of dual
variables $y_{i}$. In fact, $G_{i}(y_{i}),i=1,\ldots,m,$ are derived from
optimization problems in (8) whose gradients are hard to compute directly.
This poses a big challenge to solve (1) by using distributed consensus-based
optimization algorithms where the cost functions must be explicit functions of
the variables. As mentioned in Section I, several researchers have addressed
this issue by using first-order gradient distributed consensus-based
optimization algorithms over static networks with synchronous protocol under
certain assumptions.

In this section, we propose a procedure (i.e., P1-P3 below) that can be used
for applying first-order gradient algorithms of distributed consensus-based
optimization to solve distributed resource allocation optimization (1) under
some assumptions. The assumptions given here are weaker than those of the
existing results and thus offer more algorithm options for using existing
distributed consensus-based optimization (see Surveys [40]-[43]) which work
over static, time-varying, or random directed networks with or without
asynchronous protocols (see also Remark 1).

For using first-order gradient algorithms of distributed consensus-based
optimization, we are required to know the properties of first-order gradients
of $G_{i}(y_{i}),i=1,\ldots,m$. Consequently, we need to know properties of
first-order gradient of $f^{*}_{i}(y_{i}).$ Therefore, it is critical to know
relations between properties of $f_{i}(x_{i})$ and $f^{*}_{i}(y_{i})$. These
relations are given in Lemmas 1 and 2 below.

Lemma 1 [45, Cor. 4.1.4, Th. 4.2.1, Th. 4.2.2, Cor. 4.2.10]:

  * •

Let $f_{i}:\Re^{n}\longrightarrow\Re,i=1,\ldots,m,$ be strictly convex,
differentiable, and $1-$coercive. Then

(i) $f^{*}_{i}$ is likewise finite-valued on $\Re^{n},$ strictly convex,
differentiable, and $1-$coercive;

(ii) The continuous mapping $\nabla f_{i}$ is one-to-one from $\Re^{n}$ onto
$\Re^{n},$ and its inverse is continuous;

(iii) We have

$f^{*}_{i}(y_{i})=\langle y_{i},(\nabla f_{i})^{-1}(y_{i})\rangle-
f_{i}((\nabla f_{i})^{-1}(y_{i})),$ (9) $\forall y_{i}\in\Re^{n}.$

  * •

Assume that $f_{i}:\Re^{n}\longrightarrow\Re,i=1,\ldots,m,$ are $\rho-$
strongly convex where $\rho>0$. Then $domf^{*}_{i}=\Re^{n}$, and $\nabla
f^{*}_{i}$ is $\frac{1}{\rho}-$Lipschitz, i.e.,

$\|\nabla f^{*}_{i}(s_{1})-\nabla
f^{*}_{i}(s_{2})\|\leq\frac{1}{\rho}\|s_{1}-s_{2}\|,\quad{}\forall
s_{1},s_{2}\in\Re^{n}.$

  * •

Let $f_{i}:\Re^{n}\longrightarrow\Re,i=1,\ldots,m,$ be convex and have
$L-$Lipschitz gradient on $\Re^{n}$ where $L>0$. Then $f^{*}_{i}$ is
$\frac{1}{L}-$strongly convex on each subset $C\subset dom\partial f^{*}_{i}.$

  * •

Let $f_{i}:\Re^{n}\longrightarrow\Re,i=1,\ldots,m,$ be convex, twice
differentiable, and $1-$coercive. Assume, moreover, that
$\nabla^{2}f_{i}(x_{i})$ is positive definite for all $x_{i}\in\Re^{n}.$ Then
$f^{*}_{i}$ enjoys the same properties and

$\nabla^{2}f^{*}_{i}(y_{i})=[\nabla^{2}f_{i}(\nabla
f^{-1}_{i}(y_{i}))]^{-1},\quad{}\forall y_{i}\in\Re^{n}.$ (10)

Lemma 2 [51]: Let $f_{i}:\Re^{n}\longrightarrow\Re,i=1,\ldots,m,$ be strictly
convex.Then

$\nabla f^{*}_{i}(y_{i})=\arg\max_{x_{i}}(y^{T}_{i}x_{i}-f_{i}(x_{i})).$ (11)

Remark 2: Gradient and Hessian of the Fenchel conjugate
$f^{*}_{i}(y_{i}),i=1,\ldots,m,$ are given explicitly in (11) and (10),
respectively, under the assumptions in Lemma 2 and Lemma 1. One can also
compute gradient and Hessian of $f^{*}_{i}$ by using (9) under the assumptions
in Lemma 1.

Observation 1: By reformulating distributed resource allocation (1) as
distributed consensus-based optimization (6) on Lagrange’s dual variables
$y_{i},i=1,\ldots,m,$ and using properties of the Fenchel conjugate functions
$f^{*}_{i}(y_{i})$ (mentioned in Lemma 1), any first-as well as second-order
gradient distributed consensus-based optimization algorithms can be used for
solving (1) under the assumptions in Lemma 1. Here, we give more assumptions
than those of the literature such that these assumptions allow us to utilize
distributed consensus-based optimization algorithms over directed static,
time-varying, or random networks with or without asynchronous protocol under
arbitrary initialization (see also Remark 1).

It remains to show how to apply existing first-order distributed consensus-
based optimization algorithms by using Lemma 1 to derive first-order
distributed algorithm for resource allocation (1).

The procedure to solve (1) by solving (6) is given in P1-P3 as follows:

P1. Consider distributed consensus-based optimization (6) with variables
$y_{i}$ and select an existing first-order gradient distributed algorithm and
its related theorem for convergence.

P2. By using the selected distributed algorithm and conditions of the related
theorem, impose suitable assumptions on first-order gradient of
$f_{i}^{*}(y_{i})$ defined in (8) (see Remark 2); also impose suitable
assumptions on $f_{i}(x_{i})$ by using Lemmas 1 and 2; consequently, compute
the first-order gradient of $G_{i}(y_{i})$ defined in (7) by using Lemmas 1
and 2; therefore, the optimization (6) is solved by the chosen algorithm whose
conditions are satisfied, namely all agents get consensus on a dual optimal
solution $y^{*}$ of (6).

P3. Each agent obtains the primal optimal solution $x_{i}^{*},i=1,\ldots,m,$
of (1) via
$x_{i}^{*}=\displaystyle\arg\max_{x_{i}}(x_{i}^{T}y^{*}-f_{i}(x_{i}))$ if
exists. Consequently, the relevant theorem for solving (1) can be developed
from the related theorem of the selected algorithm in P1.

In the following subsection, based on Observation 1, we apply an existing
first-order gradient distributed discrete-time algorithm for solving
distributed consensus-based optimization to derive a first-order gradient
discrete-time algorithm for solving distributed resource allocation (1). We
explicitly show how to use P1-P3 above to derive such algorithms and to
develop related theorems. As we mentioned in Contribution in Section I, we
only develop first-order distributed algorithms in this paper.

###  IV-A A First-Order Totally Asynchronous Algorithm for Solving (1)

To the best of authors’ knowledge, the discrete-time algorithm presented in
this subsection is the first totally asynchronous (see footnote 2) algorithm
for solving (1).

Let the weighted matrix of the graph (see also footnote 3) be defined as
$\mathcal{W}(\omega^{*})=[\mathcal{W}_{ij}(\omega^{*})]$ with
$\mathcal{W}_{ij}(\omega^{*})=a_{ij}(\omega^{*})$ for
$j\in\mathcal{N}_{i}^{in}(\omega^{*})\cup\\{i\\}$, and
$\mathcal{W}_{ij}(\omega^{*})=0$ otherwise, where $a_{ij}(\omega^{*})>0$ is
the scalar constant weight that agent $i$ assigns to the information $x_{j}$
received from agent $j$. For instance, if
$\mathcal{W}(\mathcal{G}_{\tilde{k}})=I_{m}$, for some
$1\leq\tilde{k}\leq\bar{N}$, implies that there are no edges in
$\mathcal{G}_{k}$, or/and all nodes are not activated for communication
updates in asynchronous protocol or both. The advantage of this formulation is
that it unifies switching networks with asynchrony in a better understandable
formulation (see [44], [52] for details).

Assumption 1 [44]:

  * •

A1. The weighted graph matrix $\mathcal{W}(\omega^{*})$ is doubly stochastic
for each $\omega^{*}\in\Omega^{*}$ where $\Omega^{*}$ is defined in Section
III, i.e.,

i)
$\sum_{j\in\mathcal{N}_{i}^{in}(\omega^{*})\cup\\{i\\}}\mathcal{W}_{ij}(\omega^{*})=1,i=1,2,...,m$,

ii)
$\sum_{j\in\mathcal{N}_{i}^{out}(\omega^{*})\cup\\{i\\}}\mathcal{W}_{ij}(\omega^{*})=1,i=1,2,...,m.$

  * •

A2. The union of all of the graphs in $\Omega^{*}$ is strongly connected,
i.e.,
$Re[\lambda_{2}(\sum_{\omega^{*}\in\Omega^{*}}(I_{m}-\mathcal{W}(\omega^{*})))]>0$.

  * •

A3. There exists a nonempty subset $\tilde{K}\subseteq\Omega^{*}$ such that
$FVP(T)=\\{\tilde{z}|\tilde{z}\in\Re^{mn},\tilde{z}=T(\bar{\omega},\tilde{z}),\forall\bar{\omega}\in\tilde{K}\\}$
where $T(\omega^{*},x):=W(\omega^{*})x$, $\forall\omega^{*}\in\Omega^{*}$,
$W(\omega^{*}):=\mathcal{W}(\omega^{*})\otimes I_{n}$, and each element of
$\tilde{K}$ occurs infinitely often almost surely.

Remark 3 [44]: If the sequence $\\{\omega^{*}(t)\\}_{n=0}^{\infty}$ is
mutually independent with $\sum_{t=0}^{\infty}Pr_{t}(\bar{\omega})=\infty$
where $Pr_{t}(\bar{\omega})$ is the probability of $\bar{\omega}$ occurring at
time $t$, then part A3 in Assumption 1 is satisfied. Moreover, any ergodic
stationary sequences $\\{\omega^{*}(t)\\}_{t=0}^{\infty},Pr(\bar{\omega})>0,$
satisfy part A3 in Assumption 1. Consequently, any time-invariant Markov chain
with its unique stationary distribution as the initial distribution satisfies
part A3 in Assumption 1. Also B-connectivity (see footnote 1) assumption
satisfies part A3 in Assumption 1.

Assumption 2: $f_{i}:\Re^{n}\longrightarrow\Re,i=1,\ldots,m,$ are
$\frac{1}{K}$-strongly convex where $K>0$, and $\nabla f_{i}(x_{i})$ are
$\frac{1}{\mu}$-Lipschitz continuous where $\mu>0$.

In Theorem 1 below and its proof, we show how to apply P1-P3 and Observation 1
to derive the first totally asynchronous first-order gradient distributed
discrete-time algorithm for resource allocation (1).

Theorem 1: Consider distributed consensus-based optimization (6) with
Assumptions 1 and 2. Let $\beta\in(0,\frac{2\mu}{K^{2}})$ and
$\alpha(t)\in[0,1],t\in\mathbb{N}\cup\\{0\\},$ satisfy

(a) $\displaystyle\lim_{t\longrightarrow\infty}\alpha(t)=0,$

(b) $\sum_{t=0}^{\infty}\alpha(t)=\infty.$

Then starting from any initial points, the sequences
$\\{y_{i}(t)\\}_{t=0}^{\infty},i=1,\ldots,m,$ generated by

$\displaystyle y_{i}(t+1)$ $\displaystyle=\alpha(t)(y_{i}(t)-\beta
z_{i}(t))+(1-\alpha(t))((1-\eta)y_{i}(t)$
$\displaystyle\quad{}\quad{}+\eta\sum_{j\in\mathcal{N}_{i}^{in}(\omega^{*}(t))\cup\\{i\\}}\mathcal{W}_{ij}(\omega^{*}(t))y_{j}(t)),$
(12) $\displaystyle z_{i}(t)$ $\displaystyle=x_{i}(t)-R_{i},$ (13)
$\displaystyle x_{i}(t)$
$\displaystyle=\arg\min_{q_{i}}(f_{i}(q_{i})-y^{T}_{i}(t)q_{i}),$ (14)

where $\eta\in(0,1)$, and $\omega^{*}(t)$ is a realization of $\Omega^{*}$ at
time $t$, globally converge almost surely and in mean square to the unique
solution of dual problem (6). Moreover, the sequences
$\\{x_{i}(t)\\}_{t=0}^{\infty},i=1,\ldots,m,$ converge almost surely and in
mean square to the unique primal solution of distributed resource allocation
(1).

Proof: Let’s apply P1-P3 and Observation 1 to prove Theorem 1.

P1: We consider the first totally asynchronous algorithm proposed in [44] for
solving distributed consensus-based optimization (1) and its related theorem
for convergence (i.e., Theorem 3 in Appendix A).

The first totally asynchronous distributed discrete-time algorithm proposed in
[44] in compact form is given as follows:

$\displaystyle Y(t+1)$ $\displaystyle=\alpha(t)(Y(t)-\beta\nabla G(Y(t)))$
$\displaystyle\quad{}+(1-\alpha(t))((1-\eta)Y(t)+\eta W(\omega^{*}(t))Y(t))$
(15)

where $W(\omega^{*}):=\mathcal{W}(\omega^{*})\otimes I_{n}$, $\eta\in(0,1),$
and $\omega^{*}(t)$ is a realization of $\Omega^{*}$ at time $t$. From Theorem
2 in Appendix A, Algorithm (15) converges almost surely and in mean square to
the unique optimal solution $y^{*}$ of (6) under Assumption 1 above and
Assumption 3 in Appendix A (see Appendix A for details).

P2: We impose suitable assumptions on the first-order gradient of
$G_{i}(y_{i})$ defined in (7) and calculate it by using Lemmas 1-2.

Since, from Assumption 3 in Appendix A, we require that
$G_{i}(y_{i}),i=1,\ldots,m,$ be $\mu$-strongly convex, and $\nabla
G_{i}(y_{i})$ be $K$-Lipschitz continuous, we need from Lemma 1 to impose
Assumption 2 on $f_{i}(x_{i})$. Therefore, based on Lemma 2 and Assumption 2,
we can compute the gradient of $G_{i}(y_{i})$ as follows:

$\displaystyle\nabla
G_{i}(y_{i})=\arg\max_{x_{i}}(y^{T}_{i}x_{i}-f_{i}(x_{i}))-R_{i}.$ (16)

Now to develop Theorem 1 from Theorem 2 in Appendix A, we substitute
Assumption 2 for Assumption 3 since Assumption 2 is on primal variables
$x_{i},i=1,\ldots,m.$

P3: Each agent obtains the primal optimal solution $x_{i}^{*},i=1,\ldots,m,$
of (1) via
$x_{i}^{*}=\displaystyle\arg\max_{x_{i}}(x_{i}^{T}y^{*}-f_{i}(x_{i}))$.

We obtain the primal optimal solutions $x_{i}^{*},i=1,\ldots,m,$ as

$x_{i}^{*}=\displaystyle\arg\min_{x_{i}}(f_{i}(x_{i})-x_{i}^{T}y^{*})$ (17)

that exist from Assumption 2.

Therefore, we obtain Algorithm (12) from (15)-(17). Note that Algorithm (15)
is in a compact form and can be viewed based on local information as Algorithm
(12). As a matter of fact, the distributed algorithm (12) can converge almost
surely and in mean square to the primal optimal solutions
$x_{i}^{*},i=1,\ldots,m,$ of (1) under Assumptions 1-2. As seen in Theorem 2
in Appendix A, the convergence occurs over infinite-time horizon. Thus the
proof of Theorem 1 is complete.

Remark 4: Algorithm (12) does not require a priori B-connectivity or
distribution assumption on switching communication graphs for convergence that
results in the first totally asynchronous algorithm for solving distributed
resource allocation (1) (see also Remark 3). In general, the rate of
convergence cannot be established for a totally asynchronous algorithm.
However, determining rate of convergence of Algorithm (12) under suitable
assumptions remains a part of the future work.

Remark 5: We refer the interested reader to [48] for deriving the first
distributed second-order continuous-time algorithm for resource allocation.

##  V Solving (3) by Using Distributed Resource Allocation Algorithms

Fenchel’s dual problem [45] of distributed consensus-based optimization (3) is
the following resource allocation (see [46, Sec. III.A] for proof):

$\displaystyle\underset{y_{1},\ldots,y_{m}}{\text{min}}$
$\displaystyle\sum_{i=1}^{m}h_{i}^{*}(y_{i})$ (18) subject to
$\displaystyle\sum_{i=1}^{m}y_{i}=\textbf{0}_{n}$

where $h_{i}^{*}(y_{i})$ is Fenchel conjugate function of $h_{i}(x_{i})$ (see
Definition 2). We require to mention that similar challenge to the challenge
explained in Comment 1 holds for (18). Similar to Observation 1, we have the
following observation for solving (3) by using distributed resource
allocation’s algorithms.

Observation 2: By reformulating distributed consensus-based optimization (3)
as distributed resource allocation (18) on Fenchel’s dual variables
$y_{i},i=1,\ldots,m,$ and using properties of the Fenchel conjugate functions
$h^{*}_{i}(y_{i})$ presented in Lemmas 1-2, any first-as well as second-order
gradient distributed resource allocation algorithms can be used for solving
(3) under the assumptions of Lemma 1. Indeed, Remark 1 is also valid here.

As stated previously, the process for deriving first-order gradient
distributed consensus-based optimization algorithms from distributed resource
allocation algorithms is similar to the procedures of the proof of Theorems 1
in the previous section.

##  VI Conclusions

In this paper, we have proposed a procedure to utilize distributed consensus-
based optimization algorithms for solving resource allocation optimization and
vice versa over switching networks with/without synchronous protocol. We have
shown that first-order gradient algorithms of distributed consensus-based
optimization can be used to find an optimal solution of distributed resource
allocation under weaker assumptions than those of the literature over static
networks under synchronous protocol. This offers more choices for distributed
consensus-based optimization algorithms to be utilized for solving distributed
resource allocation problem that can overcome difficulties of existing
results. We have also shown that distributed resource allocation algorithms
can be employed to derive distributed consensus-based optimization algorithms.
This paper has presented the first totally asynchronous algorithm for solving
distributed resource allocation by using the proposed procedure. These results
can also be utilized to derive distributed algorithms for static, time-
varying, or random directed networks with or without synchronous protocol with
arbitrary initialization.

## Appendix A

The following theorem has been given in [44] for distributed optimization (6).

Assumption 3 [44]: $G_{i}(y_{i}),i=1,\ldots,m,$ is $\mu$-strongly convex, and
$\nabla G_{i}(y_{i})$ is $K$-Lipschitz continuous.

Theorem 2 [44, Cor. 1-2]: Consider dual problem (6) with Assumptions 1 and 3.
Let $\beta\in(0,\frac{2\mu}{K^{2}})$ and
$\alpha_{n}\in[0,1],n\in\mathbb{N}\cup\\{0\\}$, satisfy (a) and (b) in Theorem
1. Then starting from any initial point, the sequence generated by the
distributed algorithm (15) globally converges almost surely and in mean square
to the unique solution of (6).

Remark 6: An example of the diminishing step size $\alpha_{n}$ satisfying (a)
and (b) in Theorem 1 is $\alpha(t):=\frac{1}{(1+t)^{\zeta}}$ where
$\zeta\in(0,1]$.

Remark 7: As seen from part A3 in Assumption 1, Algorithm (15) does not
require a priori B-connectivity or distribution assumption on switching
communication graphs for convergence that results in the first totally
asynchronous algorithm for solving distributed consensus-based optimization
(6) (see [44] for details).

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