# Accelerated Distributed Primal-Dual Dynamics using Adaptive Synchronization

P. A. Bansode
&K. C. Kosaraju
&S. R. Wagh
&R. Pasumarthy
&N. M. Singh P. A. Bansode is with department of Instrumentation Engineering,
Ramrao Adik Institute of Technology, Mumbai, 400706 India.
<EMAIL_ADDRESS>C. Kosaraju is with faculty of Science and
Engineering, University of Groningen, AG Groningen, 9747 The Netherlands.S. R.
Wagh is with department of Electrical Engineering, Veermata Jijabai
Technological Institute, Mumbai, 400019 India.R. Pasumarthy is with department
of Electrical Engineering, Indian Institute of Technology Madras, Madras,
600036 India.N. M. Singh is with department of Electrical Engineering,
Veermata Jijabai Technological Institute, Mumbai, 400019 India.

###### Abstract

This paper proposes an adaptive primal-dual dynamics for distributed
optimization in multi-agent systems. The proposed dynamics incorporates an
adaptive synchronization law that reinforces the interconnection strength
between the coupled agents. By strengthening the synchronization between the
primal variables of the coupled agents, the given law accelerates the
convergence of the proposed dynamics to the saddle-point solution. The
resulting dynamics is represented as a feedback-interconnected networked
system that proves to be passive. The passivity properties of the proposed
dynamics are exploited along with the LaSalle’s invariance principle for
hybrid systems, to establish asymptotic convergence and stability of the
saddle-point solution. Further, the primal dynamics is analyzed for the rate
of convergence and stronger convergence bounds are established, it is proved
that the primal dynamics achieve accelerated convergence under the adaptive
synchronization. The robustness of the proposed dynamics is quantified using
$L_{2}$-gain analysis and the correlation between the rate of convergence and
robustness of the proposed dynamics is presented. The effectiveness of the
proposed dynamics is demonstrated by applying it to solve distributed least
squares and distributed support vector machines problems.

_K_ eywords Distributed optimization $\cdot$ Networked control system $\cdot$
Primal-dual dynamics $\cdot$ Adaptive synchronization

##  1 Introduction

Distributed optimization techniques have been the subject of substantial
research for many years. Their applications include wireless sensor networks
[1, 2, 3], power networks [4], large scale support vector machines [5, 6] etc.
An exhaustive survey of these techniques can be found in [7]. Mainly,
distributed optimization techniques are categorized as either decomposition
based distributed optimization (see, [8] and references therein) or consensus-
based distributed optimization. The consensus based distributed optimization
techniques have been significantly explored lately [9, 5, 4, 6, 10, 11, 12],
which is the prime subject of this paper.

Many algorithms have been proposed to solve consensus-based distributed
optimization problems arising in networked systems, such as the seminal work
on distributed sub-gradient methods[13], distributed primal-dual dynamical
algorithms[4], distributed gradient descent algorithms [14, 10] etc. Out of
these, the distributed primal-dual dynamics based algorithms deserve special
attention because of their rich systems and control theoretic properties [15,
16, 17, 18, 19] and ability to obtain simultaneously both primal as well as
dual optimal solutions. The seminal work on the primal-dual dynamics or the
saddle point dynamics dates back to late $1950$s [20, 21]. While their
application for solving optimization problems over a network first appeared in
[15] with the focus on asymptotic convergence and stability of these
algorithms. This framework is later extended to distributed optimization over
a network of communicating nodes in [22, 4]. The primal-dual dynamics in [22]
combine the decomposition and the consensus-based methods to propose
proportional-integral distributed optimization for equality constrained
optimization problems and achieves a globally asymptotically stable saddle-
point solution. The primal-dual gradient-based algorithm proposed in [4]
achieves asymptotic convergence for a consensus-based distributed optimization
problem with local inequality constraints and implements the algorithm for
load-sharing control in power networks. The notion of asymptotic convergence
and stability of the (distributed) primal-dual dynamics for distributed
optimization has been well established.

From the perspectives of online optimization, it is necessary to certify the
distributed optimization algorithms on the basis of their rate of convergence
as well as stability. The asymptotic convergence of the primal-dual dynamics
implies that the trajectories will converge to the saddle-point solution as
$t\rightarrow\infty$ which is not sufficient as a notion when the algorithm
solves the distributed optimization problem online. Lately, the algorithms
such as distributed gradient (sub-gradient) methods have been widely re-
studied with the objective of improvement in the rate of convergence, see [14,
23, 24, 10, 25]. However, the distributed primal-dual dynamics are not yet
explored with the same objective which could limit their application to large-
scale distributed optimization problems. While the existing methods on
improving the rate convergence of the primal-dual dynamics rely upon
increasing the convexity of the objective function by using quadratic penalty
terms (augmented Lagrangian techniques)[18], their usage for solving
distributed optimization problems will destroy the distributed structure of
the objective function. Thus, increasing convexity by using quadratic
penalties may not pose as a suitable way of improving the rate of convergence
of the distributed primal-dual dynamics. As an alternative route to this could
be to exploit the graph-Laplacian properties of the underlying network and use
adaptive coupling gains between the nodes to improve the convergence results.
Addressing this issue, the present work primarily contributes to the
accelerated convergence of the distributed primal-dual dynamics.

###  1.1 Relevant literature and contributions

The work proposed in this paper is in the same spirit with the recent articles
[4, 19]. In [4], the framework of primal-dual dynamics for network utility
maximization [26] which uses Krasovskii type Lyapunov function to derive
asymptotic convergence, is extended for distributed optimization with
application to load sharing control in power systems. Our contribution
significantly differs from [4] in the sense that the proposed dynamics is
first analyzed using passivity tools of dynamical systems which then lead to
its asymptotic stability when combined with the LaSalle’s invariance principle
of hybrid systems [27]. The advantage of passivity-based stability analysis is
that the proposed dynamics can be realized as a negative feedback
interconnection of the primal and the dual subsystems. This also facilitates
to understand the interaction between the primal and the dual dynamical
subsystems through their inputs and outputs. Thus each subsystem also enjoys
$L_{2}$ stability properties of feedback connected dynamical systems. This
feature later comes to the aid of robustness analysis of the proposed dynamics
using $L_{2}$-gains. The fundamental results on passivity-based stability
analysis of the primal-dual dynamics are established in [19]. Our work, in a
way, extends these results for the consensus-based distributed optimization
problems. However, the primal dynamical subsystem derived in this paper does
not use Brayton-Moser framework [28] to arrive at the optimal solution.

The central theme of the paper, that is the adaptively coupled primal-dual
dynamics is derived by integrating the consensus protocol in the distributed
primal-dual dynamics with the adaptive coupling laws motivated from the
results in [29]. In [29], the adaptive synchronization technique has been
proved to guarantee the synchronization between the trajectories of
diffusively coupled agents of a multiagent system. This technique is
essentially based upon modifying the coupling weights of the diffusively
coupled agents in accordance with the synchronization error between them.
Larger values of synchronization errors result in increasing the coupling
weights and vice-a-versa. In this paper, it is shown that the adaptation in
the coupling weights strengthens the synchronization of the primal variables
of the coupled agents. With this, the proposed work establishes results on an
accelerated convergence of the proposed dynamics to the saddle point solution.
While the adaptive synchronization has proved to accelerate the convergence,
it is shown that it affects the robustness of the proposed dynamics. By
introducing exogenous inputs in the interconnected network dynamics of the
primal-dual subsystems, the $L_{2}$-gain of the proposed dynamics is analyzed
and worst case $L_{2}$-gain is quantified in correlation with the rate of
convergence. Although it is well known that the interconnected network of
passive dynamical systems is inherently robust to exogenous inputs [30], our
results have quantified the $L_{2}$-gain margins and established a relation
between these margins and the rate of convergence.

To summarize, the proposed work envelopes the following key points:

  1. 1.

The proposed algorithm, designated hereafter as the adaptively synchronized
distributed primal-dual dynamics (ADPDD), ensures synchronization of the
network-wide primal variables to a common trajectory which is then driven to
the optimal solution.

  2. 2.

The ADPDD is posed as a negative feedback interconnection of the primal
dynamical subsystem and the dual dynamical subsystems. It is proved that these
subsystems remain individually passive, which subsequently, ensures the
passivity and the asymptotic stability of the proposed dynamics.

  3. 3.

The convergence rate of the ADPDD is established and it is proved that the
ADPDD has an accelerated convergence than the distributed primal-dual dynamics
(DPDD).

  4. 4.

By allowing time-scale separation between the adaptive coupling laws and the
primal-dual dynamics, the adaptively coupled distributed primal dynamics is
proved to have an accelerated convergence to the optimal solution than the
conventional distributed primal dynamics.

  5. 5.

The $L_{2}$-gain analysis of the proposed dynamics against the exogenous
disturbances is presented to show the correlation between the rate of
convergence and the robustness of the proposed algorithm.

  6. 6.

Applicability of the proposed algorithm to solve distributed least squares,
distributed support vector machines problems is discussed.

###  1.2 Notations and Preliminaries

The set $\mathbb{R}$ (respectively $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{>0}$)
is the set of real (respectively non-negative or positive) numbers. $I_{n}$ is
the $n\times n$ identity matrix. $\mathbf{0}$ is a zero vector of appropriate
dimensions. For a square matrix $A\in\mathbb{R}^{n\times n}$,
$\mathrm{eig}(A)=\\{\lambda_{1}(A),\lambda_{2}(A),\ldots,\lambda_{n}(A)\\}\in\mathbb{R}$
represents eigenvalues of $A$ in an ascending order. The smallest eigenvalue
of $A$ is given by $\lambda_{1}(A)$ and the second smallest eigenvalue is
given by $\lambda_{2}(A)$. If $B\in\mathbb{R}^{m\times n}$ and
$C\in\mathbb{R}^{p\times q}$ are real matrices, then $B\otimes
C\in\mathbb{R}^{mp\times nq}$ is a block matrix that defines the Kronecker
product of $B$ and $C$.

The interaction topology in a multi-agent system is represented using an
undirected graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$ with
$\mathcal{N}=\\{1,2,\ldots,n\\}$ as the set of agents and
$\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}$ as the set of edges. The
neighbor set of the $i^{th}$ agent is
$\mathcal{N}_{i}=\\{q\in\mathcal{N}|(q,i)\in\mathcal{E}\\}$, where
$i\in\mathcal{N}$. The number of agents $n$ is the cardinality of
$\mathcal{G}$. Let $D\in\mathbb{R}^{n\times n}$ be the degree matrix of
$\mathcal{G}$ and $A\in\mathbb{R}^{n\times n}$ be the adjacency matrix of
$\mathcal{G}$, with elements $a_{iq}=a_{qi}>0,\forall(i,q)\in\mathcal{E}$,
then $L=D-A$ is the Laplacian matrix of $\mathcal{G}$. By definition,
$L\in\mathbb{R}^{n\times n}$ is a symmetric positive semidefinite matrix that
encodes the connectivity of the agents and their interaction topology in
$\mathcal{G}$.

If $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is continuously differentiable in
$x\in\mathbb{R}^{n}$, then
$\nabla_{x}f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is the gradient of $f$
with respect to $x$. If $f$ is twice continuously differentiable and strictly
convex in $x$ then $\mathbb{H}=\nabla^{2}_{x}f\in\mathbb{R}^{n\times n}_{>0}$
is a symmetric positive definite matrix of second-order partial derivatives of
$f$ with respect to $x$.

Consider the following dynamical system

$\displaystyle\dot{x}=F(x,u),y=G(x,u),$ (1)

where state $x\in\mathbb{R}^{n}$, input $u\in\mathbb{R}^{m}$, and output
$y\in\mathbb{R}^{m}$, with $F,G$ (of appropriate dimensions) sufficiently
smooth and satisfying $F(0)=G(0)=0$.

######  Definition 1.1 ([31]).

The system (1) is said to be passive if there exists a positive semidefinite
storage function (Lyapunov function) $V:\mathbb{R}^{n}\rightarrow\mathbb{R}$,
continuously differentiable in $x$ such that $\dot{V}\leq u^{T}y$.

For scalars $x,y$, $[x]^{+}_{y}:=x$ if $y>0$ or $x>0$, and $[x]^{+}_{y}:=0$
otherwise.

The remainder of the paper is mainly divided into two sections. Section 2
discusses the main results of the paper and Section 3 presents examples to
validate the proposed work. Subsection 2 is divided as follows: Section 2.1
describes the consensus-based distributed optimization problem. In Subsection
2.2.1 the adaptive synchronization technique is elaborated. Subsection 2.2.2
formulates the adaptive distributed primal-dual dynamical algorithm to solve
distributed optimization problem proposed in Subsection 2.1. Subsections 2.3
and 2.4 present passivity and stability analysis of the proposed dynamics. In
Subsection 2.5 the convergence bounds of the proposed algorithm are obtained
and the proof for an accelerated convergence of the same is provided.
Subsection 2.6 provides $L_{2}$-gain analysis of the proposed dynamics and
establishes a correlation between both robustness and rate of convergence of
the same. Section 3 presents the application of the proposed dynamics to the
distributed least squares and the distributed support vector machines
problems. Some numerical examples of academic interests are also discussed.
Section 4 concludes the paper.

##  2 Problem Formulation and main results

###  2.1 Distributed Optimization

Consider the following distributed optimization problem

$\displaystyle\begin{aligned} \min_{x\in\mathbb{R}^{ln}}&\leavevmode\nobreak\
f(x)=\sum_{i=1}^{n}f_{i}(x_{i})\\\ \mathrm{subject\leavevmode\nobreak\
to}&\leavevmode\nobreak\ x_{ik}=x_{qk},\leavevmode\nobreak\
\forall^{l}_{k=1},\forall i,q\in\mathcal{N},\\\ &\leavevmode\nobreak\
g_{j}(x_{ik})\leq 0,\leavevmode\nobreak\
\forall^{m^{ik}_{g}}_{j=1},\forall^{l}_{k=1},\forall
i\in\mathcal{N},\end{aligned}$ (2)

where $x_{i}=[x_{i1},\ldots,x_{il}]^{T}\in\mathbb{R}^{l}$ and
$x=[x^{T}_{1},\ldots,x^{T}_{n}]^{T}\in\mathbb{R}^{ln}$. It is assumed that the
functions $f_{i}:\mathbb{R}^{l}\rightarrow\mathbb{R}$ is twice differentiable
and strongly convex, and $g_{j}:\mathbb{R}\rightarrow\mathbb{R}$ is convex.
The optimization problem (2) can be decomposed into $n$ subproblems wherein
each subproblem minimizes the cost $f_{i}(x_{i})$ subject to the consensus
constraint $x_{ik}=x_{qk}$ and inequality constraints $g_{j}(x_{ik})\leq 0$.
The problem (2) can not be fully decoupled into a set of $n$ subproblems
because of the consensus constraints, but it can be addressed as a network-
based multiagent optimization problem using graph theory as a tool. Let an
undirected and connected graph $\mathcal{G}(\mathcal{N},\mathcal{E})$ describe
the communication topology of the underlying network, where $\mathcal{N}$
denotes the set of agents or subproblems, and $\mathcal{E}$ denotes the set of
communication links. Each agent minimizes a local cost function $f_{i}(x_{i})$
subject to the consensus constraints $x_{ik}=x_{qk},\forall^{l}_{k=1},\forall
q\in\mathcal{N}_{i}$ and the local inequality constraints $g_{j}(x_{ik})\leq
0,\forall^{l}_{k=1}$. The global consensus corresponds to the optimal solution
of (2), when $x^{*}_{1}=x^{*}_{2}=\ldots=x^{*}_{n}=x^{*}$. The index
$m^{ik}_{g}$ is the number of inequality constraints associated with the
scalar $x_{ik}$.

The strong duality of (2) is subject to the convexity of $f$ and the
constraint satisfaction given by the Slater’s condition (see, [32]), which is
as follows: Assuming that there exists an $x\in\mathrm{relint}\mathcal{D}$
such that $g_{j}(x_{ik})<0,x_{ik}=x_{qk},\forall^{l}_{k=1},\forall
q\in\mathcal{N}_{i},\forall^{n}_{i=1}$, then $x$ is strictly feasible, where
$\mathcal{D}$ is the domain of (2) defined as $\mathcal{D}=\mathrm{dom}f$. The
convexity of $f$ strongly imply the uniqueness of its optimal solution
$x^{*}$.

######  Assumption 1.

$f$ is $\mu-$strongly convex and $\kappa$-smooth, i.e., for all
$x_{1},x_{2}\in\mathbb{R}^{ln}$,

$\displaystyle\mu\|x_{1}-x_{2}\|^{2}\leq\langle\nabla f(x_{1})-\nabla
f(x_{2}),x_{1}-x_{2}\rangle\leq\kappa\|x_{1}-x_{2}\|^{2}.$ (3)

The Lagrangian function ${\mathcal{L}}$ of the problem (2) is given by: where
$\alpha_{ik}\in\mathbb{R}$ is a Lagrange multiplier associated with the
consensus constraint $x_{ik}=x_{qk}$ and
$\theta^{ik}_{j}\in\mathbb{R}_{+}=\\{\theta^{ik}_{j}\in\mathbb{R}|\theta^{ik}_{j}\geq
0,\leavevmode\nobreak\ \forall^{l}_{k=1},\forall^{m^{ik}_{g}}_{j=1},\forall
i\in\mathcal{N}\\}$ is a Lagrange multiplier associated with the inequality
constraint $g_{j}(x_{ik})\leq 0$. The vector notations of the respective
Lagrange multipliers are $\theta\in\mathbb{R}^{m^{ik}_{g}ln}_{+}$ and
$\alpha\in\mathbb{R}^{ln}$.

######  Remark 1.

Assuming that the Slater’s condition is satisfied and a strong duality holds,
the saddle-point $(x^{*},\alpha^{*},\theta^{*})$ satisfies the Karush-Kuhn-
Tucker (KKT) conditions derived the Lagrangian (LABEL:dlag1), as follows:

$\displaystyle\nabla_{x^{*}_{ik}}f_{i}(x^{*}_{ik})+\sum_{q\in\mathcal{N}_{i}}a_{iq}(\alpha^{*}_{ik}-\alpha^{*}_{qk})$
$\displaystyle+\sum_{j=1}^{m^{ik}_{g}}(\theta^{ik}_{j})^{*}\nabla_{x^{*}_{ik}}g_{j}(x^{*}_{ik})=0,\forall^{l}_{k=1},\forall
i\in\mathcal{N},$ $\displaystyle g_{j}(x^{*}_{ik})\leq
0,(\theta^{ik}_{j})^{*}\geq
0,\forall^{l}_{k=1},\forall^{m^{ik}_{g}}_{j=1},\forall i\in\mathcal{N},$
$\displaystyle(\theta^{ik}_{j})^{\ast}g_{j}(x^{*}_{ik})=0,\forall^{l}_{k=1},\forall^{m^{ik}_{g}}_{j=1},\forall
i\in\mathcal{N},$ $\displaystyle x^{*}_{ik}=x^{*}_{qk},\leavevmode\nobreak\
\forall^{l}_{k=1},\forall(i,q)\in\mathcal{N}.$ (4)

In order to ensure the global consensus of the states $x_{i},\forall
i\in\mathcal{N}$, the Lagrangian function defined in (LABEL:dlag1) is
augmented with the term $x^{T}(L\otimes I_{l})x$. The augmented Lagrangian
function is defined below:

$\displaystyle\begin{aligned}
\bar{\mathcal{L}}(x,\alpha,\theta)={\mathcal{L}}(x,\alpha,\theta)+x^{T}(L\otimes
I_{l})x.\end{aligned}$ (5)

######  Remark 2.

Note that augmenting the Lagrangian (LABEL:dlag1) with $x^{T}(L\otimes
I_{l})x$ does not affect its convexity-concavity properties. This owes to the
fact that $x^{T}(L\otimes I_{l})x$ is a positive semidefinite function of the
primal variable $x$. Thus the saddle-point satisfying (4) also satisfies the
following KKT conditions for the Lagrangian (5):

$\displaystyle\nabla_{x^{*}_{ik}}f_{i}(x^{*}_{ik})+\sum_{q\in\mathcal{N}_{i}}a_{iq}(x^{*}_{ik}-x^{*}_{qk})+\sum_{q\in\mathcal{N}_{i}}a_{iq}(\alpha^{*}_{ik}-\alpha^{*}_{qk})$
$\displaystyle+\sum_{j=1}^{m^{ik}_{g}}(\theta^{ik}_{j})^{*}\nabla_{x^{*}_{ik}}g_{j}(x^{*}_{ik})=0,\forall^{l}_{k=1},\forall
i\in\mathcal{N},$ $\displaystyle g_{j}(x^{*}_{ik})\leq
0,(\theta^{ik}_{j})^{*}\geq
0,\forall^{l}_{k=1},\forall^{m^{ik}_{g}}_{j=1},\forall i\in\mathcal{N},$
$\displaystyle(\theta^{ik}_{j})^{\ast}g_{j}(x^{*}_{ik})=0,\forall^{l}_{k=1},\forall^{m^{ik}_{g}}_{j=1},\forall
i\in\mathcal{N},$ $\displaystyle x^{*}_{ik}=x^{*}_{qk},\leavevmode\nobreak\
\forall^{l}_{k=1},\forall(i,q)\in\mathcal{N}.$ (6)

Using the augmented Lagrangian (5), the primal-dual dynamics is derived as
follows:

$\displaystyle\begin{aligned}
\dot{x}_{ik}&=-\nabla_{x_{ik}}\bar{\mathcal{L}}(x,\alpha,\theta),\dot{\alpha}_{ik}=\nabla_{\alpha_{ik}}\bar{\mathcal{L}}(x,\alpha,\theta),\\\
\dot{\theta}^{ik}_{j}&=[\nabla_{\theta^{ik}_{j}}\bar{\mathcal{L}}(x,\alpha,\theta)]^{+}_{\theta^{ik}_{j}},\leavevmode\nobreak\
\forall^{l}_{k=1};\forall^{m^{ik}_{g}}_{j=1};\forall
i\in\mathcal{N}.\end{aligned}$ (7)

With the primal-dual dynamics derived as given in (7), the following
subsection develops the ADPDD.

###  2.2 Adaptively Synchronized Distributed Primal-dual dynamics

The following subsection presents the adaptive synchronization mechanism which
is later integrated with the dynamics defined in (7) to arrive at ADPDD.

####  2.2.1 Adaptive synchronization

The adaptive synchronization mechanism has been widely used in multi-agent
systems to guarantee synchronization between the agents with respect to their
state variables [29, 33], which is explained subsequently.

The primal variables associated with each agent evolve according to

$\displaystyle\dot{x}_{ik}=-\nabla_{x_{ik}}\bar{\mathcal{L}}(x,\alpha,\theta)$
(8)

as described in (7). By performing gradient descent on (5), the primal
dynamics (8) can be further derived as:

$\displaystyle\dot{x}_{ik}$
$\displaystyle=-\nabla_{x_{ik}}f(x)-\sum_{q\in\mathcal{N}_{i}}a_{iq}(x_{ik}-x_{qk})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\sum_{q\in\mathcal{N}_{i}}a_{iq}(\alpha_{ik}-\alpha_{qk})-\sum_{j=1}^{m^{ik}_{g}}\theta^{ik}_{j}\nabla_{x_{ik}}g_{j}(x_{ik}).$
(9)

Let $u_{x_{ik}}\in\mathbb{R}$ corresponds to the following term in (9):

$\displaystyle\begin{aligned}
u_{x_{ik}}=-\sum_{q\in\mathcal{N}_{i}}a_{iq}(x_{ik}-x_{qk}),\forall
q\in\mathcal{N}_{i},\end{aligned}$ (10)

where the interconnection strength or the coupling weight $a_{iq}$ belongs to
the adjacency matrix $A$ such that

$\displaystyle a_{iq}=a_{qi}=\begin{cases}\mathrm{a\leavevmode\nobreak\
positive\leavevmode\nobreak\ scalar,}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ (q,i)\in\mathcal{E},\\\
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
(q,i)\notin\mathcal{E}.\end{cases}$

The equation (10) is regarded widely as the consensus protocol or the
consensus law[29, 34]. Define further $u_{x_{i}}\in\mathbb{R}^{l}$, the
consensus protocol (10) can be modified to accommodate
$x_{i}\in\mathbb{R}^{l}$ as given below:

$\displaystyle\begin{aligned}
u_{x_{i}}=-\sum_{q\in\mathcal{N}_{i}}a_{iq}(x_{i}-x_{q}),\forall
q\in\mathcal{N}_{i},\end{aligned}$ (11)

Similarly,

$\displaystyle u_{x}=-(L\otimes I_{l})x$ (12)

is a compact form representation of (11).

If $i$ and $q$ are neighbors in $\mathcal{G}$ with $e_{iq}=x_{i}-x_{q}$
defined as the local synchronization error, then the coupling weight can be
represented as a function of $e_{iq}$, i.e. $\dot{a}_{iq}=h_{i}(e_{iq})$,
where $h_{i}:\mathbb{R}^{l}\rightarrow\mathbb{R}$ monotonically increases in
$e_{iq}$. It yields a stronger synchronization between the primal variables of
the coupling agents which motivates to incorporate adaptive synchronization to
address the convergence rate of the distributed primal-dual dynamics. In line
with this, the following coupling weight update rule is proposed:

$\displaystyle\dot{a}_{iq}=d_{iq}(e^{T}_{iq}e_{iq}+\dot{e}^{T}_{iq}\dot{e}_{iq}),$
(13)

where $d_{iq}=d_{qi}>0$ is the adaptive gain constant.

######  Remark 3.

Represent (13) in the form $\dot{a}_{iq}=h_{i}(e_{iq},\dot{e}_{iq})$,
throughout the rest of the paper it is assumed that the real valued function
$h_{i}:\mathbb{R}^{ln}\rightarrow\mathbb{R}$ is Lipschitz continuous.

The dynamics (13) addresses two questions, viz. how far from each other the
local primary variables are and how fast they can be synchronized to a common
trajectory. The quadratic appearance of $e_{iq}$ and $\dot{e}_{iq}$ in (13)
ensures that it is monotonically increasing in $\mathbb{R}$.

Figure 1: Interconnected networked dynamics of $H_{1},H_{2}$ and $H_{3}$.

####  2.2.2 Integrating the adaptive coupling law (13) with the primal-dual
dynamics (7)

By integrating the adaptive coupling law (13) with the PDD (7) and
partitioning the resulting dynamics into three interconnected subsystems i.e.,
$H_{1}$ (primal partition), $H_{2}$ (consensus dual partition), and $H_{3}$
(inequality dual partition) as shown in Fig. 1, yields:

$\displaystyle H_{1}$
$\displaystyle:\begin{cases}\dot{x}&=-\nabla_{x}f(x)+u_{x}+u_{H_{1}},\\\
\dot{a}_{iq}&=d_{iq}(e^{T}_{iq}e_{iq}+\dot{e}^{T}_{iq}\dot{e}_{iq}),\forall
i\in\mathcal{N},\forall q\in\mathcal{N}_{i},\\\ y_{H_{1}}&=x.\end{cases}$ (14)
$\displaystyle H_{2}$ $\displaystyle:\begin{cases}\dot{\alpha}&=u_{H_{2}},\\\
y_{H_{2}}&=\alpha.\end{cases}$ (15)

The system $H_{3}$ represents the $\theta^{ik}_{j}$ dynamics in the stacked
vector form with $u_{H_{3}}$ and $y_{H_{3}}$ as its input and output
respectively, as given below:

$\displaystyle H_{3}$
$\displaystyle:\begin{cases}\dot{\theta}^{ik}_{j}&=[g_{j}(x_{ik})]^{+}_{\theta^{ik}_{j}},\leavevmode\nobreak\
\forall^{l}_{k=1};\forall^{m^{ik}_{g}}_{j=1};\forall i\in N,\\\
y_{H_{3}}&=\sum_{j=1}^{m^{ik}_{g}}\theta^{ik}_{j}\nabla_{x_{ik}}g_{j}(x_{ik}),\forall^{l}_{k=1};\forall^{m^{ik}_{g}}_{j=1};\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\in
N,\end{cases}$ (16)

where $y_{H_{1}},y_{H_{2}},y_{H_{3}}\in\mathbb{R}^{ln}$ and
$u_{H_{1}}=-(L\otimes I_{l})y_{H_{2}}-y_{H_{3}},u_{H_{2}}=(L\otimes
I_{l})y_{H_{1}}$, and $u_{H_{3}}=y_{H_{1}}$.

The ADPDD (14)-(16) has been characterized as the feedback interconnected
networked system as shown in the Fig. 1. Each agent in the underlying network
is diffusively coupled with its neighboring agents by virtue of the
communication topology that defines the interaction between such agents on the
graph $\mathcal{G}(\mathcal{N},\mathcal{E})$. It can be noted that the network
representation in Fig. 1 is independent of the graph parameters such as
communication topology, number of agents, and interaction links. Irrespective
of such parameters, if the graph $\mathcal{G}(\mathcal{N},\mathcal{E})$ is
connected, one can arrive at the stability results of the underlying network
by only verifying its passivity properties. Towards this end, the following
subsection first motivates the passivity analysis of the network shown in Fig.
1 which further leads to its closed loop stability and robustness analysis.

###  2.3 Passivity based stability analysis of ADPDD

This section begins with passivity analysis of the subsystems $H_{1}$,
$H_{2}$, $H_{3}$ and their feedback interconnection shown in Fig. 1 and then
moves towards the stability and robustness analysis of the said feedback
interconnection. The Krasovskii type storage function has been defined for
each subsystem (see, [15]) which has led to a new passivity property with
differentiation at both ports[35, Proposition 2]. The intuition behind this
proposition is to define the Krasovskii type storage function $V(x)$ for the
dynamical system defined in (1), such that $\dot{V}\leq\dot{u}^{T}\dot{y}$,
where $\dot{u}$ and $\dot{y}$ are considered as port variables. This
inequality shows that the map from the port input $\dot{u}$ to the port output
$\dot{y}$ is passive. Motivated by this result, subsequently it is shown that
the ADPDD is a passive system.

####  2.3.1 $H_{1}$ is passive

######  Proposition 2.1.

Assuming that the graph $G$ is connected and $f$ is strictly convex in $x$, if
there exists $x_{eq}\in\mathbb{R}^{ln}$ that satisfies (4), then the subsystem
$H_{1}$ is passive with port variables $(\dot{{y}}_{H_{1}},\dot{u}_{H_{1}})$.

###### Proof.

Let $\tilde{a}_{iq}=a_{iq}-a^{*}_{iq}$ with $a^{*}_{iq}>0$ defined as follows:

$\displaystyle a^{*}_{iq}=\begin{cases}a^{*}\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{if $i$ and $q$ are neighbors in
$\mathcal{G}$},\\\ 0\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{if $i$
and $q$ are not neighbors in $\mathcal{G}$},\end{cases}$ (17)

where $a^{*}$ is a parameter to be selected.

Consider the following storage function for the update law (13) [29].

$\displaystyle
W=\frac{1}{2}\sum_{i=1}^{p}\sum_{q=1}^{p}\frac{1}{d_{iq}}\tilde{a}^{2}_{iq}.$
(18)

Differentiating (18) with respect to time yields the following:

$\displaystyle\dot{W}=\sum_{i=1}^{p}\sum_{q=1}^{p}\tilde{a}_{iq}(e^{T}_{iq}e_{iq}+\dot{e}^{T}_{iq}\dot{e}_{iq}).$
(19)

Acknowledging the graph symmetry and substituting for $e_{iq}=x_{i}-x_{q}$,
(19) modifies to

$\displaystyle\begin{aligned} \dot{W}&=\dot{x}^{T}(L\otimes
I_{l})\dot{x}-a^{*}\dot{x}^{T}(L\otimes I_{l})\dot{x}\\\ &\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +x^{T}(L\otimes
I_{l})x-a^{*}x^{T}(L\otimes I_{l})x,\\\ &=(1-a^{*})\dot{x}^{T}(L\otimes
I_{l})\dot{x}+(1-a^{*})x^{T}(L\otimes I_{l})x.\end{aligned}$ (20)

Now, consider the following storage function for $H_{1}$, which is a sum of
Krasovskii-type storage function of $x$ and (18):

$\displaystyle\begin{aligned}
V_{H_{1}}(x)&=\frac{1}{2}\dot{x}^{T}\dot{x}+W.\end{aligned}$ (21)

Differentiating (21) with respect to time and using (20) yields,

$\displaystyle\begin{aligned}
\dot{V}_{H_{1}}(x)&=-\dot{x}^{T}\mathbb{H}\dot{x}-\dot{x}^{T}(L\otimes
I_{l})\dot{x}+(1-a^{*})\dot{x}^{T}(L\otimes I_{l})\dot{x}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+(1-a^{*}){x}^{T}(L\otimes I_{l}){x}+\dot{x}^{T}\dot{u}_{H_{1}},\\\
&=-\dot{x}^{T}\mathbb{H}\dot{x}-a^{*}\dot{x}^{T}(L\otimes
I_{l})\dot{x}+(1-a^{*}){x}^{T}(L\otimes I_{l}){x}\\\ &\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +\dot{x}^{T}\dot{u}_{H_{1}},\\\
&\leq-\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}\|\dot{y}_{H_{1}}\|^{2}\\\ &\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +(1-a^{*})\lambda_{2}(L\otimes
I_{l})\|y_{H_{1}}\|^{2}+\dot{{y}}_{H_{1}}^{T}\dot{u}_{H_{1}}.\end{aligned}$
(22)

Notice that $y_{H_{1}}=x$ and choosing $a^{*}>1$ makes the term
$(1-a^{*})\lambda_{2}(L\otimes I_{l})\|y_{H_{1}}\|^{2}$ in (22) negative
definite. Since $\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})>0$ for a non-negative value of $a^{*}$, the inequality (22) implies
that the subsystem $H_{1}$ is output strictly passive (“OSP”[30]) with respect
to the port variables $\dot{u}_{H_{1}}$ and $\dot{y}_{H_{1}}$. ∎

####  2.3.2 $H_{2}$ is passive

######  Proposition 2.2.

Assuming that the graph $G$ is connected and $f$ is strictly convex in $x$, if
there exists $\alpha_{eq}\in\mathbb{R}^{ln}$ satisfying (4), then the
subsystem $H_{2}$ is passive with port variables
$(\dot{y}_{H_{2}},\dot{u}_{H_{2}})$.

###### Proof.

Consider a Krasovskii-type storage function for $H_{2}$ as given below:

$\displaystyle{V}_{H_{2}}(\alpha)=\frac{1}{2}\dot{\alpha}^{T}\dot{\alpha}.$
(23)

Differentiating (23) with respect to time yields,

$\displaystyle\dot{V}_{H_{2}}(\alpha)$ $\displaystyle=$
$\displaystyle\dot{\alpha}^{T}(L\otimes I_{l})\dot{x},$ (24) $\displaystyle=$
$\displaystyle\dot{y}_{H_{2}}^{T}\dot{u}_{H_{2}}.$

(24) yields the following inequality,

$\displaystyle
V_{H_{2}}(\alpha(\tau))-V_{H_{2}}(\alpha(0))\leq\int_{0}^{\tau}\dot{y}_{H_{2}}^{T}\dot{u}_{H_{2}}dt.$
(25)

Hence, the subsystem $H_{2}$ is passive with respect to port variables
$\dot{u}_{H_{2}}$ and $\dot{y}_{H_{2}}$. ∎

####  2.3.3 $H_{3}$ is passive

In the following, $H_{3}$ is modeled as a switched dynamical system.

The dynamics in (16) becomes discontinuous when $\theta^{ik}_{j}=0$ and
$g_{j}(x_{ik})<0$. The value of $g_{j}(x_{ik})^{+}$ switches from
$g_{j}(x_{ik})$ to $0$. To further clarify that, (16) is reformulated below as
given in Kose [20].

$\displaystyle\dot{\theta}^{ik}_{j}=\begin{cases}g_{j}(x_{ik}),\leavevmode\nobreak\
\mathrm{if}\leavevmode\nobreak\ \theta^{ik}_{j}>0\leavevmode\nobreak\
\mathrm{or}\leavevmode\nobreak\ g_{j}(x_{ik})>0,\\\ 0.\end{cases}$ (26)

From (26), the projection is seen to be active for the second case. Let
$\mathcal{I}_{i}=\\{1,\ldots,lm^{ik}_{g}\\}$ and
$\sigma_{i}:[0,\infty)\rightarrow\mathcal{I}_{i},\forall
k=1,\ldots,l;j\in\mathcal{I}_{i}$ be an arbitrary switching signal. Then

$\displaystyle\sigma_{i}(t)=\\{j|\theta^{ik}_{j}=0,g_{j}(x_{ik})\leq 0,\forall
k;\forall j\in\mathcal{I}_{i}\\},$ (27)

represents the switching time instances when there is an active projection.
Considering (27), the inequality constraint dynamics given in (16) takes the
form of a switched system:

$\displaystyle\dot{\theta}^{ik}_{j}=\begin{cases}g_{j}(x_{ik}),\forall
k;j\notin\sigma_{i}(t),\\\ 0,\forall k;j\in\sigma_{i}(t),\end{cases}$ (28)

where $\sigma_{i}(t)\subset\sigma(t),\forall^{n}_{i=1}$. Let $V_{H_{3}}$ be
the Lyapunov function associated with $H_{3}$. It is defined as given below:

$\displaystyle{V}_{H_{3}}(\theta)=\frac{1}{2}\sum_{i=1}^{n}\sum_{k=1}^{l}\sum_{j\notin\sigma_{i}(t)}(\dot{\theta}^{ik}_{j})^{2}.$
(29)

######  Proposition 2.3.

The subsystem $H_{3}$ is passive with port input $\dot{u}_{H_{3}}$, and port
output $\dot{y}_{H_{3}}$ for each pair of switching time instances
$(\tau^{+}_{\sigma_{i}},\tau^{-}_{\sigma_{i}})$ corresponding to (28) where
$\tau^{-}_{{\sigma_{i}}}<\tau^{+}_{{\sigma_{i}}}$ such that
$\sigma_{i}(\tau^{+}_{\sigma_{i}})=\sigma(\tau^{-}_{\sigma_{i}})=\sigma_{i}\in\mathcal{I}_{i}$
and $\sigma_{i}(\tau^{\prime})\neq\sigma_{i}$ for
$\tau^{-}_{{\sigma_{i}}}<\tau<\tau^{+}_{\sigma_{i}}$.

###### Proof.

Differentiating (29) with respect to time yields,

$\displaystyle\dot{V}_{H_{3}}(\theta)$
$\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{l}\sum_{j\notin\sigma_{i}(t)}\dot{\theta}^{ik}_{j}\ddot{\theta}^{ik}_{j},$
$\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{l}\sum_{j\notin\sigma_{i}(t)}\dot{\theta}^{ik}_{j}\nabla
g_{j}(x_{ik})\dot{x}_{ik},$
$\displaystyle\leq\sum_{i=1}^{n}\sum_{k=1}^{l}\sum_{j=1}^{m^{ik}_{g}}\dot{\theta}^{ik}_{j}\nabla
g_{j}(x_{ik})\dot{x}_{ik},$
$\displaystyle\leq\sum_{i=1}^{n}\sum_{k=1}^{l}\dot{x}_{ik}\sum_{j=1}^{m^{ik}_{g}}\dot{\theta}^{ik}_{j}\nabla
g_{j}(x_{ik}).$ (30)

Using $u_{H_{3}}$ and $y_{H_{3}}$ from (16) in (30),

$\displaystyle\dot{V}_{H_{3}}(\theta)\leq\dot{y}_{H_{3}}^{T}\dot{u}_{H_{3}}.$
(31)

Thus,

$\displaystyle\sum_{i=1}^{n}\sum_{k=1}^{l}\sum_{j\notin\sigma_{i}(t)}V_{k}(\theta^{ik}_{j}(\tau^{+}_{\sigma_{i}}))-V_{k}(\theta^{ik}_{j}(\tau^{-}_{\sigma_{i}}))\leq\int_{\tau^{-}_{\sigma_{i}}}^{\tau^{+}_{\sigma_{i}}}\dot{y}_{H_{3}}^{T}\dot{u}_{H_{3}}dt.$
(32)

(32) ensures that the switched system (28) represents a finite family of
passive systems. However, it must be ensured that the Lyapunov function
$V_{H_{3}}$ does not increase during the switching events. In line with this,
the following two cases have been considered:

  1. 1.

It may happen for some $x_{ik}$ in (28), that the function $g_{j}(x_{ik})$
goes from negative to positive through $0$. This will cause the Lyapunov
function to change from $V_{k}(\theta^{ik}_{j}(\tau^{-}_{\sigma_{i}}))$ to
$V_{k}(\theta^{ik}_{j}(\tau^{+}_{\sigma_{i}}))$. If that happens, the
Lagrangian multiplier $\theta^{ik}_{j}>0$ will add a new term to
$V_{k}(\theta^{ik}_{j}(\tau_{\sigma_{i}}))$. Since,
$V_{k}(\theta^{ik}_{j}(\tau_{\sigma_{i}}))$ is continuous in time, (32) holds
for $\tau>\tau^{-}_{\sigma_{i}}$ as well as $\tau<\tau^{+}_{\sigma_{i}}$.
Hence,
$V_{k}(\theta^{ik}_{j}(\tau^{+}_{\sigma_{i}}))=V_{j}(\theta^{ik}_{j}(\tau^{-}_{\sigma_{i}}))$.

  2. 2.

In this case the projection of $k^{th}$ constraint for a given $j$ becomes
active, i.e., $\theta^{ik}_{j}$ reaches to $0$ from a positive value for the
$k^{th}$ constraint of the $i^{th}$ machine. Hence, the corresponding $k^{th}$
term of the Lyapunov function $V_{k}(\theta^{ik}_{j})$ will disappear. In
turn, the following inequality will be satisfied.
$V_{k}(\theta^{ik}_{j}(\tau^{+}_{\sigma_{i}}))<V_{k}(\theta^{ik}_{j}(\tau^{-}_{\sigma_{i}}))$.

Hence, in both the cases, the Lyapunov function $V_{k}(\theta^{ik}_{j}(\tau))$
will be non-increasing. ∎

###  2.4 Stability analysis of the feedback interconnection shown in Fig 1.

######  Proposition 2.4.

Let $a^{*}>1$ then the interconnected network dynamics (14)-(16) is passive
from the input $u_{x}$ to the output $y_{H_{1}}$.

###### Proof.

Let $V$ be the candidate Lyapunov function for the interconnected system
represented in Fig. 1 such that

$\displaystyle V=V_{H_{1}}+V_{H_{2}}+V_{H_{3}}.$ (33)

Differentiating (33) and using (22), (24), (31) yields

$\displaystyle\dot{V}$
$\displaystyle=\dot{V}_{H_{1}}+\dot{V}_{H_{2}}+\dot{V}_{H_{3}},$
$\displaystyle\leq-\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}\|\dot{y}_{H_{1}}\|^{2}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +(1-a^{*})\lambda_{2}(L\otimes
I_{l})\|y_{H_{1}}\|^{2}+\dot{{y}}_{H_{1}}^{T}\dot{u}_{H_{1}}+\dot{u}_{H_{2}}^{T}\dot{y}_{H_{2}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\dot{u}_{H_{3}}^{T}\dot{y}_{H_{3}}$ (34)
$\displaystyle\leq-\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}\|\dot{y}_{H_{1}}\|^{2}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +(1-a^{*})\lambda_{2}(L\otimes
I_{l})\|y_{H_{1}}\|^{2}.$ (35)

Thus, the interconnected network dynamics (14)-(16) is passive, if $a^{*}>1$
strictly holds.

The following result establishes the boundedness of the trajectories of
(14)-(16).

######  Proposition 2.5.

The trajectories of (14)-(16) are bounded for any finite initial conditions.

###### Proof.

To show that the trajectories of (14)-(16) are bounded, consider the following
storage function:

$\displaystyle\bar{V}=\frac{1}{2}\|x-x^{*}\|^{2}+\frac{1}{2}\|\alpha-\alpha^{*}\|^{2}+\frac{1}{2}\|\theta-\theta^{*}\|^{2}+W.$
(36)

where $W$ is the storage function defined in (18). Differentiating (36) with
respect to time yields

$\displaystyle\dot{\bar{V}}$
$\displaystyle=-\nabla_{x}\bar{\mathcal{L}}(x,\alpha,\theta)^{T}(x-x^{*})+\nabla_{\alpha}\bar{\mathcal{L}}(x,\alpha,\theta)^{T}(\alpha-\alpha^{*})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+(\theta-\theta^{*})^{T}\nabla_{\theta}[\bar{\mathcal{L}}(x,\alpha,\theta)]^{+}_{\theta}+\dot{W},$
$\displaystyle\leq-\nabla_{x}\bar{\mathcal{L}}(x,\alpha,\theta)^{T}(x-x^{*})+\nabla_{\alpha}\bar{\mathcal{L}}(x,\alpha,\theta)^{T}(\alpha-\alpha^{*})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\sum_{i}\sum_{k}\sum_{j}(\theta^{ik}-(\theta^{ik}_{j})^{*})[g_{j}(x_{ik})]^{+}_{\theta^{ik}_{j}}+\dot{W},$
$\displaystyle\leq-\nabla_{x}\bar{\mathcal{L}}(x,\alpha,\theta)^{T}(x-x^{*})+\nabla_{\alpha}\bar{\mathcal{L}}(x,\alpha,\theta)^{T}(\alpha-\alpha^{*})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+(\theta-\theta^{*})^{T}g(x)+\dot{W}.$ (37)

Note that $(\theta^{ik}_{j}-(\theta^{ik}_{j})^{*})g_{j}(x_{ik})\geq 0,\forall
j\in\sigma_{i}(t)$ because $g_{j}(x_{ik})<0$ and $\theta^{ik}_{j}=0$ as
confirmed by (28). Using first order condition of convexity-concavity of the
Lagrangian function (5) and replacing $\dot{W}$ by right-hand side of (20),
(38) modifies to the following:

$\displaystyle\dot{\bar{V}}$
$\displaystyle\leq-[\bar{\mathcal{L}}(x,\alpha,\theta)-\bar{\mathcal{L}}(x^{*},\alpha,\theta)]+[\bar{\mathcal{L}}(x,\alpha,\theta)-\bar{\mathcal{L}}(x,\alpha^{*},\theta)]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+[\bar{\mathcal{L}}(x,\alpha,\theta)-\bar{\mathcal{L}}(x,\alpha,\theta^{*})]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+(1-a^{*})\dot{x}^{T}(L\otimes I_{l})\dot{x}+(1-a^{*})x^{T}(L\otimes I_{l})x.$
(38)

Since $(x^{*},\alpha^{*},\theta^{*})$ is the saddle-point of (5), with
$a^{*}>1$ yields the following

$\displaystyle\dot{\bar{V}}\leq 0.$ (39)

which is sufficient to ensure that the trajectories of (14)-(16) are bounded.
∎

In what follows, the asymptotic stability of the saddle-point solution of
(14)-(16) is established. To this end, the underlying networked dynamics is
represented as a hybrid system wherein $H_{1}$, $H_{2}$ are represented as
continuous-time dynamical systems and $H_{3}$ is represented as a system with
right-hand side discontinuity. The framework of LaSalle’s invariance principle
for hybrid dynamical systems (see, [27]) is stated below, which in our case
provides a useful result on the convergence of (14)-(16) to the saddle point
solution that satisfies (4).

######  Proposition 2.6.

Consider the hybrid networked dynamics (14)-(16) and let
$z=[x^{T},\alpha^{T},\theta^{T}]^{T}\in\mathcal{X}\subseteq\mathbb{R}^{ln(2+m^{ik}_{g})}$,
and $\Psi\subseteq\mathcal{X}$ be compact and positively invariant. Assuming
that the Lyapunov function $V$ defined in (33) is continuously differentiable
and $\dot{V}\leq 0$ along the trajectories of $z(t)\in\Psi$, every trajectory
in $\Psi$ converges to $\epsilon$, where $\epsilon\subset\Psi$ is a maximal
positive invariant set of $\Psi$ such that

  1. 1.

$\dot{V}=0$ for a fixed $\sigma$.

  2. 2.

$V_{k}(\theta^{ik}_{j}(\tau^{+}_{\sigma_{i}}))=V_{k}(\theta^{ik}_{j}(\tau^{-}_{\sigma_{i}}))$
for a switching instance $\tau$ between $\tau^{-}_{\sigma_{i}}$ and
$\tau^{+}_{\sigma_{i}}$.

∎

Proposition 2.6 gives the next result on the convergence of (14)-(16) to the
saddle point solution that satisfies the conditions in (4).

######  Proposition 2.7.

The hybrid network dynamics (14)-(16) converges to the saddle point solution
$x^{*},\alpha^{*},\theta^{*}$ satisfying (4).

###### Proof.

From Proposition 2.6, for a fixed $\sigma$, $\dot{V}=0$. Thus the primal as
well as dual dynamics in (14)-(16) converge to the saddle point solution
contained within the set $\epsilon$. If $g_{j}(x^{*}_{ik})<0$ then
$(\theta^{ik}_{j})^{*}=0$. However, if $g_{j}(x^{*}_{ik})>0$, then
$(\theta^{ik}_{j})^{*}$ will penalize the constraint violation by rising to a
large value. Since all trajectories are bounded, it contradicts the continuity
of $V$, thus $\dot{\theta}^{ik}_{j}=0$. To this end, the solutions of
(14)-(16) also satisfy the KKT conditions (4) and yield the saddle point
solution $(x^{*},\alpha^{*},\theta^{*})$. ∎

Choosing $a^{*}>1$ and using (12), (35) modifies to

$\displaystyle\dot{V}$
$\displaystyle\leq-\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}\|\dot{y}_{H_{1}}\|^{2}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +(1-a^{*})\lambda_{2}(L\otimes
I_{n})\|y_{H_{1}}\|^{2}\leq 0.$ (40)

######  Proposition 2.8.

The saddle point solution of (5) is asymptotically stable.

###### Proof.

The proof is straightforward from Proposition 2.4 and Proposition 2.7 and
(40). ∎

In the recent article [36] the global asymptotic stability of the primal-dual
dynamics is proved by using the Lyapunov function similar to that of the sum
of Krasovskii-type Lyapunov function (33) and the Lyapunov function defined in
(36). This result can be extended to the globally asymptotic stability of the
saddle-point of (5).

######  Remark 4.

Let
$\tilde{V}:\mathbb{R}^{ln}\times\mathbb{R}^{ln}\times\mathbb{R}^{lnm^{ik}_{g}}\rightarrow\mathbb{R}$
denote the candidate Lyapunov function for the ADPDD (14)-(16), given as sum
of the candidate Lyapunov functions (33) and (36) as follows:

$\displaystyle\tilde{V}=V+\bar{V}.$ (41)

If Assumption 1 holds then the trajectories of (14)-(16) converge to the
saddle-point $(x^{*},\alpha^{*},\theta^{*})$ which is globally asymptotically
stable. The proof of the Remark would be similar to proof the of [36, Theorem
5.1]. Hence it is omitted from here to avoid repetition.

With the global asymptotic stability of the proposed dynamics (14)-(16)
established, the subsequent section addresses its rate of convergence and its
comparison with the rate of convergence with the primal-dual dynamics without
adaptive weights.

###  2.5 Accelerated convergence using ADPDD

Let $\mathcal{A}\subseteq\mathbb{R}^{|\mathcal{E}|}_{>0}$ define the set of
coupling weights, and $|\mathcal{E}|$ define the cardinality of the edge set
$\mathcal{E}$. Then, in view of its definition, the Laplacian matrix $L\otimes
I_{l}$ is a parameter varying, real and symmetric matrix, which is
differentiable and uniformly continuous on $\mathcal{A}$. As a consequence,
the following hold:

######  Statement 1.

There exists $\Lambda>0$ such that the spectral norm $\|L\otimes
I_{l}\|<\Lambda,\forall a_{iq}\in\mathcal{A},\forall
q\in\mathcal{N}_{i},\forall i\in\mathcal{N}$.

######  Statement 2.

The gradient of $L\otimes I_{l}$ with respect to $a_{iq}$ is bounded above by
some scalar $\eta$, $\left\|\nabla L\otimes
I_{l}\right\|\leq\eta,a_{iq}\in\mathcal{A}$.

Let $L_{0}\otimes I_{l}$ be the Laplacian matrix of $\mathcal{G}$ whose
coupling weights are constant parameters, then $L_{0}\otimes I_{l}$ results in
a constant matrix.

######  Proposition 2.9.

If the coupling weights evolve according to the law (13), then the following
holds $\forall t>t_{0}$ :

$\displaystyle\lambda_{2}(L\otimes I_{l})>\lambda_{2}(L_{0}\otimes I_{l}).$
(42)

###### Proof.

If $E$ is the incidence matrix of the undirected graph $\mathcal{G}$, then the
Laplacian matrices $L_{0}$ and $L\otimes I_{l}$ can be written as:

$\displaystyle L_{0}\otimes I_{l}$ $\displaystyle=EE^{T}$ (43) $\displaystyle
L\otimes I_{l}$ $\displaystyle=EC(t)E^{T}$ (44)

where $C(t)$ is a diagonal matrix containing the coupling weights. To prove
(42), it is first proved that $x^{T}EC(t)E^{T}x\geq x^{T}EE^{T}x$.

$\displaystyle x^{T}EC(t)E^{T}x-x^{T}EE^{T}x$
$\displaystyle=x^{T}(EC(t)E^{T}-EE^{T})x$
$\displaystyle=x^{T}[E(C(t)-I)E^{T}]x.$ (45)

For an undirected graph $\mathcal{G}$, $a_{iq}(t_{0})\geq 1,\forall
q\in\mathcal{N}_{i},\forall i\in\mathcal{N}$. Then
$\forall(q,i)\in\mathcal{E}$, $C(t)\geq I_{|\mathcal{E}|}$. Hence,

$\displaystyle C(t)-I\geq 0,$ (46)

in fact, $C(t_{0})$ is a diagonal matrix with the coupling weights
$a_{iq}(t_{0})$, thus $C(t)\geq C(t_{0}),\forall t>t_{0}$. Thus from the above
reasoning, and (46),

$\displaystyle x^{T}EC(t)E^{T}\geq x^{T}EE^{T}x$

From (43) and (44),

$\displaystyle x^{T}L\otimes I_{l}x\geq x^{T}L_{0}\otimes I_{l}x.$ (47)

Let $\lambda_{i}$ be the $i^{th}$ eigenvalue in the ordered-pair of
eigenvalues represented below:

$\displaystyle\lambda_{2}\leq\ldots\leq\lambda_{i}\leq\ldots\leq\lambda_{n}.$
(48)

Then according to Courant-Fischer theorem [37],

$\displaystyle\lambda_{i}(EE^{T})$ $\displaystyle=\min_{x\neq 0,x\perp
v_{1}}\frac{x^{T}EE^{T}x}{x^{T}x}$ $\displaystyle\leq\min_{x\neq 0,x\perp
v_{1}}\frac{x^{T}EC(t)E^{T}x}{x^{T}x}=\lambda_{i}(EC(t)E^{T})$ (49)

where $v_{1}$ is the eigenvector (vector of all ones) corresponding to the
eigenvalue $\lambda_{1}=0$. Thus for $i=2$,

$\displaystyle\lambda_{2}(EE^{T})\leq\lambda_{2}(EC(t)E^{T})$
$\displaystyle\lambda_{2}(L_{0}\otimes I_{l})\leq\lambda_{2}(L\otimes I_{l})$
$\displaystyle\lambda_{2}(L\otimes I_{l})>\lambda_{2}(L_{0}\otimes
I_{l}),\forall t>t_{0}.$ (50)

∎

######  Proposition 2.10.

If the coupling weights evolve according to (13), then the following always
hold:

$\displaystyle\lambda_{2}(L\otimes I_{l})\leq\frac{\lambda_{n}(L\otimes
I_{l})}{\lambda_{n}(L_{0}\otimes I_{l})}\lambda_{2}(L_{0}\otimes I_{l}).$ (51)

###### Proof.

The proof simply follows from the inequality (48). Taking the ratio of the
ordered pair of eigenvalues of $L\otimes I_{l}$ and $L_{0}\otimes I_{l}$,
yields the following:

$\displaystyle\frac{\lambda_{2}(L\otimes I_{l})}{\lambda_{2}(L_{0}\otimes
I_{l})}\leq\frac{\lambda_{n}(L\otimes I_{l})}{\lambda_{n}(L_{0}\otimes
I_{l})},$ (52)

But, for $t>t_{0}$, the inequality (52) strictly holds. Thus

$\displaystyle\lambda_{2}(L\otimes I_{l})\leq\frac{\lambda_{n}(L\otimes
I_{l})}{\lambda_{n}(L_{0}\otimes I_{l})}\lambda_{2}(L_{0}\otimes I_{l}).$ (53)

∎

Proposition 2.9 and 2.10 can be further used to prove that the adaptive primal
dual dynamics has an accelerated, yet bounded convergence rate as compared to
the conventional primal-dual dynamics.

######  Proposition 2.11.

If the inequality (50) holds, then the primal dynamic in (14), under the
adaptive coupling law (13), achieves accelerated convergence.

###### Proof.

Below a timescale separation is enforced in the dynamics of the primal
subsystem $H_{1}$,

$\displaystyle\dot{x}$ $\displaystyle=-\nabla_{x}f(x)+u_{x}+u_{H_{1}},$ (54)
$\displaystyle\dot{a}_{iq}$ $\displaystyle=\epsilon
d_{iq}(e^{T}_{iq}e_{iq}+\dot{e}^{T}_{iq}\dot{e}_{iq}),\forall
i\in\mathcal{N},\forall q\in\mathcal{N}_{i},$ (55)

with $\epsilon<<1$ ensuring that the primal variable $x_{i}$ evolves faster
than the coupling weights $a_{iq}$.

The primal subsystem has two control inputs $u_{x}$, to study the primal
dynamics with respect to $u_{x}$ in (12), let us analyze the primal subsystem
$H_{1}$ when $u_{H_{1}}$ is at steady state or equal to $0$. With the
assumption that the coupling weight dynamics is much slower, the primal
dynamics is re-written as:

$\displaystyle\dot{x}$ $\displaystyle=-\nabla_{x}f(x)-(L\otimes
I_{l})x+u_{H_{1}},$ $\displaystyle=-F(x)+u_{H_{1}},$ (56)

where $F(x)=\nabla_{x}f(x)+(L\otimes I_{l})x$ .

Using Assumption 1 it can be proved that the primal dynamics (56) is strongly
monotone for all $x\in\mathbb{R}^{ln}$ by evaluating the Jacobian of $F(x)$,
i.e. $\nabla F(x)=\mathbb{H}+L\otimes I_{l}\geq\mu I$, where $\mu$ is the
modulus of convexity of $f$ (from Assumption 1. Since $\mu>0$, the Jacobian
$\nabla F(x)$ is symmetric and positive definite $\forall
x\in\mathbb{R}^{ln}$. It proves that $F(x)$ is strongly monotone by virtue of
which the primal dynamics (56) converges to the unique optimizer $x^{*}$. Thus
uniqueness of the primal optimizer remains invariant under the adaptive
coupling law (13).

The following result establishes the accelerated convergence of (56) with
respect to the unique optimizer $x^{*}$. Let $V_{H_{1}}$ define the Lyapunov
candidate function as given below:

$\displaystyle V_{H_{1}}=\frac{1}{2}\dot{x}^{T}\dot{x}.$ (57)

Differentiating $V_{H_{1}}$ with respect to time $t$,

$\displaystyle\dot{V}_{H_{1}}$
$\displaystyle\leq-(\lambda_{min}(\mathbb{H})+\lambda_{2}(L\otimes
I_{l}))\dot{x}^{T}\dot{x}$ (58) $\displaystyle\leq-\lambda_{m}V_{H_{1}}$ (59)

where $\lambda_{m}=2(\lambda_{min}(\mathbb{H})+\lambda_{2}(L\otimes I_{l}))$.
Therefor,

$\displaystyle V_{H_{1}}(x(t))\leq
V_{H_{1}}(x(t_{0}))\exp\\{-\lambda_{m}t\\},\forall t\geq t_{0}.$ (60)

or

$\displaystyle\|x-x^{*}\|\leq\sqrt{2V_{H_{1}}(x(t_{0}))}\exp\\{-0.5\lambda_{m}t\\},\forall
t\geq t_{0}.$ (61)

Further, since the primal-dual dynamics has a bounded convergence with respect
to the saddle point solution (see Proposition 39), using Assumption 3, and
Remark 4, every initial condition $x(t_{0})\in\mathbb{R}^{ln}$ approaches the
optimal solution $x^{*}$ faster than the usual. Thus the accelerated
convergence holds globally. Considering the upper bound on
$\lambda_{2}(L\otimes I_{l})$ as given in (53), let $\lambda_{2}(L\otimes
I_{l})=\frac{\lambda_{n}(L\otimes I_{l})}{\lambda_{n}(L_{0}\otimes
I_{l})}\lambda_{2}(L_{0}\otimes I_{l})$ and
$\lambda_{m_{0}}=2(\lambda_{min}(\mathbb{H})+\lambda_{2}(L_{0}\otimes
I_{l}))$. Then it is seen that
$\lambda_{m}=\lambda_{min}(\mathbb{H})+\frac{\lambda_{n}(L\otimes
I_{l})}{\lambda_{n}(L_{0}\otimes I_{l})}\lambda_{2}(L_{0}\otimes
I_{l})>>\lambda_{m_{0}}$. Hence proved. ∎

######  Remark 5.

The ADPDD (14)-(16) achieves accelerated convergence to the saddle point
solution that satisfies the KKT conditions (2).

###### Proof.

The proof follows from Proposition 2.7 and Proposition 2.11. The occurrence of
the primal optimizer $x^{*}$ and the dual optimizers is simultaneous.

Recall from (14)-(16) that $y_{H_{1}}=x,y_{H_{2}}=\alpha$, and
$y_{H_{3}}=\sum_{j=1}^{m^{ik}_{g}}\theta^{ik}_{j}\nabla_{x_{ik}}g_{j}(x_{ik}),\forall^{l}_{k=1};\forall^{m^{ik}_{g}}_{j=1}$.
$y_{H_{1}}=x^{*}\implies\dot{y}_{H_{1}}=0$ which also implies that
$\dot{\alpha}=0$ and $\dot{\theta}=0$. Thus accelerated convergence of (14) to
the primal optimizer $x^{*}$ implies the accelerated convergence of both (15)
and (16) converge to the dual optimizers $(\alpha^{*},\theta^{*})$. ∎

######  Remark 6.

Since the adaptive synchronization in the primal variables results in an
accelerated convergence to the primal optimizer $x^{*}$, the synchronization
error $x_{i}-x_{j},\forall j\in\mathcal{N}_{i},\forall i\in\mathcal{N}$
remains lower than that of DPDD for all time. This significantly reduces the
consensus constraint violation and thus the dual variable $\alpha$ pertaining
to the consensus constraints does not become unnecessarily large for the ADPDD
problem. The uniqueness of the primal optimizer for both Lagrangian functions
(5) and (LABEL:dlag1) owes to the strongly monotone property of $F(x)$ as
discussed in the Proposition 2.11. However, the dual dynamics (15) (which
solely a function of the adaptively coupled primal variables) is not strongly
monotone, which implies that the Lagrangian (5) and (LABEL:dlag1) are not
strongly concave with respect to $\alpha$. Thus, the dual dynamics (15) under
the effect of adaptively synchronized primal variables and the one without
adaptively synchronized primal variables settle to different equilibrium
states. This further indicates that there exists a unique primal optimizer for
both Lagrangian functions (5) and (LABEL:dlag1) but the same does not hold for
the dual optimizer $\alpha^{*}$. Since the dual variable $\theta^{*}$ pertains
to the local inequality constraints, it remains unaffected by the adaptive
synchronization in primal variables. Hence, the dual optimizer $\theta^{*}$ is
also unique for both (5) and (LABEL:dlag1).

The convergence rate of the distributed primal-dual dynamics is improved under
the influence of adaptive synchronization. However, it may adversely affect
the robustness of the proposed dynamics. Thus, there arises a necessity to
quantify the robustness of the proposed dynamics with respect to the rate of
convergence. The analysis presented below obtains a relation between the
convergence rate of the proposed dynamics and its $L_{2}$-gain.

###  2.6 Robustness analysis of the network dynamics with respect to the
exogenous inputs

Before proceeding with the robustness analysis of this section, it is worth
noting the following remark on robustness property of the passive dynamical
systems.

######  Remark 7.

From the inequalities (22), (25), and (32), it is apparent that the
interconnected network dynamics comprising (14)-(16) is passive, and
inherently robust to the perturbations arising in the primal and dual
variables [see, Proposition 4.3.1, Remark 4.3.3 of [30]].

Remark 7 states the qualitative behavior of the proposed dynamics with respect
to the notion of robustness. In the following, the robustness of the proposed
dynamics against exogenous inputs is quantified in terms of the $L_{2}$-gain.

Consider without loss of generality, the new inputs to (14)-(16) as

$\displaystyle\begin{aligned} H_{1}:\tilde{u}_{H_{1}}=u_{H_{1}}+\bigtriangleup
u_{H_{1}},\\\ H_{2}:\tilde{u}_{H_{2}}=u_{H_{2}}+\bigtriangleup u_{H_{2}},\\\
H_{3}:\tilde{u}_{H_{3}}=u_{H_{3}}+\bigtriangleup u_{H_{3}},\end{aligned}$ (62)

respectively, where $\bigtriangleup u_{(.)}$ corresponds to the perturbations
in the input $u_{(.)}\in\mathbb{R}^{ln}$. As discussed in [38],
$\bigtriangleup u_{(.)}$ represent additive uncertainties or disturbances such
as the numerical error accumulated in the corresponding variables. In what
follows, the robustness of the ADPDD is quantified using $L_{2}$-gain analysis
of dynamical systems. Let
${\tilde{u}}=[{\tilde{u}}^{T}_{H_{1}},{\tilde{u}}^{T}_{H_{2}},{\tilde{u}}^{T}_{H_{3}}]^{T}$
and ${y}=[{y}^{T}_{H_{1}},{y}^{T}_{H_{2}},{y}^{T}_{H_{3}}]^{T}$.

######  Proposition 2.12.

The interconnected network dynamics (14), (15), and (16) with $a_{iq}$ updated
according to (13), remains $L_{2}$ stable with the $L_{2}$-gain,
$\gamma\leq\frac{1}{\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})}$, if $a^{*}>1$ strictly holds.

###### Proof.

Replacing the inputs in (14)-(16) by the new ones as defined in (62), the time
differential of the Lyapunov function (33) modifies to the following:

$\displaystyle\dot{V}$
$\displaystyle\leq-\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}\|\dot{y}_{H_{1}}\|^{2}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +(1-a^{*})\lambda_{2}(L\otimes
I_{n})\|y_{H_{1}}\|^{2}+\dot{{y}}^{T}_{H_{1}}\dot{\tilde{u}}_{H_{1}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\dot{y}_{H_{2}}^{T}\dot{\tilde{u}}_{H_{2}}+\dot{y}_{H_{3}}^{T}\dot{\tilde{u}}_{H_{3}}.$
(63)

Acknowledging that $y_{H_{1}}=x$ and using (12) in (63) further yields

$\displaystyle\dot{V}$
$\displaystyle\leq-\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}\|\dot{y}_{H_{1}}\|^{2}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-(1-a^{*})y_{H_{1}}^{T}u_{x}+\dot{\tilde{u}}^{T}\dot{y},$ (64)

where $\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes I_{l})>0$ since
$\mathbb{H}$ is positive definite. With $a^{*}>1$, the $L_{2}$-gain of the
interconnected network dynamics, from the port input $\dot{\tilde{u}}$ to the
port output $\dot{y}$ can be calculated by setting $u_{x}$ to $0$. From
inequality (64), the map from the input $\dot{\tilde{u}}$ to the output
$\dot{y}$ remains finite $L_{2}$-gain stable around the saddle point
$x^{*},\alpha^{*},\theta^{*}$, when the corresponding $L_{2}$-gain, satisfies

$\displaystyle\gamma\leq\frac{1}{\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\lambda_{2}(L\otimes
I_{l})\big{)}}.$ (65)

∎

The inequality (65), clearly indicates that the $L_{2}$ gain corresponding to
the adaptive distributed primal-dual dynamics reduces in margin as compared to
the $L_{2}$ gain corresponding to the distributed primal-dual dynamics
(without adaptive synchronization). Using (53), one can obtain the following
expression for the $L_{2}$-gain in the worst case:

$\displaystyle\underline{\gamma}=\frac{1}{\big{(}\lambda_{min}({\mathbb{H}})+a^{*}\frac{\lambda_{n}(L\otimes
I_{l})}{\lambda_{n}(L_{0}\otimes I_{l})}\lambda_{2}(L_{0}\otimes
I_{l})\big{)}}.$ (66)

Comparing (65) and (66), it can be found out that the $L_{2}$-gain for the
ADPDD has a reduced margin than that of the DPDD. Thus the algorithm calls for
trade-off between the robustness and the accelerated convergence of the
proposed dynamics. While the adaptive synchronization improves the rate of
convergence of the primal-dual dynamics, it simultaneously degrades the
robustness of the proposed algorithm wherein the worst-case $L_{2}$-gain is
quantified by $\underline{\gamma}(<\gamma)$ in (66).

##  3 Applications and Numerical Examples

This section discusses the application of the proposed dynamics to the
distributed optimization problems concerning least squares[7, 39] and support
vector machines[40]. These problems are solved online over a network of
wireless sensors or computing devices, in such premises the rate of
convergence is a vital factor. In the following, the proposed dynamics
(14)-(16) is employed to solve the distributed least squares[41] and
distributed support vector machines[5, 6] problems.

###  3.1 Distributed Least Squares

Distributed least squares problems have been widely studied over recent
years[42, 43, 12]. These techniques have found applications in parameter
estimation over wireless sensor networks [44], estimation of electro-
mechanical oscillation modes of large power system networks [41, 45] etc. Each
agent in the network is given a task to simultaneously and iteratively compute
the same least squares solution to the linear equation $Ax=b$ where
$A\in\mathbb{R}^{r_{1}\times r_{2}}$ with $r_{1}>r_{2}$ and
$b\in\mathbb{R}^{r_{1}\times 1}$.

Formally, the least squares problem is defined as given below[46]:

$\displaystyle\min_{x}\frac{1}{2}\|Ax-b\|^{2}.$ (67)

####  3.1.1 Data partitioning

It is assumed that each agent in the network adheres to $n_{r}=r_{1}/n$
consecutive rows of $A$ and $b$. For the sake of simplicity, equal
partitioning of the rows of $A$ is considered. However, the proposed approach
would hold even if the partitioning is uneven.

$\displaystyle A=\begin{bmatrix}A_{1}\\\ A_{2}\\\ \vdots\\\
A_{n}\end{bmatrix},b=\begin{bmatrix}b_{1}\\\ b_{2}\\\ \vdots\\\
b_{n}\end{bmatrix}.$ (68)

where $A_{i}\in\mathbb{R}^{n_{r}\times{l}}$ and
$A_{i}\in\mathbb{R}^{n_{r}\times 1}$.

####  3.1.2 Distributed formulation of least squares problem

The consensus-based distributed optimization formulation of (67) would require
the local estimates $x_{1},x_{2},\ldots,x_{n}$ to reach consensus on the
global optimizer $x^{*}$. With data partitioning as defined above, the
distributed version of the least squares problem (67)[41] is defined as

$\displaystyle\min_{x}\sum_{i=1}^{n}\frac{1}{2}\|A_{i}x-b_{i}\|^{2}$
$\displaystyle\mathrm{subject\leavevmode\nobreak\ to}\leavevmode\nobreak\
x_{i}=x_{j},\forall j\in\mathcal{N}_{i}.$ (69)

####  3.1.3 Solution to the distributed least squares problem (69) using ADPDD

The Lagrangian problem corresponding to (69) can be defined as

$\displaystyle\mathcal{L}(x,\alpha)=\sum_{i=1}^{n}\|A_{i}x-b_{i}\|^{2}+\alpha^{T}L\otimes
I_{l}x+x^{T}L\otimes I_{l}x.$ (70)

Similarly to (7), the proposed dynamics can be derived from (70) as given
below:

$\displaystyle H_{1}$
$\displaystyle:\begin{cases}\dot{x}&=-A^{T}(Ax-b)-(L\otimes
I_{l})x+u_{H_{1}},\\\
\dot{a}_{iq}&=d_{iq}(e^{T}_{iq}e_{iq}+\dot{e}^{T}_{iq}\dot{e}_{iq}),\forall
i\in\mathcal{N},\forall q\in\mathcal{N}_{i}\\\ y_{H_{1}}&=x.\end{cases}$ (71)
$\displaystyle H_{2}$ $\displaystyle:\begin{cases}\dot{\alpha}&=u_{H_{2}},\\\
y_{H_{2}}&=\alpha.\end{cases}$ (72)

where $u_{H_{1}}=-(L\otimes I_{l})y_{H_{2}}$ and $u_{H_{2}}=(L\otimes
I_{l})y_{H_{1}}$.

####  3.1.4 Simulations

The simulation parameters are randomly generated matrix
$A\in\mathbb{R}^{100\times 80}$ and vector $b\in\mathbb{R}^{100\times 1}$. The
network with a cyclic graph topology is assumed to comprise of $\mathrm{4}$
agents wherein each agent holds $A_{i}\in\mathbb{R}^{25\times 80}$ component
of $A$ as well as the respective $b_{i}$. Each agent in the network computes
$x\in\mathbb{R}^{80}$ local estimates and reaches consensus over the global
solution $x^{*}$ as shown in the Fig. 2. The simulations were carried out
using $d_{iq}=0.1$, the rate of convergence of (71) is compared with that of
the non-adaptive version of the distributed primal-dual dynamics employed to
solve the problem (69). The rate of convergence is significantly improved as
shown in the Fig. 3. The global solution to (69) is also compared with the
solution of the least square solver $\mathrm{lsqlin}$ in $\mathrm{MATLAB}$.
The global optimizer $x^{*}_{1}=x^{*}_{2}=x^{*}_{3}=x^{*}_{4}$ obtained using
the proposed algorithm coincides with the optimal solution $x^{*}$ obtained
using $\mathrm{lsqlin}$ as shown in the Fig. 4.

Figure 2: Convergence of (71) to the global solution $x^{*}$. Figure 3:
Eigenvalue comparison of $L\otimes I_{l}$ and $L_{0}\otimes I_{l}$ for the
problem (71). Figure 4: Comparison of the global optimizers of (69) with the
optimal solution of the $\mathrm{lsqlin}$ solver.

###  3.2 Quadratic-inequality Constrained Distributed Least Squares

A box-constrained linear least squares problem is the one in which the upper
and lower bounds on the estimated values are incorporated to handle
limitations of the physical system. These methods are studied with
applications to GPS positioning [47], geodesic applications [48, 49, 50] etc.
The box-constrained least squares problem is generally defined as follows:

$\displaystyle\min_{x}\|Ax-b\|^{2},\mathrm{subject\leavevmode\nobreak\
to}\leavevmode\nobreak\ x_{l}\leq x\leq x_{u},$ (73)

where $x_{l}$ and $x_{u}$ are the upper and lower bounds of the variable $x$.
It is known that a quadratic constraint formulation of the box constrained
least square problem is an efficient approach to obtain the optimal solution
of (73) [39]. The quadratic-constrained equivalent formulation of the box-
constrained least square problem (74) is given as:

$\displaystyle\min_{x}\|Ax-b\|^{2},\mathrm{subject\leavevmode\nobreak\
to}\leavevmode\nobreak\
(x_{i}-\bar{x}_{i})^{2}\leq\rho^{2}_{i},\forall^{n}_{i=1}.$ (74)

where $\bar{x}_{i}$ is the midpoint of the interval $[x_{l},x_{u}]$. It is
computed as $\bar{x}_{i}=(x_{l}+x_{u})/2$ with $\rho_{i}=(x_{u}-x_{l})/2$.

A distributed framework for the quadratic-constrained least squares problem
(74) can be obtained as:

$\displaystyle\min_{x}\sum_{i=1}^{n}\|A_{i}x-b_{i}\|^{2},$
$\displaystyle\mathrm{subject\leavevmode\nobreak\ to}\leavevmode\nobreak\
(x_{ik}-\bar{x}_{ik})^{2}\leq\rho^{2}_{ik},\forall^{l}_{k=1},\forall^{n}_{i=1}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
x_{i}=x_{j},\forall j\in\mathcal{N}_{i}.$ (75)

The ADPDD formulation of the problem (75) is similar to that of the proposed
dynamics (14)-(16). Hence, it is omitted to avoid repetition of the equations.

####  3.2.1 Simulations

For the sake of simplicity and readability of the simulation results, a small
problem of the form (75) is taken as a proof of concept with the parameters
$A\in\mathbb{R}^{20\times 4}$ and $b\in\mathbb{R}^{20\times 1}$. A network
with a cyclic graph topology containing $\mathrm{4}$ agents is considered
wherein each agent holds on to $A_{i}\in\mathbb{R}^{5\times 4}$ component of
the matrix $A$. All agents iteratively reach the global consensus of the
optimizer value $x^{*}$ with $d_{iq}=2$, as shown in the Fig. 5. It can be
observed that the trajectories $x_{1},x_{2},x_{3}$, and $x_{4}$ synchronize to
respective common trajectories at around $t\approxeq 0.03\leavevmode\nobreak\
\mathrm{seconds}$. The result is also compared with the solution of
$\mathrm{lsqlin}$ and it can be seen from the Fig. 6 that the global optimizer
of (75) coincides with the solution obtained using $\mathrm{lsqlin}$. The
accelerated convergence of the proposed algorithm employed to solve (75) is
evident from the Fig. 7.

######  Remark 8.

A strong synchronization between the trajectories of the agents imply
guaranteed convergence to the global optimizer under sparse communication
events. It is also indicative of the fact that the communication between the
agents need not be periodic. The proposed algorithm can be augmented with the
event-triggered control framework.

Figure 5: Convergence of the solutions of (75) to the global optimizer
$x^{*}$. Figure 6: Comparison of the global optimizers of (75) with the
optimal solution of the $\mathrm{lsqlin}$ solver. Figure 7: Eigenvalue
comparison of $L\otimes I_{l}$ and $L_{0}\otimes I_{l}$ for the problem (71).

###  3.3 Distributed Support Vector Machines

Support vector machines (SVMs) are supervised learning based paradigms in the
machine learning domain, used for classification and regression analysis on
raw data, (see [40]). For applications with a huge amount of data, there are
often limitations with respect to bandwidth requirement, data storage and
processing capability of the computing machine, response time, etc. As it
turns out, a single computing machine is inefficient in dealing with the SVM
algorithm with large datasets. Distributed versions of support vector machines
have been proposed as an alternative method to overcome these limitations, as
discussed in [5, 6]. With the aim of enabling accelerated convergence to the
optimal solution, the distributed SVM problem is formulated in terms of the
adaptive primal-dual dynamics. However, due to the complexity involved with
simulations of large-scale SVM problems, the present work only considers the
mathematical formulation and does not provide the simulation results for the
same.

A problem formulation of the support vector machines for the case of non-
separable data is given below:

$\displaystyle\begin{aligned} &\min\leavevmode\nobreak\
\frac{1}{2}\|w\|^{2}+pC\sum_{j=1}^{m}\xi_{j}\\\
&\mathrm{s.t.}\leavevmode\nobreak\ y_{j}(w^{T}x_{j}+b)\geq
1-\xi_{j},\forall^{m}_{j=1},\end{aligned}$ (76)

where $\frac{1}{\|w\|}$ is the margin that separates positive and negative
observations, $(x_{j},y_{j})\in S$ is a paired observation sample, and $w,b$
are weight and bias variables, respectively. $1-\xi_{j}-y_{j}(w^{T}x_{j}+b)$
is called as a hinge loss function. C is used to trade off the sum over all
slack variables $\xi$ against the size of the margin. $p>0$ is the scaling
factor.

####  3.3.1 Data Partitioning

It is assumed that the set of observations $S$ is horizontally partitioned and
distributed among computing nodes in $\mathcal{G}(\mathcal{N},\mathcal{E})$
[6], where now $\mathcal{N}=\\{1,\ldots,n\\}$ represents the computing nodes
and the set of edges $\mathcal{E}$ describes communication links between them.
Assuming that the graph is connected and enabling only one-hop neighborhood
communication, each node $i$ communicates with its neighbors belonging to
$\mathcal{N}_{i}$. Each node $i\in\mathcal{N}$ stores a sample set of labeled
observations, denoted by
$S_{i}=\\{(x_{i1},y_{i1}),\ldots,(x_{im_{i}},y_{im_{i}})\\}$. Note that:

  1. 1.

$S_{i}$ is a set of labeled observations allocated to $i^{th}$ computing node,
$S_{i}\in S$, where $S$ is a superset of the labeled observations.

  2. 2.

$x_{i}\in\mathbb{R}^{m_{i}\times 1}$.

  3. 3.

$y_{ij}\in\\{-1,+1\\}$ is a class label.

In what follows, an adaptive primal-dual dynamics based formulation of
distributed support vector machines is provided.

####  3.3.2 ADPDD formulation of Distributed Support Vector Machines

A distributed version of the support vector machines problem (76) is
formulated as given below (see, [5]):

$\displaystyle\begin{aligned} \min\leavevmode\nobreak\
&\frac{1}{2}\sum_{i=1}^{n}\|w_{i}\|^{2}+pC\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}\xi_{ij}\\\
\mathrm{s.t.}&\leavevmode\nobreak\ y_{ij}(w_{i}x_{ij}+b_{i})\geq
1-\xi_{ij},\xi_{ij}\geq 0,\forall i\in\mathcal{N},\forall^{m_{i}}_{j=1},\\\
&\leavevmode\nobreak\ w_{i}=w_{q},b_{i}=b_{q},\leavevmode\nobreak\ \forall
i\in\mathcal{N},q\in\mathcal{N}_{i}.\end{aligned}$ (77)

The objective function in (77) is a differentiable $(C^{2})$ and strongly
convex in $w$. The decision (primal) variables are $w,b\in\mathbb{R}^{m}$,
where $w_{i}=w_{q},b_{i}=b_{q}$ are the consensus constraints with $q$ as a
neighbor of $i$ if and only if $q\in\mathcal{N}_{i}$. Let
$h_{ij}(\xi_{ij},w_{i},b_{i})=1-\xi_{ij}-y_{ij}(w_{i}x_{ij}+b_{i})$.

The Lagrangian formulation of the problem (77) is given by

$\displaystyle\begin{aligned}
\mathcal{L}(w,b,\xi,\theta,\mu,\alpha,\beta)=&\frac{1}{2}\|w\|^{2}+pC\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}\xi_{ij}\\\
&+\alpha^{T}Lw+\beta^{T}Lb\\\
&+\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}\theta_{ij}h_{ij}(\xi_{ij},w_{i},b_{i})\\\
&+\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}\mu_{ij}\xi_{ij}+\frac{1}{2}w^{T}Lw+\frac{1}{2}b^{T}Lb,\end{aligned}$
(78)

where $\theta_{ij},\mu_{ij}$ are the Lagrange multipliers associated with
inequality constraints $h_{ij}(\xi_{ij},w_{i},b_{i})$ and $\xi_{ij}\geq 0$, of
$i^{th}$ computing node, and $\alpha_{i},\beta_{i}$ are the Lagrange
multipliers associated with coupling constraints of $i^{th}$ and
$q^{th},\forall q\in\mathcal{N}_{i}$ nodes. $L$ is the Laplacian matrix of the
undirected graph $G$.

Let $z=[w^{T},b^{T}]^{T}$ (with $z_{i}=[w_{i},b_{i}]$, $l=2$) then,
$e_{iq}=z_{i}-z_{q}$. The interconnected network dynamics for the distributed
support vector machines problem (77) is represented as follows:

$\displaystyle H_{1}$
$\displaystyle:\begin{cases}\dot{w}&=-w-Lw-L\alpha-\zeta,\\\
\dot{b}&=-Lb-L\beta-\eta,\\\
\dot{a}_{iq}&=d_{iq}(e^{T}_{iq}e_{iq}+\dot{e}^{T}_{iq}\dot{e}_{iq}),\forall
i\in\mathcal{N},\forall q\in\mathcal{N}_{i},\\\ u_{H_{1}}&=-(L\otimes
I_{l})y_{H_{2}}-y_{H_{3}},\\\ y_{H_{1}}&=z.\end{cases}$ (79)

The subsystem $H_{2}$ contains only consensus-dual variables, with $u_{H_{2}}$
and $y_{H_{2}}$ as its input and output respectively, as given below:

$\displaystyle H_{2}$ $\displaystyle:\begin{cases}\dot{\alpha}&=Lw,\\\
\dot{\beta}&=Lb,\\\ u_{H_{2}}&=(L\otimes I_{l})y_{H_{1}},\\\
y_{H_{2}}&=[\alpha^{T},\beta^{T}]^{T}.\end{cases}$ (80)

The subsystem $H_{3}$ contains the slack variable, and the dual variables
corresponding to the inequality constraints, with $u_{H_{3}}$ and $y_{H_{3}}$
as its input and output respectively, as given below:

$\displaystyle H_{3}$
$\displaystyle:\begin{cases}\dot{\theta}_{ij}&=[h_{ij}(\xi_{ij},w_{i},b_{i})]^{+}_{\theta_{ij}}\forall^{m_{i}}_{j=1},\forall^{n}_{i=1},\\\
\dot{\mu}_{ij}&=[\xi_{ij}]^{+}_{\mu_{ij}}\forall^{m_{i}}_{j=1},\forall^{n}_{i=1},\\\
\dot{\xi}_{ij}&=[-pC-\mu_{ij}+\theta_{ij}]^{+}_{\xi_{ij}}\forall^{m_{i}}_{j=1},\forall^{n}_{i=1},\\\
u_{H_{3}}&=y_{H_{1}},\\\ y_{H_{3}}&=[\zeta^{T},\eta^{T}]^{T},\end{cases}$ (81)

where $\zeta,\eta,\mu\in\mathbb{R}^{n}$, and
$\zeta_{i}=\sum_{j=1}^{m_{i}}\theta_{ij}(-y_{ij}x_{ij})$ with
$\eta_{i}=\sum_{j=1}^{m_{i}}\theta_{ij}(-y_{ij})$.

Thus, the proposed dynamics can be implemented for solving the distributed
support vector machines problem (77) as shown in (79)-(81). The solution of
the underlying dynamics will correspond to the saddle-point solution of (78),
wherein the primal solution is the optimal solution of (77).

In the following, two different formulations of (2) are considered and the
results of the proposed dynamics are compared with that of the non-adaptive
version of the distributed primal-dual dynamics.

###  3.4 Numerical Example 1

Consider the following distributed optimization problem consisting
$\mathrm{3}$ agents having more than one variable and convex inequality
constraints.

$\displaystyle\begin{aligned} &\min_{x\in\mathbb{R}^{6}}\leavevmode\nobreak\
\sum_{i=1}^{3}f_{i}(x_{i}),\\\ &\mathrm{subject\leavevmode\nobreak\
to}\leavevmode\nobreak\ x_{i}=x_{q},\leavevmode\nobreak\ g_{i}(x_{i})\leq
0,\leavevmode\nobreak\ \forall i,q\in\mathcal{N}.\end{aligned}$ (82)

where the objective function associated with each agent is given below

$\displaystyle f_{1}(x_{1})=(x_{11}-x_{12})^{2}+(x_{11}-1)^{2},$ (83)
$\displaystyle f_{2}(x_{2})=\frac{1}{3}(x_{21}-x_{22})^{2}+(x_{21}-3)^{2},$
(84) $\displaystyle
f_{3}(x_{3})=\frac{1}{3}(x_{31}-x_{32})^{2}+(x_{31}-6)^{2},$ (85)

with the following local inequality constraints

$\displaystyle g_{1}(x_{1})=6x_{11}^{2}+3x_{12}^{2}-11,$ (86) $\displaystyle
g_{2}(x_{2})=7x_{11}^{2}+11x_{12}^{2}-7,$ (87) $\displaystyle
g_{3}(x_{3})=2x_{11}^{2}+9x_{12}^{2}-20.$ (88)

The graph connectivity is assumed to be as follows: $\mathcal{N}_{1}=1$,
$\mathcal{N}_{2}=2$, and $\mathcal{N}_{3}=1$. The ADPDD algorithm is employed
to solve the problem (82), and the corresponding trajectories are shown in
Fig. 8. The primal optimizers are $\mathrm{(1.4099,0.8966)}$. Besides that, in
Fig. 9 the steady state eigenvalues of $L\otimes I_{l}$ are plotted along with
the eigenvalues of $L_{0}\otimes I_{l}$. The eigenvalue $\lambda_{2}(L\otimes
I_{l})$ at steady state is equal to $\mathrm{192.8079}$ as compared to the
eigenvalue corresponding to a non-adaptive DPDD, $\lambda_{2}(L_{0}\otimes
I_{l})=1$. From Proposition 2.9 and Proposition 2.11, it can be seen that the
adaptive synchronization has sought to increase the rate of convergence of the
ADPDD.

Figure 8: Convergence of the ADPDD (Example 1). Figure 9: Eigenvalues of
$L_{0}\otimes I_{l}$ and $L\otimes I_{l}$ (Example 1).

###  3.5 Numerical Example 2

In this subsection, the local inequality constraints associated with each
agent are relaxed and the following optimization problem is considered on a
random graph with $\mathrm{10}$ agents as shown in Fig. 10. Note that the
degree of each agent is selected randomly.

$\displaystyle\begin{aligned} &\min_{x\in\mathbb{R}^{10}}\leavevmode\nobreak\
\sum_{i=1}^{10}f_{i}(x_{i}),\\\ &\mathrm{subject\leavevmode\nobreak\
to}\leavevmode\nobreak\ x_{i}=x_{q},\leavevmode\nobreak\ \forall
i,q\in\mathcal{N}.\end{aligned}$ (89)

with a randomly generated Hessian
$\mathbb{H}=\mathrm{diag}([136,439,355,298,302,350,327,398,353,294])$. The
proposed dynamics is employed to solve (89), first considering $d_{iq}=0.001$
and then $d_{iq}=0.01$. Fig. 11 and Fig. 12 correspond to the case of
$d_{iq}=0.001$ while Fig. 13 and Fig. 14 correspond to the case of
$d_{iq}=0.01$. It can be seen that for the latter case the convergence is much
faster. This owes to the difference between the resulting eigenvalues, i.e.,
for the case of $d_{iq}=0.001$, the second smallest eigenvalue
$\lambda_{2}(L\otimes I_{l})$ yields to be $\mathrm{10.72}$ whereas the same
for the case of $d_{iq}=0.01$ increases to $\mathrm{33310}$. The eigenvalue
results for both values of $d_{iq}$ are shown in the Fig. 12 and the Fig. 14.

Figure 10: A random graph $\mathcal{G}$ containing $\mathrm{10}$ nodes
(Example 2). Figure 11: Convergence of the ADPDD ($d_{iq}=0.001$) (Example 2).
Figure 12: Eigenvalues of $L_{0}\otimes I_{l}$ and $L\otimes I_{l}$
($d_{iq}=0.001$) (Example 2). Figure 13: Convergence of the ADPDD
($d_{iq}=0.01$) (Example 2). Figure 14: Eigenvalues of $L_{0}\otimes I_{l}$
and $L\otimes I_{l}$ ($d_{iq}=0.01$) (Example 2).

##  4 Conclusions

In this paper, an adaptive distributed primal-dual dynamics is proposed to
solve inequality and consensus constrained distributed optimization problems.
The adaptive synchronization of the primal variables is brought into play by
allowing the coupling weights to update according to the difference between
the local trajectories (trajectories belonging to the neighboring nodes or
agents) as well as the difference between the rate of change of the local
trajectories respectively. It is proved that the proposed dynamics represents
a network of feedback-interconnected passive dynamical systems which are
asymptotically stable. Further, by allowing a time-scale separation between
the adaptive coupling law and primal dynamics, stronger convergence bounds for
the primal dynamic are derived, and it is proved that the adaptively coupled
primal dynamics converges to the unique primal optimizer.

The performance of the proposed dynamics is quantified in terms of the induced
$L_{2}$-gain from the disturbance input to the output. The effect of adaptive
synchronization on the $L_{2}$-gain is discussed and it is established that
the adaptive distributed primal-dual dynamics are comparatively less robust to
the exogenous input disturbances than the distributed primal-dual dynamics. On
the other hand, the analysis also revealed that in order to achieve
accelerated convergence to the saddle-point solution, the proposed algorithm
must call for a trade-off between the convergence and the robustness
parameters.

The future scope of the work will be directed towards improving the rate of
convergence of the proposed dynamics without compromising its robustness
properties. Its applications to large-scale distributed optimization problems
such as distributed support vector machines [6], distributed least squares
[12] etc will be considered.

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