# A Proximal Linearization-based Decentralized Method for Nonconvex Problems
with Nonlinear Constraints

Yu Yang, Guoqiang Hu, and Costas J. Spanos This work was supported by the
Republic of Singapore’s National Research Foundation through a grant to the
Berkeley Education Alliance for Research in Singapore (BEARS) for the
Singapore-Berkeley Building Efficiency and Sustainability in the Tropics
(SinBerBEST) Program. BEARS has been established by the University of
California, Berkeley as a center for intellectual excellence in research and
education in Singapore.Yu Yang is with SinBerBEST, Berkeley Education Alliance
for Research in Singapore, Singapore 138602 e-mail: (yu.yang@bears-
berkeley.sg).Guoqiang Hu is with the School of Electrical and Electronic
Engineering, Nanyang Technological University, Singapore, 639798 e-mail:
(gqhu@ntu.edu.sg).Costas J. Spanos is with the Department of Electrical
Engineering and Computer Sciences, University of California, Berkeley, CA,
94720 USA email: (spanos@berkeley.edu).

###### Abstract

Decentralized optimization for non-convex problems are now demanding by many
emerging applications (e.g., smart grids, smart building, etc.). Though
dramatic progress has been achieved in convex problems, the results for non-
convex cases, especially with non-linear constraints, are still largely
unexplored. This is mainly due to the challenges imposed by the non-linearity
and non-convexity, which makes establishing the convergence conditions
bewildered. This paper investigates decentralized optimization for a class of
structured non-convex problems characterized by: (i) nonconvex global
objective function (possibly nonsmooth) and (ii) coupled nonlinear constraints
and local bounded convex constraints w.r.t. the agents. For such problems, a
decentralized approach called Proximal Linearization-based Decentralized
Method (PLDM) is proposed. Different from the traditional (augmented)
Lagrangian-based methods which usually require the exact (local) optima at
each iteration, the proposed method leverages a proximal linearization-based
technique to update the decision variables iteratively, which makes it
computationally efficient and viable for the non-linear cases. Under some
standard conditions, the PLDM’s global convergence and local convergence rate
to the $\epsilon$-critical points are studied based on the Kurdyka-Łojasiewicz
property which holds for most analytical functions. Finally, the performance
and efficacy of the method are illustrated through a numerical example and an
application to multi-zone heating, ventilation and air-conditioning (HVAC)
control.

###### Index Terms:

decentralized optimization, nonconvex problems, coupled nonlinear constraints,
proximal linearization, augmented Lagrangian-based method.

##  I Introduction

The Internet of Things (IoT) and smart sensors have promoted the emergence of
large-scale networked systems, which are eager for scalable and efficient
computation. Consequently, the past decades have witnessed the revived
interests and dramatic progress in decentralized optimization, especially for
convex problems (see for examples, [1, 2, 3]). Nevertheless, the presence of
complex dynamic systems and big data (see [5, 6] and references therein)
nowadays requests for decentralized approaches that work for non-convex and
non-linear context, which still remains an open question and has not been well
established. This is mainly due to the intrinsic challenges imposed by the
non-linearity and non-convexity, which lead to the NP-hard complexity as the
existence of multiple optima, the lack of structure properties to guarantee
the presence of optima (e.g., convexity and strong convexity), and the deficit
of conditions to investigate convergence (e.g., sufficient global optimality
conditions).

###  I-A Related Works

In general, the available results on (decentralized) non-convex optimization
can be categorized based on the problem structures: (i) unconstrained non-
convex problems, (ii) linearly constrained non-convex problem, and (iii) non-
linearly constrained non-convex problems. Unless specified, we refer the
constraints above to the coupled constraints among different agents. For
unconstrained problems, the most straightforward approaches are (sub-)gradient
methods, whose convergence has been early established for convex problems
(e.g., see [8]). Recently, their extension [9] and variations, such as Frank-
Wolf algorithm [10], proximal gradient method [11], have been discussed for
non-convex cases. The broad results are that some local optima (i.e., critical
points) can be approached with diminishing step-sizes. Generally, the above
works are mainly focused on investigating the convergence of such methods for
non-convex situations rather than achieving decentralized computation. On that
basis, some decentralized paradigms by combining (proximal) gradient-based
methods with alternating minimization technique (i.e, Gauss-Seidel update)
have been proposed, which include proximal alternating linearized minimization
(PALM) [12, 13], inexact proximal gradient methods (IPG) [14] and some
variations [15, 17].

As the noteworthy performance of the methods of multipliers (MMs) or augmented
Lagrangian-based methods, such as alternating direction multiplier method
(ADMM) [3], has been thoroughly observed and understood in tackling coupled
(linear) constraints in convex context, their extension to linearly coupled
non-convex problems seems natural. The recent years have witnessed the
widespread discussions on decentralized optimization for non-convex problems
subject to coupled linear constraints under the Lagrangian-based framework,
especially ADMM and its variations (see [18, 19, 20] and the references
therein). In principle, ADMM can leverage the fast convergence feature of
augmented Lagrangian methods and the separable structure of dual decomposition
based on the alternating minimization technique (i.e, Gauss-Seidel update).
Nevertheless, in contrast to convex cases (see [18, 19, 20]), it’s not
straightforward or trivial to achieve their performance guarantee in non-
convex context. Though favorable performance of ADMM and its variations does
have been observed and reported in various applications involving non-
convexity (see for examples, [21, 22]), the theoretical understandings are
still fairly limited and deficient except for [23, 24, 25, 26], where some
results have been established for special structured problems.

With the prevalence of complex multi-agent dynamic systems, growing demand has
been raised for decentralized non-convex optimization in the presence of
general coupled non-linear constraints. Such instances are ubiquitous
nowadays, including the optimal power flow (OPF) problem in smart grids [27]
and the multi-zone heating, ventilation and air-conditioning (HVAC) control in
buildings [28], etc. The non-linear constraints generally arise from the
coupled complex system dynamics, which are pivotal while designing
decentralized controllers. However, such non-linear couplings further compound
the difficulties and challenges to develop decentralized computing paradigms.
The complexities mainly stem from _i)_ the lack of standard framework to deal
with such general non-linear constraints; _ii)_ the difficulties to ensure
feasibility of the coupled nonlinear constraints while performing
decentralized computing; _iii)_ the challenges to investigate the convergence
of a specific decentralized algorithm without any structure properties (e.g.,
convexity, strong convexity, etc). Though the augmented Lagrangian methods are
appealing in deal with coupled constraints, their extension to general non-
linear constraints are not straightforward or well-founded. Moreover, the
existing standard augmented Lagrangian-based framework generally require the
exact optimization (e.g.. local optima) at each iteration, which are not
viable for nonlinear and nonconvex cases practically. Surprisingly, though
lack of theoretical foundations, the favorable performance of some Lagrangian-
based methods also has been observed in some applications (e.g. matrix
completion and factorization [22], optimal power flow [21, 29]). This has
inspired some recent exploratory studies on their theoretical understandings
[31, 32, 33]. For example, [31] studied the direct extension of ADMM to two-
block convex problems with coupled non-linear but separable constraints. [34]
proposed a tailored penalty dual decomposition (PPD) method by combining
penalty method and augmented Lagrangian method to tackle non-convex problems
with non-linear constraints. Except those, there exist another two excellent
recent works that have shed some light on such situations. One is [33] which
_i)_ thoroughly and systematically investigated the intrinsic challenges to
establish convergence guarantee for ADMM in such situations; and _ii)_
resorted to a two-level nested framework as a remedy. However, [33] generally
requires the joint optimization of multi-block non-linear problem at each
iteration in the inner loop, which are not viable or attainable in practice.
Another noteworthy work is [16] which investigated the general conditions for
Lagrangian-based framework to achieve global convergence guarantee in non-
convex context. Rather than proposing a specific algorithm, [16] seems more
focused on establish a general framework and leaves the algorithm design open.

Overall, two key points from the status quo that may necessitate our
attention. _First_ , the above recent progress on non-convex optimization are
mainly attributed to the establishment of Kurdyka-Łojasiewicz properties hold
by many analytical functions, which was first proposed in [35, 36] and later
extended in [37, 38]. The KL properties are powerful as they make it possible
to characterize sequence around critical points without convexity. _Second_ ,
the key ideas to investigate the methods’ convergence are mainly twofolds:
_i)_ studying the convergence of primal and dual sequences by inspecting a
tailored Lyapunov function; _ii)_ investigating the local convergence of the
algorithm based on the KL properties. However, a typical and difficult problem
is that the augmented Lagrangian function generally oscillates in the non-
convex context, which makes the use and design of Lyapunov functions
particularly difficult [16].

###  I-B Our Contributions

Motivated by the recent progress on non-convex optimization, this paper seeks
to investigate decentralized optimization for a class of structured problems
with _(i)_ nonconvex global objective function (possibly nonsmooth) and _(ii)_
local bounded convex constraints and coupled nonlinear constraints w.r.t. the
agents.

In general, to develop a viable decentralized method, we need to overcome two
main challenges. _First_ , we need to realize the difficulties of calculating
local optima of non-linear problems at each iteration as required by most
existing decentralized paradigms [34]. _Second_ , we need to figure out the
convergence conditions to achieve performance guarantee. To address such
issues, this paper proposes a _Proximal Linearization-based Decentralized
Method_ (PLDM). The main ideas are twofolds. _First_ , considering the
difficulties to ensure the feasibility of the non-linear coupled constraints,
we first introduce some consensus variables to eliminate the nonlinear
couplings. _Second_ , to overcome the intrinsic challenges to guarantee
convergence as explained in [33], we solve a relaxed non-convex problems in a
decentralized manner by combing the augmented Lagrangian-based framework, the
_alternating minimization_ (i.e., i.e, Gauss-Seidel), and _proximal
linearization_ [12, 39] to approximate the solutions of the original problem.
In particular, different from the traditional MMs and augmented Lagrangian-
based methods where the exact local optima of the problems are required at
each iteration, the proposed method leverages proximal linearization-based
technique to update decision variables at each iteration, which makes it
computationally efficient and viable for the nonconvex and nonlinear cases.
The main contributions of this paper are outlined, i.e.,

  * •

We propose a PLDM for a class of structured nonconvex problems subject to
coupled nonlinear constraints and local bounded convex constraints.

  * •

The global convergence and local convergence rate of the method to the
$\epsilon$-critical points are studied by inspecting a tailored Lyapunov
function and the Łojasiewicz property of the AL function.

  * •

The performance of the decentralized method is illustrated by presenting a
numeric example and an application to multi-zone heating, ventilation and air-
conditioning (HVAC) control.

The remainder of this paper is structured. In Section II, the main notations
and the problem are presented. In Section III, the PLDM is introduced. In
Section IV, the global convergence and local convergence rate of the method
are investigated. In Section IV, the performance of the method is evaluated
through a numeric example and an application. In Section V, we briefly
conclude and discuss this paper.

##  II The Problem and Main Notations

###  II-A Notations

Throughout this paper, we use $\mathbb{N}$, $\mathbb{R}$, $\mathbb{R}^{n}$
($\mathbb{R}^{n}_{+}$), and $\mathbb{R}^{m\times n}$ to denote the spaces of
integers, reals, $n$-dimensional (positive) real vectors, and $m\times
n$-dimensional real matrices, respectively. The superscript $T$ denotes the
transpose operator. $\bm{I}_{n}$ and $\bm{O}_{n}$ denote the $n$-dimensional
identify and zero matrices. Without specification, $\|\cdot\|$ denotes
$\ell_{2}$ norm. $P_{\mathcal{X}}[\cdot]$ represents the projection operation
on the set $\mathcal{X}$. We use $\nabla f=\big{(}\frac{\partial f}{\partial
x_{1}},\frac{\partial f}{\partial x_{2}},\cdots,\frac{\partial f}{\partial
x_{n}}\big{)}^{T}$ to denote the gradients of
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ w.r.t its entities. If
$\bm{h}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, i.e.,
$\bm{h}(\bm{x})=(h_{1}(\bm{x}),h_{2}(\bm{x}),\cdots,h_{m}(\bm{x}))^{T}$ with
$h_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$, we have
$\nabla\bm{h}=\big{(}\nabla h_{1},\nabla h_{2},\cdots,\nabla
h_{m}\big{)}^{T}$. $\prod_{i=1}^{N}\mathcal{X}_{i}$ denotes the Cartesian
product of the sets $\mathcal{X}_{i}$ ($i=1,2,\cdots,N$). The other notations
are standard and follow the literature [34, 11].

###  II-B The Problem

This paper focuses on a class of problems given by

$\begin{split}\min_{\bm{x}_{i},i=1,2,\cdots,N}&\sum_{i=1}^{N}f_{i}(\bm{x}_{i})+\sum_{i=1}^{N}\phi_{i}(\\{x_{j}\\}_{j\in\mathcal{N}_{i}})\\\
s.t.&\quad h_{i}(\\{x_{j}\\}_{j\in\mathcal{N}_{i}})=0,\quad~{}\forall
i\in\mathcal{N}.\\\
&\quad\bm{x}_{i}\in\bm{\mathcal{X}}_{i},\quad\quad\quad\quad\quad~{}\forall
i\in\mathcal{N}.\end{split}$ (1)

where $i$ denotes the index of the agents from the set
$\mathcal{N}=\\{1,2,\cdots,N\\}$. Here $\bm{x}_{i}\in\mathbb{R}^{n_{i}}$
denotes the local decision component of Agent $i$. $\mathcal{N}_{i}$ is
alluded to the collection of agent $i$ and its neighbors.

Note that problem (1) has global objective function which is composed by the
separable parts $f_{i}:\mathbb{R}^{n_{i}}\rightarrow\mathbb{R}$ and the
composite parts $\phi_{i}:\mathbb{R}^{\bar{n}_{i}}\rightarrow\mathbb{R}$
($\bar{n}_{i}=\sum_{i\in\mathcal{N}_{i}}n_{i}$) w.r.t. agents. Wherein the
objective function $f_{i}$ and $\phi_{i}$ may be nonconvex (possibly
nonsmooth). $h_{i}:\mathbb{R}^{\bar{n}_{i}}\rightarrow\mathbb{R}^{m_{i}}$
denotes the coupled nonlinear constraints pertaining to Agent $i$, which are
smooth and differentiable. As inequality constraints could be transformed to
equality constraints by introducing slack variables, this paper only
investigate equality constraints. Besides, there exist local bounded convex
constraints represented by $\mathcal{X}_{i}$ ($i\in\mathcal{N}$) for the
agents. In addition, we make the following assumptions for problem (1) in our
analysis, i.e.,

  1. (A1)

The equality constraints $h_{i}$ are continuously differentiable over
$\mathcal{X}_{i}$ ($h_{i}$ and $\nabla h_{i}$ are Lipschitz continuous with
constants $L_{h_{i}}$ and $M_{h_{i}}$).

  2. (A2)

The separable parts $f_{i}$ and $\nabla f_{i}$ are Lipschitz continuous with
constants $L_{f_{i}}$ and $M_{f_{i}}$ over $\mathcal{X}_{i}$, respectively.

  3. (A3)

The coupled parts $\phi_{i}$ and $\nabla\phi_{i}$ are Lipschitz continuous
with constants $L_{\phi_{i}}$ and $M_{\phi_{i}}$ over
$\prod_{j\in\mathcal{N}_{i}}\mathcal{X}_{i}$, respectively.

###  II-C Problem Reformulation

As aforementioned, it’s generally challenging to tackle problem (1) with non-
linearity and non-convexity both in the objective and the constraints with
performance guarantee. Therefore, this part presents some reformulations of
the problem as a necessary preparation for the following study. First, to
handle the coupled nonlinear couplings, we introduce a block of consensus
variables
$\bm{Z}=((z_{1})^{T},(z_{2})^{T},\cdots,(z_{N})^{T})^{T}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}n_{i}}$,
which represents the hypothetical copy of the collected decision components
for all the agents. As displayed in Fig. 1, each Agent $i$ will hold a local
copy of the augmented decision variables denoted by
$\bm{X}_{i}\\!=\\!(x_{i},\\{x^{j}_{i}\\}_{j\in\mathcal{N}_{i}\setminus\\{i\\}})^{T}\\!\in\\!\\!\mathbb{R}^{\bar{n}_{i}}$
($\bar{n}_{i}=\sum_{j\in\mathcal{N}_{i}}n_{j}$), in which $x^{j}_{i}$ denotes
the local copy of the decision component for its neighboring Agent $j$.
Meanwhile, we assume there is a virtual Agent $0$, who is obligated to manage
the block of consensus decision variables $\bm{Z}$. Intuitively, for an
algorithm to converge, we require

$\displaystyle x_{i}=z_{i},~{}~{}x^{j}_{i}=z_{j}.$ (2a) $\displaystyle
x^{i}_{j}=z_{i},~{}~{}x_{j}=z_{j}.$ (2b)

Figure 1: The local and global (consensus) decision variables.

In this case, problem (1) can be restated as

$\displaystyle\min_{\bm{X}_{i},i=1,2,\cdots,N.\bm{Z}}$
$\displaystyle\quad\sum_{i=1}^{N}\tilde{f}_{i}(\bm{X}_{i})+\sum_{i=1}^{N}\phi_{i}(\bm{X}_{i})$
(3) $\displaystyle s.t.\quad$ $\displaystyle
h_{i}(\bm{X}_{i})=\bm{0},~{}\forall i\in\mathcal{N}.$ (3a)
$\displaystyle\bm{X}_{i}\\!=\\!\bm{E}_{i}\bm{Z},~{}~{}~{}\forall
i\in\mathcal{N}.$ (3b) $\displaystyle\bm{Z}\in\bm{\mathcal{X}}.$ (3c)

where we have $\bm{\mathcal{X}}=\prod_{i\in\mathcal{N}}\mathcal{X}_{i}$.
$\tilde{f}_{i}:\mathbb{R}^{\bar{n}_{i}}\rightarrow\mathbb{R}$ denotes the
extended function of $f_{i}$.
$\bm{E}_{i}=diag\\{\bm{I}_{n_{j}}\\}_{j\in\mathcal{N}_{i}}\in\mathbb{R}^{\bar{n}_{i}\times\sum\limits_{j\in\mathcal{N}}n_{j}}$
are constant matrices. (3b) denotes the compact form of the equality
constraints (2a) and (2b).

Observe (3), we note that the objective function is now decomposable w.r.t.
the agents. However, there appear three types of constraints: (i) local
nonlinear and nonconvex constraints (3a), (ii) coupled linear constraints
(3b), and (iii) local bounded convex constraints (3c). There exist two blocks
of decision variables, i.e., the primal and consensus decision variables
$\bm{X}\\!=\\![\bm{X}^{T}_{i}]_{i\in\mathcal{N}}^{T}$ and $\bm{Z}$.
Nevertheless, we note that the local constraints are now not regular as
$[(\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i}))^{T},\bm{I}_{\bar{n}_{i}}]^{T}$ is not
full row rank. In fact, this presents one of the intrinsic challenges to
investigate the convergence of a general decentralized methods for such
problems (this may be comprehended in the rigorous analysis later). Here this
may be understood that even a particular (decentralized) algorithm is able to
force $\bm{Z}$ to some (local) optima $\bm{Z}^{*}$, no $\bm{X}^{*}$ that
satisfies the constraints (3a) and (3c) simultaneously can be found. To
address such a challenging issue, we introduce some slack variables
$\bm{Y}_{i}\in\mathbb{R}^{\bar{n}_{i}}$ ($i\in\mathcal{N}$) and restate
problem (3) as

$\displaystyle\min_{\bm{\bar{X}}_{i},i\in\mathcal{N}.\bm{Z}}$
$\displaystyle\quad\sum_{i=1}^{N}\tilde{f}_{i}(\bm{X}_{i})+\sum_{i=1}^{N}\phi_{i}(\bm{X}_{i})$
(4) $\displaystyle s.t.\quad$ $\displaystyle
h_{i}(\bm{X}_{i})=\bm{0},~{}~{}~{}~{}\forall i\in\mathcal{N}.$ (4a)
$\displaystyle\bm{A}_{i}\bm{\bar{X}}_{i}=\\!\bm{E}_{i}\bm{Z},~{}~{}\forall
i\in\mathcal{N}.$ (4b)
$\displaystyle\bm{Y}_{i}=\bm{0},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
i\in\mathcal{N}.$ (4c) $\displaystyle\bm{Z}\in\bm{\mathcal{X}},$ (4d)

where
$\bm{A}_{i}=[\bm{I}_{\bar{n}_{i}},\bm{I}_{\bar{n}_{i}}]\in\mathbb{R}^{\bar{n}_{i}\times
2\bar{n}_{i}}$.
$\bm{\bar{X}}_{i}=\big{(}(\bm{X}_{i})^{T},(\bm{Y}_{i})^{T}\big{)}^{T}$
($i\in\mathcal{N}$) denotes the extended local decision variable hold by Agent
$i$.

Note that problem (3) is equivalent to (4) as the slack variables $\bm{Y}_{i}$
are forced to be _zero_ in the constraints. However, the local constraints
w.r.t the extended local decision variables $\bm{\bar{X}}_{i}$
($i\in\mathcal{N}$) are still not regular, therefore we have to resort to the
following relaxed problem:

$\displaystyle\min_{\bm{\bar{X}}_{i},i\in\mathcal{N}.\bm{Z}}$
$\displaystyle\quad\sum_{i=1}^{N}\tilde{f}_{i}(\bm{X}_{i})+\sum_{i=1}^{N}\phi_{i}(\bm{X}_{i})+\sum_{i=1}^{N}M_{i}\|\bm{Y}_{i}\|^{2}$
(5) $\displaystyle s.t.\quad$ $\displaystyle
h_{i}(\bm{X}_{i})=\bm{0},~{}~{}~{}~{}\forall i\in\mathcal{N}.$ (5a)
$\displaystyle\bm{A}_{i}\bm{\bar{X}}_{i}=\\!\bm{E}_{i}\bm{Z},~{}~{}\forall
i\in\mathcal{N}.$ (5b) $\displaystyle\bm{Z}\in\bm{\mathcal{X}},$ (5c)

where $M_{i}>0$ ($i\in\mathcal{N}$) denotes some positive penalty parameter.
In contrast of (5) with (4), one may note that the constraints
$\bm{Y_{i}}=\bm{0}$ for the slack variables have been softened by appending
some penalty terms $M_{i}\|\bm{Y}_{i}\|^{2}$ in the global objective function.
As _i)_ problem (5) contains all the feasible solutions of (4); _ii)_
$P^{(5),*}\leq P^{(4),*}$ ($P^{(\cdot),*}$ denotes the optima of the
problems), problem (5) can be regarded as an relaxation of problem (4).
Moreover, we note that with $M_{i}\rightarrow 0$ ($i\in\mathcal{N}$), the
well-posedness optimal solutions of (5) (with bounded objective value) are
exactly those for the original problem (4) as we will have
$\bm{Y}^{*}_{i}\rightarrow\bm{0}$ ($\forall i\in\mathcal{N}$) otherwise
$P^{(5),*}\rightarrow\infty$.

The main ideas of this paper are twofolds. _First_ , we investigate
decentralized method for problem (5) with performance guarantee. _Second_ , we
prove that the attained solutions are the $\epsilon$-critical points of (4)
under some mild conditions. Before that, we first make the following extra
assumptions.

  * (A4)

$M_{i}>0$ ($i\in\mathcal{N}$) are sufficiently large.

######  Remark 1.

Generally, rigorous analysis requires $M_{i}\rightarrow+\infty$ to guarantee
the equivalence of problem (4) and (5). However, a sufficiently large positive
value is enough to guarantee the sub-optimality in practice.

  * (A5)

$\bm{F}_{i}(\bm{\bar{X}}_{i})\\!\\!=\\!\\!\left(\begin{array}[]{ccc}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})&\\!\\!\\!\\!&\bm{O}_{n_{i}}\\\
&\bm{A}_{i}&\\\ \end{array}\right)$ is uniformly regular with constant
$\theta$ over the bounded set $\bm{\mathcal{\bar{X}}}^{\eta}_{i}$ (see
Definition 3), where
$\bm{\mathcal{\bar{X}}}_{i}^{\eta}=P_{\mathbb{R}^{\bar{n}_{i}}}[\bm{\mathcal{X}}^{\eta}]$,
with
$\bm{\mathcal{X}}^{\eta}=\\{\bar{\bm{X}}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}2\overline{n}_{i}}$
$|\sum_{i=1}^{N}\|h_{i}(\bm{X}_{i})\|\leq\eta\\}$ ($\eta$ is a positive
threshold).

######  Remark 2.

(A5) is standard and can generally be satisfied through regulating the
dimension of the slack variables.

##  III Proximal Linearization-Based Decentralized Method

This section first presents a decentralized method called PLDM for problem (5)
and gives the main results on its convergence. After that we prove that the
attained solutions are the $\epsilon$-critical points of the original problem
(4). In particular, the proposed method is established under the general
augmented Lagrangian-based framework for nonconvex context (see [40, 16] for
examples ) and contains the three standard steps: (i) primal update based on
proximal linearization, (ii) dual update and (iii) adaptive step for penalty
factors.

To handle the local nonlinear constraints (4a) and the coupled linear
constraints (4b), we define the Augmented Lagrangian (AL) function as

$\displaystyle\begin{split}\mathbb{L}_{\rho}&(\bm{\bar{X}},\bm{\lambda},\bm{\mu})=\sum_{i=1}^{N}\tilde{f}_{i}(\bm{X}_{i})+\sum_{i=1}^{N}\phi_{i}(\bm{X}_{i})+\sum_{i=1}^{N}M_{i}\|\bm{Y}_{i}\|^{2}\\\
&+\sum_{i=1}^{N}(\bm{\lambda}_{i})^{T}h_{i}(\bm{X}_{i})+\sum_{i=1}^{N}\frac{\rho}{2}\big{\|}h_{i}(\bm{X}_{i})\big{\|}^{2}\\\
&+\sum_{i=1}^{N}\big{(}\bm{\mu}_{i}\big{)}^{T}\\!(\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z})\\!+\\!\sum_{i=1}^{N}\\!\frac{\rho}{2}\big{\|}\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z}\big{\|}^{2}\end{split}$
(6)

where
$\bm{\bar{X}}=[\bm{\bar{X}}_{i}]_{i\in\mathcal{N}}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}2\bar{n}_{i}}$
denotes the augmented decision variable for the problem.
$\bm{\lambda}=[\bm{\lambda}_{i}]_{i\in\mathcal{N}}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}m_{i}}$
and
$\bm{\mu}=[\bm{\mu}_{i}]_{i\in\mathcal{N}}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}2\bar{n}_{i}}$
are Lagrangian multipliers. $\rho>0$ denotes the penalty factor.

For given Lagrangian multipliers $\bm{\lambda}$ and $\bm{\mu}$, the primal
problem needs to be solved is given by

$\begin{split}\min_{\bm{\bar{X}},\bm{Z}}~{}~{}&\mathbb{L}_{\rho}(\bm{\bar{X}},\bm{\lambda},\bm{\mu})\\\
s.t.~{}~{}&\bm{Z}\in\bm{\mathcal{X}}.\\\ \end{split}$ (7)

Primal Update: observe that (7) has two blocks of decision variables
(i.e.,$\bm{\bar{X}}$ and $\bm{Z}$). For general MMs, a joint optimization of
the two decision variable blocks is usually required [41]. However, the primal
problem (7) is nonlinear and nonconvex, even obtaining a local optima is
difficult with existing approaches. To handle this, we achieve primal update
at each iteration in a decentralized manner by performing two steps : (i)
_proximal linearization_ of the nonlinear subproblems and (ii) Gauss-seidal
update w.r.t. the decision variable blocks (i.e., $\bm{\bar{X}}$, $\bm{Z}$).
Observing that AL function (6) is decomposable w.r.t. the agents with the
extended local decision variable $\bm{\bar{X}}_{i}$ ($\forall
i\in\mathcal{N}$), we define the subproblems for each Agent $i$ as

$\begin{split}&\underset{\bm{\bar{X}}_{i}}{\operatorname{min}}\quad\mathbb{L}^{i}_{\rho}(\bm{\bar{X}}_{i},\bm{Z},\bm{\lambda}_{i},\bm{\mu}_{i})=\tilde{f}_{i}(\bm{X}_{i})+\phi_{i}(\bm{X}_{i})+M_{i}\|\bm{Y}_{i}\|^{2}\\\
&\quad+(\bm{\lambda_{i}})^{T}h_{i}(\bm{X}_{i})+\frac{\rho}{2}\big{\|}h_{i}(\bm{X}_{i})\big{\|}^{2}+\big{(}\bm{\mu}_{i}\big{)}^{T}(\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z})\\\
&\quad+\frac{\rho}{2}\big{\|}\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z}\big{\|}^{2},~{}~{}\forall
i\in\mathcal{N}.\end{split}$ (8)

By performing _proximal linearization_ on the nonlinear and nonconvex
subproblems (8) at each iteration $k$, we have

$\begin{split}&\min_{\bm{\bar{X}}_{i}}\mathbb{\tilde{L}}^{i}_{\rho_{k}}(\bm{\bar{X}}_{i},\bm{\bar{X}}^{k}_{i},\bm{Z}^{k},\bm{\lambda}^{k}_{i},\bm{\mu}_{i}^{k})=(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z}^{k+1})\\\
&+\\!\frac{\rho_{k}}{2}\|\bm{A}_{i}\bm{\bar{X}}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\!+\\!\langle\nabla_{\bm{X}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}_{i}^{k},\rho_{k}),\bm{\bar{X}}_{i}\\!-\\!\bm{\bar{X}}^{k}_{i}\rangle\\\
&+\frac{c^{k}_{i}}{2}\|\bm{\bar{X}}_{i}-\bm{\bar{X}}^{k}_{i}\|^{2},~{}~{}\forall
i\in\mathcal{N}.\end{split}$

where $c^{k}_{i}$ denotes the step-size for subproblem $i$ at iteration $k$.
Besides, we define
$g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}_{i},\rho)\\!=\\!\tilde{f}_{i}(\bm{X}_{i})\\!+\\!\phi_{i}(\bm{X}_{i})\\!+\\!(\bm{\lambda_{i}})^{T}h_{i}(\bm{X}_{i})+\frac{\rho}{2}\|h_{i}(\bm{X}_{i})\|^{2}+M_{i}\|\bm{Y}_{i}\|^{2}$.

######  Remark 3.

If the objective function $\tilde{f}_{i}$ or $\phi_{i}$ is non-smooth w.r.t
the local decision variable $\bm{\bar{X}}_{i}$, we can remove it from
$g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k}_{i},\rho_{k})$ and keep them in
$\mathbb{\tilde{L}}^{i}_{\rho_{k}}(\bm{\bar{X}}_{i},\bm{\bar{X}}^{k}_{i},\bm{Z}^{k},\bm{\lambda}^{k}_{i},\bm{\mu}^{k}_{i})$.

For the consensus variable $\bm{Z}$, the subproblem is given by

$\begin{split}&\underset{\bm{Z}}{\operatorname{min}}\quad\mathbb{L}_{\rho}(\bm{Z},\bm{\bar{X}},\bm{\mu})\\!=\\!\sum_{i=1}^{N}\big{(}\bm{\mu}_{i}\big{)}^{T}\big{(}\bm{A}_{i}\bm{\bar{X}}_{i}\\!-\\!\bm{E}_{i}\bm{Z}\big{)}\\\
&\quad+\sum_{i=1}^{N}\frac{\rho}{2}\big{\|}\bm{A}_{i}\bm{\bar{X}}_{i}-\\!\bm{E}_{i}\bm{Z}\big{\|}^{2},~{}~{}s.t.~{}~{}\bm{Z}\in~{}{\bm{\mathcal{X}}}.\end{split}$
(9)

We note that subproblem (9) is a quadratic programming (QP), which can be
solved efficiently.

Dual update & Adaptive step : the dual variables are updated following the
standard augmented Lagrangian methods (see [42] for example). However, we
introduce an adaptive step to dynamically update the penalty factor $\rho$. In
particular, we pre-define a sub-feasible region regarding the non-linear
constraints of problem (5), i.e.,

$\bm{\mathcal{\bar{X}}}^{\eta}\triangleq\big{\\{}\bar{\bm{X}}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}2\overline{n}_{i}}|\sum_{i=1}^{N}\|h_{i}(\bm{X}_{i})\|\leq\eta\big{\\}}.$

We iteratively increase $\rho_{k}$ with an increment $\delta$ until
$\bm{\bar{X}}^{k}$ is forced into the pre-defined sub-feasible region
$\bm{\mathcal{\bar{X}}}^{\eta}$.

The main steps of the proposed PLDM method for solving problem (5) are
summarized in Algorithm 1. The algorithm starts by initializing the Lagrangian
multipliers, penalty factor, and decision variables. Afterwards, the main
steps include the alternative update of the two decision blocks
($\bm{\bar{X}}$, $\bm{Z}$) (Step 3-5), the Lagragnain multipliers
($\bm{\lambda}$, $\bm{\mu}$) (Step 6), and the penalty factor ($\rho$) (Step
7). We note that as the primal problem (7) is decomposable w.r.t the agents,
the update of the primal decision variable block ($\bm{\bar{X}}$) can be
performed in parallel by the agents. In Algorithm 1, the stopping criterion is
defined as the residual error bound of the constraints, i.e.,

$\begin{split}R(k)=\sum_{i=1}^{N}\Big{\\{}\|h_{i}(\bm{X}^{k+1}_{i})\|+\|\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1}\|\Big{\\}}\leq\epsilon\end{split}$

where $\epsilon$ is a constant threshold.

The algorithm iterates until the stopping criterion is reached. Still, this
does not mean the convergence of the algorithm. This needs to be studied in
greater detail later.

Algorithm 1 Proximal Linearization-based Decentralized Method (PLDM) for
Nonconvex and Nonlinear Problems

1:Initialization: $\bm{\lambda}^{0}$, $\bm{\mu}^{0}$, $\bm{\bar{X}}^{0}$,
$\bm{Z}^{0}$ and $\rho_{0}$, and set $k\rightarrow 0$.

2:Repeat:

3:Primal Update:

4: Update the consensus variables $\bm{Z}$, i.e.,
$\begin{split}\bm{Z}^{k+1}=\arg\min_{\bm{Z}\in\mathcal{X}}\mathbb{L}_{\rho_{k}}(\bm{Z},\bm{\bar{X}}^{k},\bm{\mu}^{k}).\end{split}$
(10)

5: Update the primal decision variables $\bm{\bar{X}}$ in parallel, i.e.,
$\begin{split}\bm{\bar{X}}_{i}^{k+1}\\!=\\!\arg\min_{\bm{\bar{X}}_{i}}\mathbb{\tilde{L}}^{i}_{\rho_{k}}(\bm{\bar{X}}_{i},&\bm{\bar{X}}^{k}_{i},\bm{Z}^{k+1},\bm{\lambda}^{k}_{i},\bm{\mu}^{k}_{i}),\\\
&\forall i\in\mathcal{N}.\\\ \end{split}$ (11)

6:Dual update: Update Lagrangian multipliers $\bm{\lambda}$, $\bm{\mu}$, i.e.,
$\begin{split}&\bm{\lambda}_{i}^{k+1}=\bm{\lambda}_{i}^{k}+\rho_{k}h_{i}(\bm{X}^{k+1}_{i}),\\\
&\bm{\mu}^{k+1}_{i}=\bm{\mu}^{k}_{i}+\rho_{k}\big{(}\bm{A}_{i}\bm{\bar{X}}_{i}^{k+1}\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\big{)},\\\
&\quad\quad\quad\quad\quad\forall i\in\mathcal{N}.\end{split}$ (12)

7:Adaptive step: Update the penalty factor $\rho$, i.e., if
$\bm{\bar{X}}^{k+1}\in\bm{\mathcal{X}}^{\eta}$, set
$\rho_{k+1}=\rho_{k}+\delta$, otherwise $\rho_{k+1}=\rho_{k}$.

8:If $R(k)\\!\leq\\!\epsilon$ stop, otherwise set $k\\!=\\!k\\!+\\!1$ and go
to Step 2\.

##  IV Performance and Convergence Analysis of PLDM

This section discusses the performance and convergence of PLDM. First, we
present the additional assumptions, basic definitions and propositions
required. Then we illustrate the main results in Theorem 1 and Theorem 2.

###  IV-A Definitions and Lemmas

######  Lemma 1.

(Descent lemma) (see [8], Proposition A.24) Let
$h:\mathbb{R}^{d}\\!\rightarrow\\!\mathbb{R}$ be $M_{h}$-Lipschitz gradient
continuous, we have

$\begin{split}h(u)\leq h(v)\\!+\\!\langle u-v,\nabla
h(u)\rangle\\!+\\!&\frac{M_{h}}{2}\|u-v\|^{2},~{}\forall
u,v\in\mathbb{R}^{d}.\end{split}$

######  Lemma 2.

(Sufficient decrease property) (see [12], Lemma 2) Let
$h:\mathbb{R}^{d}\\!\rightarrow\\!\mathbb{R}$ be $M_{h}$-Lipschitz gradient
continuous and $\sigma:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a proper and
lower semicontinuous function with $\inf_{\mathbb{R}^{d}}\sigma>-\infty$. If
$prox_{t}^{\sigma}=\arg\min\\{\sigma(u)+\frac{t}{2}\|u-x\|^{2}\\}$ denotes the
proximal map associated with $\sigma$, then for any fixed $t>M_{h}$, $u\in$
dom $\sigma$, and $u^{+}$ defined by

$u^{+}\in{prox}^{\sigma}_{t}\Big{(}u-\frac{1}{t}\nabla h(u)\Big{)}$ (13)

we have $h(u^{+})+\sigma(u^{+})\leq
h(u)+\sigma(u)-\frac{1}{2}(t-M_{h})\big{\|}u^{+}-u\big{\|}^{2}$.

######  Definition 1.

(Normal cone) (see [40]) Let
$\mathcal{\bm{X}}\\!\\!\subseteq\\!\\!\mathbb{R}^{d}$ be a convex set, the
normal cone of $\mathcal{\bm{X}}$ is the set-valued mapping

$\begin{split}\mathcal{N}_{\mathcal{\bm{X}}}(\bar{x})=\left\\{\begin{array}[]{lcl}\big{\\{}g\in\mathbb{R}^{d}|\forall
x\in\mathcal{\bm{X}},g^{T}(x-\bar{x})\leq
0\big{\\}},{}{}&\textrm{if}~{}\bar{x}\in\mathcal{\bm{X}}{}{}\\\
\varnothing,&\textrm{if}~{}\bar{x}\notin\mathcal{\bm{X}}.{}{}\end{array}\right.\end{split}$

######  Definition 2.

(Critical point) (see [43, 44, 45]) Considering the following problem (P):

$\begin{split}&(\textbf{{P}})~{}~{}\Big{\\{}\min_{x\in\mathbb{R}^{d}}f(x)~{}|~{}\bm{g}(x)\leq\bm{0},\bm{h}(x)=\bm{0}.\Big{\\}}\end{split}$

where the objective $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$, and the
constraints $\bm{g}=(g_{1},g_{2},\cdots,g_{m})$ with
$g_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$,
$\bm{h}=(h_{1},h_{2},\cdots,h_{\ell})$ with
$h_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ are continuously differentiable.
The critical points (i.e., KKT points) of problem (P) denote its feasible
points satisfying first-order optimality condition described by

$\begin{split}&\textrm{crit}(\bm{P})\\!=\\!\left\\{\begin{array}[]{c}x\in\mathbb{R}^{d}\\\
\bm{\lambda}\in\mathbb{R}_{+}^{m}\\\ \bm{\mu}\in\mathbb{R}^{\ell}\\\
\end{array}\Bigg{|}\begin{array}[]{c}\nabla
f(x)\\!+\\!(\nabla\bm{g}(x))^{T}\bm{\lambda}\\!+\\!(\nabla
h(x))^{T}\bm{\mu}\\!=\\!\bm{0}.\\\ \bm{g}(x)\geq 0,~{}\bm{h}(x)=0.\\\
\bm{\lambda}_{i}g_{i}(x)=0,~{}i=1,2,\cdots,m.\\\ \end{array}\right\\}\\\
\end{split}$

As a direct extension, we can define the collection of $\epsilon$-critical
points $\textrm{crit}_{\epsilon}(\bm{P})$ by replacing $\bm{0}$ on the right-
hand side of the equalities (inequalities) with $\epsilon$.

######  Remark 4.

For convex problems, the critical points (KKT points) are exactly the global
optima. However, without convexity, a critical point can be a global optima, a
local optima, or a “saddle point”. For problem (5) discussed in this paper,
the critical points can be described by

$\begin{split}&\textrm{crit \big{(}~{}problem (5) \big{)}}=\\\
&\left\\{\begin{array}[]{c}\bm{\bar{X}}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}\bar{n}_{i}}\\\
\bm{\lambda}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}m_{i}}\\\
\bm{\mu}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}2\bar{n}_{i}}\\\
\end{array}\Bigg{|}\begin{array}[]{c}\nabla_{\bm{\bar{X}}}\mathbb{L}_{\rho}(\bm{\bar{X}},\bm{Z},\bm{\lambda},\bm{\mu})=\bm{0}.\\\
\nabla_{\bm{\bar{X}}}\mathbb{L}_{\rho}(\bm{\bar{X}},\bm{Z},\bm{\lambda},\bm{\mu})+\mathcal{N}_{\mathcal{\bm{X}}}(\bm{Z})=\bm{0}.\\\
\bm{h}_{i}(\bm{\bar{X}_{i}})=\bm{0},~{}\forall i\in\mathcal{N}.\\\
\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z}=0,~{}\forall
i\in\mathcal{N}.\end{array}\right\\}\end{split}$

######  Definition 3.

(Uniform Regularity) (see [16]) Let $\mathcal{\bm{X}}\subseteq\mathbb{R}^{m}$
and $h:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ be continuously
differentiable, we claim $h$ as uniformly regular over $\mathcal{\bm{X}}$ with
a positive constant $\theta$ if

$\begin{split}\big{\|}(\nabla
h(x))^{T}\bm{v}\big{\|}\geq\theta\|\bm{v}\|,\forall
x\in\mathcal{\bm{X}},\bm{v}\in\mathbb{R}^{n}\end{split}$ (14)

######  Remark 5.

For a given $x\in\mathcal{X}$, asserting that $\nabla h(x)$ is uniformly
regular with $\theta$ ($\theta>0$) is equivalent to

$\gamma(F,x)=\min_{\|\bm{v}\|=1}\Big{\\{}\|(\nabla
h(x))^{T}\bm{v}\|\Big{\\}}>0.$

Equivalently, the mapping $(\nabla h(x))^{T}$ is supposed to be surjective or
$\nabla h(x)(\nabla h(x))^{T}$ is positive definite. one may note that $\nabla
h(x)(\nabla h(x))^{T}$ is always positive semidefinite. Therefore, the uniform
regularity of $\nabla h(x)$ requires that the minimum eigenvalue of $\nabla
h(x)(\nabla h(x))^{T}$ is positive. Besides, we note that if $\nabla h(x)$ has
full row rank, $\nabla h(x)$ will be uniformly regular with an existing
positive constant.

The next definition is a property that many analytical functions hold and
plays a central role in the convergence analysis of nonconvex optimization
(see [12, 46]).

######  Definition 4.

(Kurdyka-Łojasiewicz (KŁ) property) (see [12]) We say function
$h:\mathbb{R}^{p}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ has the KŁ property
at $x^{*}\in\textrm{dom}~{}\partial h$, if there exist $\eta\in(0,+\infty)$, a
neighborhood $U$ of $x^{*}$, and a continuous concave function
$\varphi:[0,\eta)\rightarrow\mathbb{R}_{+}$ such that:

(i) $\varphi(0)=0$ and $\varphi$ is differentible on $(0,\varphi)$;

(ii) $\forall s\in(0,\eta)$, $\varphi^{{}^{\prime}}(s)>0$;

(iii) $\forall s\in U\cap\big{\\{}x:h(x^{*})<h(x)<h(x^{*})+\eta\big{\\}}$.
Then the following KŁ inequality holds:

$\varphi^{{}^{\prime}}\big{(}h(x)-h(x^{*})\big{)}\cdot\textrm{dist}\big{(}0,\partial
h(x)\big{)}\geq 1,$

and function $\varphi$ is called a desingularizing function of $h$ at $x^{*}$.

For Algorithm 1, we need two additional assumptions to guarantee convergence.

  * (A6)

The Lagrangian multipliers sequences
$\\{\bm{\lambda}_{i}^{k}\\}_{k\in\mathbb{N}}$ ($i\in\mathcal{N}$) and
$\\{\bm{\bm{\mu}_{i}^{k}}\\}_{k\in\mathbb{N}}$ ($i\in\mathcal{N}$) generated
by Algorithm 1 are bounded by $M_{\bm{\lambda}_{i}}$ and $M_{\bm{\mu}_{i}}$,
respectively.

  * (A7)

The original problem (1) is well-defined. Thus for a proper penalty factor
$\rho<+\infty$, the primal problem in (7) is lower bounded as

$\begin{split}\inf_{\bm{\bar{X}},\bm{Z}\in\bm{\mathcal{X}}}\mathbb{L}_{\rho}(\bm{\bar{X}},\bm{Z},\bm{\lambda},\bm{\mu})>-\infty.\end{split}$
(15)

###  IV-B PLDM Convergence

In this subsection, we first illustrate Propositions 1-9 that required to
prove the convergence of the PLDM. After that we present the main results in
Theorem 1.

######  Proposition 1.

Let
$\bm{\mathcal{\bar{X}}}_{i}^{\eta}=P_{\mathbb{R}^{\bar{n}_{i}}}[\bm{\mathcal{\bar{X}}}^{\eta}]$
($i\in\mathcal{N}$), there exists a finite iteration $\underline{k}$ such that

  * (a)

$\\{\bm{\bar{X}}^{k}_{i}\\}_{k\in\mathbb{N},k\geq\underline{k}}\in\bm{\mathcal{\bar{X}}}^{\eta}_{i}~{}(\forall
i\in\mathcal{N}).$

  * (b)

$\rho_{k}=\rho_{\underline{k}},~{}~{}\forall k\geq\underline{k}.$

Proof: Refer to Appendix A.

######  Remark 6.

Proposition 1(a) illustrates that for any pre-defined sub-feasible region for
the non-linear constraints (5a), the local decision variables
($\bm{\bar{X}}^{k}_{i}$) for the agents will be forced into the sub-feasible
region ($\bm{\mathcal{\bar{X}}}_{i}^{\eta}$) within finite iterations
($\underline{k}$). Meanwhile, the penalty factor ($\rho$) will stop increasing
as described in Proposition 1(b).

######  Proposition 2.

$g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}^{k}_{i},\rho_{k})$ ($\forall
k\in\mathbb{N}$) is Lipschitz continuous w.r.t $\bm{\bar{X}}_{i}$ over
$\bm{\mathcal{\bar{X}}}_{i}^{\eta}$ with constant
$L_{g_{i}}=\max\\{L_{f_{i}}+L_{\phi_{i}}+M_{{\lambda}_{i}}L_{h_{i}}+\rho_{\underline{k}}C_{h_{i}},2M_{i}\\}$
($C_{h_{i}}$ denotes the upper bound of
$\|(\nabla_{\bm{X}_{i}}\\!h_{i}(\bm{X}_{i}))^{T}h_{i}(\bm{X}_{i})\|$ over
$\bm{\mathcal{\bar{X}}}^{\eta}_{i}$).

Proof: Refer to Appendix B.

######  Remark 7.

Proposition 2 presents the smoothness of the function
$g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}^{k}_{i},\rho_{k})$ ($\forall
k\in\mathbb{N}$) over the bounded sub-feasible region
$\bm{\mathcal{\bar{X}}}_{i}^{\eta}$.

######  Proposition 3.

Let $\\{\bm{\bar{X}}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ and
$\\{\bm{Z}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ be the sequences
generated by Algorithm 1, then we have

$\begin{split}&\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})\leq\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})\\\
&\quad\quad-\sum_{i=1}^{N}\frac{1}{2}(c_{i}^{k}-L_{g_{i}})\|\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i}\|^{2}\end{split}$
(16)

Proof: Refer to Appendix C.

######  Remark 8.

Proposition 3 provides the lower bound for the “decrease” of the AL function
w.r.t. the primal updates. One may note that the AL function is non-increasing
w.r.t the primal update at each iteration with a step-size $c_{i}^{k}\geq
L_{g_{i}}$.

######  Proposition 4.

Let $\\{\bm{\bar{X}}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ and
$\\{\bm{Z}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ be the sequances
generated by Algorithm 1, we have

$\begin{split}\big{\|}\nabla_{\bm{\bar{X}}_{i}}&\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})\big{\|}\\\
&\leq\big{(}L_{g_{i}}+c_{i}^{k}\big{)}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|,~{}\forall
i\in\mathcal{N}.\\\ \end{split}$ (17)

Proof: Refer to Appendix D.

######  Remark 9.

Proposition 4 provides the upper bound for the subgradients of the AL
functions after primal update at each iteration.

######  Proposition 5.

Let $\\{\bm{\bar{X}}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ be the
sequence generated by Algorithm 1, we have

$\displaystyle\begin{split}\|\bm{\gamma}^{k+1}_{i}-\bm{\gamma}^{k}_{i}\|\leq\Omega^{i}_{1}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|+&\Omega^{i}_{2}\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k-1}_{i}\|,\\\
&~{}\forall i\in\mathcal{N}.\end{split}$ (18)

where
$\bm{\gamma}_{i}=\big{(}(\bm{\lambda}_{i})^{T},(\bm{\mu}_{i})^{T}\big{)}^{T}$
denoting the augmented Lagrangian multipliers for Agent $i$. We have
$\Omega^{i}_{1}=\big{(}L_{g_{i}}+c_{i}^{k}+L_{f_{i}}+L_{\phi_{i}}+M_{h_{i}}M_{\bm{\gamma}_{i}}\big{)}/\theta$
and $\Omega^{i}_{2}=\big{(}L_{g_{i}}+c_{i}^{k\\!-\\!1})/\theta$
($i\in\mathcal{N}$).

Proof: Refer to Appendix E.

######  Remark 10.

Proposition 5 provides the upper bound for the difference of the Lagrangian
multipliers over two successive iterations.

To illustrate the convergence of the proposed PLMD, we need to resort to a
Lyapunov function defined as

$\begin{split}\Phi_{\beta}(\bm{\bar{X}},\bm{Z},\bm{\lambda},\bm{\mu},\bm{U})&=\mathbb{L}_{\rho}(\bm{\bar{X}},\bm{Z},\bm{\lambda},\bm{\mu})+\beta\|\bm{\bar{X}}-\bm{U}\|^{2},\\\
\end{split}$

where $\beta>0$ and $\bm{U}\in\mathbb{R}^{\sum_{i\in\mathcal{N}}\bar{n}_{i}}$.
One may note that the Lyapunov function closely relates to the AL function
except for the extra term $\beta\|\bm{\bar{X}}-\bm{U}\|^{2}$. For the Lyapunov
function under the sequences generated by Algorithm 1, we have the following
proposition.

######  Proposition 6.

Let
$\Big{\\{}\bm{W}^{k}\\!=\\!\big{(}\bm{\bar{X}}^{k},\\!\bm{Z}^{k},\\!\bm{\lambda}^{k},\\!\bm{\mu}^{k},\bm{\bar{X}}^{k-1}\big{)}\Big{\\}}_{k\in\mathbb{N},k\geq\underline{k}}$
be the sequence generated by Algorithm 1, we have

$\begin{split}\Phi_{\beta_{k}}&(\bm{W}^{k})-\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\\\
&\geq
b_{1}^{k}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|^{2}+b^{k}_{2}\|\bm{\bar{X}}^{k}-\bm{\bar{X}}^{k-1}\|^{2},\\\
\end{split}$ (19)

where
$\Phi_{\beta_{k}}(\bm{W}^{k})\\!=\\!\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})+\beta_{k}\|\bm{\bar{X}}^{k}-\bm{\bar{X}}^{k-1}\|^{2}$,
with $\beta_{k}$ a positive parameter. Besides, we have

$\begin{split}&b^{k}_{1}=\min\limits_{i}\Big{\\{}\frac{1}{2}(c_{i}^{k}-L_{g_{i}})-\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}-\beta_{k+1}\Big{\\}},\\\
&b^{k}_{2}=\min\limits_{i}\Big{\\{}\beta_{k}-\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}}\Big{\\}}.\end{split}$

Proof: Refer to Appendix F.

######  Remark 11.

From Proposition 1, we have
$\\{\bm{\bar{X}}_{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}\in\bm{\mathcal{\bar{X}}}^{\eta}_{i}$.
To illustrate the convergence of PLDM over $k\geq\underline{k}$, we resort to
the Lyaponov function $\Phi_{\beta_{k}}(\bm{W}^{k})$. Proposition 6 provides
the lower bound for the decrease of the Lyapunov function over successive
iterations after iteration $\underline{k}$.

######  Proposition 7.

Let $\\{\bm{\lambda}^{k}\\}_{k\in\mathbb{N}}$ and
$\\{\bm{\bar{X}}^{k}\\}_{k\in\mathbb{N}}$ be the sequance generated by
Algorithm 1, then we have

$\begin{split}&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\Big{\|}\rightarrow
0.\\\
&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\Big{\|}\rightarrow
0,\\\
&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\lambda}_{i}^{k+1}-\bm{\lambda}_{i}^{k}\Big{\|}\rightarrow
0,\\\
&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\mu}_{i}^{k+1}-\bm{\mu}_{i}^{k}\Big{\|}\rightarrow
0,~{}\forall i\in\mathcal{N},\\\ \end{split}$ (20)

provided with Condition (a):

$\begin{split}&b^{k}_{1}=\min\limits_{i}\Big{\\{}\frac{1}{2}(c_{i}^{k}-L_{g_{i}})-\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}-\beta_{k+1}\Big{\\}}>0,\\\
&b^{k}_{2}=\min\limits_{i}\Big{\\{}\beta_{k}-\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}}\Big{\\}}>0.\end{split}$

is satisfied.

Proof: Refer to Appendix G.

######  Remark 12.

Proposition 7 illustrates the boundedness of primal and dual sequences
generated by Algorithm 1 under Condition (a), which can be satisfied by
selecting $\beta_{k}$ that
$\max_{i}\\{\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}}\\}<\beta_{k}<\min_{i}\\{\frac{1}{2}(c_{i}^{k-1}-L_{g_{i}})-\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}\\}$.

######  Proposition 8.

Let $\\{\bm{W}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ be the sequance
generated by Algorithm 1, we have

$\begin{split}\|\nabla\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\|&\leq
b^{k}_{3}\sum_{i=1}^{N}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|\\\
&+b^{k}_{4}\sum_{i=1}^{N}\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k-1}_{i}\|\\\
\end{split}$ (21)

where we have
$B=\sup_{k\geq\underline{k}}\|\bm{F}_{i}(\bm{\bar{X}}_{i}^{k+1})\|$.
$b^{k}_{3}=\max\limits_{i}\big{\\{}L_{g_{i}}+c^{k}_{i}+\Omega^{i}_{1}B+4\beta_{k+1}+\rho_{\underline{k}}+\frac{\Omega^{i}_{1}}{\rho_{\underline{k}}}\big{\\}}$
and
$b^{k}_{4}=\max\limits_{i}\big{\\{}\Omega^{i}_{2}B+\frac{\Omega^{i}_{2}}{\rho_{\underline{k}}}\big{\\}}$.

Proof: Refer to Appendix H.

######  Remark 13.

Proposition 8 provides the upper bound for the subgradient of the Lyapunov
function $\Phi_{\beta_{k}}$.

Based on Proposition1-9, we have the following results for the convergence of
the PLDM.

######  Theorem 1.

(Convergence) The PLDM described in Algorithm 1 converges to the critical
points of problem (5) provided with a step-size $c_{k}^{i}$ satisfying
Condition (a) in Proposition 7\.

###### Proof.

Based on _Proposition_ 7 and _Proposition_ 8, we have

$\lim_{k\rightarrow+\infty}\big{\|}\nabla\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\big{\|}\rightarrow
0.$ (22)

Specifically, we have

$\begin{split}\nabla_{\bm{\bar{X}}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})=&\\!\nabla_{\bm{\bar{X}}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\mu}^{k+1})\\\
&+2\beta_{k+1}(\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k})\\\
\nabla_{\bm{Z}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})=&\\!\nabla_{\bm{Z}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\mu}^{k+1})\\\
&\quad+\mathcal{N}_{\mathcal{\bm{X}}}(\bm{Z}^{k+1})\\\
\nabla_{\bm{\gamma}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})=&\\!\nabla_{\bm{\gamma}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\mu}^{k+1})\\\
\nabla_{\bm{U}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})=&2\beta_{k+1}(\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k})\\\
\end{split}$ (23)

By combining (22) with (23), we have

$\begin{split}&\lim_{k\rightarrow+\infty}\nabla_{\bm{\bar{X}}}~{}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1}.\bm{\lambda}^{k+1},\bm{\mu}^{k+1})=0.\\\
&\lim_{k\rightarrow+\infty}\nabla_{\bm{Z}}~{}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\mu}^{k+1})+\mathcal{N}_{\mathcal{\bm{X}}}(\bm{Z}^{k+1})=0.\\\
&\lim_{k\rightarrow+\infty}h_{i}(\bm{\bar{X}_{i}}^{k+1})=0,~{}\forall
i\in\mathcal{N}.\\\
&\lim_{k\rightarrow+\infty}\big{(}\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1}\big{)}=0,~{}\forall
i\in\mathcal{N}.\\\ \end{split}$

According to Definition 2, the above implies that PLDM will converge to the
critical points of problem (4).

∎

######  Theorem 2.

The critical points of problem (5) obtained from Algorithm 1 are the
$\epsilon$-critical points of problem (4).

###### Proof.

We first denote the critical points of problem (5) as
$\bm{W}^{*}=\big{(}\bm{\bar{X}}^{*},\\!\bm{Z}^{*},\\!\bm{\lambda}^{*},\\!\bm{\mu}^{*},\bm{\bar{X}}^{*}\big{)}$.
Based on Theorem 1, we have $\bm{W}^{k}\rightarrow\bm{W}^{*}$ with
$k\rightarrow+\infty$. Based on the definition of $\epsilon$-critical points
(see Definition 2), we only need to prove $\|\bm{Y}_{i}^{*}\|\leq\epsilon$ to
illustrate the results.

This can be illustrated in twofolds.

_i)_ according to _proposition_ 6, we have $\Phi_{\beta_{k}}(\bm{W}^{k})$ is
non-increasing w.r.t the iteration $k$. Therefore, we have
$\Phi_{\bar{\beta}}(\bm{W}^{*})\leq\Phi_{\beta_{0}}(\bm{W}^{0})$ with
$k\in\mathbb{N}$ (bounded), where we denote $\beta_{k}\rightarrow\bar{\beta}$
with $k\rightarrow+\infty$.

_ii)_ Based on Theorem 1, we have
$\Phi_{\bar{\beta}}(\bm{W}^{*})=\sum_{i=1}^{N}\tilde{f}_{i}(\bm{X}^{*}_{i})+\sum_{i=1}^{N}\phi_{i}(\bm{X}^{*}_{i})+\sum_{i=1}^{N}M_{i}\|\bm{Y}^{*}_{i}\|^{2}$
as $h_{i}(\bm{\bar{X}_{i}}^{*})=0$ and
$\big{(}\bm{A}_{i}\bm{\bar{X}}^{*}_{i}-\bm{E}_{i}\bm{Z}^{*}\big{)}=\bm{0}$
($\forall i\in\mathcal{N}$).

Thus by combining _i)_ and _ii)_ , we imply

$\|\bm{Y}^{*}_{i}\|\leq\sqrt{\frac{\Phi_{\beta_{0}}(\bm{W}^{0})-\sum_{i=1}^{N}\tilde{f}(\bm{X}^{*}_{i})-\sum_{i=1}^{N}\phi_{i}(\bm{X}^{*}_{i})}{\min\limits_{i}M_{i}\cdot
N}}\leq\epsilon,$

$~{}\forall i\in\mathcal{N}$ with $M_{i}$ sufficiently large.

∎

###  IV-C Convergence Rate

With the global convergence of the PLDM studied, this section discusses the
local convergence rate of the method. Before we present the main results, we
illustrate the following propositions that referred to.

######  Proposition 9.

Let $\\{\bm{W}^{k}\\}_{k\in\mathbb{N},k\geq\underline{k}}$ be the sequance
generated by Algorithm 1, then we have

$\begin{split}\Big{[}\textrm{dist}&\big{(}\nabla\Phi_{\beta_{k+1}}(\bm{W}^{k+1}),0\big{)}\Big{]}^{2}\\\
&\\!\leq\\!{2N}{\nu_{k}}\big{(}\Phi_{\beta_{k}}(\bm{W}^{k})-\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\big{)}\end{split}$
(24)

provided with the step-size $c^{k}_{i}$ and the parameter $\beta_{k}$
satisfying Condition (b):

$\nu_{k}=\frac{(b^{k}_{4})^{2}}{b^{k}_{2}}\leq\frac{(b^{k}_{3})^{2}}{b^{k}_{1}}$

where $\nu_{k}$ is the positive parameter that need to be decided for
Algorithm 1.

######  Remark 14.

Proposition 9 gaps the subgradients of the Lyapunov function
$\Phi_{\beta_{k+1}}(\bm{W}^{k+1})$ in terms of its value decrease over
successive iterations. To guarantee Condition (b), the step-size $c^{k}_{i}$
and the parameter $\beta_{k}$ can be selected following the procedures below:

Based on Proposition 6 and Proposition 8, we have

$\begin{split}&b^{k}_{1}=\min\limits_{i}\Big{\\{}\frac{1}{2}(c_{i}^{k}-L_{g_{i}})-\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}-\beta_{k+1}\Big{\\}}\\\
&\quad\quad=\frac{1}{2}c^{k}_{i}-\max\limits_{i}\Big{\\{}\frac{1}{2}L_{g_{i}})+\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}+\beta_{k+1}\Big{\\}}\\\
&\quad\quad=\frac{1}{2}c^{k}_{i}-M\\\ \end{split}$
$\begin{split}&b^{k}_{2}=\min\limits_{i}\Big{\\{}\beta_{k}-\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}}\Big{\\}}\\\
&b^{k}_{3}=\max\limits_{i}\Big{\\{}L_{g_{i}}\\!+\\!c^{k}_{i}\\!+\\!\Omega^{i}_{1}B+4\beta_{k+1}\\!+\\!\rho_{\underline{k}}\\!+\\!\frac{\Omega^{i}_{1}}{\rho_{\underline{k}}}\Big{\\}}\\\
&\quad\quad=c_{i}^{k}+\max\limits_{i}\Big{\\{}L_{g_{i}}\\!+\\!\Omega^{i}_{1}B+4\beta_{k+1}\\!+\\!\rho_{\underline{k}}\\!+\\!\frac{\Omega^{i}_{1}}{\rho_{\underline{k}}}\Big{\\}}\\\
&\quad\quad=c_{i}^{k}+L\\\
&b^{k}_{4}=\max\limits_{i}\big{\\{}\big{(}B+\frac{1}{\rho_{\underline{k}}}\big{)}\Omega^{i}_{2}\big{\\}}\end{split}$

where we have
$M=\max\limits_{i}\Big{\\{}\frac{1}{2}L_{g_{i}}+\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}+\beta_{k+1}\Big{\\}}$
and
$L=\max\limits_{i}\Big{\\{}L_{g_{i}}\\!+\\!\Omega^{i}_{1}B+4\beta_{k+1}\\!+\\!\rho_{\underline{k}}\\!+\\!\frac{\Omega^{i}_{1}}{\rho_{\underline{k}}}\Big{\\}}$.

We first select a large enough positive parameter $\nu_{k}>0$. By letting
$\nu_{k}=\frac{(b^{k}_{4})^{2}}{b^{k}_{2}}=\frac{\max\limits_{i}\big{\\{}\big{(}B+\frac{1}{\rho_{\underline{k}}}\big{)}^{2}(\Omega^{i}_{2})^{2}\big{\\}}}{\min\limits_{i}\Big{\\{}\beta_{k}-\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}}\Big{\\}}}$,
we can set

$\begin{split}\beta_{k}&=2\Big{(}\frac{(B+\frac{1}{\rho_{\underline{k}}})^{2}}{\nu_{k}}+\frac{1}{\rho_{\underline{k}}}\Big{)}\max_{i}\big{\\{}(\Omega^{i}_{2})^{2}\big{\\}}\\\
&=2\frac{(1+B\rho_{\underline{k}})^{2}+\nu_{k}}{\nu_{k}\rho_{\underline{k}}}\max_{i}\big{\\{}(\Omega^{i}_{2})^{2}\big{\\}}.\end{split}$

Besides, if we select $\nu_{k}=\nu_{\underline{k}}$, we have
$\beta_{k}=\beta_{\underline{k}}~{}~{}\forall k\geq\underline{k}.$

Afterwards, by letting $\nu_{k}\geq(b^{k}_{3})^{2}/b^{k}_{1}$, i.e.,

$\begin{split}\nu_{k}\geq\frac{(c^{k}_{i}+L)^{2}}{\frac{1}{2}c^{k}_{i}-M},\end{split}$
(25)

we can select the step-size $c^{k}_{i}$ at each iteration $k$ as
$c^{k}_{i}\in\Big{[}\frac{(\nu_{k}-4M)-\sqrt{(\nu_{k})^{2}-(8M+16L)\nu_{k}}}{4},$
$\frac{(\nu_{k}-4M)+\sqrt{(\nu_{k})^{2}-(8M+16L)\nu_{k}}}{4}\Big{]}$
$\cap(0,+\infty)$. Besides, to ensure that at least one positive step-size
exists, the parameter $\nu_{k}$ should be selected as $\nu_{k}\geq(8M+16L)$
($\forall k\geq\underline{k}$).

######  Theorem 3.

(Convergence rate) Suppose the Lynapunov function
$\Phi_{\beta_{k}}(\bm{W}^{k})$ satisfy the KL property111Note that KL property
is general for most analytical functions  with the desingularising function of
the form $\varphi(t)=\frac{C}{\theta}t^{\theta}$ for some $C>0$,
$\theta\in(0,1]$ ([46]) and $c^{k}_{i}$, $\beta_{k}$ are selected satisfying
Condition (b) in Proposition 9, we have:

  * (i)

If $\theta=1$, Algorithm 1 terminates in finite iterations.

  * (ii)

If $\theta\in(0,1/2)$, $\exists~{}k_{0},k_{1},k_{2}\in\mathbb{R}$, such that

$\forall\max\\{k_{0},k_{1}\\}\leq k\leq k_{2}$, we have

$\Delta^{k}\leq\Big{(}\frac{2NC^{2}\nu_{k-1}}{1+2NC^{2}\nu_{k-1}}\Big{)}^{\frac{1}{1+2\theta}}\Delta^{k-1}.$

$\forall k\geq\max\\{k_{0},k_{1},k_{2}\\}$, we have

$\Delta^{k}\leq\big{(}\frac{2NC^{2}\nu_{k-1}}{1+2NC^{2}\nu_{k-1}}\big{)}^{\frac{1}{2-2\theta}}\big{(}\Delta^{k-1}\big{)}^{\frac{1}{2-2\theta}}.$

  * (iii)

If $\theta\in[1/2,1)$, $\exists~{}k_{3}\in\mathbb{N}$, $\forall k\geq k_{3}$,
such that

$\Delta^{k}\leq\frac{2NC^{2}\nu_{k-1}}{1+2NC^{2}\nu_{k-1}}\cdot\Delta^{k-1}$

where $\Delta^{k}=\Phi_{\beta_{k}}(\bm{W}^{k})-\Phi^{*}$ denotes the
“suboptimality” (critial points or KKT points) at iteration $k$, with
$\Phi^{*}$ as the “optimal” value of the Lynapunov function.

###### Proof.

We define the set of critical point for the Lyapunov function
$\Phi_{\beta_{k}}(\bm{W}^{k})$ as $\bm{\omega}(\bm{W}^{0})$ with
$\bm{W}^{0}=(\bm{X}^{0},\bm{X}^{0},\bm{Z}^{0},\bm{\lambda}^{0},\bm{\mu}^{0})$
as any given start point.

According to Theorem 1, we conclude that the sequence
$\\{\bm{W}^{k}\\}_{k\in\mathbb{N}}$ generated by Algorithm 1 from any given
initial point $\bm{W}^{0}$ will converge to the set of critical points denoted
by $\omega(\bm{W}^{0})\neq\varnothing$.

Based on Proposition 7, $\\{\bm{W}^{k}\\}_{k\in\mathbb{N}}$ is bounded
(convergent), and for any given initial point $\bm{W}^{0}$, $k\in+\infty$,
there exist a limit point
$\overline{\bm{W}}=(\bm{\bar{X}},\bm{\bar{Z}},\bm{\bar{\lambda}},\bm{\bar{\mu}},\bm{\bar{X}})$.

Since the Lyapunov function is continuous, we have

$\begin{split}\lim_{k\rightarrow+\infty}\Phi_{\beta_{k}}(\bm{W}^{k})=\Phi_{\bar{\beta}}(\bm{\overline{W}})\end{split}$
(26)

As illustrated, $\Phi_{\beta_{k}}(\bm{W}^{k})$ is lower bounded, i.e.,
$\lim\limits_{k\rightarrow+\infty}\Phi_{\beta_{k}}(\bm{W}^{k})\\!\\!>\\!\\!\\!-\infty$
and non-increasing w.r.t the iteration $k$ (Proposition 6), thus we have

$\begin{split}\lim_{k\rightarrow+\infty}\Phi_{\beta_{k}}(\bm{W}^{k})=\Phi_{\bar{\beta}}(\bm{\bar{W}})=\underline{\Phi}\end{split}$
(27)

where $\underline{\Phi}$ denotes the limit of $\Phi_{\beta_{k}}(\bm{W}^{k})$
from the initial point $\bm{W}^{0}$.

The above implies that $\Phi_{\beta_{k}}(\bm{W}^{k})$ is constant on
$\omega(\bm{W}^{0})$.

From Theorem 1, we have

$\lim_{k\rightarrow+\infty}\textrm{dist}\big{(}\nabla\Phi_{\beta_{k}}(\bm{W}^{k}),0\big{)}=0$

That means $\forall\varepsilon$, $\exists k_{0}\in\mathbb{N}$ that $\forall
k\geq k_{0}$,

$\textrm{dist}\Big{(}\nabla\Phi_{\beta_{k}}(\bm{W}^{k}),0\Big{)}\leq\epsilon$

For notation, we denote $\Phi^{*}$ as the limit of
$\Phi_{\beta_{k}}(\bm{W}^{k})$ on $\omega(\bm{W}^{0})$.
$\Phi_{\beta_{k}}(\bm{W}^{k})$ is non-increasing w.r.t. the iteration $k$,
thus $\forall\eta\geq 0$, there exist $k_{1}\in\mathbb{N}$ such that $\forall
k\geq k_{1}$, we have

$\Phi_{\beta_{k}}(\bm{W}^{k})\leq\Phi^{*}+\eta$

Further, $\forall k\geq\max\big{\\{}k_{0},k_{1}\big{\\}}$, we have

$\begin{split}\bm{W}^{k}\in&\Big{\\{}\bm{W}:\textrm{dist}\big{(}\nabla\Phi_{\beta_{k}}(\bm{W}^{k}),0\big{)}\leq\epsilon\Big{\\}}\\\
&\quad\cap\Big{\\{}\bm{W}:\Phi^{*}<\Phi_{\beta_{k}}(\bm{W}^{k})<\Phi^{*}+\eta\Big{\\}}=\Omega_{\varepsilon,\eta}\end{split}$
(28)

Since the Lyapunov function $\Phi_{\beta_{k}}(\bm{W}^{k})$ posess the KL
property, based on Definition 4, we have

$\begin{split}1\leq\Big{[}\varphi^{{}^{\prime}}\big{(}\Phi_{\beta_{k}}(\bm{W}^{k})-\Phi^{*}\big{)}\Big{]}^{2}\Big{[}\textrm{dist}\big{(}0,\nabla\Phi_{\beta_{k}}(\bm{W}^{k})\big{)}\Big{]}^{2}\end{split}$
(29)

By combining (29) with Proposition 9, we have

$\begin{split}1\\!\leq\\!\Big{[}\varphi^{{}^{\prime}}\\!\big{(}\Phi_{\beta_{k}}(\bm{W}^{k})\\!-\\!\Phi^{*}\big{)}\Big{]}^{2}\\!{2N}{\nu_{k\\!-\\!1}}\Big{[}\Phi_{\beta_{k}}(\bm{W}^{k\\!-\\!1})\\!-\\!\Phi_{\beta_{k\\!+\\!1}}(\bm{W}^{k})\Big{]}\end{split}$
(30)

Rearrange (30) based on $\Delta^{k}=\Phi_{\beta_{k}}(\bm{W}^{k})-\Phi^{*}$, we
have

$\begin{split}1&\leq[\varphi^{{}^{\prime}}(\Delta^{k})]^{2}\cdot{2N}{\nu_{k-1}}\big{[}\Delta^{k-1}-\Delta^{k}\big{]}\end{split}$
(31)

Regarding the desingularising function, we have
$\varphi^{{}^{\prime}}(t)=Ct^{\theta-1}$. Therefore, we can derive from (31)
that

$1\leq{2NC^{2}}{\nu_{k-1}}(\Delta^{k})^{2(\theta-1)}\cdot\big{[}\Delta^{k-1}-\Delta^{k}\big{]}.$

_(i)_ If $\theta=1$, we have

$\begin{split}1\leq{2NC^{2}}{\nu_{k-1}}\cdot\big{[}\Delta^{k-1}-\Delta^{k}\big{]}\leq
0\end{split}$ (32)

This above inequality is a contradiction and implies that the set
$\Omega_{\varepsilon,\eta}=\varnothing$. In other word, $\forall
k\geq\max\\{k_{0},k_{1}\\}$, we have $\Delta^{k}=0$, i.e., Algorithm 1
terminates in finite iterations.
_(ii)_ $\theta\in(0,1/2)$, we have $1<2-2\theta<2$.

$\begin{split}[\Delta^{k}]^{2}&\leq{2NC^{2}}{\nu_{k-1}}\cdot[\Delta^{k-1}-\Delta^{k}]\cdot(\Delta^{k})^{2\theta}\\\
&\leq{2NC^{2}}{\nu_{k-1}}\Delta^{k-1}(\Delta^{k})^{2\theta}-{2NC^{2}}{\nu_{k-1}}(\Delta^{k})^{1+2\theta}\\\
\end{split}$ (33)

Thus, we have

$\begin{split}[\Delta^{k}]^{2}&+{2NC^{2}}{\nu_{k-1}}(\Delta^{k})^{1+2\theta}\\\
&\leq{2NC^{2}}{\nu_{k-1}}\Delta^{k-1}\cdot(\Delta^{k})^{2\theta}\\\
\end{split}$ (34)

Since $\Delta^{k}$ is non-increasing (Proposition 6), there exist
$k_{3}\in\mathbb{R}$ that $\forall k\leq k_{2}$, we have
$\Delta^{k-1}\geq\Delta^{k}\geq 1$.

In this case, it’s easy to figure out that

$1<1+2\theta<2,~{}~{}0<2\theta<1$
$(\Delta^{k})^{2}\geq(\Delta^{k})^{1+2\theta}$
$(\Delta^{k})^{2\theta}\leq(\Delta^{k-1})^{2\theta}$

Therefore, we can derive from (34) that

$\begin{split}&(1+{2NC^{2}}{\nu_{k-1}})(\Delta^{k})^{1+2\theta}\leq\\\ &\
[\Delta^{k}]^{2}+{2NC^{2}}{\nu_{k-1}}(\Delta^{k})^{1+2\theta}\leq{2NC^{2}}{\nu_{k-1}}\Delta^{k-1}(\Delta^{k})^{2\theta}\\\
&\quad\quad\quad\leq{2NC^{2}}{\nu_{k-1}}(\Delta^{k-1})^{1+2\theta}\\\
\end{split}$

In this case, $\forall\max\\{k_{0},k_{1}\\}\leq k\leq k_{2}$, we have

$\begin{split}\Delta^{k}\leq\Big{(}\frac{2NC^{2}\nu_{k-1}}{1+2NC^{2}\nu_{k-1}}\Big{)}^{\frac{1}{1+2\theta}}\Delta^{k-1}\end{split}$
(35)

This implies Algorithm 1 shows linear convergence rate.

As mentioned, $\forall k\geq k_{2}$, we have $0\leq\Delta_{k}<1$. In this
case, it’s straight forward that

$1<1+2\theta<2$ $(\Delta^{k})^{2}\leq(\Delta^{k})^{1+2\theta}$

Similarly, we can derive from (34) that

$\begin{split}&(1+{2NC^{2}}{\nu_{k-1}})(\Delta^{k})^{2}\leq[\Delta^{k}]^{2}+{2NC^{2}}{\nu_{k-1}}(\Delta^{k})^{1+2\theta}\\\
&\quad\quad\quad\leq{2NC^{2}}{\nu_{k-1}}\Delta^{k-1}(\Delta^{k})^{2\theta}\\\
\end{split}$ (36)

This implies that

$\begin{split}&(1+{2NC^{2}}{\nu_{k-1}})(\Delta^{k})^{2-2\theta}\leq{2NC^{2}}{\nu_{k-1}}\Delta^{k-1}\\\
\end{split}$ (37)

Equivalently, $\forall k\geq\max\\{k_{0},k_{1},k_{2}\\}$, we have

$\begin{split}&\Delta^{k}\leq\Big{(}\frac{2NC^{2}\nu_{k-1}}{1++2NC^{2}\nu_{k-1}}\Big{)}^{\frac{1}{2-2\theta}}\big{(}\Delta^{k-1}\big{)}^{\frac{1}{2-2\theta}}\\\
\end{split}$ (38)

_(iii)_ $\theta\in[1/2,1)$, we have $0\\!<\\!2-2\theta\\!\leq\\!1$. As
$\Delta^{k}$ is non-increasing, there exist $k_{3}\in\mathbb{R}$ that $\forall
k\geq k_{3}$, we have

$(\Delta^{k})^{2(1-\theta)}\geq\Delta^{k}$

Therefore, $\forall k\geq\max\\{k_{0},k_{1},k_{3}\\}$, we have

$\Delta^{k}\leq{2NC^{2}}{\nu_{k-1}}\cdot\big{(}\Delta^{k-1}-\Delta^{k}\big{)}$

i.e.,

$\Delta^{k}\leq\frac{2NC^{2}\nu_{k-1}}{1+2NC^{2}\nu_{k-1}}\cdot\Delta^{k-1}$

In this case, Algorithm 1 presents linear convergence rate.

∎

According to Theorem 1, the step-size should be selected to guarantee
Condition (a). Nevertheless this is a challenging task to achieve efficiently
in practice. To reduce computation and facilitate implementation, we can
select the step-size by the backward linesearch procedures in Algorithm 2.
$\alpha$ denotes a small positive value that can be selected adaptively
according to the problems.

Algorithm 2 Backward Linesearch for Stepsize

1:Initialization: $c_{i}^{0}$.

2:Repeat:

3:If
$\displaystyle\begin{split}&g_{i}(\bm{\bar{X}}_{i}^{k+1},\bm{\lambda}^{k},\rho_{k})\\!+\\!\alpha\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}_{i}^{k}\|^{2}\\!>\\!g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k},\rho_{k})\\\
&\\!+\\!\langle\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\\!\bm{\lambda}^{k},\\!\rho_{k}),\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{\bar{X}}_{i}^{k}\rangle\\!+\\!\frac{c_{i}^{k}}{2}\big{\|}\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{\bar{X}}_{i}^{k}\big{\|}^{2}\\\
\end{split}$ (39) set $c^{k+1}_{i}=c^{i}_{k}/\delta$ and go to Step 2,
otherwise stop.

4:Output $c^{k}_{i}$.

######  Corollary 1.

Suppose the step-size $c^{k}_{i}$ at each iteration $k$ is selected according
to Algorithm 2, the PLDM converges to the critical points of Problem (5).

###### Proof.

Refer to Appendix C. ∎

##  V Numeric Experiments

This section illustrates the PLDM’s performance on (i) a simple numerical
example with 2 agents and (ii) the application to multi-zone HVAC control. The
numerical example helps illustrate the theoretical analysis and the
application demonstrates its practical capabilities.

###  V-A A Numerical Example

We consider a nonlinear and non-convex example with 2 agents given by

$\begin{split}\min_{x_{1},x_{2}}~{}~{}x_{1}&+x_{2}+\frac{1}{2}x_{1}\cdot
x_{2}^{2}+\frac{1}{2}x_{1}\cdot x_{2}^{2}\\\ s.t.&~{}~{}x_{1}\cdot x_{2}=2.\\\
&~{}~{}~{}x_{1}^{2}+x_{2}^{2}=5.\\\ &~{}~{}~{}0\leq x_{1}\leq 4.\\\
&~{}~{}~{}0\leq x_{2}\leq 5.\end{split}$ (40)

The problem is well-posed with the (local) optima $(2,1)$. The problem is
solved by using PLDM in a decentralized manner and our analysis is presented
here.

$\begin{split}\bm{F}_{i}(\bm{\bar{X}}_{i})&=\left(\begin{array}[]{ccc}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})&\\!\\!&\bm{O}_{n_{i}}\\\
&\bm{A}_{i}&\\\ \end{array}\right)\\\
&=\left(\begin{array}[]{cccc}\bm{X}_{i}(2)&\bm{X}_{i}(1)&0&0\\\
2\bm{X}_{i}(1)&2\bm{X}_{i}(2)&0&0\\\ 1&0&1&0\\\ 0&1&0&1\\\
\end{array}\right),~{}i=\\{1,2\\}.\end{split}$

Note that $\bm{F}_{i}(\bm{\bar{X}}_{i})$ has row full rank and regular with
$\bm{X}_{i}\neq\bm{0}$. This implies the satisfaction of assumption _(A4)_ ,
which closely related to the convergence. The trajectory of the local decision
variable sequence $\\{\bm{X}_{1}\\}_{k\in\mathbb{R}}$ is shown in Fig. 2. Fig.
3 shows the subgradients of AL function, i.e.,
$\|\nabla\mathbb{L}_{\rho_{k}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})\|$
and the Lyapunov function, i.e., $\|\nabla\Phi_{\beta_{k}}(\bm{W}^{k})\|$
converging to zero w.r.t the iteration $k$. This additionally demonstrates the
PLDM’s convergence property illustrated in Theorem 1.

In addition, we study the PLDM’s convergence rate by closely inspecting the
Lyapunov function $\Phi_{\beta_{k}}(\bm{W}^{k})$ w.r.t the iteration $k$. As
shown in Fig. 4, we see that the Lyapunov function
$\Phi_{\beta_{k}}(\bm{W}^{k})$ approximately decrease w.r.t the iteration $k$
until the optima $\Phi^{*}$ is reached.

Figure 2:  the trajectory of the sequence $\\{\bm{X}_{1}\\}_{k\in\mathbb{N}}$
(the feasible points include $(1,2)$ and $(2,1)$ with the (local) optima
$(2,1)$).

Figure 3:  (a) the subgradients $\|\nabla\Phi_{\beta_{k}}(\bm{W}^{k})\|$ w.r.t
iteration k. (b) the subgradients
$\|\nabla\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})\|$
w.r.t iteration $k$.

Figure 4:  (a) the residual error $R(k)$ w.r.t iteration k. (b) the Lynapunov
function $\Phi_{\beta_{k}}(\bm{W}^{k})$ w.r.t iteration $k$.

###  V-B Application: Multi-zone HVAC Control

This section presents the PLDM’s application to multi-zone HVAC control, which
has raised extensive discussion among the research communities. The general
formulation for designing a model predictive controller (MPC) to minimize the
HVAC’s energy cost while guaranteeing thermal comfort is exhibited in (41)
with constraints imposed by _(i)_ zone thermal dynamics (coupled, nonlinear
and nonvex) (41a), _(ii)_ comfortable temperature ranges (41b), and _(iii)_
operation limits of the local variable air box (41c). Readers are referred to
[47, 28, 48] for the detailed problem formulation and notations.

$\displaystyle\min_{m^{zi}_{t},T^{i}_{t}}J=\sum_{t=0}^{H-1}c_{t}\cdot\Big{\\{}c_{p}(1-d_{r})\sum_{i=1}^{I}m^{zi}_{t}(T^{o}_{t}-T^{c}_{t})\\!\quad\quad\quad$
$\displaystyle\quad+c_{p}d_{r}\sum_{i=1}^{I}m^{zi}_{t}(T^{i}_{t}-T^{c}_{t})+\kappa_{f}\cdot(\sum_{i=1}^{I}{m^{zi}_{t}})^{3}\Big{\\}}\cdot\Delta_{t}$
(40) $\displaystyle s.t.~{}$ $\displaystyle
T^{i}_{t+1}\\!=\\!A^{ii}T^{i}_{t}\\!+\\!\sum_{j\in\mathcal{N}_{i}}A^{ij}T^{j}_{t}\\!+\\!C^{ii}m^{zi}_{t}(T^{i}_{t}\\!-\\!T^{c}_{t})+D^{ii}_{t},$
(41a) $\displaystyle\quad\underline{T}^{i}\leq T^{i}_{t}\leq\overline{T}^{i},$
(41b) $\displaystyle\quad\underline{m}^{zi}\leq
m^{zi}_{t}\leq\overline{m}^{zi},\quad\forall
i\in\mathcal{I},~{}t=0,1,\cdots,H.$ (41c)

One can see that the control of the multi-zone HVAC system requires solving a
nonlinear and nonconvex optimization problem with coupled nonlinear
constraints. Centralized optimization methods are not scalable or
computationally viable and therefore decentralized methods become imperative.
However, the existing decentralized methods can not be adapted due to the non-
linearity and non-convexity both from the objective function and the
constraints.

In this part, we resort to the proposed PLDM. We first consider a building
with $I=10$ zones and randomly generate a network to denote the thermal
couplings among the different zones. The comfortable zone temperature ranges
and the maximum zone mass flow rate are set as $[24,26]^{o}C$ and
$\bar{m}^{zi}=0.5kg/s$. The other parameter settings can refer to [47]. When
the PLDM method is applied, the zone temperature and zone mass flow rates for
two randomly picked zones (Zone 1 and 8) are shown in Fig. 5 (a) and 5 (b). We
note that both the zone temperature and zone mass flow rates are maintained
within the setting ranges. This implies the feasibility of the solutions.
Besides, by observing the residual error w.r.t the iterations in Fig. 5 (c),
we can imply the fast convergence rate of the PLDM. Further, to evaluate the
sub-optimality of local optima under the PLDM with some initial start points,
we compare the results with a centralized method in case studies with
$I\in\\{10,20\\}$ zones. In the centralized method, the optimal solutions can
be obtained by solving the non-linear and non-convex problem (41) using Ipopt
solver. As shown in Fig. 5 (d), while benchmarking the centralized method
(optimal cost is $1$), the sub-optimality of the local optima under the PLDM
is less than $10$%. This implies that the PLDM can approach a satisfactory
local optima for non-linear and non-convex problems for some randomly picked
initial points. In principle, the global optima for the non-linear problems
can be approached by scattering enough initial points. However, some local
optima with favorable performance and feasibility guarantee are generally
enough in practice.

Figure 5:  (a)Zone temperature. (b) Zone mass flow rate. (c) Residual error of
the constraints. (d) Comparison of PLDM vs. Centralized Method.

##  VI Conclusion

This paper investigated decentralized optimization on a class of non-convex
problems structured by: (i) nonconvex global objective function (possibly
nonsmooth) and (ii) local bounded convex constraints and coupled nonlinear
constraints. Such problems arise from a variety of applications (e.g., smart
grid and smart buildings, etc), which demand for efficient and scalable
computation. To solve such challenging problems, we proposed a _Proximal
Linearization-based Decentralized Method_ (PLDM) under the augmented
Lagrangian-based framework. Deviating from conventional augmented Lagrangian-
based methods which require exact joint optimization of different decision
variable blocks (local optima), the proposed method capitalized on _proximal
linearization_ technique to update the decision variables at each iteration,
which makes it computationally efficient in the presence of non-linearity.
Both the global convergence and the local convergence rate of the PLDM’s to
the $\epsilon$-critical points of the problem were discussed based on the
Kurdyka-Łojasiewicz property and some standard assumptions. In addition, the
PLDM’s performance was illustrated on academic and application examples in
which the convergence was observed. Applying PLDM to emerging applications and
eliminating consensus variables are the future work to be explored.

##  Appendix A Proof of Proposition 1-8

###  A-A Proof of Proposition 1

###### Proof.

We illustrate the proposition by contradiction. We assume $\forall
k\in\mathbb{N}$, we have

$\begin{split}&\sum_{i=1}^{N}\|h_{i}(\bm{X}^{k}_{i})\|>\eta>0\end{split}$ (42)

In this case, we have $\rho_{k}$ kept increasing, i.e,
$\rho_{k}\rightarrow+\infty$. Based on
$\bm{\lambda}^{k}_{i}=\bm{\lambda}^{k-1}_{i}+\rho_{k-1}h_{i}(\bm{X}^{k}_{i})$,
we have

$\begin{split}&\sum_{i=1}^{N}\|h_{i}(\bm{X}^{k}_{i})\|\\!=\\!\frac{1}{\rho_{k-1}}\sum_{i=1}^{N}\|\bm{\lambda}^{k}_{i}-\bm{\lambda}^{k-1}_{i}\|\leq\frac{\sum_{i=1}^{N}M_{\bm{\lambda}_{i}}}{\rho_{k-1}}\end{split}$

($\\{\bm{\lambda}^{k}_{i}\\}_{k\in\mathcal{N}}$ is assumed bounded, see (A6))

The above implies that

$\lim_{k\rightarrow+\infty}\sum_{i=1}^{N}\|h_{i}(\bm{X}^{k}_{i})\|\rightarrow
0$ (43)

One may note that (42) and (43) contradict with each other. Thus, we conclude
that there exist $\underline{k}$ such that

$\begin{split}&\sum_{i=1}^{N}\|h_{i}(\bm{X}^{k}_{i})\|\leq\frac{\sum_{i=1}^{N}M_{\bm{\lambda}_{i}}}{\rho_{k-1}}\leq\eta,~{}~{}\forall
k\geq\underline{k}.\\\
&\\{\bm{\bar{X}}^{k}_{i}\\}_{k\in\mathbb{N},k\geq\underline{k}}\in\bm{\mathcal{\bar{X}}}^{\eta}_{i}~{}(i\in\mathcal{N}).\end{split}$

Further, based on the adaptive procedure of PLDM, we have

$\rho_{k}=\rho_{\underline{k}},~{}~{}\forall k\geq\underline{k}.\\\ $

∎

###  A-B Proof of Proposition 2

###### Proof.

Since $h_{i}$ is twice differentiable, we have $h_{i}(\bm{X}_{i})$ and
$\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})$ are continuous and bounded over the
bounded set $\bm{\mathcal{\bar{X}}}^{\eta}_{i}$. In addition, we have
$\rho_{k}\leq\rho_{\underline{k}}$ ($\forall k\in\mathbb{N}$), thus

$\big{\|}\rho_{k}\big{(}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})\big{)}^{T}h_{i}(\bm{X}_{i})\big{\|}\leq\rho_{\underline{k}}C_{h_{i}}\leq+\infty~{}~{}(\forall
k\in\mathbb{N})$

where $C_{h_{i}}$ is upper bound of
$\|(\nabla_{\bm{X}_{i}}\\!h_{i}(\bm{X}_{i}))^{T}h_{i}(\bm{X}_{i})\|$ over
$\bm{\mathcal{\bar{X}}}^{\eta}_{i}$.

Besides, we have

$\begin{split}&\nabla_{\bm{X}_{i}}g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}^{k}_{i},\rho_{k})=\nabla_{\bm{X}_{i}}\tilde{f}_{i}(\bm{X}_{i})+\nabla_{\bm{X}_{i}}\phi_{i}(\bm{X}_{i})\\\
&\quad+\big{(}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})\big{)}^{T}\bm{\lambda}^{k}_{i}+\rho_{k}\big{(}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})\big{)}^{T}h_{i}(\bm{X}_{i}).\\\
&\nabla_{\bm{Y}_{i}}g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}^{k}_{i},\rho_{k})=2M_{i}\bm{Y}_{i}.\end{split}$
(44)

As $\tilde{f}_{i}$, $\phi_{i}$ and $h_{i}$ are Lipschitz continuous w.r.t
$\bm{X}_{i}$ (or $\bm{\bar{X}}_{i}$) with constant $L_{f_{i}}$, $L_{\phi_{i}}$
and $L_{h_{i}}$, $\\{\bm{\lambda}^{k}_{i}\\}_{k\in\mathbb{N}}$ upper bounded
by $M_{\bm{\lambda}_{i}}$ (see _(A6)_), we conclude that

$\|\nabla_{\bm{X}_{i}}g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}^{k}_{i},\rho^{k})\|\leq
L_{g_{i}}~{}\textrm{over}~{}\bm{\mathcal{\bar{X}}}^{\eta}_{i},~{}\forall
k\in\mathbb{N}.$

where we have
$L_{g_{i}}=\max\\{L_{f_{i}}+L_{\phi_{i}}+M_{{\lambda}_{i}}L_{h_{i}}+\rho_{\underline{k}}C_{h_{i}},2M_{i}\\}$.

That is, $g_{i}(\bm{\bar{X}}_{i},\bm{\lambda}^{k}_{i},\rho_{k})$ ($\forall
k\in\mathbb{N}$) is Lipschitz continuous w.r.t $\bm{\bar{X}}_{i}$ over
$\bm{\mathcal{\bar{X}}}^{\eta}_{i}$ with constant $L_{g_{i}}$. ∎

###  A-C Proof of Proposition 3

###### Proof.

Based on subproblem (10), it’s straightforward that

$\displaystyle\begin{split}&\sum_{i=1}^{N}\big{(}\bm{\mu}^{k}_{i}\big{)}^{T}\big{(}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\big{)}\\!\\!+\\!\\!\sum_{i=1}^{N}\frac{\rho_{\underline{k}}}{2}\Big{\|}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k\\!+\\!1}\Big{\|}^{2}\\\
&\leq\sum_{i=1}^{N}\big{(}\bm{\mu}^{k}_{i}\big{)}^{T}\big{(}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k}\big{)}\\!\\!+\\!\\!\sum_{i=1}^{N}\frac{\rho_{\underline{k}}}{2}\Big{\|}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k}\Big{\|}^{2}\\\
\end{split}$ (45)

Subproblem (11) is equivalent to

$\begin{split}\bm{\bar{X}}^{k+1}_{i}\in{prox}^{\sigma_{i}}_{c^{k}_{i}}\Big{(}\bm{\bar{X}}^{k}_{i}-\frac{1}{c^{k}_{i}}\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})\Big{)}\end{split}$

where we have
$\sigma_{i}(\bm{\bar{X}}_{i})=(\bm{\mu}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}\bm{Z}^{k+1})+\frac{\rho_{\underline{k}}}{2}\|\bm{A}_{i}\bm{\bar{X}}_{i}-\bm{E}_{i}{\bm{Z}}^{k+1}\|^{2}$.

Thus, by invoking Lemma 2, we have

$\begin{split}&g_{i}(\bm{\bar{X}}^{k+1}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})+(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1})\\\
&\quad\quad+\frac{\rho_{\underline{k}}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\\
&\leq
g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})+(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{X}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1})\\\
&\quad\quad+\frac{\rho_{\underline{k}}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\\
&\quad\quad\\!-\\!\frac{1}{2}\big{(}c_{i}^{k}\\!-\\!L_{g_{i}}\big{)}\|\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i}\|^{2},~{}\forall
i\in\mathcal{N}.\\\ \end{split}$ (46)

By summing up (45) and (46) for $\forall i\in\mathcal{N}$, we have

$\begin{split}&\sum_{i=1}^{N}g_{i}(\bm{\bar{X}}^{k+1}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})\\!+\\!\\!\sum_{i=1}^{N}(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1})\\\
&\\!+\\!\sum_{i=1}^{N}\\!\frac{\rho_{\underline{k}}}{2}\\!\|\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}\\!\\!-\\!\\!\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\!\\!+\\!\\!\sum_{i=1}^{N}\\!\big{(}\bm{\mu}^{k}_{i}\big{)}^{T}\\!\\!\big{(}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!\\!-\\!\\!\bm{E}_{i}\bm{Z}^{k+1}\big{)}\\\
&+\sum_{i=1}^{N}\frac{\rho_{\underline{k}}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\\
&\leq\sum_{i=1}^{N}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})+\sum_{i=1}^{N}(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1})\\\
&\\!+\\!\sum_{i=1}^{N}\\!\\!\frac{\rho_{\underline{k}}}{2}\\!\|\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\!+\\!\sum_{i=1}^{N}\big{(}\bm{\mu}^{k}_{i}\big{)}^{T}\big{(}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k}\big{)}\\\
&+\sum_{i=1}^{N}\frac{\rho_{\underline{k}}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\\!\bm{E}_{i}\bm{Z}^{k}\|^{2}\\\
&+\sum_{i=1}^{N}\frac{1}{2}(c^{k}_{i}-L_{g_{i}})\|\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i}\|^{2}\\\
\end{split}$ (47)

Thus, Proposition 3 can be concluded by removing the same terms from both
sides of (47). ∎

###  A-D Proof of Proposition 4

###### Proof.

As subproblem (11) is QP, based on the first-order optimality condition and
$\bm{\mu}^{k+1}_{i}=\bm{\mu}^{k}_{i}+\rho_{k}(\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1})$,
we have

$\begin{split}\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},&\bm{\lambda}_{i}^{k},\rho_{\underline{k}})\\!+\bm{\mu}^{k+1}_{i}+c^{k}_{i}\big{(}\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{\bar{X}}^{k}_{i}\big{)}=0\\\
\end{split}$ (48)

Besides, we have

$\begin{split}\nabla_{\bm{\bar{X}}_{i}}\mathbb{L}_{\rho_{\underline{k}}}&(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})\\\
&=\nabla_{\bm{X}_{i}}g_{i}(\bm{\bar{X}}^{k+1}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})+\bm{\mu}^{k+1}_{i}\\\
\end{split}$ (49)

By combining (48) and (49), we have

$\begin{split}&\|\nabla_{\bm{\bar{X}}_{i}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})\|\\\
&=\|\nabla_{\bm{X}_{i}}g_{i}(\bm{\bar{X}}^{k+1}_{i},\bm{\lambda}_{i}^{k},\rho_{\underline{k}})-\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}_{i}^{k},\rho_{\underline{k}})\\\
&\quad-c^{k}_{i}\big{(}\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\big{)}\|\\\
&\leq\|\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k+1}_{i},\bm{\lambda}_{i}^{k},\rho_{\underline{k}})-\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}_{i}^{k},\rho_{\underline{k}})\|\\\
&\quad+c^{k}_{i}\|\big{(}\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\big{)}\|\\\
&\leq\big{(}L_{g_{i}}+c_{i}^{k}\big{)}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|\\\
\end{split}$ (50)

where the last inequality is concluded from Proposition 2. ∎

###  A-E Proof of Proposition 5

###### Proof.

For notation, we define

$\begin{split}\Delta^{\bm{\lambda}_{i}}_{k}&\\!=\\!\big{(}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}^{k+1}_{i})~{}\bm{O}_{\bar{n}_{i}}\big{)}^{T}{\bm{\lambda}}^{k+1}_{i}\\!-\\!\big{(}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}^{k}_{i})~{}\bm{O}_{\bar{n}_{i}}\big{)}^{T}{\bm{\lambda}}^{k}_{i}\\\
\Delta^{\bm{\mu}^{i}}_{k}&=(\bm{A}_{i})^{T}\bm{\mu}^{k+1}_{i}-(\bm{A}_{i})^{T}\bm{\mu}^{k}_{i}\\\
\end{split}$

First, we have

$\begin{split}\|&\Delta^{\bm{\lambda}_{i}}_{k}\\!+\\!\Delta_{k}^{\bm{\mu}^{i}}\|\\!=\\!\big{\|}\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k+1}_{i})\big{)}^{T}\bm{\gamma}^{k+1}_{i}\\!-\\!\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k}_{i})\big{)}^{T}\bm{\gamma}^{k}_{i}\big{\|}\\\
&=\|\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k+1}_{i})\big{)}^{T}\big{(}\bm{\gamma}^{k+1}_{i}-\bm{\gamma}^{k}_{i}\big{)}\\\
&\quad+\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k+1}_{i})\big{)}^{T}-\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k}_{i})\big{)}^{T}\bm{\gamma}^{k}_{i}\big{\|}\\\
&\geq\big{\|}\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k+1}_{i})\big{)}^{T}\big{(}\bm{\gamma}^{k+1}_{i}-\bm{\gamma}^{k}_{i}\big{)}\big{\|}\\\
&\quad-\big{\|}\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k+1}_{i})\big{)}^{T}-\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k}_{i})\big{)}^{T}\bm{\gamma}^{k}_{i}\big{\|}\\\
\end{split}$ (51)

where we have
$\bm{F}_{i}(\bm{\bar{X}}_{i})=\left(\begin{array}[]{ccc}\nabla_{\bm{X}_{i}}h_{i}(\bm{X}_{i})&\\!\\!&\bm{O}_{n_{i}}\\\
&\bm{A}_{i}&\\\ \end{array}\right)$ and
$\bm{\gamma}_{i}=\big{(}(\bm{\lambda}_{i})^{T}~{}(\bm{\mu}_{i})^{T})^{T}$ as
defined.

According to (A6), we have
$\|\bm{\gamma}^{k}_{i}\|^{2}=\|\bm{\lambda}^{k}_{i}\|^{2}+\|\bm{\mu}^{k}_{i}\|^{2}\leq(M_{\bm{\lambda}_{i}})^{2}+(M_{\bm{\mu}_{i}})^{2}$
($\forall k\in\mathbb{N}$). Thus

$\|\bm{\gamma}^{k}_{i}\|\leq M_{\bm{\gamma}_{i}}~{}\forall k\in\mathbb{N}$
(52)

with
$M_{\bm{\gamma}_{i}}=\sqrt{(M_{\bm{\lambda}_{i}})^{2}+(M_{\bm{\mu}_{i}})^{2}}.$

Besides, according to (A1), we have

$\displaystyle\begin{split}&\|\bm{F}_{i}(\bm{X}^{k+1}_{i})\\!\\!-\\!\\!\bm{F}_{i}(\bm{X}^{k}_{i})\|\\!\\!=\\!\\!\|\nabla_{\bm{X}_{i}}h_{i}(\bm{X}^{k+1}_{i})\\!\\!-\\!\\!\nabla_{\bm{X}_{i}}h_{i}(\bm{X}^{k}_{i})\|\\\
&\leq M_{h_{i}}\|\bm{X}^{k+1}_{i}-\bm{X}^{k}_{i}\|\leq
M_{h_{i}}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|,~{}\forall
k\geq\underline{k}.\end{split}$ (53)

Based on (52), (53) and (A5), we can derive from (51) that

$\begin{split}\|&\Delta^{\bm{\lambda}_{i}}_{k}\\!+\\!\Delta_{k}^{\bm{\mu}^{i}}\|\\!\geq\theta\|\bm{\gamma}^{k+1}_{i}\\!-\\!\bm{\gamma}^{k}_{i}\|-M_{h_{i}}M_{\bm{\gamma}_{i}}\|\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{\bar{X}}^{k}_{i}\|\end{split}$
(54)

Meanwhile, we have

$\begin{split}\|&\Delta^{\bm{\lambda}_{i}}_{k}\\!+\\!\Delta_{k}^{\bm{\mu}^{i}}\|\\!\\!=\\!\\!\big{\|}\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k+1}_{i})\big{)}^{T}\bm{\gamma}^{k+1}_{i}\\!-\\!\big{(}\bm{F}_{i}(\bm{\bar{X}}^{k}_{i})\big{)}^{T}\bm{\gamma}^{k}_{i}\big{\|}\\\
&=\|\big{(}\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k+1}_{i},\bm{\lambda}^{k}_{i},\rho_{\underline{k}})+\big{(}\bm{A}_{i}\big{)}^{T}\bm{\mu}^{i,k+1}\big{)}\\\
&\quad-\big{(}\nabla_{\bm{X}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k-1}_{i},\rho_{\underline{k}})+\big{(}\bm{A}_{i}\big{)}^{T}\bm{\mu}^{i,k}\big{)}\\\
&\quad+\big{(}\nabla_{\bm{X}_{i}}\tilde{f}_{i}(\bm{X}^{k}_{i})-\nabla_{\bm{X}_{i}}\tilde{f}_{i}(\bm{X}^{k+1}_{i})\big{)}\\\
&\quad+\big{(}\nabla_{\bm{X}_{i}}\phi_{i}(\bm{X}^{k}_{i})-\nabla_{\bm{X}_{i}}\phi_{i}(\bm{X}^{k+1}_{i})\big{)}\|\\\
\end{split}$
$\begin{split}&\leq\|\nabla_{\bm{\bar{X}}_{i}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})\|\\\
&\quad+\|\nabla_{\bm{\bar{X}}_{i}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k-1},\bm{\mu}^{k-1})\|\\\
&\quad+\|\nabla_{\bm{X}_{i}}\tilde{f}_{i}(\bm{X}^{k}_{i})-\nabla_{\bm{X}_{i}}\tilde{f}_{i}(\bm{X}^{k+1}_{i})\|\\\
&\quad+\|\nabla_{\bm{X}_{i}}\phi_{i}(\bm{X}^{k}_{i})-\nabla_{\bm{X}_{i}}\phi_{i}(\bm{X}^{k+1}_{i})\|\\\
&\leq\big{(}L_{g_{i}}+c_{i}^{k}+L_{f_{i}}+L_{\phi_{i}}\big{)}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|\\\
&\quad+\big{(}L_{g_{i}}+c^{i}_{k\\!-\\!1}\big{)}\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k\\!-\\!1}_{i}\|\\\
\end{split}$ (55)

where the last equality is derived from Proposition 4 and (A2)-(A3) .

By combining (54) and (55), we have

$\begin{split}\|\bm{\gamma}^{k+1}_{i}\\!\\!-\\!\\!\bm{\gamma}^{k}_{i}\|\leq\Omega^{i}_{1}\|\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{\bar{X}}^{k}_{i}\|\\!+\\!\Omega^{i}_{2}\|\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{\bar{X}}^{k-1}_{i}\|\\\
\end{split}$

where
$\Omega^{i}_{1}=\big{(}L_{g_{i}}+c_{i}^{k}+L_{f_{i}}+L_{\phi_{i}}+M_{h_{i}}M_{\bm{\gamma}_{i}}\big{)}/\theta$
and $\Omega^{i}_{2}=\big{(}L_{g_{i}}+c_{i}^{k\\!-\\!1})/\theta$
($i\in\mathcal{N}$). ∎

###  A-F Proof of Proposition 6

###### Proof.

First, based on
$h_{i}(\bm{X}^{k+1}_{i})=\frac{1}{\rho_{k}}(\bm{\lambda}^{k+1}_{i}-\bm{\lambda}^{k}_{i})$
and
$(\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1})=\frac{1}{\rho_{k}}(\bm{\mu}^{k+1}_{i}-\bm{\mu}^{k}_{i})$,
we have

$\begin{split}&\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\\!\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\mu}^{k+1})\\!-\\!\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\\!\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})\\\
&=\frac{1}{\rho_{\underline{k}}}\Big{(}\sum_{i=1}^{N}\|\bm{\lambda}^{k+1}_{i}-\bm{\lambda}_{i}^{k}\|^{2}+\sum_{i=1}^{N}\|{\bm{\mu}_{i}}^{k+1}-\bm{\mu}_{i}^{k}\|^{2}\Big{)}\\\
&=\frac{1}{\rho_{\underline{k}}}\Big{(}\sum_{i=1}^{N}\|\bm{\gamma}^{k+1}_{i}-\bm{\gamma}^{k}_{i}\|^{2}\Big{)}\\\
&\leq\sum_{i=1}^{N}\Big{\\{}\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}\|\bm{\bar{X}}^{k+1}_{i}\\!\\!-\\!\\!\bm{\bar{X}}^{k}_{i}\|^{2}\\!\\!+\\!\\!\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}}\|\bm{\bar{X}}^{k}_{i}\\!\\!-\\!\\!\bm{\bar{X}}^{k-1}_{i}\|^{2}\Big{\\}}\end{split}$
(56)

where the last inequality is derived by invoking Proposition 5 and the
Cauchy–Schwarz inequality $(a+b)^{2}\leq 2(a^{2}+b^{2})$.

Further, by combining (16) with (56), we have

$\begin{split}&\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})-\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\mu}^{k+1})\\\
&\geq\sum_{i=1}^{N}\Big{(}\frac{1}{2}(c_{i}^{k}-L_{g_{i}})-\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}\Big{)}\|\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i}\|^{2}\\\
&+\sum_{i=1}^{N}(-\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}})\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k-1}_{i}\|^{2}\\\
\end{split}$

Thus, we have

$\begin{split}&\Phi_{\beta_{k}}(\bm{W}^{k})\\!-\\!\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\geq\\!\\!\sum_{i=1}^{N}(\beta_{k}\\!-\\!\frac{2(\Omega^{i}_{2})^{2}}{\rho_{\underline{k}}})\|\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{\bar{X}}^{k-1}_{i}\|^{2}\\\
&+\sum_{i=1}^{N}\Big{(}\frac{1}{2}(c_{i}^{k}\\!-\\!L_{g_{i}})\\!-\\!\\!\frac{2(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}\\!-\\!\beta_{k+1}\Big{)}\|\bm{\bar{X}}_{i}^{k+1}\\!-\\!\bm{\bar{X}}^{k}_{i}\|^{2}\\\
\end{split}$

Proposition 6 is concluded. ∎

###  A-G Proof of Proposition 7

###### Proof.

According to Proposition 6 and by summing up (19) for
$k\in[\underline{k},\bar{k}-1]$ ($\bar{k}\in\mathbb{R}$,
$\bar{k}>\underline{k}$), we have

$\begin{split}&\Phi_{\beta_{\underline{k}}}(\bm{W}^{\underline{k}})-\Phi_{\beta_{\bar{k}}}(\bm{W}^{\bar{k}})\\\
&\geq\sum_{k=\underline{k}}^{\bar{k}-1}b^{k}_{1}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|^{2}+\sum_{k=\underline{k}}^{K}b^{k}_{2}\|\bm{\bar{X}}^{k}-\bm{\bar{X}}^{k-1}\|^{2}\\\
&\geq
b_{1}\sum_{k=\underline{k}}^{\bar{k}-1}\|\bm{\bar{X}}^{k}-\bm{\bar{X}}^{k-1}\|^{2}+b_{2}\sum_{k=\underline{k}}^{\bar{k}-1}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|^{2}\\\
\end{split}$ (57)

where $b_{1}=\min\limits_{k\in[\underline{k},\bar{k}-1]}b^{k}_{1}$ and
$b_{2}=\min\limits_{k\in[\underline{k},\bar{k}-1]}b^{k}_{2}$.

Since
$\Phi_{\beta_{k}}(\bm{W}^{k})\geq\inf_{\bm{\bar{X}},\bm{Z}}\mathbb{L}_{\rho_{k}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})>-\infty$
(see (A6)), let $\bar{k}\rightarrow+\infty$, we have

$\begin{split}&b_{1}\sum_{k=\underline{k}}^{+\infty}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|^{2}+b_{2}\sum_{k=\underline{k}}^{+\infty}\|\bm{\bar{X}}^{k}-\bm{\bar{X}}^{k-1}\|^{2}\\\
&\leq\Phi_{\beta_{\underline{k}}}(\bm{W}^{\underline{k}})-\lim_{k\in+\infty}\Phi_{\beta_{k}}(\bm{W}^{k})<+\infty\end{split}$
(58)

If Condition (a) is satisfied, we have $b_{1},b_{2}>0$, thus

$\begin{split}\sum_{k=\underline{k}}^{+\infty}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|^{2}<\infty\end{split}$
(59)

This above implies

$\begin{split}&\lim_{k\rightarrow+\infty}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|\rightarrow
0\\\
&\lim_{k\rightarrow+\infty}\|\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}_{i}^{k}\|\rightarrow
0,\forall i\in\mathcal{N}.\\\ \end{split}$ (60)

Further, based on Proposition 5, we have

$\begin{split}&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\gamma}_{i}^{k+1}-\bm{\gamma}_{i}^{k}\Big{\|}\rightarrow
0,\\\
&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\lambda}_{i}^{k+1}-\bm{\lambda}_{i}^{k}\Big{\|}\rightarrow
0,\\\
&\lim_{k\rightarrow+\infty}\Big{\|}\bm{\mu}_{i}^{k+1}-\bm{\mu}_{i}^{k}\Big{\|}\rightarrow
0,~{}~{}\forall i\in\mathcal{N}.\\\ \end{split}$ (61)

∎

###  A-H Proof of Proposition 8

###### Proof.

_(i)_ the subgradients of $\Phi_{\beta_{k+1}}(\bm{W}^{k+1})$ w.r.t
$\bm{\bar{X}}_{i}$:

$\displaystyle\begin{split}&\|\nabla_{\bm{\bar{X}}_{i}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\|\\\
&\\!=\\!\|\nabla_{\bm{\bar{X}}_{i}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\bm{\mu}^{k+1}})\\!+\\!2\beta_{k+1}(\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i})\|\\\
&\\!=\\!\|\nabla_{\bm{\bar{X}}_{i}}\mathbb{L}_{\rho_{\underline{k}}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\bm{\mu}^{k}})\\!\\!+\\!\\!\big{(}\bm{F}_{i}(\bm{\bar{X}}_{i}^{k+1}))^{T}\big{(}\bm{\gamma}_{i}^{k+1}\\!\\!-\\!\\!\bm{\gamma}_{i}^{k}\big{)}\\\
&\quad+2\beta_{k+1}(\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i})\|\\\
&\leq(L_{g_{i}}+c^{i}_{k})\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|\\\
&\quad+B\|\bm{\gamma}_{i}^{k+1}-\gamma_{i}^{k}\|+2\beta_{k+1}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|\\\
&\leq\big{(}L_{g_{i}}+c^{i}_{k}+\Omega^{i}_{1}B+2\beta_{k+1}\big{)}\|\bm{X}^{k+1}_{i}-\bm{X}^{k}_{i}\|\\\
&\quad+\Omega^{i}_{2}B\|\bm{X}^{k}_{i}-\bm{X}^{k-1}_{i}\|\\\ \end{split}$ (62)

where the first inequality is derived from Proposition 4 and the definition
$B=\sup_{k\geq\underline{k}}\|\bm{F}_{i}(\bm{\bar{X}}_{i}^{k+1})\|$. The
second inequality is derived by invoking Proposition 5.

_(ii)_ the subgradients of $\Phi_{\beta_{k+1}}(\bm{W}^{k+1})$ w.r.t. $\bm{Z}$:

$\begin{split}&\nabla_{\bm{Z}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\\\
&=\nabla_{\bm{Z}}\mathbb{L}_{\rho_{k}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k+1},\bm{\bm{\mu}^{k+1}})+\mathcal{N}_{\mathcal{\bm{X}}}(\bm{Z})\\\
&=-\sum_{i=1}^{N}\big{(}\bm{E}_{i}\big{)}^{T}\bm{\mu}^{k+1}_{i}+\mathcal{N}_{\mathcal{X}}(\bm{Z}^{k+1})\end{split}$
(63)

Based on the first-order optimality condition of subproblem (10), we have

$\begin{split}&-\\!\sum_{i=1}^{N}\big{(}\bm{E}_{i}\big{)}^{T}\bm{\mu}^{k}_{i}\\!-\\!\sum_{i=1}^{N}\rho_{k}\big{(}\bm{E}_{i}\big{)}^{T}\big{(}\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\big{)}\\\
&+\mathcal{N}_{\mathcal{X}}(\bm{Z}^{k+1})=0\end{split}$ (64)

Further, based on
$\bm{\mu}^{k+1}_{i}\\!=\\!\bm{\mu}^{k}_{i}\\!+\\!\rho_{k}\big{(}\bm{A}_{i}\bm{X}_{i}^{k+1}\\!\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\big{)}$
and by rearranging (64), we have

$\begin{split}&-\sum_{i=1}^{N}\big{(}\bm{E}_{i}\big{)}^{T}\bm{\mu}^{k+1}_{i}+\mathcal{N}_{\mathcal{X}}(\bm{Z}^{k+1})\\\
&=\sum_{i=1}^{N}\rho_{\underline{k}}\big{(}\bm{E}_{i}\big{)}^{T}\big{(}\bm{A}_{i}\bm{X}^{k+1}_{i}-\bm{A}_{i}\bm{X}^{k}_{i}\big{)}\end{split}$
(65)

By combining (63) with (65), we have

$\begin{split}&\|\nabla_{\bm{Z}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\|\\!=\\!\|\sum_{i=1}^{N}\rho_{\underline{k}}\big{(}\bm{E}_{i}\big{)}^{T}\\!\bm{A}_{i}\big{(}\bm{X}^{k+1}_{i}\\!-\\!\bm{X}^{k}_{i}\big{)}\|\\\
&\leq\rho_{\underline{k}}\sum_{i=1}^{N}\|\big{(}\bm{E}_{i}\big{)}^{T}\bm{A}_{i}(\bm{X}^{k+1}_{i}-\bm{X}^{k}_{i})\|\\\
&\leq\rho_{\underline{k}}\lambda_{\max}((\bm{E}_{i}\big{)}^{T}\bm{A}_{i})\sum_{i=1}^{N}\|\bm{X}^{k+1}_{i}-\bm{X}^{k}_{i}\|\end{split}$
(66)

where $\lambda_{\max}((\bm{E}_{i}\big{)}^{T}\bm{A}_{i})$ denotes the maximum
eigenvalue of matrix $(\bm{E}_{i}\big{)}^{T}\bm{A}_{i}$.

_(iii)_ the subgradients of $\Phi_{\beta_{k+1}}(\bm{W}^{k+1})$ w.r.t. the
Lagrangian multipliers $\bm{\lambda}$ and $\bm{\mu}$:

$\begin{split}&\nabla_{\bm{\lambda}_{i}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\\!\\!=\\!\rho_{\underline{k}}h_{i}(\bm{X}^{k+1}_{i})=\frac{\bm{\lambda}_{i}^{k+1}\\!-\\!\bm{\lambda}_{i}^{k}}{\rho_{\underline{k}}}\\\
&\nabla_{\bm{\mu}_{i}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\\!\\!=\\!\\!\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}\\!\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\\!=\\!\frac{\bm{\mu}^{k+1}_{i}-\bm{\mu}^{k}_{i}}{\rho_{\underline{k}}}\end{split}$
(67)

Based on Proposition 5, we have

$\begin{split}&\|\nabla_{\bm{\gamma}_{i}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\|=\|\frac{\bm{\gamma}_{i}^{k+1}\\!-\\!\bm{\gamma}_{i}^{k}}{\rho_{\underline{k}}}\|\\\
&\leq\frac{\Omega^{i}_{1}}{\rho_{\underline{k}}}\|\bm{\bar{X}}^{k+1}_{i}\\!\\!-\\!\\!\bm{\bar{X}}^{k}_{i}\|\\!\\!+\\!\\!\frac{\Omega^{i}_{2}}{\rho_{\underline{k}}}\|\bm{\bar{X}}^{k}_{i}\\!\\!-\\!\\!\bm{\bar{X}}^{k-1}_{i}\|\end{split}$
(68)

_(iv)_ the subgradients of $\Phi_{\beta_{k+1}}(\bm{W}^{k+1})$ w.r.t. $\bm{U}$:

$\begin{split}&\|\nabla_{\bm{U}}\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\|\\!=\\!2\beta_{k+1}\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|\end{split}$
(69)

By combining (62), (66), (68) and (69), we have

$\begin{split}&\|\nabla\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\|\leq\sum_{i=1}^{N}\big{(}L_{g_{i}}+c^{k}_{i}+\Omega^{i}_{1}B\\\
&\quad\quad\quad+4\beta_{k+1}+\rho_{\underline{k}}+\frac{\Omega^{i}_{1}}{\rho_{\underline{k}}}\big{)}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|\\\
&\quad+\sum_{i=1}^{N}\big{(}\Omega^{i}_{2}B+\frac{\Omega^{i}_{2}}{\rho_{\underline{k}}}\big{)}\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k-1}_{i}\|\\\
\end{split}$ (70)

Proposition 8 is concluded. ∎

##  Appendix B Proof of Proposition 9

###### Proof.

By invoking Proposition 8 and the Cauchy–Schwarz inequality
$(\frac{\sum_{i=1}^{N}x_{i}}{N})^{2}\leq\frac{1}{N}\sum_{i=1}^{N}(x_{i})^{2}$,
we have

$\begin{split}\Big{[}\textrm{dist}&\big{(}\nabla\Phi_{\beta_{k+1}}(\bm{W}^{k+1}),0\big{)}\Big{]}^{2}\\!\leq\\!\\!2N(b^{k}_{3})^{2}\sum_{i=1}^{N}\|\bm{X}^{k+1}_{i}\\!-\\!\bm{X}^{k}_{i}\|^{2}\\\
&\quad\quad+2N(b^{k}_{4})^{2}\sum_{i=1}^{N}\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k-1}_{i}\|^{2}\\\
\end{split}$ (71)

Besides, according to _Proposition_ 6, we have

$\begin{split}&\Phi_{\beta_{k}}(\bm{W}^{k})-\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\\\
&\geq
b^{k}_{1}\sum_{i=1}^{N}\|\bm{\bar{X}}_{i}^{k+1}-\bm{\bar{X}}^{k}_{i}\|^{2}+b^{k}_{2}\sum_{i=1}^{N}\|\bm{\bar{X}}^{k}_{i}-\bm{\bar{X}}^{k-1}_{i}\|^{2}\\\
\end{split}$ (72)

Thus, by combining (71), (72) and Condition (b), we have

$\begin{split}\Big{[}\textrm{dist}\big{(}\nabla&\Phi_{\beta_{k+1}}(\bm{W}^{k+1}),0\big{)}\Big{]}^{2}\\\
&\\!\leq\\!{2N}{\nu_{k}}\big{(}\Phi_{\beta_{k}}(\bm{W}^{k})-\Phi_{\beta_{k+1}}(\bm{W}^{k+1})\big{)}\end{split}$
(73)

∎

##  Appendix C Proof of Corollary 1

###### Proof.

If the stepsize $c^{k}_{i}$ is selected according to Algorithm 1, we have

$\begin{split}&g_{i}(\bm{\bar{X}}_{i}^{k+1},\bm{\lambda}^{k},\rho_{k})+\alpha\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}_{i}^{k}\|^{2}\\\
&\quad\leq
g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k},\rho_{k})+\langle\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k},\rho_{k}),\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\rangle\\\
&\quad\quad\quad\quad\quad+\frac{c_{i}^{k}}{2}\big{\|}\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}_{i}^{k}\big{\|}^{2}\end{split}$
(74)

Based on subproblem (11), we have

$\begin{split}&(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1})+\frac{\rho_{k}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\!\\\
&g_{i}(\bm{\bar{X}}_{i}^{k+1},\bm{\lambda}^{k},\rho_{k})+a\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}_{i}^{k}\|^{2}\\\
&\leq(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1})+\frac{\rho_{k}}{2}\|\bm{\bm{A}_{i}\bar{X}}^{k}_{i}\\!-\\!\bm{E}^{i}\bm{Z}^{k+1}_{j}\|^{2}\\\
&+g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}^{k},\rho_{k})\end{split}$ (75)

By combining (74) with (75), we have

$\begin{split}&(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}-\bm{E}_{i}\bm{Z}^{k+1})+\\!\frac{\rho_{k}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{E}^{i}_{j}\bm{Z}^{k+1}_{j}\|^{2}\\!\\\
&+\\!\langle\nabla_{\bm{\bar{X}}_{i}}g_{i}(\bm{\bar{X}}^{k}_{i},\bm{\lambda}_{i}^{k},\rho_{k}),\bm{\bar{X}}^{k+1}_{i}\\!-\\!\bm{\bar{X}}^{k}_{i}\rangle+\frac{c^{k}_{i}}{2}\|\bm{\bar{X}}^{k+1}_{i}-\bm{\bar{X}}^{k}_{i}\|^{2}\\\
&\leq(\bm{\mu}^{k}_{i})^{T}(\bm{A}_{i}\bm{\bar{X}}^{k}_{i}-\bm{E}_{i}\bm{Z}^{k+1})+\\!\frac{\rho_{k}}{2}\|\bm{A}_{i}\bm{\bar{X}}^{k}_{i}\\!-\\!\bm{E}_{i}\bm{Z}^{k+1}\|^{2}\\\
\end{split}$ (76)

By summing up (76) $\forall i\in\mathcal{N}$, we have

$\begin{split}&\mathbb{L}_{\rho_{k}}(\bm{\bar{X}}^{k+1},\bm{Z}^{k+1},\bm{\lambda}^{k},\bm{\mu}^{k})+\alpha\|\bm{\bar{X}}^{k+1}-\bm{\bar{X}}^{k}\|^{2}\\\
&\quad\quad\leq\mathbb{L}_{\rho_{k}}(\bm{\bar{X}}^{k},\bm{Z}^{k},\bm{\lambda}^{k},\bm{\mu}^{k})\end{split}$
(77)

In this case, the convergence of PLDM can be illustrate analogously following
Theorem 1 by replacing $\frac{1}{2}(c^{k}_{i}-L_{g_{i}})$ with $\alpha$. Also,
to guarantee Condition (a): $\alpha$ should be selected and satisfy
$\alpha\geq\frac{4(\Omega^{i}_{1})^{2}}{\rho_{\underline{k}}}+4\beta_{k+1}$.

∎

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