# A Robust Optimization Approach to Network Control
Using Local Information Exchange

Georgios Darivianakis  ENEXAN, Zurich, Switzerland Angelos Georghiou
Department of Business and Public Administration, University of Cyprus, Cyprus
Soroosh Shafiee  Operations Research and Information Engineering, Cornell
University, United States John Lygeros  Automatic Control Laboratory, ETH
Zurich, Switzerland

###### Abstract

Designing policies for a network of agents is typically done by formulating an
optimization problem where each agent has access to state measurements of all
the other agents in the network. Such policy designs with _centralized
information exchange_ result in optimization problems that are typically hard
to solve, require establishing substantial communication links, and do not
promote privacy since all information is shared among the agents. Designing
policies based on arbitrary communication structures can lead to non-convex
optimization problems which are typically NP-hard. In this work, we propose an
optimization framework for decentralized policy designs. In contrast to the
centralized information exchange, our approach requires only _local
communication exchange_ among the neighboring agents matching the physical
coupling of the network. Thus, each agent only requires information from its
direct neighbors, minimizing the need for excessive communication and
promoting privacy amongst the agents. Using robust optimization techniques, we
formulate a convex optimization problem with a loosely coupled structure that
can be solved efficiently. We numerically demonstrate the efficacy of the
proposed approach in energy management and supply chain applications. We show
that the proposed approach leads to solutions that closely approximate those
obtained by the centralized formulation only at a fraction of the
computational effort.

Keywords: robust optimization; network control; decentralized policy design;
local information; state forecast sets; decision rules.

##  1 Introduction

Controlling physical networks of interconnected systems remains an active
field of research due to its high impact on real-world applications, e.g.,
regulation of power networks [71], energy management of building districts,
[16] and supply chains [68]. For large-scale systems, designing and deploying
policies that have a centralized communication structure can be challenging.
In the design phase, computational limitations can restrict the size of the
problem that can be tackled, while in the deployment phase, the centralized
nature of the policy requires excessive communication which does not promote
privacy between the interconnected systems. In such cases, it is desirable to
design policies with local information exchange that ideally rely on local
computational resources to compute their policies.

Synthesizing policies based on an arbitrary communication structure can lead
to non-convex infinite-dimensional optimization problems which are typically
NP-hard [69]. For that reason, several studies have been devoted to
identifying specific communication structures under which the designing
policies can be cast as a convex problem [39, 45]. For instance, if the
communication network admits a partially nested structure [30], then affine
policies are known to be optimal for decentralized linear systems with
quadratic costs and additive Gaussian noise [30, 55, 56]. Similar results
exist for communication structures that are spatially invariant [2, 2, 48],
including delays in information sharing [34, 49, 50].

Recent advances shifted research interest in identifying information
communication structures that allow for the optimal distributed controllers
synthesis problem to be formulated as a convex optimization problem [1, 17,
20, 46, 54]. These structures usually possess properties such as quadratic
invariance [58, 65] and funnel causality [1] which essentially eliminates the
incentive of signaling among the interconnected systems. For general network
structures, the usual practice is to resort to linear matrix inequality
relaxations [35, 72] or semidefinite programming relaxations [37, 22] to
obtain a suboptimal design with performance guarantees.

A downside of the aforementioned design approaches is the inability to cope
with state and input constraints in the systems. Optimization based
approaches, often referred to as Model Predictive Control, are well-suited for
the control of constrained systems [47]. In this setting, control designs are
usually categorized into cooperative or non-cooperative [59] referring to the
level of information exchange amongst the interconnected systems. As before,
cooperative approaches require substantial communication infrastructure and
computation resources in the design phase since a system-wide optimization
problem is formulated and solved [71, 62, 25]. On the other hand, non-
cooperative approaches, though computationally simple and effective in
practice, can be conservative in presence of strong coupling [57, 32, 66]. In
addition, non-cooperative schemes typically require a centralized offline
design phase, thus suffering from similar complexity and privacy concerns as
the centralized designs.

There is a stream of literature that tries to address these issues by
developing schemes that rely on local computational resources and information
structures [12, 19, 21, 43, 67]. This is commonly achieved by each system
individually considering the worst-case effect of its neighbors as a bounded
exogenous uncertainty to its own system. Nevertheless, this can lead to
conservative designs as the sets of bounded exogenous uncertainties are
_calculated off-line_ , thus disregarding the possibility of adapting their
size based on the dynamical evolution of the system.

Recent advances in robust optimization techniques deal with exogenous
uncertainties in a computationally efficient way, allowing to address both
static and multistage problems [8, 18, 27]. In contrast, problems with
endogenous uncertainties usually referred to as problems with decision-
dependent uncertainty sets, are typically computationally intractable [52].
However, by exploiting structural characteristics of the problem such as
right-hand side uncertainty in the constraints, the work of [31, 73, 9]
proposed approximations that restrict the space of admissible uncertainty sets
to those that exhibit an affine dependence on fixed sets. By doing so, the
resulting approximate problems can be equivalently cast as robust optimization
problems with exogenous uncertainties, and hence they can be efficiently
solved using existing methods. In this paper, we leverage similar techniques
to design control policies with local information structure, by _designing on-
line_ the bounded sets of exogenous uncertainties which model the worst-case
effect of the neighbors for a given system.

In this paper, we address communication and computation issues of policy
designs for physical networks of interconnected systems. The contributions of
this paper are:

  * $\diamond$

We propose a new modeling paradigm for designing policies with local
communication exchanges which exactly match the physical coupling of the
network. In contrast to centralized and partially nested models where the
states are explicitly communicated amongst the neighboring agents as functions
of the uncertain parameters, the proposed approach communicates compact sets
which we refer to as the _state forecast set_ , implicitly encompassing the
states of neighboring systems. As the agents are only coupled through the
forecast sets, the proposed structure addresses privacy concerns in the
information exchange and allows for a natural interpretation of the resulting
policy. The framework generalizes the proof-of-concept conference paper [15]
in which a simple version of the idea was applied for the efficient energy
management of building districts.

  * $\diamond$

We show that the proposed paradigm directly relates to centralized and
partially nested information exchange policy designs. In particular, we show
that the optimal local communication policy constitutes a conservative
approximation to the partially nested information design problem, while the
optimal partially nested information policy constitutes a conservative
approximation to the centralized information design problem. In addition, we
identify network structures for which the optimal values of these problems
coincide. Hence, the proposed approach fits well within the established
modeling paradigms.

  * $\diamond$

We propose a tractable approximation that restricts the functional form of the
state forecast sets to an affine transformation of a fixed set, in a similar
spirit as [31, 73, 9]. Under this restriction, we show that the problem can be
cast as a multistage robust optimization problem, which is a well studied
class of problems and multiple solution methods exist to either approximate or
solve the problem explicitly. By construction, the resulting problem has a
decoupled structure making the problem highly scalable with respect to the
number of interconnected systems in the network. The efficacy of the proposed
approach is demonstrated in three numerical experiments where we study the
quality of the solution with respect to the state forecast sets approximation,
the scalability properties of the approach, as well as how the problem can be
solved in an almost decentralized manner using the alternating direction
method of multipliers algorithm.

The remainder of this paper is organized as follows. Section 2 provides the
problem formulation and briefly reviews the centralized and partially nested
information exchange policies. Section 3 presents the new modeling paradigm
and discusses the relationship between the centralized and partially nested
policy designs. In Section 4 we derive tractable approximations and show how
the problem can be cast as a multistage robust optimization problem. In
Section 5 we present two numerical examples: (i) an illustrative example that
showcases the proposed method and discusses the effect of the state forecast
sets approximation, and (ii) a cooperative energy management system. All
proofs are found in the Appendix A. We also provide an additional numerical
examples in Appendix B that examines a contract design problem for a supply
chain.

Notation: The calligraphic letters $\mathcal{M},\mathcal{P},\mathcal{T}$ are
reserved for finite index sets with cardinalities $M,P,T$, that is,
$\mathcal{M}=\\{1,\dots,M\\}$ etc. The subscript $+$ in $\mathcal{T}_{+}$
indicates that the index set $\mathcal{T}$ additionally includes $T+1$, that
is, $\mathcal{T}_{+}=\mathcal{T}\bigcup\\{T+1\\}$. Concatenated vectors are
represented in boldface. Dimensions of matrices and concatenated vectors are
assumed clear from the context. For given vectors $v_{i}\in\mathbb{R}^{k_{i}}$
with $k_{i}\in\mathbb{N}$, $i\in\mathcal{M}$, we define
$\bm{v}_{\scriptscriptstyle\mathcal{M}\scriptstyle}=[v_{i}]_{i\in\mathcal{M}}=[v_{1}^{\top}\ldots
v_{M}^{\top}]^{\top}\in\mathbb{R}^{k}$ with $k=\sum_{i=1}^{M}k_{i}$ as their
vector concatenation. Given time dependent vectors
$\nu_{i,t}\in\mathbb{R}^{\ell_{i}}$ with $i\in\mathcal{M}$, $t\in\mathcal{T}$
and $\ell_{i}\in\mathbb{N}$, we define
$\bm{\nu}_{{\scriptscriptstyle\mathcal{M}\scriptstyle},t}=[\nu_{i,t}]_{i\in\mathcal{M}}$
as the concatenated vector at time $t$,
$\bm{\nu}_{i}^{t}=[\nu_{i,1}^{\top}\ldots\nu_{i,t}^{\top}]^{\top}$ as the
history of the $i$-th vector up to time $t$, and
$\bm{\nu}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t}=[\bm{\nu}_{i}^{t}]_{i\in\mathcal{M}}$
as the history of the concatenated vector up to time $t$. We denote by
$\text{ext}(\Xi)$ the set of extreme points of set $\Xi$. An extended notation
section summarizing the major notation can be found in Appendix C.

##  2 Problem formulation

We consider a physical network comprising $M$ interconnected systems,
henceforth referred to as agents. We assume that the agents are coupled
through the dynamics. We describe these interactions through a directed graph
in which an arc connecting agent $j$ to agent $i$, with $i,j\in\mathcal{M}$,
indicates that the states of the $j$-th agent affect the dynamics of the
$i$-th agent. We refer agent $j$ as the preceding neighbor to agent $i$,
henceforth neighbor, and we define the set $\mathcal{N}_{i}\subset\mathcal{M}$
to include all the neighbors of the $i$-th agent. Figure 1 illustrates a
system of $M=5$ agents where the neighbors of agent 3 are given by
$\mathcal{N}_{3}=\\{2,5\\}$. In the sequel, we refer to the physical network
depicted in Figure 1 as the “working example” and use it to streamline the
presentation of key ideas in the paper.

Figure 1: Working example: _Physical coupling graph of 5 agents_. Solid line
arrows represented the direction of the interaction.

###  2.1 System dynamics, constraints, and objective function

In this paper, we study finite horizon problems with $T$ stages. We use linear
dynamics to model the state evolution of the agent $i$ at time instant
$t\in\mathcal{T}$, as

$x_{i,t+1}=A_{i,t}x_{i,t}+B_{i,t}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},t}+D_{i,t}u_{i,t}+E_{i,t}\xi_{i,t}.$
(1)

Here $x_{i,t}\in\mathbb{R}^{n_{x,i}}$ denotes the states, with the initial
state $x_{i,1}$ known. The interaction of agent $i$ and its neighbors is
captured through the term
$B_{i,t}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},t}$. Vector
$u_{i,t}\in\mathbb{R}^{n_{u,i}}$ models the inputs and
$\xi_{i,t}\in\mathbb{R}^{n_{\xi,i}}$ captures the exogenous uncertainties
affecting the system dynamics. The time-varying system matrices $A_{i,t}$,
$B_{i,t}$, $D_{i,t}$ and $E_{i,t}$ are assumed known with proper dimensions
and of full column rank. To economize on notation, we now compactly rewrite
(1) as

$\bm{x}_{i}=f_{i}(\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}):=A_{i}x_{i,1}+B_{i}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}+D_{i}\bm{u}_{i}+E_{i}\bm{\xi}_{i},$

where $\bm{x}_{i}:=[x_{i,t}]_{t\in\mathcal{T}_{+}}$,
$\bm{u}_{i}:=[u_{i,t}]_{t\in\mathcal{T}}$,
$\bm{\xi}_{i}:=[\xi_{i,t}]_{t\in\mathcal{T}}$ and
$\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}:=[\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},t}]_{t\in\mathcal{T}}$.
The system matrices $A_{i}$, $B_{i}$, $D_{i}$ and $E_{i}$ used to define the
function $f_{i}(\cdot)$ are directly constructed by the problem data given in
(1) (see, e.g., [28] for such a derivation). The $i$-th agent is subject to
linear constraints

$\mathcal{O}_{i}=\big{\\{}(\bm{x}_{i},\bm{u}_{i})\,:\,H_{x,i}\bm{x}_{i}+H_{u,i}\bm{u}_{i}\leq
h_{i}\big{\\}},$ (2)

where the matrices $H_{x,i}$, $H_{u,i}$ and $h_{i}$ are assumed known and of
proper dimensions. Note that this compact constraint formulation allows the
consideration of time-varying linear operational constraints with time-stage
coupling. In addition, constraints involving neighboring states or exogenous
uncertainties can always be included by appropriately extending the state
space of the $i$-th system. The objective associated with $i$-th agent is
given by

$J_{i}(\bm{x}_{i},\bm{u}_{i})=\sum_{t=1}^{T}\left(\|Q_{i}x_{i,t}\|_{q}+\|R_{i}u_{i,t}\|_{q}\right),$
(3)

where $q\in\\{\infty,1\\}$ allows for different objective formulation. The
penalization matrices $Q_{i}$, $R_{i}$ are assumed known and of proper
dimensions.

In the following, we assume that the exogenous uncertainties affecting agent
$i$ reside in the nonempty, convex and compact polyhedral uncertainty set
$\Xi_{i}=\\{\bm{\xi}_{i}:W\bm{\xi}_{i}\geq w\\}$ where matrix $W$ and vector
$w$ are known and of proper dimensions. We will be making the simplifying
assumption that the joint uncertainty set of all agents in the system has a
decoupled structure, i.e.,
$\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}\in\Xi_{\scriptscriptstyle\mathcal{M}\scriptstyle}=\bigtimes_{i\in\mathcal{M}}\Xi_{i}$,
which essentially precludes the existence of coupled uncertainties amongst the
agents. This assumption can be relaxed at the expense of further case
distinctions in what follows.

###  2.2 Designing policies with centralized information exchange

A common assumption in designing policies is to assume that at time $t$, each
agent has access to the states from all the other agents in the network up to
and including period $t$ [28, 29]. We will refer to this communication as the
_centralized information exchange_ , depicted in Figure 2 for the working
example. In this context, we denote the _causal state feedback_ policies for
agent $i$ at time $t\in\mathcal{T}$ as
$\pi_{i,t}:\mathbb{R}^{n_{x}^{t}}\rightarrow\mathbb{R}^{n_{u,i}}$ where
$n_{x}^{t}=t\left(\sum_{j\in\mathcal{M}}n_{x,j}\right)$, such that its input
at time $t$ is given as
$u_{i,t}=\pi_{i,t}(\bm{x}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t})$.
We write
$\bm{\pi}_{i}(\bm{x}_{\scriptscriptstyle\mathcal{M}\scriptstyle})=[\pi_{i,t}(\bm{x}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t})]_{t\in\mathcal{T}}$
to denote the policy concatenation over the horizon.

Figure 2: Working example: _Centralized information exchange_. Dotted line
arrows represent the communication links between the agents, with
$\mathcal{M}=\\{1,2,3,4,5\\}$.

The optimization problem for designing policies with centralized information
exchange which minimize the sum of worst-case individual objectives is
formulated as

$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\mathcal{M}}\in\Xi_{\mathcal{M}}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\pi}_{i}(\bm{x}_{\scriptscriptstyle\mathcal{M}\scriptstyle}):=[\pi_{i,t}(\bm{x}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\end{array}\right\\}\forall\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}\in\Xi_{\scriptscriptstyle\mathcal{M}\scriptstyle},\;\forall
i\in\mathcal{M},\end{array}$ (4a) where the optimization variables are the
policies $\bm{\pi}_{i}(\cdot)$ for all $i\in\mathcal{M}$. As shown in [28,
29], the state feedback structure of the policies induces a non-convex
feasible region. To deal with this, they propose the design of _strictly
causal uncertainty feedback policies_
$\Pi_{i,t}:\mathbb{R}^{n_{\xi}^{t}}\rightarrow\mathbb{R}^{n_{u,i}}$ where
$n_{\xi}^{t}=(t-1)\sum_{i\in\mathcal{M}}n_{\xi,i}$, such that the input at
each time step is given by
$u_{i,t}=\Pi_{i,t}(\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t-1})$.
We write
$\bm{\Pi}_{i}(\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle})=[\Pi_{i,t}([\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t-1}]_{t\in\mathcal{T}})$
to denote the policy concatenation over the horizon. This formulation leads to
the infinite dimensional linear optimization
$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\mathcal{M}}\in\Xi_{\mathcal{M}}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\Pi}_{i}(\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}):=[\Pi_{i,t}(\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}^{t-1})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\end{array}\right\\}\forall\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}\in\Xi_{\scriptscriptstyle\mathcal{M}\scriptstyle},\;\forall
i\in\mathcal{M}.\end{array}$ (4b)

Using the fact that matrices $A_{i,t}$, $B_{i,t}$ and $E_{i,t}$ are full
column rank, there is a one-to-one relationship between state and uncertainty
feedback policies in terms of feasibility and optimality. Restricting the
admissible policies to have an affine structure and reformulating using robust
optimization techniques reduces the problem to a finite-dimensional linear
optimization problem which can be solved with off-the-shelf optimization
solvers. Furthermore, due to the one-to-one relationship between state and
uncertainty feedback policies, there is also a unique mapping that translates
the resulting affine uncertainty policies to an equivalent affine state
feedback policy which allows being implemented locally by each agent using the
centralized communication network.

Although theoretically appealing, policies based on centralized information
exchange are hard to design and implemented in practice for large-scale
systems. This is partially the case since for large networks $(i)$ solving the
linear optimization problem resulting from the affine policy approximation can
be computationally challenging due to its monolithic structure; while $(ii)$
the excessive communication and the centralized physical network required to
allow each agent to evaluate its policy, does not promote privacy since the
exact policy/constraints of individual agents are eventually revealed to the
rest of the network. We will demonstrate the former through numerical
experiments in Section 5.

###  2.3 Designing policies with partially nested information exchange

In an attempt to address the computational and privacy issues, researchers
have proposed a number of policy designs that consider only partial
communication among the agents. For an arbitrary information exchange network
the design phase typically results in a non-convex problem that is
computationally intractable. A notable exception is the work of [40] which
assumes a _partially nested information exchange_ , leading to convex
formulations. Roughly speaking this communication exchange implies that agent
$i$ has access to information coming from all of its _precedent agents_ in a
non-anticipative manner. In this setting, agent $j$ is named a _precedent to
agent $i$_, if the input of agent $j$ at time $t^{\prime}$ can affect the
local information available to agent $i$ at some time $t>t^{\prime}$ in the
future [40, Definition 1]. The partially nested information exchange
associated with the working example is depicted in Figure 3, where we see that
although agent 1 is not a physical neighbor of agents 3 and 4, its actions can
affect the future states of both these agents, thus it is a precedent agent.

Figure 3: Working example: _Partially nested information exchange_. Dotted
line arrows represent the established communication links between local
controllers, with $\overline{\mathcal{N}}_{1}=\\{1\\}$,
$\overline{\mathcal{N}}_{2}=\\{1,2\\}$,
$\overline{\mathcal{N}}_{3}=\\{1,2,3,5\\}$,
$\overline{\mathcal{N}}_{4}=\\{1,2,3,4,5\\}$ and
$\overline{\mathcal{N}}_{5}=\\{5\\}$.

In a partially nested communication, the policy is designed as follows. We
denote by $\overline{\mathcal{N}}_{i}\subseteq\mathcal{M}$ the set that
includes agent $i$ and all its precedent agents. At time $t$, the $i$-th agent
measures its own states and the states of its precedent agents, denoted by
$\bm{x}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle},t}=[x_{j,t}]_{j\in\overline{\mathcal{N}}_{i}}$.
Using all measurements from stage 1 up to time $t$, it designs a _causal
partial state feedback_ policy
$\phi_{i,t}:\mathbb{R}^{\overline{n}_{x,i}^{t}}\rightarrow\mathbb{R}^{n_{u,i}}$,
where
$\overline{n}_{x,i}^{t}=t\left(\sum_{j\in\overline{\mathcal{N}}_{i}}n_{x}^{j}\right)$.
The input is now given as
$u_{i,t}=\phi_{i,t}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t})$.
We write
$\bm{\phi}_{i}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})=[\phi_{i,t}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}$
to denote the policy concatenation over the time horizon.

The optimization problem to design policies with partially nested information
exchange that minimize the sum of worst-case individual objectives is
formulated as

$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\phi}_{i}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}):=[\phi_{i,t}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\end{array}\right\\}\forall\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}},\;\forall
i\in\mathcal{M},\end{array}$ (5a) where
$\Xi_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}=\bigtimes_{j\in\overline{\mathcal{N}}_{i}}\Xi_{j}$
and the optimization variables are the policies $\bm{\phi}_{i}(\cdot)$ for all
$i\in\mathcal{M}$. Problem (5a) is typically non-convex due to the state
feedback structure of the policies. Similar to the centralized case, [40]
proposes the use of _strictly causal partial nested uncertainty feedback
policies_
$\Phi_{i,t}:\mathbb{R}^{\overline{n}_{\xi,i}^{t}}\rightarrow\mathbb{R}^{n_{u,i}}$
where
$\overline{n}_{\xi,i}^{t}=(t-1)\left(\sum_{j\in\overline{\mathcal{N}}_{i}}n_{\xi}^{j}\right)$,
such that the input at each time step is given by
$u_{i,t}=\Phi_{i,t}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t-1})$.
We write
$\bm{\Phi}_{i}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}})=[\Phi_{i,t}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t-1})]_{t\in\mathcal{T}}$
to denote the policy concatenation, leading to the infinite dimensional linear
optimization
$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\Phi}_{i}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}):=[\Phi_{i,t}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t-1})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\end{array}\right\\}\forall\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}},\;\forall
i\in\mathcal{M}.\end{array}$ (5b)

If these uncertainty feedback policies are restricted to admit an affine
structure and using robust optimization to reformulate the resulting semi-
infinite constraints, then Problem (5b) becomes a finite-dimensional linear
optimization problem. As with the centralized case, there is a one-to-one
relationship between the state and uncertainty feedback policies, both for the
infinite-dimensional and affine restriction. This allows the agents in the
network to evaluate their policies based on the established partially nested
communication.

The following theorem establishes the connection between centralized and
partially nested information design problems and will be used in the following
section to demonstrate the relationship between the proposed approach to the
centralized and partially nested policy structures.

######  Theorem 1.

Problem (5b) is a conservative approximation of Problem (4b) in the following
sense: every feasible solution of Problem (5b) is feasible in Problem (4b),
and the optimal value of Problem (5b) is larger or equal to the optimal value
of Problem (4b).

Partially nested information exchange slightly reduces the communication
requirements compared to the centralized problem (see Figures 2 and 3 of the
working example). This has a positive impact on the solution time needed to
design affine feedback policies, however, the resulting linear program
inherits in large part the monolithic structure of the centralized problem due
to the absence of a non-sparse structure. Most importantly, even in the simple
model of our working example, the partially nested communication requires
three additional links compared to the physical links. In strongly connected
physical networks, i.e., in networks where there is a directed path from any
node to every other node in the network, the minimum number of communications
links needed to ensure partially nested information coincides with the
centralized information exchange, see the example presented in Section 5.2
with Figures 12 and 13. Therefore, the synthesis of policies with partially
nested information inherits, to a large extent, the drawbacks of the
centralized problem, both from a computational and privacy standpoint.

##  3 Designing policies with local information exchange

In this paper, we propose a decentralized policy structure that relies on
local information exchange among the agents. The proposed policy aims to
address both the computational and privacy concerns discussed so far. While in
the previous section the information flow had to be sufficiently complex to
capture a partially nested structure, in this section we will assume that the
information flow can be as simple as the physical network, as this is
illustrated in Figure 4 for the working example.

Figure 4: Working example: _Local information exchange_. Dotted line arrows
represent the established communication links between agents,
$\mathcal{N}_{1}=\emptyset$, $\mathcal{N}_{2}=\\{1\\}$,
$\mathcal{N}_{3}=\\{2,5\\}$, $\mathcal{N}_{4}=\\{3\\}$ and
$\mathcal{N}_{5}=\emptyset$

In contrast to the previous section where agent $j\in\mathcal{N}_{i}$
communicates to agent $i$ explicitly communicates its states which are
functions of the uncertain parameters, in the proposed framework agent $j$
communicates a compact set $\mathcal{X}_{j}\subseteq\mathbb{R}^{N_{x,i}}$
where $N_{x,i}=\sum_{t\in\mathcal{T}_{+}}n_{x,i}$, henceforth referred as the
_state forecast set_ , that contains possible evolution of its states, i.e.,
$\bm{x}_{j}\in\mathcal{X}_{j}$. In this framework, the shape of
$\mathcal{X}_{j}$ is a decision quantity for agent $j$. Upon receiving these
state forecast sets from all of its neighbors, agent $i$ treats these states
as uncertain quantities that affect its dynamics in a similar way as exogenous
uncertainties. To emphasize this, we denote by
$\zeta_{j,t}\in\mathbb{R}^{n_{x,j}}$ the _belief_ of agent $i$ about the
states of agent $j$ at time $t$, such that
$\bm{\zeta}_{j}=[\zeta_{j,t}]_{t\in\mathcal{T}_{+}}\in\mathcal{X}_{j}$. In
this context, the policy of agent $i$ at time $t$ is based on the information
from its own states $x_{i,t}$ and on the belief states
$\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},t}$, from stage 1
up to $t$. This leads us to design _causal local state/uncertainty feedback
policies_
$\psi_{i,t}:\mathbb{R}^{\hat{n}_{x,i}^{t}}\rightarrow\mathbb{R}^{n_{u}^{i}}$
where
$\hat{n}_{x,i}^{t}=t\left(n_{x,i}+\sum_{j\in\mathcal{N}_{i}}n_{x,j}\right)$,
such that the input of agent $i$ at time $t$ is given by
$u_{i,t}=\psi_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})$.
We denote by
$\bm{\psi}_{i}(\bm{x}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})=[\psi_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}$
the policy concatenation over the time horizon.

In this decentralized setting, the robust optimization problem to design
policies with local information exchange which minimize the sum of worst-case
individual objectives is formulated as

$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\psi}_{i}(\bm{x}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}):=[\psi_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\\\
~{}&\bm{x}_{i}\in\mathcal{X}_{i},\;\mathcal{X}_{i}\in\mathcal{F}(\mathcal{S}_{i})\end{array}\right\\}\begin{array}[]{@{}l}\forall\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\\\
\forall\bm{\xi}_{i}\in\Xi_{i}\end{array}\;\forall i\in\mathcal{M},\end{array}$
(6a) where
$\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}=\bigtimes_{j\in\mathcal{N}_{i}}\mathcal{X}_{j}$,
$\mathcal{F}(\mathcal{S}_{i})$ denotes the field of sets generated by all the
subsets of the power set of $\mathcal{S}_{i}$, where
$\mathcal{S}_{i}\subseteq\mathbb{R}^{N_{x,i}}$. The optimization variables are
the policies $\bm{\psi}_{i}(\cdot)$ and sets $\mathcal{X}_{i}$ for all
$i\in\mathcal{M}$. Since agent $i$ treads the beliefs
$\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}$ as exogenous
uncertainties, its decisions are taken in view of the worst-case both with
respect to $\bm{\xi}_{i}\in\Xi_{i}$ and
$\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}$.
This is also reflected in the construction of the objective function. Notice
that agent $i$ is not directly affected by the uncertainties
$\bm{\xi}_{j}\in\Xi_{j}$ of its neighbors. Rather, the effect of
$\bm{\xi}_{j}\in\Xi_{j}$ of agent $j\in\mathcal{N}_{i}$ is been translated
into set $\mathcal{X}_{j}$, which in turn affects agent $i$.

Problem (6a) can be interpreted as a method for finding a compromise between
the agents as this is represented through sets $\mathcal{X}_{i}$. If we focus
attention on two agents, agents $i$ and $j$ with $j\in\mathcal{N}_{i}$, on the
one hand, agent $j$ will benefit the most if the set $\mathcal{X}_{j}$ is as
large as possible. By doing so, set $\mathcal{X}_{j}$ will not impose any
additional constraints on its states, thus agent $j$ will individually achieve
the lowest objective value contribution. On the other hand, agent $i$ will
benefit the most if it receives from agent $j$ the smallest possible set
$\mathcal{X}_{j}$ (preferable $\mathcal{X}_{j}$ is a singleton) which will
reduce the uncertainty on its dynamics, and will have a positive effect in
individually achieving the lowest objective value contribution. Since the
objective of Problem (6a) is to minimize the equally weighted sum of
individual worst-case costs, the resulting policy/set pair finds the trade-off
among the agents, achieving the lowest network-wide objective value, while
being robustly feasible with respect to all
$\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}\in\Xi_{\scriptscriptstyle\mathcal{M}\scriptstyle}$.

Problem (6a) diminishes the privacy concerns in the following ways. First, the
local communication network ensures that the information exchange is the
minimum among the agents as it is the same as the physical network. Second,
focusing again on agents $i$ and $j$ with $j\in\mathcal{N}_{i}$, agent $j$
does not directly reveal its states which are functions of the uncertain
parameters to agent $i$, which is the case in both the centralized and
partially nested information exchange. Rather, the future state trajectories,
are “masks” by set $\mathcal{X}_{j}$, thus reducing exposure to agent $i$ who
might want to leverage the knowledge gained about the constraints and dynamics
of agent $j$. If additionally, the set $\mathcal{X}_{j}$ has a simple
structure, e.g., $\mathcal{X}_{j}$ is rectangular, agent $j$ will reveal the
bare minimum information to its neighbor. This will be studied further in the
next section.

The state feedback nature of policies induces a non-convex optimization
problem. As in [29, 40], we now focus on purely uncertain feedback policies
$\Psi_{i,t}:\mathbb{R}^{\hat{n}_{\xi,i}^{t}}\rightarrow\mathbb{R}^{n_{u,i}}$
where $\hat{n}_{\xi,i}^{t}=(t-1)n_{\xi,i}+t\sum_{j\in\mathcal{N}_{i}}n_{x,j}$,
such that the input at time $t$ is
$u_{i,t}=\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})$.
We write
$\bm{\Psi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})=[\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}$
to denote the policy concatenation over time. Note that the policy is
“strictly causal” to the uncertain vector $\bm{\xi}_{i}^{t-1}$ which the
policy is allowed to depend up to stage $t-1$, while the policy is “causal” in
state beliefs $\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t}$
and is allowed to depend up to stage $t$. The following theorem, establishes
the equivalence between the proposed state/uncertainty and uncertainty
feedback policies and the corresponding optimization problems.

$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\Psi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}):=[\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\\\
~{}&\bm{x}_{i}\in\mathcal{X}_{i},\;\mathcal{X}_{i}\in\mathcal{F}(\mathcal{S}_{i})\end{array}\right\\}\begin{array}[]{@{}l}\forall\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\\\
\forall\bm{\xi}_{i}\in\Xi_{i}\end{array}\;\forall i\in\mathcal{M},\end{array}$
(6b)

######  Theorem 2.

Problem (6a) is a equivalent to Problem (6b) in the following sense: Given a
feasible state/uncertainty feedback policy $\bm{\psi}_{i}(\cdot)$ for Problem
(6a), a feasible uncertainty feedback policy $\bm{\Psi}_{i}(\cdot)$ for
Problem (6b) can be constructed that achieves the same objective value, and
vice versa.

The key difference between the partially nested information Problem (5a) and
the local information Problem (6a) is that the synthesis phase of the latter
requires that each agent communicates only with its direct neighbors rather
than with all its precedent agents in the network (compare Figure 3 to Figure
4 for the working example). This minimum communication exchange is sufficient
to address Problem (6a) since the coupling among agents in the network only
appears through sets $\mathcal{X}_{i}$. This however introduces a level of
conservativeness which is formalized in the following theorem.

######  Theorem 3.

Problem (6b) is a conservative approximation of Problem (5b) in the following
sense: every feasible solution of Problem (6b) is feasible in Problem (5b),
and the optimal value of Problem (6b) is equal or larger than the optimal
value of Problem (5b).

Theorem 3 indicates that for a general network topology, Problem (6b) is a
conservative approximation of Problem (5b). Nonetheless, there are specific
topologies where the optimal values of these two problems coincide. This
occurs when the local and partially nested information networks are the same.
Such a scenario is realized when the underlying physical network is a directed
acyclic graph with depth 1. By definition, this implies that it is a directed
acyclic bipartite network, as illustrated in Figure 5. The result is
formalized in Proposition 1.

Figure 5: Directed acyclic bipartite graph: An example of a directed acyclic
bipartite graph where $\overline{\mathcal{N}}_{1}=\\{1\\}$,
$\overline{\mathcal{N}}_{2}=\\{2\\}$, $\overline{\mathcal{N}}_{3}=\\{3\\}$,
$\overline{\mathcal{N}}_{4}=\\{1,4\\}$ and
$\overline{\mathcal{N}}_{5}=\\{1,2,3,5\\}$.

######  Proposition 1.

If the physical network is a directed acyclic bipartite graph, then Problem
(6b) coincides with problem Problem (5b) in the following sense: feasible
solutions in Problem (6b) are mapped to feasible solutions in Problem (5b),
and their optimal values are equal.

In contrast to Problems (4) and (5) in which the structure of the problem is a
multistage robust optimization _with exogenous uncertainty_ , Problem (6a)
falls in the category of multistage robust problem _with endogenous
uncertainty_ also referred to as problems with _decision-dependent uncertainty
sets_ [51, 52, 61]. Although sets
$\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}$ do not model
exogenous uncertainty, they affect the dynamics of the corresponding agent in
an adversarial manner. Problem (6b) is computationally intractable because
$(i)$ the optimization of the policies is performed over the infinite space of
strictly causal functions; $(ii)$ the optimization of the state forecast sets
$\mathcal{X}_{i}$ is performed over arbitrary sets, and $(iii)$ the
constraints must be satisfied robustly for every uncertain realization. In the
next section, we propose tractable approximations making the problem amenable
to off-the-shelf optimization algorithms.

We conclude this section by discussing how the optimal policies will be
implemented in practice, highlighting the main differences between
centralized, partially nested, and local information exchange problems. Assume
that we have solved Problems (4a), (5a) and (6a), and we have obtained the
optimal policies $\pi_{i}^{\star}(\bm{x}_{\mathcal{M}})$,
$\bm{\phi}_{i}^{\star}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})$
and
$\bm{\psi}_{i}^{\star}(\bm{x}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})$
for all agents $i\in\mathcal{M}$, respectively. Consider first the centralized
information problem. At time $t$, all states
$\bm{x}_{\mathcal{M}}^{t}=(\bm{x}_{\mathcal{M},1},\ldots,\bm{x}_{\mathcal{M},t})$
have been realized and communicated to all agents $\mathcal{M}$ in the system.
Consequently, the input $u_{i,t}=\pi^{\star}_{i,t}(\bm{x}_{\mathcal{M}}^{t})$
can be evaluated deterministically by each agent. Next, we examine the
partially nested information problem. At time $t$, all states
$\bm{x}_{\mathcal{M}}^{t}=(x_{\mathcal{M},1},\ldots,x_{\mathcal{M},t})$ have
been realized. In contrast to the centralized information problem, however,
each agent $i$ receives just the states
$\bm{x}_{\overline{\mathcal{N}}_{i}}^{t}$ from its precedent agents
$\overline{\mathcal{N}}_{i}$. As such, each agent $i$ can deterministically
calculate the input
$u_{i,t}=\pi_{i,t}^{\star}(\bm{x}_{\overline{\mathcal{N}}_{i}}^{t})$. Finally,
we study the local information problem. Similar to the previous cases, at time
$t$, all states
$\bm{x}_{\mathcal{M}}^{t}=(\bm{x}_{\mathcal{M},1},\ldots,\bm{x}_{\mathcal{M},t})$
have been realized. In this case, however, each agent $i$ receives just the
states $\bm{x}_{{\mathcal{N}}_{i}}^{t}$ from its neighboring agents
$\overline{\mathcal{N}}_{i}$. Using this information, each agent $i$ can
deterministically calculate the input
$u_{i,t}=\psi^{\star}_{i,t}(\bm{x}_{i}^{t},\bm{x}_{\mathcal{N}_{i}}^{t})$.
Notice that agent $i$ substitutes the uncertain vector
$\bm{\zeta}_{\mathcal{N}_{i}}^{t}$ with the realized vector
$\bm{x}_{\mathcal{N}_{i}}^{t}$ when calculating the input $u_{i,t}$. Recall
that $\mathcal{N}_{i}\subseteq\overline{\mathcal{N}}_{i}\subseteq\mathcal{M}$
for all $i\in\mathcal{M}$. This indicates that each agent in the local
information exchange problem requires less communication to implement and
calculate its input compared to the centralized and partially nested
information exchange problems. An alternative implementation procedure
typically referred to as _rolling horizon_ , only evaluates the policy at
$t=1$ and applies the input. After each agent observes the realization of its
uncertain vector $\xi_{i,1}$, the states of the period $t=2$ are
deterministically determined, i.e., $x_{i,2}$, and communicated to the
neighbors. The optimization problem is subsequently re-optimized and the
process is repeated. The rolling horizon is typically used when the Problems
(4a), (5a) and (6a) are solved approximately as re-optimization has been
observed to improve the solution quality.

##  4 State forecast set and policy approximation

In this section, we discuss appropriate restrictions to Problem (6b) that
allow obtaining a computationally tractable approximation. We start with the
state forecast sets. For agent $i\in\mathcal{M}$, a natural choice is to
restrict $\mathcal{X}_{i}$ to admit convex conic structure. However, for
arbitrary conic structures the problem is NP-hard even if we restrict the
policies to be static (or _open loop_ as denoted in the control literature).
This is the case as the optimization problem reduces the class of problems
with decision-depend uncertainty as discussed in [52, Theorem 3.2].

A simpler approximation is to restrict $\mathcal{X}_{i}$ to sets that can be
represented through an affine transformation of a fixed set, in a similar
spirit as [9, 31, 73]. Considering agent $i$. For a given convex conic compact
set $\mathcal{S}_{i}\subseteq\mathbb{R}^{N_{x,i}}$, $\mathcal{X}_{i}$ is
restricted to

$\begin{array}[]{l}\mathcal{X}_{i}(Y_{i},z_{i})=\Big{\\{}\bm{x}_{i}\in\mathbb{R}^{N_{x,i}}\,:\,\exists\bm{s}_{i}\in\mathcal{S}_{i}\text{
s.t. }\bm{x}_{i}=Y_{i}\bm{s}_{i}+z_{i}\Big{\\}}\end{array}$ (7a) where the
decision variables that define the shape of the state forecast set is the
positive semi-definite matrix $Y_{i}\in\mathbb{S}_{+}^{N_{x,i}}$ and vector
$z_{i}\in\mathbb{R}^{N_{x,i}}$. $\mathcal{S}_{i}$ is expressed through given
matrices $G_{i}\in\mathbb{R}^{\ell_{i}\times N_{x,i}}$, vectors
$g_{i}\in\mathbb{R}^{\ell_{k}}$ and convex cones $\mathcal{K}_{i}$, as follows
$\begin{array}[]{l}\mathcal{S}_{i}=\Big{\\{}\bm{s}_{i}\in\mathbb{R}^{N_{x,i}}\;:\;G_{i}\bm{s}_{i}\preceq_{\mathcal{K}_{i}}g_{i}\Big{\\}}.\end{array}$
(7b)

Matrix $Y_{i}$ allows scaling and rotation of set $\mathcal{S}_{i}$, while
vector $z_{i}$ is responsible for translating $\mathcal{S}_{i}$.

In the following, we denote by
$\mathcal{F}_{{\scriptscriptstyle\mathcal{AT}\scriptstyle}}(\mathcal{S}_{i})$
the field of bounded convex conic sets that can be represented through an
affine transformation of a fixed set $\mathcal{S}_{i}$ given in equation (7).
Restricting $\mathcal{F}(\mathcal{S}_{i})$ to
$\mathcal{F}_{{\scriptscriptstyle\mathcal{AT}\scriptstyle}}(\mathcal{S}_{i})$
in Problem (6b) gives rise to the following optimization problem:

$\begin{array}[]{@{}l}\text{~{}minimize~{}~{}}\displaystyle\sum_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in{\mathcal{X}}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}(Y_{i},z_{i})}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~~}l@{}}\text{subject
to}~{}~{}&\bm{u}_{i}=\bm{\Psi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}):=[\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}\\\
~{}~{}&Y_{i}\in\mathbb{S}_{+}^{N_{x,i}},\,z_{i}\in\mathbb{R}^{N_{x,i}}\\\
~{}~{}&\bm{x}_{i}=f_{i}\big{(}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\\\
~{}~{}&\bm{x}_{i}\in{\mathcal{X}}_{i}(Y_{i},z_{i}),\,{\mathcal{X}}_{i}(\cdot)\in\mathcal{F}_{{\scriptscriptstyle\mathcal{AT}\scriptstyle}}(\mathcal{S}_{i})\end{array}\right\\}\begin{array}[]{@{}l}\forall\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in{\mathcal{X}}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}(Y_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},z_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle})\\\
\forall\bm{\xi}_{i}\in\Xi_{i}\end{array}\forall i\in\mathcal{M},\end{array}$
(8a) where for each $i\in\mathcal{M}$ we define
${\mathcal{X}}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}(Y_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},z_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle})=\bigtimes_{j\in\mathcal{N}_{i}}{\mathcal{X}}_{j}(Y_{j},z_{j})$.
The optimization variables are policies $\bm{\Psi}_{i}(\cdot)$, matrices
$Y_{i}$ and vectors $z_{i}$ for all $i\in\mathcal{M}$. Notice that by
construction Problem (8a) belongs in the class of problems with a decision-
dependent uncertainty set, thus is in general hard to solve [52]. The affine
restriction
$\mathcal{F}_{{\scriptscriptstyle\mathcal{AT}\scriptstyle}}(\mathcal{S}_{i})$
provides a conservative approximation of Problem (6a) which is stated without
proof in the following corollary.

######  Corollary 1.

Problem (8a) is a conservative approximation of Problem (6a) in the following
sense: every feasible solution of Problem (8a) is feasible in Problem (6a),
and the optimal value of Problem (8a) is equal or larger than the optimal
value of Problem (6a).

By taking advantage of the simple structure of (7) we can reformulate the
problem as a multistage robust optimization problem with decision-independent
uncertainty sets. In the spirit of [9, 31, 73], we design strictly causal
feedback policies
$\Gamma_{i,t}:\mathbb{R}^{\hat{n}_{i}^{t}}\rightarrow\mathbb{R}^{n_{u,i}}$
such that the control input at each time step is given by
$u_{i,t}=\Gamma_{i,t}(\bm{\xi}_{i}^{t-1},\bm{s}^{t}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle})$.
We write
$\bm{\Gamma}_{i}(\bm{\xi}_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})=[\Gamma_{i,t}(\bm{\xi}_{i}^{t-1},\bm{s}^{t}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle})]_{t\in\mathcal{T}}$
to denote the policy concatenation over time. In this context, the counterpart
of Problem (8a) is given as

$\begin{array}[]{@{}l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{S}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\Gamma}_{i}(\bm{\xi}_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}):=[\Gamma_{i,t}(\bm{\xi}_{i}^{t-1},\bm{s}^{t}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle})]_{t\in\mathcal{T}}\\\
~{}&Y_{i}\in\mathbb{S}_{+}^{N_{x,i}},\,z_{i}\in\mathbb{R}^{N_{x,i}}\\\
~{}&\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}=Y_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}\bm{s}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}+z_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\\\
~{}&\bm{x}_{i}\in{\mathcal{X}}_{i}(Y_{i},z_{i}),\,{\mathcal{X}}_{i}(\cdot)\in\mathcal{F}_{{\scriptscriptstyle\mathcal{AT}\scriptstyle}}(\mathcal{S}_{i})\end{array}\right\\}\begin{array}[]{@{}l}\forall\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{S}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\\\
\forall\bm{\xi}_{i}\in\Xi_{i}\end{array}\forall i\in\mathcal{M},\end{array}$
(8b)

where
$\mathcal{S}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}=\bigtimes_{j\in\mathcal{N}_{i}}\mathcal{S}_{j}$,
and with a slight abuse of notation
$Y_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}\bm{s}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}+z_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}=\left[Y_{i}\bm{s}_{j}+z_{j}\right]_{j\in\mathcal{N}_{i}}$
for each $i\in\mathcal{M}$. The decision variables are
$\bm{\Gamma}_{i}(\cdot)$, $Y_{i}$ and $z_{i}$ for all $i\in\mathcal{M}$.
Problem (8b) can be classified as a multistage robust optimization problem
with exogenous uncertainty [4].

In the following, we investigate the relationship between Problem (8a) and
Problem (8b). To do so, we first define the linear mapping
$R_{i,t}:\mathbb{R}^{n_{x,i}}\rightarrow\mathbb{R}^{n_{x,i}}$, $s_{i,t}\mapsto
x_{i,t}$, as,

$R_{i,t}(s_{i,t})=Y_{i,t}s_{i,t}+z_{i,t},$ (9a) and the linear mapping,
$L_{i,t}:\mathbb{R}^{n_{x,i}}\rightarrow\mathbb{R}^{n_{x,i}}$, $x_{i,t}\mapsto
s_{i,t}$, as, $L_{i,t}(x_{i,t})=Y_{i,t}^{+}(x_{i,t}-z_{i,t})$ (9b)

where $Y^{+}_{i,t}:=(Y_{i,t}^{\top}Y_{i,t})^{-1}Y^{\top}_{i,t}$ is the pseudo-
inverse of the positive semi-definite matrix $Y_{i,t}$. Note that the
$L_{i,t}$ may not be unique because of the pseudo-inverse $Y^{+}_{i,t}$.
Moreover, $R_{i,t}$ can be viewed as a “left inverse” of the operator
$L_{i,t}$, i.e., it satisfies $R_{i,t}\big{(}L_{i,t}(x_{i,t})\big{)}=x_{i,t}$.
Using (9), the following theorem establishes equivalence between the two
optimization problems.

######  Theorem 4.

Problem (8a) is equivalent to Problem (8b) in the following sense: feasible
solutions in Problem (8a) are mapped to feasible solutions in Problem (8b),
and their optimal values are equal.

Policies
$\bm{\Gamma}_{i}(\bm{\xi}_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})$
are designed using (8b), however, in practice agent $i$ will require policies
$\Psi_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})$
since $\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}$ will be
the observed states of its neighbors, while
$\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}$ is in some sense
the mathematical construct that allows us to convexify the problem. Theorem
(4) allows us to construct a mapping between the two policies. For agent $i$,
given $u_{i,t}=\Gamma_{i,t}(\cdot)$ as the optimal policy of (8b), then
$u_{i,t}=\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})=\Gamma_{i,t}(\bm{\xi}_{i}^{t-1},[R_{j}^{t}(\bm{\zeta}_{j}^{t})]_{j\in\mathcal{N}_{i}})$
for all
$\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\,\bm{\xi}_{i}\in\Xi_{i}$
is an optimal policy in problem (8a). One can go a step further and construct
the optimal state feedback policy
$\bm{\psi}_{i}(\bm{x}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})$
in a similar spirit as in [29, 28], however, we omit this derivation in the
interest of space.

Despite the exogenous uncertainty structure of Problem (8b), the problem is in
general hard to solve due to the non-convex constraint (7a) as both $Y_{i}$
and $\bm{s}_{i}$ are optimization variables resulting in bilinear terms. The
problem can be efficiently approximated in practice by performing block-
coordinate descent [6] on $Y_{i}$ and $\bm{s}_{i}$ in a sequential manner
until the parameters converge. In the following, we present an additional
restriction that results in a linear formulation of the problem.

Consider the case where $Y_{i}=y_{i}$ where $y_{i}\in\mathbb{R}_{+}$, thus
(7a) takes the form

$\begin{array}[]{l}\mathcal{X}_{i}(y_{i},z_{i})=\displaystyle\left\\{\bm{x}_{i}\in\mathbb{R}^{N_{x,i}}\,:\,\exists\bm{s}_{i}\in\mathcal{S}_{i}\text{
s.t. }\bm{x}_{i}=y_{i}\bm{s}_{i}+z_{i}\right\\}\end{array}$ (10a) This
restriction allows for the scaling of $\mathcal{S}_{i}$ but does not allow for
the rotating of the set. By taking advantage of the fact that
$\mathcal{S}_{i}$ is compact, $\mathcal{X}_{i}(y_{i},z_{i})$ can be expressed
as the following convex set.
$\mathcal{\widehat{X}}_{i}(y_{i},z_{i})=\displaystyle\left\\{\bm{x}_{i}\,:\,\exists\bm{\nu}_{i}\in\mathbb{R}^{N_{x,i}}\text{
s.t. }\displaystyle\bm{x}_{i}=\bm{\nu}_{i,k}+z_{i},\;\;\\\
G_{i}\bm{\nu}_{i}\preceq_{\mathcal{K}_{i}}y_{i}g_{i}\right\\}$ (10b)

where $\bm{\nu}_{i}$ are auxiliary variables. By replacing constraint
$\bm{x}_{i}\in\mathcal{X}(y_{i},z_{i})$ in Problem (8b) with
$\bm{x}_{i}\in\widehat{\mathcal{X}}(y_{i},z_{i})$ the optimization problem
reduces to a multistage _linear_ optimization problem with exogenous
uncertainty. The relationship between sets (10a) and (10b) is summarized in
the following proposition.

######  Proposition 2.

Set
$\mathcal{X}_{{\scriptscriptstyle\mathcal{FS}\scriptstyle}}=\\{(\bm{x}_{i},y_{i},z_{i})\,:\,\bm{x}_{i}\in\mathcal{X}_{i}(y_{i},z_{i})\\}$
is equivalent to
$\widehat{\mathcal{X}}_{{\scriptscriptstyle\mathcal{FS}\scriptstyle}}=\\{(\bm{x}_{i},y_{i},z_{i})\,:\,\bm{x}_{i}\in\mathcal{\widehat{X}}_{i}(y_{i},z_{i})\\}$
in the following sense: there exist a unique mapping between feasible points
in ${\mathcal{X}}_{{\scriptscriptstyle\mathcal{FS}\scriptstyle}}$ and
$\widehat{\mathcal{X}}_{{\scriptscriptstyle\mathcal{FS}\scriptstyle}}$.

######  Remark 1.

Approximation (10) can be very restrictive as it controls the scaling of
$\mathcal{X}_{i}$ through a single parameter, namely $y_{i}$. Regardless, the
idea can be further extended to allow for the scaling of each (or groups) of
the coordinate of $\mathcal{X}_{i}$ independently from the rest. Consider the
case in which $\mathcal{X}_{i}$ can be represent as
$\mathcal{X}_{i}=\mathcal{X}_{i,1}\times\cdots\times\mathcal{X}_{i,K_{i}}$ for
some $K_{i}\leq N_{x,i}$. Then, for each $j\in\\{1,\ldots,K_{i}\\}$, we
restrict each $\mathcal{X}_{i,j}$ to admit a representation as in (10). For
example, if $K_{i}=N_{x,i}$, and setting
$\mathcal{S}_{i,j}=\\{s\in\mathbb{R}:|s|\leq 1\\}$ then we have

$\begin{array}[]{{l@{\,}l}}\mathcal{X}_{i,j}(y_{i,j},z_{i,j})&=\displaystyle\left\\{\bm{x}_{i,j}\in\mathbb{R}\,:\,\exists
s\in\mathcal{S}_{i,j}\text{ s.t. }\bm{x}_{i,j}=y_{i,j}s+z_{i,j}\right\\}\\\
&=\displaystyle\left\\{\bm{x}_{i,j}\in\mathbb{R}\,:\,z_{i,j}-y_{i,j}\leq\bm{x}_{i,j}\leq
z_{i,j}+y_{i,j}\right\\}\\\ \end{array}$

thus allowing to control the width and position of the communication set for
each of the components of $\bm{x}_{i}$. This approximation is natural in many
applications like the contract design example discuss in Section B, in which
this modeling choice is prescribed by the problem itself and does not
constitute an additional approximation to the problem.

As previously stated, problem (8b) can be classified as a multistage robust
optimization problem. There is a plethora of schemes that try to solve this
class of problems either exactly using dynamic programming techniques [24], or
approximately by restricting the functional form of the admissible policies to
admit prescribed structures, see [18, 23] for a thorough survey on the topic.
To simplify exposition, in the remainder of the paper we restrict the policies
to admit an affine structure. This will also allow for direct comparisons with
the work in [28] and [40], which we discussed in Sections 2.2 and 2.3,
respectively.

Problem (8b) retains the decoupled structure of Problem (6a) since agent $i$
only needs to communicate to its direct neighbors $(Y_{i},z_{i})$ that control
the shape of its state forecast set. This loosely coupled structure allows
employing distributed computation algorithms such as the alternating direction
method of multipliers (ADMM) algorithm [10, § 7]. Such schemes will allow the
agents to solve Problem (8b) in an almost decentralized manner, i.e., each
agent solves the subproblem of Problem (8b) involving its own objective and
constraints while seeking to achieve consensus for the variables $y_{i}$ and
$z_{i}$ with its neighbors. In Appendix B we illustrate how the ADMM algorithm
can be implemented in a supply chain example.

We next demonstrate that if the physical network forms an arborescence, it is
possible to explicitly establish an upper bound on the complexity of the
approximation required for the optimal values of Problem (8b) and Problem (6b)
to coincide. For reference, an arborescence, sometimes referred to as a
directed rooted tree, is a directed network in which there is a unique
directed path from a root node to any other node. To achieve this, we can
adjust the approximation (7) to include not only scaling and rotation of
$\mathcal{S}_{i}$ but also projections. This adjustment involves selecting a
higher dimensional set
$\mathcal{S}_{i}\subseteq\mathbb{R}^{\widetilde{N}_{x,i}}$ with
$\widetilde{N}_{x,i}\geq N_{x,i}$ and allowing for rectangular matrix
$Y_{i}\in\mathbb{R}^{N_{x,i}\times\widetilde{N}_{x,i}}$. Thus the affine
mapping $\bm{x}_{i}=Y_{i}\bm{s}_{i}+z_{i}$ where
$\bm{s}_{i}\in\mathcal{S}_{i}$ projects the high dimensional set
$\mathcal{S}_{i}$ to the lower dimension of the state $\bm{x}_{i}$. A
practical choice of $\mathcal{S}_{i}$ is the simplex set since by increasing
its dimension, one can systematically increase the complexity of the projected
set. We note that other choices are also possible. The result is summarized in
the following corollary.

######  Corollary 2.

If the physical network is an arborescence, then the optimal value of Problems
(6b) and (8b) coincide if approximation (7) is constructed such that for each
$i\in\mathcal{M}$ the set $\mathcal{S}_{i}$ is chosen to be the simplex of
dimension
$\prod_{j\in{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}|\text{ext}(\Xi_{j})|$.

We end this section with Figure 6 which summarizes the relationship between
the different problems discussed in the paper so far.

Figure 6: Illustration of the relationships among the various problems
discussed in the paper. The optimal values of these problems are non-
decreasing in the directions of the arcs. Dashed arcs indicate special cases
of networks, while solid arcs represent inequalities that hold under general
network topologies.

##  5 Numerical experiments

We now investigate the efficacy of the proposed approach in two examples: an
illustrative which allows us to understand numerically and graphically the
connection between the shape of the primitive sets $\mathcal{S}_{i}$ and the
solution quality and a stylized contract design for supply chains that are
typically operated in a distributed fashion. All optimization problems were
solved with Gurobi in MATLAB using the RSOME [13] and YALMIP [41] interfaces
on a computer equipped with $8$ GB RAM and $2.4$ GHz quad-core Intel
processor.

###  5.1 Illustrative example

We consider a single stage problem composed by two agents with states
$x_{1},x_{2}\in\mathbb{R}^{2}$, inputs $u_{1},u_{2}\in\mathbb{R}$ and
uncertainties
$\xi_{1},\xi_{2}\in\Xi=\\{\xi\in\mathbb{R}^{2}:\|\xi\|_{\infty}\leq 1\\}$,
respectively. The network structure along with the objective functions and
constraint sets of each agent are shown in Figure 7. The system matrices are
given by

$c=\begin{bmatrix}1\\\ -1\end{bmatrix},\;B=\begin{bmatrix}1&0\\\
0&-2\end{bmatrix},\;D=\begin{bmatrix}1\\\ 0.8\end{bmatrix}\text{ and
}E=\begin{bmatrix}1&-1\\\ -1&1\end{bmatrix}.$ (11)

Figure 7: Physical and information structure of the two agents in the system.

We start by computing the optimal affine policy associated for the centralized
communication and the partially nested communication, which are reminiscent of
Problem (4a) and Problem (5a), respectively. In the centralized communication
both agents observe the uncertain variables $\xi_{1}$ and $\xi_{2}$, while in
the partially nested communication agent $1$ only observes $\xi_{1}$ whereas
agent $2$ observes both $\xi_{1}$ and $\xi_{2}$ as dictated by the structure
of the network. In other words, the second agent’s policy is of the form
$u_{2}(\xi_{1},\xi_{2})=\Gamma_{2,1}\xi_{1}+\Gamma_{2,2}\xi_{2}+\gamma_{2}$ in
both cases, and the first agent’s policy is of the form
$u_{1}(\xi_{1},\xi_{2})=\Gamma_{1,1}\xi_{1}+\Gamma_{1,2}\xi_{2}+\gamma_{1}$ in
the centralized communication and
$u_{1}(\xi_{1},\xi_{2})=\Gamma_{1,1}\xi_{1}+\gamma_{1}$ in the partially
nested communication. Figure 8 illustrates information about the states of
agent $1$. With red we depict the feasible region of $x_{1}$, while blue
depicts region
$x_{1}(\xi_{1},\xi_{2})=Du_{1}^{\star}(\xi_{1},\xi_{2})+E\xi_{1}$ for all
$\xi_{1}\in\Xi_{1},\,\xi_{2}\in\Xi_{2}$, where $u_{1}^{\star}$ is the optimal
affine policy of the agent $1$. Figure 8(a) and (b) illustrate these regions
in the case of centralized and partially nested communications, respectively.
In addition, we report the resulting objective values in the title of the
respective figure as “obj. = $J_{1}(x_{1})+J_{2}(x_{2})$”. We observe that the
information restriction imposed on the partially nested policies results in a
larger objective value compared to the centralized solution.

Figure 8: Illustrating information for the state of agent $1$ for the (a)
centralized and (b) partially nested communication exchange. The red region
represents the feasible set for $x_{1}$ and the blue region represents all
possible values for $x_{1}$ under the optimal policy.

Next, we compute affine policies with local information exchange, which is
reminiscence of Problem (8b). In the local information exchange, the agents’
policies are of the form
$u_{1}(\xi_{1},\xi_{2})=\Gamma_{1,1}\xi_{1}+\gamma_{1}$ and
$u_{2}(s_{1},\xi_{2})=\Gamma_{2,1}s_{1}+\Gamma_{2,2}\xi_{2}+\gamma_{2}$,
respectively, where $s_{1}\in\mathcal{S}_{1}$ represents the auxiliary
uncertainty introduced in Section 4. We start with investigating how the shape
of set $\mathcal{S}_{1}$ together with the restriction of $\mathcal{X}_{1}$
affects the quality of the solution in terms of objective value for three
cases. In the first case (flexible), we choose
$\mathcal{S}_{1}=\\{s\in\mathbb{R}^{2}:\|s\|_{\infty}\leq 1\\}$ and restrict
$\mathcal{X}_{1}$ as in equation (7a). In the second case (rectangle), we
choose $\mathcal{S}_{1,1}=S_{1,2}=\\{s\in\mathbb{R}:|s|\leq 1\\}$ and restrict
$\mathcal{X}_{1}=\mathcal{X}_{1,1}\times\mathcal{X}_{1,2}$ as in Remark 1, and
in the third case (circle), we choose
$\mathcal{S}_{1}=\\{s\in\mathbb{R}^{2}:\|s\|_{2}\leq 1\\}$ and set
$\mathcal{X}_{1}=\\{\bm{x}_{1}\in\mathbb{R}^{2}:\exists\bm{s}\in\mathcal{S}_{1}\text{
s.t. }\bm{x}_{1}=\bm{s}y_{1}+z_{1}\\}$.

The robust counterpart in the first case results in a bilinear optimization
problem which we solve using block-coordinate descent, the second case results
in a linear program, while the third case results in a semidefinite program
due to the use of the celebrated S-Lemma [3, § 3.5] for reformulating the
robust constraints. The flexible approximation exactly recovers the optimal
solution of partially nested communication exchange, i.e., Figure 8 (b). The
result coincides with Corollary 2 and Proposition 1 as the physical network is
both and arborescence and a directeed acyclic bipartite graph, and
additionally the approximation has enough degrees of freedom to adequately
represent the optimal state forecast set. This is not the case for the
rectangular and circular approximation, whose performance is depicted in
Figure 9.

Figure 9: Illustrating information for the state of agent $1$ using
$\mathcal{S}_{1}$ as (a) box, and (b) circle. The green region represents the
feasible set for $x_{1}$, the yellow represents $x_{1}(\xi_{1})$ under the
optimal policy, while red denotes $\mathcal{X}_{1}$. Black stars depict the
binding scenarios for agent $2$ in the state forecast set $\mathcal{X}_{1}$.

As in the first experiment, red area represents the feasible region of
$x_{1}$, while the blue region depicts
$x_{1}(\xi_{1})=Du_{1}^{\star}(\xi_{1})+E\xi_{1}$ for all $\xi_{1}\in\Xi_{1}$.
Additionally, the green area denotes the optimal forecast set
$\mathcal{X}_{1}$ communicated to agent $2$. Finally, the black stars depict
the binding scenarios for agent $2$ in the state forecast set
$\mathcal{X}_{1}$. We observe that the optimal region of agent $1$ changes
with the different primitive sets in an attempt to cooperate with agent $2$.
This cooperative behavior is also identified in the objective values as agent
$1$ “sacrifices” some of its optimality for the good of agent $2$.
Interestingly, part of the state forecast set $\mathcal{X}_{1}$ may lie
outside the feasible region of the problem which adds conservativeness to the
system as can be verified by inspecting the position of the binding scenarios.
Moreover, since the binding scenarios are not necessarily placed at the corner
points defined by the optimal region, agent $1$ retains some of its privacy
since the behavior of its optimal policy is not revealed to agent $2$.

To quantify the importance of the primitive set orientation in space, in the
following experiment we use as primitive sets $(i)$ a rotated rectangular set
whose major and minor axis can be scaled independently, i.e., using the same
notation as before and with slide abuse of notation we have
$\mathcal{X}_{1}=A(\mathcal{X}_{1,1}\times\mathcal{X}_{1,2})$ where
$A\in\mathbb{R}^{2\times 2}$ is a fixed rotation matrix, and $(ii)$ a rotated
ellipsoid for which the major axis is forced to be $1.5$ times larger than its
minor axis, i.e, we set
$\mathcal{X}_{1}=\\{\bm{x}_{1}\in\mathbb{R}^{2}:\exists\bm{s}\in\mathcal{S}_{1}\text{
s.t. }\bm{x}_{1}=A(\bm{s}y_{1}+z_{1})\\}$ where again $A\in\mathbb{R}^{2\times
2}$ is a fixed rotation matrix. Figure 10 illustrates information for the
state of agent $1$, where we use the same color code as in Figure 9. Comparing
the solutions from Figure 9 and Figure 10, we can conclude that both the shape
and orientation heavily affect the optimal policies as well as the optimal
value of the problems.

Figure 10: Illustrating information for the state of agent $1$ using
$\mathcal{S}_{1}$ as (a) rectangle rotated by 15 degrees and (b) ellipsoid
rotated by 15 degrees for which the major axis is forced to be $1.5$ times
larger than its minor axis. The graphs have the same interpretation as in
Figure 9.

To clarify this finding, we repeated the experiment for all possible rotations
in the range $[0,\,180]$ degrees of the rectangular and scaled ellipsoid sets
by appropriately choosing matrix $A$. The results are reported in Figure 11.
We observe that if the rotation of the communicated sets is aligned with the
set generated by the optimal affine policy, then the solution resulting from
the proposed distributed method closely approximates the partially nested and
centralized solutions. If, however, this is not the case, then the cost
considerably deviates and even leads to infeasibility.

Figure 11: Comparing the objective value achieved by affine policies using the
centralized (C), partially nested (PN), and local information exchange (L).
For the local information we set $\mathcal{S}_{1}$ to be a rotated rectangle
(L-Rect.), and a rotated ellipsoid (L-El.) for which the major axis is forced
to be $1.5,3$ and $10$ times larger than its minor axis. The graph reports the
objective values for all possible rotations in the range $[0,\,180]$.

###  5.2 Cooperative energy management system

Buildings account for approximately $40\%$ of the world’s energy consumption,
with a significant portion dedicated to enhancing living conditions through
heating, ventilation, and air conditioning to promote healthier and more
comfortable environments [36]. Motivated by these substantial energy demands,
researchers have invested considerable efforts into developing complex control
schemes that can reduce energy usage while keeping room temperatures within
predefined ranges. Among such schemes, cooperative control systems have been
instrumental for achieving energy savings by coordinating aggregated demands
across multiple users. Most existing cooperative methods assume the existence
of a central operator that is capable of controlling both the buildings
actuation systems and the energy hub devices [53, 64]. This assumption,
however, becomes restrictive for large-scale implementations due to heavy
computational burden. Additionally, privacy concerns around revealing
sensitive building information, such as occupancy patterns and comfort
preferences, arise. Consequently, a new wave of research has focused on
developing decentralized or distributed control schemes to address the
limitations associated with centralized approaches [33, 60, 63].

Figure 12: An example of a serial (left) and complete (right) energy hub
topologies with different consumers.

In this section we examine the scalability and efficiency of our proposed
method using an energy hub system with aggregated consumers. Figure 12 is an
illustration of a serial and a complete (fully-connected) energy hub
consisting of four different consumers. In this setup agent $i$ at time $t$
can buy $G_{i,t}^{+}$ units of electricity from the grid at a cost of
$c_{t}^{+}$ and can return $G_{i,t}^{-}$ units at the cost of $c_{t}^{-}$. The
agents also have the ability to share excess electricity amongst themselves,
with $U_{ij,t}$ denoting the amount of electricity agent $i$ takes from agent
$j$ at time $t$. This transfer of electricity is done at a cost of $c_{t}^{u}$
per unit since the agents need to use the existing grid infrastructure. The
state of charge of the battery of agent $i$ at time $t$ is denoted by
$I_{i,t}$. We assume that the output of the photovoltaic system is uncertain
and denoted by $R_{i,t}(1+\xi_{i,t}^{R})$ where $R_{i,t}$ denotes the average
production and $\xi_{i,t}^{R}$ denotes the uncertain percentage deviation from
the average production. Similarly, the uncertain demand for electricity of
agent $i$ at time $t$ is denoted by $D_{i,t}(1+\xi_{i,t}^{D})$ where $D_{i,t}$
is the average demand and $\xi_{i,t}^{D}$ denotes the uncertain percentage
deviation from the average demand. Finally we define
$\xi_{i,t}=(\xi_{i,t}^{D},\xi_{i,t}^{R})$. We express the battery dynamics of
agent $i$ through

$\displaystyle
I_{i,t+1}=I_{i,t}+G^{+}_{i,t}-G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}U_{ij,t}-\sum_{j\in\mathcal{N}_{i}}U_{ji,t}+R_{i,t}(1+\xi_{i,t}^{p})-D_{i,t}(1+\xi_{i,t}^{d})\quad\forall\bm{\xi}_{i}\in\Xi_{i},\forall
t\in\mathcal{T},$

where for the illustrative example of the serial energy hub in Figure 12, we
have
$\mathcal{N}_{1}=\\{2\\},\,\mathcal{N}_{2}=\\{1,3\\},\,\mathcal{N}_{3}=\\{2,4\\},\,\mathcal{N}_{4}=\\{3\\}$,
and for the complete energy hub we have
$\mathcal{N}_{1}=\\{2,3,4\\},\,\mathcal{N}_{2}=\\{1,3,4\\},\,\mathcal{N}_{3}=\\{1,2,4\\},\,\mathcal{N}_{4}=\\{1,2,3\\}$.
In addition, at any $t\in\mathcal{T}_{+}$, we want to ensure that constraint
$0\leq I_{i,t}\leq B_{i}$ is robustly satisfied for some $B_{i}>0$ that
represents the capacity of the battery.

The communication links needed to formulate the serial energy hub in Figure 12
are depicted in Figure 13(a). Since the serial network is a strongly connected
network, as discussed in Section 2.3, the number of links required to
formulate the partially nested Problem (5a) is the same as in the centralized
Problems (4a), effectively leading to the same problem. On the other hand, the
links needed by the local information exchange matches the physical coupling
of the network thus establishing communication only between adjacent agents in
the system, see Figure 13(b). This demonstrates the modeling benefits of the
proposed approach and highlights the limitations of the partially nested
information exchange on certain network topologies.

Figure 13: Communication links need to achieve (a) centralized and partially
nested information exchange and (b) local information exchange for the serial
energy hub system.

In the centralized information exchange problem, the agents communicate the
state of their batteries to all agents in the grid. We formulate the
centralized problem as an instance of Problem (4b) as follows, where the
objective minimizes the total cost incurred by all agents.

$\displaystyle\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum_{i=1}^{M}\max\limits_{\bm{\xi}_{\mathcal{M}}\in\Xi_{\mathcal{M}}}\sum_{t=1}^{T}c_{t}^{+}G^{+}_{i,t}+\sum_{t=1}^{T}c_{t}^{-}G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}\sum_{t=1}^{T}c_{t}^{u}U_{ij,t}\\\
\left.\begin{array}[]{@{}r@{~}ll}\text{subject
to}~{}&\bm{G}_{i}^{+}=\bm{\Pi}_{i}^{+}(\bm{\xi}_{\mathcal{M}}),~{}\bm{G}_{i}^{-}=\bm{\Pi}_{i}^{-}(\bm{\xi}_{\mathcal{M}})\\\
~{}&\bm{U}_{ij}\in\bm{\Pi}_{ij}(\bm{\xi}_{\mathcal{M}}),~{}\bm{U}_{ji}\in\bm{\Pi}_{ji}(\bm{\xi}_{\mathcal{M}})&\forall
j\in\mathcal{N}_{i}\\\ ~{}&G_{i,t}^{+}\geq 0,~{}G_{i,t}^{-}\geq
0,~{}U_{ij,t}\geq 0,~{}U_{ji,t}\geq 0&\forall j\in\mathcal{N}_{i},\forall
t\in\mathcal{T}\\\ ~{}&\displaystyle
I_{i,t+1}=I_{i,t}+G^{+}_{i,t}-G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}U_{ij,t}-\sum_{j\in\mathcal{N}_{i}}U_{ji,t}\\\\[-9.00002pt]
~{}&&\forall t\in\mathcal{T}\\\\[-9.00002pt] ~{}&\hskip
45.00008pt+R_{i,t}(1+\xi_{i,t}^{p})+D_{i,t}(1+\xi_{i,t}^{d})\\\
~{}&\displaystyle 0\leq I_{i,t}\leq B_{i}&\forall
t\in\mathcal{T}_{+}\end{array}\right\\}\begin{array}[]{l}\forall
i\in\mathcal{M}\\\
\forall\bm{\xi}_{\mathcal{M}}\in\Xi_{\mathcal{M}}\end{array}\end{array}$ (23)

Consider now the case in which instead of communicating the state of their
batteries to all agents in the grid, agents are restricted to communicate an
interval that represents the amount of electricity they can supply their
neighbors at each time period. The interval
$[\underline{b}_{ji,t},\overline{b}_{ji,t}]$ denotes the amount of electricity
agent $i$ commits to supply agent $j$ at time $t$. Of course, agent $i$ does
not have to use all the electricity available from agent $j$ hence
$U_{ij,t}\in[\underline{b}_{ij,t},\overline{b}_{ij,t}]$. In this case, the
battery dynamics of agent $i$ at time $t$ is expressed as

$I_{i,t+1}=I_{i,t}+G^{+}_{i,t}-G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}U_{ij,t}-\sum_{j\in\mathcal{N}_{i}}\zeta_{ji,t}+R_{i,t}(1+\xi_{i,t}^{p})-D_{i,t}(1+\xi_{i,t}^{d})\quad\forall\zeta_{\mathcal{N}_{i},i,t}\in\underset{j\in\mathcal{N_{i}}}{\times}[\underline{b}_{ji,t},\overline{b}_{ji,t}]$

where $\zeta_{\mathcal{N}_{i},i,t}$ denotes all the electricity that can
potentially be drawn from agent $i$ by its neighbors $\mathcal{N}_{i}$ at time
$t$. This problem gives rise to a local information exchange problem which is
an instance of Problem (6a) and can be written as follows.

$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}\in\mathcal{U}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}}\sum_{t=1}^{T}c_{t}^{+}G^{+}_{i,t}+\sum_{t=1}^{T}c_{t}^{-}G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}\sum_{t=1}^{T}c_{t}^{u}U_{ij,t}\\\
\left.\begin{array}[]{@{}r@{~}ll}\text{subject
to}~{}&\bm{G}_{i}^{+}=\bm{\psi}_{i}^{+}(\bm{I}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}),~{}\bm{G}_{i}^{-}=\bm{\psi}_{i}^{-}(\bm{I}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i})\\\
~{}&\bm{U}_{ij}\in\bm{\psi}_{ij}(\bm{I}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}),~{}\underline{\bm{b}}_{ij},\,\overline{\bm{b}}_{ij}\in\mathbb{R}_{+}^{T\times
P}&\forall j\in\mathcal{N}_{i}\\\
~{}&\bm{U}_{ij}\in\mathcal{U}_{i,j}=[\underline{b}_{ij,1},\,\overline{b}_{ij,1}]\times\dots\times[\underline{b}_{ij,T},\,\overline{b}_{ij,T}]&\forall
j\in\mathcal{N}_{i}\\\ ~{}&G_{i,t}^{+}\geq 0,~{}G_{i,t}^{-}\geq
0,~{}U_{ij,t}\geq 0&\forall j\in\mathcal{N}_{i},\forall t\in\mathcal{T}\\\
~{}&\displaystyle
I_{i,t+1}=I_{i,t}+G^{+}_{i,t}-G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}U_{ij,t}-\sum_{j\in\mathcal{N}_{i}}\zeta_{j,i,t}\\\\[-9.00002pt]
~{}&&\forall t\in\mathcal{T}\\\\[-9.00002pt] ~{}&\hskip
45.00008pt+R_{i,t}(1+\xi_{i,t}^{p})+D_{i,t}(1+\xi_{i,t}^{d})\\\
~{}&\displaystyle 0\leq I_{i,t}\leq B_{i}&\forall
t\in\mathcal{T}_{+}\end{array}\right\\}\begin{array}[]{l}\forall\bm{\xi}_{i}\in\Xi_{i}\\\
\forall\bm{\zeta}_{\mathcal{N}_{i},i}\in\mathcal{U}_{\mathcal{N}_{i},i}\\\
\forall i\in\mathcal{M},\end{array}\end{array}$ (24)

By construction of Problem (24), each set
$\mathcal{U}_{i,j}=\mathcal{U}_{i,j,1}\times\cdots\times\mathcal{U}_{i,j,T}$
is a hyper-rectangles which can be controlled coordinate wise. Hence, it can
be exactly represented by the primitive sets
$\mathcal{S}_{i,j,t}=\\{s\in\mathbb{R}:|s|\leq 1\\}$ and
$\mathcal{U}_{i,j,t}(y_{i,j,t},z_{i,j,t})=\displaystyle\left\\{U_{ij,t}\in\mathbb{R}\,:\,\exists
s_{i,j,t}\in\mathcal{S}_{i,j,t}\text{ s.t.
}U_{ij,t}=s_{i,j,t}y_{i,j,t}+z_{i,j,t}\right\\},$ as discussed in Remark 1.
Applying Theorem 4, Problem (24) can thus be reformulated as

$\displaystyle\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}\in\mathcal{S}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}}\sum_{t=1}^{T}c_{t}^{+}G^{+}_{i,t}+\sum_{t=1}^{T}c_{t}^{-}G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}\sum_{t=1}^{T}c_{t}^{u}U_{ij,t}\\\
\left.\begin{array}[]{@{}r@{~}ll}\text{subject
to}~{}&\bm{G}_{i}^{+}=\bm{\Gamma}_{i}^{+}(\bm{\xi}_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}),~{}\bm{G}_{i}^{-}=\bm{\Gamma}_{i}^{-}(\bm{\xi}_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i})\\\
~{}&\bm{U}_{i,j}\in\bm{\Gamma}_{i,j}(\bm{\xi}_{i},\bm{s}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},i}),~{}\underline{\bm{b}}_{i,j},\,\overline{\bm{b}}_{i,j},\,\bm{y}_{i,j},\bm{z}_{i,j}\in\mathbb{R}_{+}^{T\times
P}&\forall j\in\mathcal{N}_{i}\\\
~{}&\underline{\bm{b}}_{i,j}=\bm{z}_{i,j}-\bm{y}_{i,j},\,\overline{\bm{b}}_{i,j}=\bm{z}_{i,j}+\bm{y}_{i,j}&\forall
j\in\mathcal{N}_{i}\\\
~{}&\bm{U}_{i,j}\in\mathcal{U}_{i,j}=[\underline{b}_{i,j,1},\,\overline{b}_{i,j,1}]\times\dots\times[\underline{b}_{i,j,T},\,\overline{b}_{i,j,T}]&\forall
j\in\mathcal{N}_{i}\\\ ~{}&G_{i,t}^{+}\geq 0,~{}G_{i,t}^{-}\geq
0,~{}U_{i,j,t}\geq 0&\forall j\in\mathcal{N}_{i},\forall t\in\mathcal{T}\\\
~{}&\zeta_{j,i,t}=y_{j,i,t}s_{j,i,t}+z_{j,i,t}&\forall
j\in\mathcal{N}_{i},\forall t\in\mathcal{T}\\\ ~{}&\displaystyle
I_{i,t+1}=I_{i,t}+G^{+}_{i,t}-G^{-}_{i,t}+\sum_{j\in\mathcal{N}_{i}}U_{i,j,t}-\sum_{j\in\mathcal{N}_{i}}\zeta_{j,i,t}\\\\[-9.00002pt]
~{}&&\forall t\in\mathcal{T}\\\\[-9.00002pt] ~{}&\hskip
45.00008pt+R_{i,t}(1+\xi_{i,t}^{p})+D_{i,t}(1+\xi_{i,t}^{d})\\\
~{}&\displaystyle 0\leq I_{i,t}\leq B_{i}&\forall
t\in\mathcal{T}_{+}\end{array}\right\\}\begin{array}[]{l}\\\ \\\
\forall\bm{\xi}_{i}\in\Xi_{i}\\\
\forall\bm{s}_{\mathcal{N}_{i},i}\in\mathcal{S}_{\mathcal{N}_{i},i}\\\ \forall
i\in\mathcal{M}.\end{array}\end{array}$ (42)

In the sequel we use the open-source dataset from [5] comprising residential
power and battery data at minute resolution. This anonymized dataset contains
real-world prosumer information on energy usage, photovoltaic power
generation, and battery characteristics. Figure 14 depicts the hourly power
consumption and generation of the first prosumer in this dataset from July 1,
2021, to October 1, 2021, using lines with low opacity. Additionally, the
solid lines represent the average data, while the dashed lines represent
observations that are one standard deviation away from the mean. These
intervals are used to calibrate the uncertainty set for each prosumer. We
assume that the initial charge of all batteries is zero. The capacity of
batteries for each prosumer is also available in the dataset. For example, the
capacity is $B_{1}=13$ for the first prosumer. Since the data is anonymized
and we lack access to the price data, we set the average hourly electricity
price as

$\displaystyle
c_{t}=18-\tanh(t)+\tanh(t-4)-2\tanh(t-6)+4\tanh(t-17)-4\tanh(t-24),$

following the typical electricity price pattern, where prices are higher at
the beginning and end of the day [70]. Problems (23) and (42) are multistage
robust linear programs which we approximate with affine decision rules.

Figure 14: Illustration of power consumption, photovoltaic power generation,
and electricity price for the first prosumer.

We conduct $10$ independent simulation. For each simulation, we pick uniformly
at random $\varepsilon_{t}\in[-0.1,0.1]$ and we set the market hourly price to
$c_{t}^{+}=(1+\varepsilon_{t})c_{t}$ for all $t\in\mathcal{T}$, i.e., the
market prices are allowed to differ by up to $10\%$ from the average price,
see Figure 14 (right). We also set $c_{t}^{-}=0.5c_{t}^{+}$ and
$c_{t}^{u}=0.2c_{t}^{+}$. We consider bi-hourly time steps, meaning that every
prosumer is allowed to request or provide electricity to others every $2$
hours. The electricity price for each $2$-hour interval is determined by the
average price, while the power consumption and generation are the sum of their
respective values. In other words, we represent a day with a horizon length of
$T=12$. As baseline, we also examine a system in which all prosumers try to
minimize their worst-case electricity costs individually, and we refer to it
as the _decoupled problem_. This can be obtained by including the constraint
$U_{ij,t}=0$ for all $t\in\mathcal{T},j\in\mathcal{N}_{i},i\in\mathcal{M}$ in
(23). We conduct simulations on serial and complete networks with an
increasing number of $M$ prosumers, using the data of the first prosumers in
[5] which have both demand and photovoltaic data available.111In particular,
we use the prosumers with identification numbers $1,2,3,9,10,11$.

Figure 15: Comparison of the cost and optimization time needed to solve the
centralized, local, and decoupled information problems for serial and complete
network. The result is based on $10$ independent simulations. (CC) Centralized
Complete, (CS) Centralized Serial, (LC) Local Complete, (LS) Local Serial, (D)
Decoupled.

The first experiment investigates the impact of the number of prosumer $M$ in
the system on the worst-case cost for centralized and local information, and
decoupled problems. Figure 15 presents the results for the serial and complete
networks across $10$ independent simulation runs. Observe that the optimal
worst-case cost for the decoupled problem is significantly higher than that of
the centralized and local information exchange problems. This is expected
because in the decoupled setting, agents do not benefit from sharing
electricity among themselves. Furthermore, the optimal worst-case cost for the
local information exchange is higher than that for the centralized approach,
as indicated by Theorems 1, 3, and Corollary 1. Additionally, the structure of
a complete network positively influences the worst-case cost outcomes when
compared to a serial network configuration.

Furthermore, our findings suggest that the restrictive nature of our proposed
local information exchange has only a minimal impact on the performance of the
control policy, as demonstrated in Figure 15 (top). Specifically, the
performance gap in terms of the worst-case cost between the local and central
controllers is approximately 2% on average. Additionally, when compared to the
decoupled problems, the centralized controller reduces costs by 15%, while the
localized controller achieves a cost reduction of 13% on average.

We also observe that the time required to solve the local information exchange
problem is less than the time needed for the centralized optimization problem,
and the computational gains are more pronounced as the number of residential
prosumers increase, as illustrated in Figure 15 (bottom). From a computational
standpoint, the optimization times for the complete network are considerably
longer than those for the serial network. This is expected due to the
quadratic increase in the number of communication links with respect to the
prosumers in the complete network compared to the linear increase in the
serial network. Specifically, optimization problems for the local controller
are solved approximately 2.5 times faster in the largest instances of the
complete network structure, while in the serial network, this ratio is about 3
times faster. Moreover, the optimization time required for the local
information problem is approximately 5 times faster for the largest instance
in the serial network compared to the complete network. As anticipated, the
computational effort increases with the increasing number of building units.

We conclude our numerical experiment by comparing the average performance of
the centralized and local information problems using a rolling horizon scheme
when $\varepsilon_{t}=0$ for all $t$ and the network structure is serial. In
this experiment, we fix all parameters as in the first experiment. We then
generate a random realization of the uncertainty $\bm{\xi}\in\Xi$. Next, we
solve the centralized, local, and decoupled information for the horizon length
$T=12$ and update the battery levels according to the realization
$\bm{\xi}_{1}$ and the first stage decisions. We repeat the process until we
reach the end of the horizon. Specifically, at any time $t=2,\ldots,T$, we
resolve each problem for the shorter horizon length $T-t$ and the initial
battery level $I_{i,t-1}$ for every $i\in\mathcal{M}$. We then update the
battery levels according to realization $\bm{\xi}_{t}$ and corresponding first
stage decisions. Figure 16 summarizes our results for $10$ randomly generated
realization of uncertainty for a serial network, where the worst-case cost of
corresponding problem with horizon $T=12$ is denoted by the marker $*$. We
observe that the average performances of the local and centralized information
problems are significantly improved compared to their worst-case performance.
The improvement is approximately around half (2 times smaller). This is
extremely valuable as in reality this problem will be repeatedly solved as
time progresses. Moreover, the performance of both the local and centralized
information controllers remains superior to that of the decoupled controller.
We exclude the result for the complete network structure as it exhibits the
same behavior.

Figure 16: Comparison of the rolling horizon cost of the centralized and local
information problems for a serial network. The result is based on $10$
independent simulations. (CS) Centralized Serial, (LS) Local Serial, (D)
Decoupled.

#### Acknowledgements.

This research was supported by the Swiss National Science Foundation under the
NCCR Automation (grant agreement 51NF40_180545) and under an Early
Postdoc.Mobility Fellowship awarded to the third author (grant agreement
P2ELP2_195149).

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##  Appendix A Proofs

###### Proof of Theorem 1.

We show that every feasible solution of Problem (5b) is a feasible solution to
Problem (4b). Let $\bm{\Phi}_{i}$ for all $i\in\mathcal{M}$ be any feasible
policy in Problem (5b). Starting with $\chi_{i,1}=x_{i,1}$, the state of agent
$i$ at time $t$ are given as

$\begin{array}[]{@{}r@{~}l}x_{i,t}~{}&=\displaystyle
A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i,\tau+1}^{t}B_{i,\tau}[\chi_{j,\tau}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle}}^{\tau-1})]_{j\in\mathcal{N}_{i}}+A_{i,\tau+1}^{t}D_{i,\tau}\Phi_{i,\tau}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}^{\tau-1})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\\
~{}&=:\chi_{i,t}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}^{t-1})\end{array}$
(A.1)

where $A_{i,\tau}^{t}=A_{i,\tau}A_{i,\tau+1}\dots A_{i,t-1}$ for $\tau<t$ and
$A_{i,t}^{t}=I$. The last implication follows from the fact that
$\overline{\mathcal{N}}_{i}\supseteq\overline{\mathcal{N}}_{j}$ for all
$j\in\mathcal{N}_{i}$ since the network admits a partially nested structure.
Given (A.1), it is easy to verify that for each agent $i$ its dynamics and
constraints in Problem (5b) only depend on
$\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}$.
Hence, any feasible solution to Problem (5b) is also feasible to the following
optimization problem:

$\begin{array}[]{l}\text{minimize}\;\;\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}\in\Xi}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\\!\\!\\!\left.\begin{array}[]{r@{~}l@{}}\text{subject
to}~{}&\bm{u}_{i}=\bm{\Phi}_{i}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}):=[\Phi_{i,t}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}^{t-1})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\end{array}\right\\}\forall\bm{\xi}_{\scriptscriptstyle\mathcal{M}\scriptstyle}\in\Xi_{\scriptscriptstyle\mathcal{M}\scriptstyle},\;\forall
i\in\mathcal{M}\end{array}$ (A.2)

Additionally, they achieve the same objective value since they share the same
objective function. This shows equivalence of Problem (5b) and Problem (A.2).
The relation between Problem (4b) and Problem (5b) stated in the theorem now
follows immediately since the two problems share the same constraints and
objective function, and the policies in Problem (5b) are restricted compared
to Problem (4b) as they are functions of
$\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}$ while
the policies in Problem (4b) are functions of $\bm{\xi}_{\mathcal{M}}$. ∎

###### Proof of Theorem 2.

The statement is proved by induction using similar theoretical tools to [29,
Prop. 2.1]. The statement holds for $t=1$ since the initial state, $x_{i,1}$,
is known for every $i\in\mathcal{M}$; therefore, functions
$\psi_{i,1}(x_{i,1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},1})$
and
$\Psi_{i,1}(\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},1})$
can always be constructed such

$\psi_{i,1}(x_{i,1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},1})=\Psi_{i,1}(\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},1})$
(A.3)

Assume now that the statement holds for all $1<\tau\leq t-1$, i.e., there
exist policies $\psi_{i}(\cdot)$ and $\Psi_{i}(\cdot)$ such that
$\psi_{i,\tau}(\bm{x}_{i}^{\tau},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{\tau})=\Psi_{i,\tau}(\bm{\xi}_{i}^{\tau-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{\tau})$.
In the sequel, we show that the statement also holds for $\tau=t$. From (1),
we have that

$\begin{array}[]{r@{}l}x_{i,t}&=\displaystyle
A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i,\tau+1}^{t}B_{i,\tau}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},\tau}+A_{i,\tau+1}^{t}D_{i,\tau}\Psi_{i,\tau}(\bm{\xi}_{i}^{\tau-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{\tau})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\\
&=:\chi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}^{t-1}),\end{array}$
(A.4)

where $A_{i,\tau}^{t}=A_{i,\tau}A_{i,\tau+1}\dots A_{i,t-1}$ for $\tau<t$ and
$A_{i,t}^{t}=I$. Moreover, it holds that

$\begin{array}[]{r@{}l}\xi_{i,t-1}&=E_{i,t-1}^{+}\big{(}x_{i,t}-A_{i,t-1}x_{i,t-1}+B_{j,t-1}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},t-1}+D_{i,t-1}\psi_{i,t-1}(\bm{x}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t-1})\big{)}\\\
&=:\rho_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t-1}),\end{array}$
(A.5)

where $E^{+}_{i,t}:=(E_{i,t}^{\top}E_{i,t})^{-1}E^{\top}_{i,t}$ is the left
inverse of $E_{i,t}$ since it is full rank.

The relation (A.4) implies that given a feasible policy $\psi_{i,t}(\cdot)$
for Problem (6a), we can construct a feasible policy for Problem (6b) as

$\psi_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})=\psi_{i,t}(\chi_{i}^{t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}^{t}),\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t}):=\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t}).$
(A.6)

The claim follows from the fact that the composition of continuous
differentiable functions is a continuous differentiable function. Hence, the
policy $\psi_{i,t}(\cdot)$ will also be feasible in Problem (6b) since the two
problems have the same pointwise constraints. Additionally, they achieve the
same objective value since they share the same objective function.

Similarly, the relation (A.5) implies that given a feasible policy
$\Psi_{i}(\cdot)$ for Problem (6b), we can construct a feasible policy for
Problem (6a) as

$\Psi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})=\Psi_{i,t}(\rho_{i}^{t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t-1}),\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t}):=\psi_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t}).$
(A.7)

The claim follows from the fact that the composition of continuous
differentiable functions is a continuous differentiable function. Hence, the
policy $\Psi_{i,t}(\cdot)$ is also feasible in Problem (6a) since the two
problems have the same pointwise constraints. Additionally, they achieve the
same objective value since they share the same objective function. ∎

###### Proof of Theorem 3.

We show that every feasible solution of Problem (6b) is feasible in Problem
(5b). Let $(\bm{\Psi}_{i},\mathcal{X}_{i})$ for all $i\in\mathcal{M}$ be
feasible in Problem (6b). Since the state of agent $i$ evolve according to
(1), we can conclude that at time $t$ we have

$\begin{array}[]{r@{}l}x_{i,t}&=\displaystyle
A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i,\tau+1}^{t}B_{i,\tau}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},\tau}+A_{i,\tau+1}^{t}D_{i,\tau}\Psi_{i,\tau}(\bm{\xi}_{i}^{\tau-1},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{\tau})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\\
&=:\chi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}^{t-1}),\end{array}$
(A.8)

where where $A_{i,\tau}^{t}=A_{i,\tau}A_{i,\tau+1}\dots A_{i,t-1}$ for
$\tau<t$ and $A_{i,t}^{t}=I$.

We note that
$\bm{\chi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})=[\chi_{i,t}(\bm{\xi}_{i}^{t-1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}^{t-1})]_{t\in\mathcal{T}_{+}}\in\mathcal{X}_{i}$
for all $\bm{\xi}_{i}\in\Xi_{i}$ and
$\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}$
due to the feasibility of Problem (6b). To show that $\bm{\Psi}_{i}$ is
feasible in Problem (5b), we first construct the state of agent $i$ which
evolves according to
$\bm{x}_{i}=f_{i}\big{(}\bm{x}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{\Psi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}),\bm{\xi}_{i}\big{)}$.
Starting with $\hat{\chi}_{i,1}=x_{i,1}$, we have that

$\begin{array}[]{r@{}l}&x_{i,t}=\displaystyle
A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i,\tau+1}^{t}B_{i,\tau}[\widehat{\chi}_{j,\tau}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle}}^{\tau-1}]_{j\in\mathcal{N}_{i}}+A_{i,\tau+1}^{t}D_{i,\tau}\Psi_{i,\tau}(\bm{\xi}_{i}^{\tau-1},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}^{\tau})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\\
&=\displaystyle
A_{i,1}^{t}x_{i,1}+\sum_{\tau=1}^{t-1}\Big{(}A_{i,\tau+1}^{t}B_{i,\tau}[\widehat{\chi}_{j,\tau}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle}}^{\tau-1}]_{j\in\mathcal{N}_{i}}+A_{i,\tau+1}^{t}D_{i,\tau}\Psi_{i,\tau}(\bm{\xi}_{i}^{\tau-1},[\widehat{\bm{\chi}}_{j}^{\tau}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle}}^{\tau-1}]_{j\in\mathcal{N}_{i}})+A_{i,\tau+1}^{t}E_{i,\tau}\xi_{i,\tau}\Big{)}\\\
&=\chi_{i,t}(\bm{\xi}_{i}^{t-1},[\widehat{\chi}^{t-1}_{j}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle}}^{t-2})]_{j\in\mathcal{N}_{i}})\\\
&=:\widehat{\chi}_{i,t}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}^{t-1}).\end{array}$
(A.9)

where the implication follows because
$\bm{\widehat{\chi}}_{i}([\bm{\xi}_{j}^{t-1}]_{j\in{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}})=[\widehat{\chi}_{i,t}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}^{t-1})]_{t\in\mathcal{T}_{+}}\in\mathcal{X}_{i}$
for all
$\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}$
and $i\in\mathcal{M}$. For each $i\in\mathcal{M}$, we consider the decision
$\bm{\hat{\Phi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})$
defined through

$\widehat{\Phi}_{i,t}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}^{t-1}):=\Psi_{i,t}(\bm{\xi}_{i}^{t-1},[\widehat{\chi}^{t}_{j}(\bm{\xi}_{{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle}}^{t-1})]_{j\in\mathcal{N}_{i}})$
(A.10)

Notice that (A.10) defines a valid policy construction since
$\bm{\widehat{\chi}}_{i}([\bm{\xi}_{j}^{t-1}]_{j\in{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}})\in\mathcal{X}_{i}$
for all
$\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}$.
It remains to show that $\bm{\Psi}_{i}$ is feasible also for the constraints
of Problem (5b). We do so using deduction, as follows:

$\begin{array}[]{rll}&\big{(}\bm{\chi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}),\bm{\Psi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})\big{)}\in\mathcal{O}_{i},&\forall\bm{\xi}_{i}\in{\Xi}_{i},\forall\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}\in\mathcal{X}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle},\\\
\implies&\big{(}\bm{\chi}_{i}(\bm{\xi}_{i},[\bm{\widehat{\chi}}_{j}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle})]_{j\in\mathcal{N}_{i}}),\bm{\Psi}_{i}(\bm{\xi}_{i},[\bm{\widehat{\chi}}_{j}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle})]_{j\in\mathcal{N}_{i}})\big{)}\in\mathcal{O}_{i},&\forall\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in{\Xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle},\\\
\implies&\big{(}\bm{\widehat{\chi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}),\bm{\widehat{\Phi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})\big{)}\in\mathcal{O}_{i},&\forall\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in{\Xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle},\end{array}$
(A.11)

The first implication follows from (A.9) and the fact that
$\bm{\widehat{\chi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})\subseteq\mathcal{X}_{i}$
for all
$\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}$,
while the second implication follows from (A.9) and (A.10). Finally, this
feasible solution attains a value for the objective function of Problem (6b)
which is equal or larger than the value attained for the objective function of
Problem (5b), that is

$\begin{array}[]{@{\,}r@{\,}l@{\,}}&\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}}J_{i}\big{(}\bm{\chi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}),\bm{\Psi}_{i}(\bm{\xi}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}})\big{)}\\\
\geq&\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}\;J_{i}\big{(}\bm{\chi}_{i}(\bm{\xi}_{i},[\bm{\widehat{\chi}}_{j}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle})]_{j\in\mathcal{N}_{i}}),\bm{\Psi}_{i}(\bm{\xi}_{i},[\bm{\widehat{\chi}}_{j}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{j}\scriptstyle})]_{j\in\mathcal{N}_{i}})\big{)}\\\
=&\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}\;J_{i}\big{(}\bm{\widehat{\chi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}),\bm{\widehat{\Phi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})\big{)},\end{array}$

where again the first implication follows from (A.9) and the fact that
$\bm{\widehat{\chi}}_{i}(\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle})\subseteq\mathcal{X}_{i}$
for all
$\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}$,
while the second implication follows from (A.9) and (A.10). ∎

###### Proof of Proposition 1.

Theorem 3 already established that a feasible solution of Problem (6b) is
feasible in Problem (5b). We will now show that a feasible solution in Problem
(5a) is feasible in Problem (6a). The result will then follow due to the
relation between Problems (5a) and (5b), see [40], and Problems (6a) to (6b),
see Theorem 2.

In a directed acyclic bipartite network, the agents can be split in two
groups, the first layer agents where $\overline{\mathcal{N}}_{i}=\\{i\\}$ and
the second layer agents where
$\overline{\mathcal{N}}_{i}=\\{\mathcal{N}_{i},i\\}$. Sets $\mathcal{FL}$ and
$\mathcal{SL}$ denote the first and second layer agents, respectively. Problem
(6a) can then be explicitly written as

$\begin{array}[]{l}\text{~{}minimize~{}~{}}\displaystyle\sum\limits_{i\in\mathcal{FL}}\max\limits_{\bm{\xi}_{i}\in\Xi_{i}}J_{i}(\bm{x}_{i},\bm{u}_{i})+\sum\limits_{i\in\mathcal{SL}}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}}J_{i}(\bm{x}_{i},\bm{u}_{i})\\\
\left.\begin{array}[]{@{}r@{~}l@{}}\text{subject to
}~{}&\bm{u}_{i}=\bm{\psi}_{i}(\bm{x}_{i}):=[\psi_{i,t}(\bm{x}_{i}^{t})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\\\
~{}&\bm{x}_{i}\in\mathcal{X}_{i},\;\mathcal{X}_{i}\in\mathcal{F}(\mathcal{S}_{i})\end{array}\right\\}\forall\bm{\xi}_{i}\in\Xi_{i},\;\forall
i\in\mathcal{FL},\\\ \left.\begin{array}[]{@{}r@{~}l@{}}\phantom{\text{subject
to
}}~{}&\bm{u}_{i}=\bm{\psi}_{i}(\bm{x}_{i},\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}):=[\psi_{i,t}(\bm{x}_{i}^{t},\bm{\zeta}_{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}^{t})]_{t\in\mathcal{T}}\\\
~{}&\bm{x}_{i}=f_{i}\big{(}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}\\\
~{}&(\bm{x}_{i},\bm{u}_{i})\in\mathcal{O}_{i}\\\
\end{array}\right\\}\begin{array}[]{@{}l}\forall\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\\\
\forall\bm{\xi}_{\mathcal{M}}\in\Xi_{\mathcal{M}}\end{array}\;\forall
i\in\mathcal{SL},\end{array}$ (A.12)

where the constraints distinguish between first and second layer agents.
Notice that agents in the first layer are the ones communicating their state
forecast sets $\mathcal{X}_{i}$ to their neighbors (constraint
$\bm{x}_{i}\in\mathcal{X}_{i}$), while the agents on the second layer only
receive these sets
($\bm{x}_{i}=f_{i}\big{(}\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}},\bm{u}_{i},\bm{\xi}_{i}\big{)}$,
$\forall\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}$).

Let $\phi_{i}$ for all $i\in\mathcal{M}$ be a feasible policy for Problem
(5a). It is easy to see that policy is feasible in
$(f_{i}\big{(}\phi_{i}(\bm{x}_{i}),\bm{\xi}_{i}\big{)},\bm{u}_{i})\in\mathcal{O}_{i}\quad\forall\bm{\xi}_{i}\in\Xi_{i},\,i\in\mathcal{FL}$
as Problems (5a) and (6a) share the same constraints. We set

$\mathcal{X}_{i}:=\\{\bm{x}_{i}\in\mathbb{R}^{N_{x,i}}\;:\;\bm{x}_{i}=f_{i}\big{(}\phi_{i}(\bm{x}_{i}),\bm{\xi}_{i}\big{)}\;\;\forall\bm{\xi}_{i}\in\Xi_{i}\\}\quad\forall
i\in\mathcal{FL},$ (A.13)

which satisfies $\mathcal{X}_{i}\in\mathcal{F}(\mathcal{S}_{i})$ where
$\mathcal{S}_{i}=\mathbb{R}^{N_{x,i}}$ for all $i\in\mathcal{FL}$. By the
feasibility of $\phi_{i}$ in Problem (5a), we have

$\begin{array}[]{lll}&\left.\begin{array}[]{l}\bm{x}_{j}=f_{j}\big{(}\phi_{j}(\bm{x}_{j}),\bm{\xi}_{j}\big{)},\;\forall\,j\in\mathcal{N}_{i}\\\
(f_{i}(\bm{x}_{\mathcal{N}_{i}},\phi_{i}(\bm{x}_{i},\bm{x}_{\mathcal{N}_{i}}),\bm{\xi}_{i}),\phi_{i}(\bm{x}_{i},\bm{x}_{\mathcal{N}_{i}}))\in\mathcal{O}_{i}\end{array}\right\\}&\forall\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}},\;\forall
i\in\mathcal{SL},\\\
\iff&\left.\begin{array}[]{l}(f_{i}(\bm{x}_{\mathcal{N}_{i}},\phi_{i}(\bm{x}_{i},\bm{x}_{\mathcal{N}_{i}}),\bm{\xi}_{i}),\phi_{i}(\bm{x}_{i},\bm{x}_{\mathcal{N}_{i}}))\in\mathcal{O}_{i}\end{array}\right\\}&\begin{array}[]{l}\forall[\bm{x}_{j}]_{j\in\mathcal{N}_{i}}=[f_{j}\big{(}\phi_{j}(\bm{x}_{j}),\bm{\xi}_{j}\big{)}]_{j\in\mathcal{N}_{i}}\\\
\forall\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}\end{array},\,\forall
i\in\mathcal{SL},\\\
\iff&\left.\begin{array}[]{l}(f_{i}(\bm{\zeta}_{\mathcal{N}_{i}},\phi_{i}(\bm{x}_{i},\bm{\zeta}_{\mathcal{N}_{i}}),\bm{\xi}_{i}),\phi_{i}(\bm{x}_{i},\bm{\zeta}_{\mathcal{N}_{i}}))\in\mathcal{O}_{i}\end{array}\right\\}&\begin{array}[]{l}\forall\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\\\
\forall\bm{\xi}_{\mathcal{M}}\in\Xi_{\mathcal{M}}\end{array},\,\forall
i\in\mathcal{SL},\\\ \end{array}$

where the first equivalence hold since for fixed policy $\phi_{i}$ the
dynamics of all agents in the first layer are uniquely determined by the
uncertain parameters $\bm{\xi}_{i}\in\Xi_{i}$, $i\in\mathcal{FL}$, while the
second equivalence holds by the definition of $\mathcal{X}_{i}$ in (A.13).
Thus, $\phi_{i}$ is feasible in Problem (6a).

We next show that $\phi_{i}$ achieves the same optimal value in both problems.

$\begin{array}[]{ll}&\displaystyle\sum\limits_{i=1}^{M}\max\limits_{\bm{\xi}_{i}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}))\\\
=&\displaystyle\sum\limits_{i\in\mathcal{FL}}\max\limits_{\bm{\xi}_{i}\in\Xi_{i}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{i}))\par+\sum\limits_{i\in\mathcal{SL}}\max\limits_{\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{i},{\bm{x}}_{{\mathcal{N}}_{i}}))\\\
=&\displaystyle\sum\limits_{i\in\mathcal{FL}}\max\limits_{\bm{\xi}_{i}\in\Xi_{i}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{i}))\par+\sum\limits_{i\in\mathcal{SL}}\max\limits_{\bm{\xi}_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle}\in\Xi_{\scriptscriptstyle\overline{\mathcal{N}}_{i}\scriptstyle},\,[\bm{x}_{j}]_{j\in\mathcal{N}_{i}}=[f_{j}\big{(}\phi_{j}(\bm{x}_{j}),\bm{\xi}_{j}\big{)}]_{j\in\mathcal{N}_{i}}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{i},{\bm{x}}_{{\mathcal{N}}_{i}}))\\\
=&\displaystyle\sum\limits_{i\in\mathcal{FL}}\max\limits_{\bm{\xi}_{i}\in\Xi_{i}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{i}))\par+\sum\limits_{i\in\mathcal{SL}}\max\limits_{\bm{\xi}_{i}\in\Xi_{i},\,\bm{\zeta}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}\in\mathcal{X}_{{\scriptscriptstyle\mathcal{N}_{i}\scriptstyle}}}J_{i}(\bm{x}_{i},\bm{\phi}_{i}(\bm{x}_{i},{\bm{\zeta}}_{{\mathcal{N}}_{i}}))\\\
\end{array}$ (A.14)

Starting from the objective of Problem (5a), the first equivalence rewrites
the objective function it in terms of the first and second layer agents. Since
for fixed $\phi_{i}$ the dynamics of the first layer agents are uniquely
determined by the uncertain parameters $\bm{\xi}_{i}\in\Xi_{i}$,
$i\in\mathcal{FL}$, the second equality re-expresses the dynamics of
${\bm{x}}_{{\mathcal{N}}_{i}}$ in terms of $\bm{\xi}_{i}\in\Xi_{i}$,
$i\in\mathcal{FL}$. Finally, in the third equality we substitute the
definition of $\mathcal{X}_{i}$ in (A.13). Notice that the uncertain
parameters governing $\mathcal{X}_{i}$ are those affective first layer agents,
thus making $\mathcal{X}_{\mathcal{FL}}$ and $\Xi_{\mathcal{SL}}$ rectangular.
The last expression in (A.14) coincides with the objective of Problem (6a). ∎

###### Proof of Proposition 2.

The recession cone of the set $\mathcal{S}_{i}$ is defined as
$\textrm{recc}(\mathcal{S}_{i})=\\{\bm{\nu}_{i}\in\mathbb{R}^{n_{x}^{i}}:\bm{s}_{i}+\lambda\bm{\nu}_{i}\in\mathcal{S}_{i},\forall\bm{s}_{i}\in\mathcal{S}_{i},\,\lambda\geq
0\\}$. The fact that $\mathcal{S}_{i}$ is bounded implies that the recession
cone of $\mathcal{S}_{i}$ is empty, i.e.,
$\textrm{recc}(\mathcal{S}_{i})=\\{0\\}$. We now show that,

$\mathcal{X}_{{\scriptscriptstyle\mathcal{FS}\scriptstyle}}=\Big{\\{}(\bm{x}_{i},y_{i},z_{i})\,:\,\exists\bm{s}_{i}\in\mathbb{R}^{N_{x,i}}\text{
s.t.
}\bm{x}_{i}=\sum_{k=1}^{K_{i}}y_{i,k}P_{i,k}\bm{s}_{i}+z_{i},\;G_{i,k}P_{i,k}\bm{s}_{i}\preceq_{\mathcal{K}_{i,k}}g_{i,k},\;k=1,\ldots,K_{i}\Big{\\}}$

is equivalent to

$\widehat{\mathcal{X}}_{{\scriptscriptstyle\mathcal{FS}\scriptstyle}}=\Big{\\{}(\bm{x}_{i},y_{i},z_{i})\,:\,\exists\bm{\nu}_{i,k}\in\mathbb{R}^{N_{x,i}}\text{
s.t.