On decentralized convex optimization in a multi-agent setting with
separable constraints and its application to optimal charging of electric
vehicles
Luca Deori, Kostas Margellos, Maria Prandini
Abstract— We develop a decentralized algorithm for multi-
agent, convex optimization programs, subject to separable
constraints, where the constraint function of each agent involves
only its local decision vector, while the decision vectors of
all agents are coupled via a common objective function. We
construct a variant of the so called Jacobi algorithm and show
that, when the objective function is quadratic, convergence
to some minimizer of the centralized problem counterpart is
achieved. Our algorithm serves then as an effective alternative
to gradient based methodologies. We illustrate its efficacy by
applying it to the problem of optimal charging of electric
vehicles, where, as opposed to earlier approaches, we show
convergence to an optimal charging scheme for a finite, possibly
large, number of vehicles.
I. INTRODUCTION
Optimization in multi-agent systems has attracted sig-
nificant attention in the control and operations research
communities, due to its applicability to different domains,
e.g., energy systems, mobility systems, robotic networks, etc.
In this paper we focus on a specific class of multi-agent
optimization programs that are convex and are subject to
constraints that are separable, i.e., the constraint function of
each agent involves only its local decision vector. The agents’
decision vectors are, however, coupled by means of a com-
mon objective function. The considered structure, although
specific, captures a wide class of engineering problems, like
the electric vehicle optimal charging problem studied in this
paper. Solving such problems in a centralized fashion would
require agents to share their local constraint functions with
each other. This would raise, however, information privacy
issues. Even if this were not an issue, solving the problem
in one shot, without exploiting the separable structure of the
constraints, would lead to an optimization program of larger
size, involving the decision variables and constraints of all
agents, and possibly pose computational challenges.
To allow for a computationally tractable solution, while
accounting for information privacy, we adopt a decentralized
perspective, where agents cooperate to obtain an optimal
solution of the centralized problem. We follow an iterative
algorithm, where at every iteration each agent solves a local
optimization problem with respect to its own local decision
Research was supported by the European Commission, H2020, under the
project UnCoVerCPS, grant number 643921.
L. Deori and M. Prandini are with the Dipartimento di Elet-
tronica Informazione e Bioingegneria, Politecnico di Milano, Piazza
Leonardo da Vinci 32, 20133 Milano, Italy, e-mail: {luca.deori,
maria.prandini}@polimi.it
K. Margellos is with the Department of Engineering Science, Univer-
sity of Oxford, Parks Road, OX1 3PJ, Oxford, United Kingdom, e-mail:
kostas.margellos@eng.ox.ac.uk
vector using the tentative solutions computed by the other
agents at the previous iteration. Agents then exchange with
each other their new tentative solutions, or broadcast them to
some central authority that sends an update to each agent; the
process is repeated on the basis of the received information.
Algorithms for the decentralized solution to convex opti-
mization problems with separable constraints can be found
in [1], [2], and references therein. Two main algorithmic
directions can be distinguished, both of them relying on
an iterative process. The first one is based on each agent
performing at every iteration a local gradient descent step,
while keeping the decision variables of all other agents fixed
to the values communicated at the previous iteration [3–5].
Under certain structural assumptions (differentiability of the
objective function and Lipschitz continuity of its gradient),
it is shown that this scheme converges to some minimizer
of the centralized problem, for an appropriate gradient step-
size. The second direction involves mainly the so called
Jacobi algorithm, which serves as an alternative to gradient
algorithms. In this framework also the Gauss-Seidel algo-
rithm which however is not of parallelizable nature unless a
coloring scheme is adopted (see [1]), and block coordinate
descent methods [6] can be considered. Under this set-up, at
every iteration, instead of performing a gradient step, each
agent minimizes the common objective function subject to
local constraints, while keeping the decision vectors of all
other agents fixed to their values at the previous iteration. It
is shown that the Jacobi algorithm converges under certain
contractiveness requirements, that are typically satisfied only
if the objective function is jointly strictly convex with respect
to the decision vectors of all agents.
An alternative research direction with a notable research
activity involves a non-cooperative treatment of the problem,
using tools from mean-field and aggregative game theory.
A complete theoretical characterization for the stochastic,
continuous-time variant of the problem, but in the absence
of constraints, is provided in [7], [8]. The deterministic,
discrete-time problem variant, accounting for the presence
of separable constraints, is treated using fixed-point theoretic
tools in [9]. In all cases, the considered algorithm is shown to
converge not to a minimizer, but to a Nash equilibrium of a
related game, in the limiting case where the number of agents
tends to infinity. Several applications in this context have
been provided, e.g., optimal power flow type of problems
[10], optimal charging of electric vehicles [11], [12], etc.
In this paper we adopt a cooperative point of view, and
construct a Jacobi-like algorithm. In contrast to the standard
Jacobi algorithm, the local minimization that each agent
solves at every iteration of the algorithm includes also an
inertial term that encompasses the solution of that agent at
the previous iteration. Our contributions can be summarized
as follows: 1) We establish an equivalence between the set
of minimizers of the problem under study and the set of
fixed-points of the mapping induced by our algorithm, for
any convex objective function. 2) For the case where the
objective function is quadratic we show that our algorithm
converges to some minimizer of the centralized problem, thus
constituting an alternative to gradient methods, and without
requiring strict convexity of the objective function as in the
standard Jacobi algorithm. This result extends the equiva-
lence between proximal operators and gradient algorithms
observed in [2] for the single-agent case, to the multi-agent
setting. 3) We apply the proposed algorithm to the problem
of optimal charging of electric vehicles, extending the results
of [9], [11], [12], and achieving convergence to an optimal
charging scheme with a finite number of vehicles.
Notation
For any a ∈ R
n
, kak denotes the Euclidean norm of a,
and kak
Q
denotes the Q-weighted Euclidean norm of a. For
a vector a, we denote by a
i
the i-th block component of
a, whereas a
−i
denotes the vector emanating from a by
removing a
i
. Similarly for matrix A, A
i,i
denotes the i-
th diagonal block of A, whereas A
−i,i
denotes the matrix
composed by the i-th block column of A and all, but the
i-th block row. For a continuously differentiable function
J(·) : R
n
→ R, ∇J(a) is the gradient of J(·) evaluated at
a ∈ R
n
, and ∇
i
J(a) is its i-th component, i = 1, . . . , n.
[a]
U
Q
denotes the projection of a vector a on the set U
with respect to the Q-weighted Eucliedean norm. 1
n×m
denotes the matrix with all entries equal to 1 with dimension
n × m, and I
c
denotes the identity matrix with appropriate
dimension, multiplied by the scalar c ∈ R.
II. PROBLEM STATEMENT
Consider the following optimization problem
P : min
{u
i
∈R
n
i
}
m
i=1
J(u
1
, . . . , u
m
) (1)
subject to u
i
∈ U
i
, for all i = 1, . . . , m, (2)
where J(·, . . . , ·) : R
n
1
×. . .×R
n
m
→ R, and U
i
⊆ R
n
i
, for
all i = 1, . . . , m. Let n =
P
m
i=1
n
i
and U = U
1
× . . . × U
m
.
We impose the following assumption throughout the paper.
Assumption 1: The function J(·, . . . , ·) : R
n
1
× . . . ×
R
n
m
→ R is continuously differentiable, and jointly convex
with respect to all arguments. Moreover, the sets U
i
⊆ R
n
i
,
i = 1, . . . , m, are non-empty, compact and convex.
Under Assumption 1, by the Weierstrass’ theorem (Propo-
sition A8, p. 625 in [1]), P admits at least one optimal
solution. Note, however, that P does not necessarily admit a
unique minimizer. With a slight abuse of notation, for each
i, i = 1, . . . , m, let J(·, u
−i
) : R
n
i
→ R be the objective
function in (1) as a function of the decision vector u
i
of agent
i, when the decision vectors of all other agents are fixed to
u
−i
∈ R
n−n
i
. We will occasionally also write J(u) instead
of J(u
1
, . . . , u
m
), for u = (u
1
, . . . , u
m
), the interpretation
will always be clear from the context. Problem P can be
thought of as a multi-agent problem, where agents have a
Algorithm 1 Decentralized algorithm
1: Initialization
2: k = 0.
3: Consider u
i
(0) ∈ U
i
, for all i = 1, . . . , m.
4: For i = 1, . . . , m repeat until convergence
5: Agent i receives u
−i
(k) from central authority.
6: u
i
(k + 1) = λu
i
(k)
+(1−λ)argmin
z
i
∈U
i
J(z
i
, u
−i
(k))+ckz
i
−u
i
(k)k
2
.
7: k ← k + 1.
local decision vector u
i
and a local constraint set U
i
, and
cooperate to determine a minimizer of J, which couples
the individual decision vectors. Motivated by the particular
structure of P with separable constraint sets, we follow a
decentralized, iterative approach described in Algorithm 1.
This allows to cope with privacy and computational issues.
Initially, each agent i, i = 1, . . . , m, starts with some
tentative value u
i
(0) ∈ U
i
, such that
u
1
(0), . . . , u
m
(0)
is feasible and constitutes an estimate of what the minimizer
of P might be (step 3, Algorithm 1). At iteration k + 1,
each agent i receives the values of all other agents u
−i
(k)
(step 5, Algorithm 1) from the central authority, and up-
dates its estimate by averaging with weight λ ∈ (0, 1) the
previous estimate and the solution of a local minimization
problem (step 6, Algorithm 1). The performance criterion in
this local problem is a linear combination of the objective
J(z
i
, u
−i
(k)), where the variables of all other agents apart
from the i-th one are fixed to their values at iteration k,
and a quadratic term, penalizing the difference between the
decision variables and the value of agent’s i own variable at
iteration k, i.e., u
i
(k). The relative importance of these two
terms is dictated by c ∈ R
+
; we defer the discussion on the
importance of the penalty term until Section IV. Note that
under Assumption 1, and due to the presence of the quadratic
penalty term, the resulting problem is strictly convex with
respect to z
i
, and hence admits a unique minimizer.
III. PRELIMINARY DEFINITIONS AND RESULTS
A. Definitions
1) Minimizers: By (1)-(2), the set of minimizers of P is
M =
u ∈ U : u ∈ arg min
{z
i
∈U
i
}
m
i=1
J(z
1
, . . . , z
m
)
. (3)
Following the discussion below Assumption 1, M is non-
empty. Note that set of optimizers of M is not necessarily a
singleton; this will be the case if J is jointly strictly convex
with respect to its arguments.
2) Fixed-points: For each i, i = 1, . . . , m, consider the
mappings T
i
(·) : U → U
i
and
e
T
i
(·) : U → U
i
, defined such
that, for any u = (u
1
, . . . , u
m
) ∈ U,
T
i
(u) = arg min
z
i
∈U
i
kz
i
− u
i
k
2
(4)
subject to J(z
i
, u
−i
) ≤ min
ζ
i
∈U
i
J(ζ
i
, u
−i
)
e
T
i
(u) = arg min
z
i
∈U
i
J(z
i
, u
−i
) + ckz
i
− u
i
k
2
. (5)
The mapping in (4) serves as a tie-break rule to select, in
case J(·, u
−i
) admits multiple minimizers, the one closer to
u
i
with respect to the Euclidean norm. Note that both the
minimizers of (4) and (5) are unique, so that both mappings
are well defined. Note also that with u(k) in place of u,
(5) implies that the update step 6 in Algorithm 1 can be
equivalently written as u
i
(k+1) = λu
i
(k)+(1−λ)
e
T
i
(u(k)).
Define also the mappings T (·) : U → U and
e
T (·) : U → U,
such that their components are given by T
i
(·) and
e
T
i
(·), re-
spectively, for i = 1, . . . , m, i.e., T (·) =
T
1
(·), . . . , T
m
(·)
and
e
T (·) =
e
T
1
(·), . . . ,
e
T
m
(·)
. The mappings T (·) and
e
T (·)
can be equivalently written as
T (u) = arg min
z∈U
m
X
i=1
kz
i
− u
i
k
2
(6)
subject to J(z
i
, u
−i
) ≤ min
ζ
i
∈U
i
J(ζ
i
, u
−i
), ∀i = 1, . . . , m
e
T (u) = arg min
z∈U
m
X
i=1
J(z
i
, u
−i
) + ckz
i
− u
i
k
2
, (7)
where the terms inside the summations are decoupled. The
set of fixed-points of T (·) and
e
T (·) is, respectively, given by
F
T
=
u ∈ U : u = T (u)
, and F
e
T
=
u ∈ U : u =
e
T (u)
.
B. Connections between minimizers and fixed-points
We report here a fundamental optimality result (e.g., see
Proposition 3.1 in [1]), that we will often use in the sequel.
Proposition 1 (Proposition 3.1 in [1]): Consider any n ∈
N
+
, and assume that J(·) : R
n
→ R is a continuously
differentiable function, and U ⊆ R
n
is non-empty, closed
and convex. It holds that i) if u ∈ U minimizes J(·) over
U, then (z − u)
>
∇J(u) ≥ 0, for all z ∈ U ; ii) if J(·)
is also convex on U, then the condition of the previous
part is also sufficient for u to minimize J(·) over U , i.e.,
u ∈ arg min
z∈U
J(z).
The following propositions show that the set of minimizers
M of P in (3) and the set of fixed-points F
T
of the mapping
T in (6) coincide, and that the set of fixed-points F
T
of T (·)
and the set of fixed-points F
e
T
of
e
T (·) coincide.
Proposition 2: Under Assumption 1, M = F
T
.
Proposition 3: Under Assumption 1, F
T
= F
e
T
.
Proof: The proofs are omitted due to space limitation,
they can be found in [13, Proposition 2 and 3].
Note that the connection between minimizers and fixed-
points, similar to the ones in Proposition 2, has been also
investigated in [14], in the context of Nash equilibria in non-
cooperative games.
IV. MAIN CONVERGENCE RESULT
In this section we strengthen Assumption 1, and focus
on convex optimization problems with a convex, quadratic
objective function.
Assumption 2: For any u ∈ U, J(u) = u
>
Qu + q
>
u,
where Q 0 and q ∈ R
n
.
Note that Q can be assumed to be symmetric (i.e., Q = Q
>
)
without loss of generality.Moreover if additional terms that
depend on the local decision vectors u
i
, i = 1, . . . , m, and
encode the utility function of each agent were present in the
objective function, they could be incorporated in the local
constraint set U
i
, i = 1, . . . , m, by means of an epigraphic
reformulation, thus bringing the cost back to be quadratic.
Under Assumption 2, the mapping
e
T (·) in (7) is given by
e
T (u) = arg min
z∈U
m
X
i=1
J(z
i
, u
−i
) + ckz
i
− u
i
k
2
(8)
= arg min
z∈U
m
X
i=1
z
>
i
(Q
i,i
+I
c
)z
i
+(2u
>
−i
Q
−i,i
−2u
>
i
I
c
+q
>
i
)z
i
= arg min
z∈U
z
>
(Q
d
+ I
c
)z + (2u
>
Q
z
− 2u
>
I
c
+ q
>
)z,
where, for all i = 1, . . . , m, Q
i,i
is the i-th block of Q, cor-
responding to the decision vector z
i
, Q
d
is a block diagonal
matrix whose i-th block is Q
i,i
, and Q
z
= Q−Q
d
. Notice the
slight abuse of notation in (8), where the weighted identity
matrix I
c
in the second and the third equality are not of the
same dimension. Let ξ(u) = (Q
d
+ I
c
)
−1
(I
c
u − Q
z
u − q/2)
denote the unconstrained minimizer of (8). Then
e
T (u) = arg min
z∈U
(z − ξ)
>
(Q
d
+ I
c
)(z − ξ) = [ξ(u)]
U
Q
d
+I
c
.
Note that Q
d
+ I
c
is always positive definite for c ∈ R
+
, so
that the projection [ξ(u)]
U
Q
d
+I
c
is well defined.
In the next proposition the non-expansive property of the
mapping
e
T (·) is proven. This property will be exploited in
Theorem 1 to establish convergence of Algorithm 1.
Proposition 4: Consider Assumptions 1 and 2. If
2Q Q
Q Q
d
+ I
c
0, (9)
the mapping
e
T (u) = [ξ(u)]
U
Q
d
+I
c
is non-expansive with
respect to || · ||
Q
d
+I
c
, namely k
e
T (u) −
e
T (v)k
Q
d
+I
c
≤ ku −
vk
Q
d
+I
c
, for all u, v ∈ U .
Proof: Any projection mapping is non-expansive (see
Proposition 3.2 in [1]). Therefore, we have that
k
e
T (u) −
e
T (v)k
Q
d
+I
c
= k [ξ(u)]
U
Q
d
+I
c
− [ξ(v)]
U
Q
d
+I
c
k
Q
d
+I
c
≤ kξ(u) − ξ(v)k
Q
d
+I
c
. (10)
We will show that, if (9) holds,
kξ(u) − ξ(v)k
Q
d
+I
c
≤ ku − vk
Q
d
+I
c
, (11)
proving that the mapping
e
T (u) is non-expansive. Replacing
in (11) the expression of ξ, and raising to the square yields
k(Q
d
+ I
c
)
−1
(I
c
− Q
z
)(u − v)k
2
Q
d
+I
c
≤ ku − vk
2
Q
d
+I
c
⇔
k(I − (Q
d
+ I
c
)
−1
Q)(u − v)k
2
Q
d
+I
c
≤ ku − vk
2
Q
d
+I
c
(12)
where I is an identity matrix of appropriate dimension, and
the equivalence follows from the fact that Q
z
= Q − Q
d
. By
bringing both terms in (12) in the left-hand side and by the
definition of || · ||
Q
d
+I
c
, (12) is satisfied if
(I − (Q
d
+ I
c
)
−1
Q)
>
(Q
d
+ I
c
)(I − (Q
d
+ I
c
)
−1
Q)
− (Q
d
+ I
c
) 0 ⇔ 2Q − Q(Q
d
+ I
c
)
−1
Q 0. (13)
where the last inequality follows after some algebraic cal-
culations and the fact that Q and Q
d
+ I
c
are symmetric.
Equation (13) can be rewritten by means of Schur’s comple-
ment finally obtaining (9).
Notice that if ¯c satisfies (9), then any c ≥ ¯c satisfies (9)
as well. To see this, take any c ≥ ¯c and rewrite (9) as
2Q Q
Q Q
d
+ I
¯c
+
0 0
0 I
˜c
0, (14)
where ˜c = c−¯c. The matrices in (14) are both positive semi-
definite and, hence, their sum is also positive semi-definite.
Condition (9) can be easily checked by means of standard
LMI solvers. In fact, it can be shown that for any Q 0
there exists c such that (9) is satisfied. Indeed condition, (9)
can be equivalently written as
u
>
v
>
2Q Q
Q Q
d
+ I
c
u
v
≥ 0, for all u, v ∈ R
n
⇔u
>
2Qu + v
>
(Q
d
+ I
c
)v + 2v
>
Qu ≥ 0. (15)
From the fact that (u + v)
>
Q(u + v) ≥ 0 it follows that
2v
>
Qu ≥ −u
>
Qu− v
>
Qv. Replacing the latter in (15), the
following sufficient condition to satisfy (9) is obtained.
u
>
Qu + v
>
(Q
d
+ I
c
− Q)v ≥ 0, for all u, v ∈ R
n
. (16)
The first term in (16) is non-negative because Q 0, while
the second term can be made non-negative exploiting the
matrix I
c
to move the eigenvalues of Q
d
− Q by c. Indeed,
letting λ
I
c
+Q
d
−Q
and λ
Q
d
−Q
denote the eigenvalues of I
c
+
Q
d
− Q and Q
d
− Q, respectively, we have
λ
I
c
+Q
d
−Q
= λ
Q
d
−Q
+ c. (17)
Hence, (16) can be satisfied by choosing c such that c ≥
λ
max
Q−Q
d
, where λ
max
Q−Q
d
denotes the maximum eigenvalue of
matrix Q − Q
d
. Note that, since Q − Q
d
is symmetric with
zero trace, its eigenvalues will be real and at least one should
be non-negative. As a result, c ≥ λ
max
Q−Q
d
≥ 0.
Theorem 1: Consider Assumptions 1 and 2. If c ∈ R
+
is
chosen so that (9) is satisfied, then Algorithm 1 converges
to a minimizer of P.
Proof: Step 6 of Algorithm 1 corresponds to the so
called Krasnoselskij iteration [15] (referred to as averaged
operator in [2]),
u(k + 1) = λu(k) + (1 − λ)
e
T (u(k)), (18)
of the mapping
e
T (·), which, according to Proposition 4, is
non-expansive if (9) is satisfied. By Theorem 3.2 in [15], we
have that for any non-expansive mapping
e
T (u) : U → U,
with U compact and convex, the Krasnoselskij iteration (18)
converges to a fixed-point of
e
T (·) for any λ ∈ (0, 1), and for
any initial condition u(0) ∈ U . Under Assumptions 1 and 2,
the mapping
e
T (·) defined in (8), satisfies the aforementioned
requirements, hence, Algorithm 1 leads to a fixed-point of
e
T (·). By Propositions 2 and 3, this fixed-point will also be
a minimizer of P.
It should be noted that the condition on c in Proposition
4 is related to the requirement imposed in [1] (Proposition
3.4, p. 214) on the step-size of a gradient based approach.
This is due to the fact that in the case where J(·) satisfies
Assumption 2, step 6 of Algorithm 1 can be shown to be
equivalent to a scaled gradient projection algorithm (see
Section 3.3.3 in [1]), with 1/c playing the role of the gradient
step-size and with (Q
d
+ I
c
)
−1
(inverse of the Hessian of
the objective function in step 6 of Algorithm 1) being the
scaling matrix. For such algorithms convergence results exist
for an appropriate step-size, which is in turn related to c. In
particular, the step-size for which convergence is guaranteed
is related to the Lipschitz constant of the gradient of the
objective function, and, under Assumption 2, it can be shown
to be equivalent with choosing c so that (9) is satisfied.
Hence, in the case of a convex quadratic objective func-
tion, both the proposed algorithm and a gradient based
approach are applicable. Our analysis, however, not only
complements the scaled gradient projection algorithm by
constituting its Jacobi-like equivalent without requiring strict
convexity of the objective function as it is usually the
case for the standard Jacobi algorithm, but also follows
a different analysis from that in [1], motivated by the
fixed-point theoretic results of [9]. Moreover, the results of
Section II are valid for any convex objective function, not
necessarily quadratic, thus opening the road for extending the
convergence results of Section III by relaxing Assumption 2;
see [13] for preliminary results.
A. Alternative implementations
We investigate the convergence properties of Algorithm 1
when the so called Krasnoselskij iteration (step 6 of Algo-
rithm 1) is replaced by the simpler Picard-Banach iteration:
u(k + 1) =
e
T (u(k)). (19)
Note that this corresponds to setting λ = 0 in step 6 of
Algorithm 1. As in the proof of Theorem 1, once convergence
is proven, then it is easily shown that the solution is a
minimizer of P due to Propositions 2 and 3.
For Algorithm 1 to converge when step 6 is replaced by
(19), the mapping
e
T (·) has to be either contractive or firmly
non-expansive [15]. We first investigate conditions under
which
e
T (·) is contractive. Following a reasoning similar to
the one in the proof of Proposition 4, it can be seen that if
2Q − (1 − α
2
)(Q
d
+ I
c
) Q
Q Q
d
+ I
c
0, (20)
is satisfied for some α ∈ [0, 1), then k
e
T (u) −
e
T (v)k
Q
d
+I
c
≤
αku − vk
Q
d
+I
c
, for all u, v ∈ U, which in turn implies that
the mapping
e
T (·) is contractive with respect to || · ||
Q
d
+I
c
[15]. Condition (20), however, can not be always satisfied
by appropriately choosing c; indeed, it is necessary that
Q is positive definite for (20) to be satisfied. The latter
is equivalent to requiring that the objective function in
Assumption 2 is strictly convex.
We now investigate conditions under which
e
T (·) is firmly
non-expansive. Motivated by the analysis of [9], we have that
k
e
T (u)−
e
T (v)k
2
Q
d
+I
c
= k [ξ(u)]
U
Q
d
+I
c
− [ξ(v)]
U
Q
d
+I
c
k
2
Q
d
+I
c
≤ (ξ(u) − ξ(v))
>
(Q
d
+ I
c
)([ξ(u)]
U
Q
d
+I
c
− [ξ(v)]
U
Q
d
+I
c
)
=(u−v)
>
(I−Q(Q
d
+I
c
)
−1
)(Q
d
+I
c
)([ξ(u)]
U
Q
d
+I
c
−[ξ(v)]
U
Q
d
+I
c
)
=(u−v)
>
(Q
d
+ I
c
− Q)([ξ(u)]
U
Q
d
+I
c
− [ξ(v)]
U
Q
d
+I
c
) (21)
where the first inequality follows from the definition of a
firmly non-expansive mapping and the fact that any projec-
tion mapping is firmly non-expansive (see Proposition 4.8 in
[16]). The second equality is due to the definition ξ(·), and
the last one follows after performing computation.
Since Q 0, then k
e
T (u) −
e
T (v)k
2
Q
d
+I
c
−Q
≤ k
e
T (u) −
e
T (v)k
2
Q
d
+I
c
. This, together with (21), implies that
k [ξ(u)]
U
Q
d
+I
c
− [ξ(v)]
U
Q
d
+I
c
k
2
Q
d
+I
c
−Q
(22)
≤ (u − v)
>
(Q
d
+ I
c
− Q)([ξ(u)]
U
Q
d
+I
c
− [ξ(v)]
U
Q
d
+I
c
).
By the definition of a firmly non-expansive mapping [15],
(22) implies that, if Q
d
+ I
c
− Q 0,
e
T (·) is firmly non-
expansive with respect to || · ||
Q
d
+I
c
−Q
. The condition Q
d
+
I
c
− Q 0 can be satisfied by choosing c as in (15)-(17),
rendering (19) an alternative to step 6 of Algorithm 1.
B. Information exchange
To implement Algorithm (1), at iteration k+1, it is needed
that some central authority, or common processing node,
collects and broadcasts the tentative solution of each agent
to all others, and that the agents have knowledge of the
common objective function J(·) so that each of them can
compute J(·, u
−i
(k)) (alternatively the central authority can
broadcast it to each agent i, i = 1, . . . , m). However, the
required amount of information that needs to be exchanged
can be significantly reduced when the objective function
exhibits some particular structure. This is the case, e.g., for
objective functions that are coupled only through the average
of some variables. The central authority needs then to collect
the solutions of all agents, but it only has to broadcast the
average value. Each agent will then be able to compute
J(·, u
−i
(k)) by subtracting from the average the value of
its local decision vector at iteration k, i.e., u
i
(k).
V. OPTIMAL CHARGING OF ELECTRIC VEHICLES
We consider the problem of optimizing the charging
strategy for a fleet of m plug-in electric vehicles (PEVs)
over a finite horizon T . We follow the formulation of [9],
[11], [12]; but, our algorithm converges to a minimizer of the
centralized problem counterpart and with a finite number of
agents/vehicles, as opposed to the aforementioned references,
where convergence to a Nash equilibrium at the limiting case
of an infinite population of agents is established. The PEV
charging problem is formulated as follows:
min
{u
i,t
}
m
i=1
T
t=0
1
m
T
X
t=0
p
t
(d
t
+
m
X
i=1
u
i,t
)
2
(23)
subject to
P
T
t=0
u
i,t
= γ
i
, ∀i=1, . . . , m
u
i,t
≤ u
i,t
≤ u
i,t
, ∀i=1, . . . , m, t= 0, . . . , T,
where p
t
∈ R is an electricity price coefficient at time t,
d
t
∈ R represents the non-PEV demand at time t, u
i,t
∈ R
is the charging rate of vehicle i at time t, γ
i
∈ R represents
a prescribed charging level to be reached by each vehicle i
at the end of the considered time horizon, and u
i,t
, u
i,t
∈ R
are bounds on the minimum and maximum value of u
i,t
,
respectively. The objective function in (23) encodes the total
electricity cost given by the demand (both PEVs and non-
PEVs) multiplied by the price of electricity, which in turn
depends linearly on the total demand through p
t
, thus giving
rise to the quadratic function in (23). This linear dependency
of price with respect to the total demand models the fact that
agents/vehicles are price anticipating authorities, anticipating
their consumption to have an effect on the electricity price
(see [17]). Problem (23) can be written in compact form as
min
u∈R
m(T +1)
(d + Au)
>
P (d + Au) (24)
subject to u
i
∈ U
i
, for all i = 1, . . . , m,
where P = (1/m)diag(p) ∈ R
(T +1)×(T +1)
, and diag(p) is
a matrix with p = (p
0
, . . . , p
T
) ∈ R
T +1
on its diagonal.
A = 1
1×m
⊗ I ∈ R
(T +1)×m
, where ⊗ denotes the
Kronecker product. Moreover, d = (d
0
, . . . , d
T
) ∈ R
T +1
,
u = (u
1
, . . . , u
m
) ∈ R
m(T +1)
with u
i
= (u
i,0
, . . . , u
i,T
) ∈
R
T +1
, and U
i
encodes the constraints of each vehicle i,
i = 1, . . . , m, in (23). Problem (24) can be solved in a
decentralized fashion by means of Algorithm 1. We compute
the value of c so that (9) is satisfied and the mapping
e
T (·) associated to problem (24) is non-expansive. Note
that the objective function in (24) is not strictly convex
as A
>
P A = 1
m×m
⊗ P , and it exhibits a structure that
allows for reduced information exchange as described in
Section IV-B. Indeed, at iteration k + 1 of Algorithm 1,
the central authority needs to collect the solution of each
agent but it only has to broadcast V (k) = d + Au(k).
Each agent i, can then compute its tentative objective as
J(z
i
, u
−i
(k)) = (V (k)−u
i
(k)+ z
i
)
>
P (V (k) − u
i
(k)+ z
i
).
Step 6 in Algorithm 1 for problem (24) reduces then to
u
i
(k + 1) = λu
i
(k) + (1 − λ)
e
T (u(k)) =
λu
i
(k)+(1−λ)argmin
z
i
∈U
i
(V (k)−u
i
(k)+z
i
)
>
P (V (k)−u
i
(k)+z
i
).
A. Simulation results
We consider first a fleet of m = 100 PEVs, each of them
having to reach a different level of charge γ
i
∈ [0.1, 0.3], i =
1, . . . , m, at the end of a time horizon T = 24, corresponding
to hourly steps. The bounds on u
i,t
are taken to be u
i,t
= 0
and u
i,t
= 0.02, for all i = 1, . . . , m, t = 0, . . . , T . The
non-PEV demand profile is retrieved from [11], whereas the
price coefficient is p
t
= 0.15, t = 0, . . . , T . Note that, as in
[12], u
i,t
corresponds to normalized charging rate, which is
then rescaled to be turned into reasonable power values. All
optimization problems are solved using CPLEX, [18].
For comparison purposes, problem (24) is solved first in
a centralized fashion, achieving an optimal objective value
J
?
= 2.67. It is then solved in a decentralized fashion
by means of Algorithm 1, setting c = 0.1 and λ = 0.4.
Note that for (9) to be satisfied, according to the analysis of
Section IV, we should have c ≥ 0.0735. After 30 iterations
the difference between the decentralized and the centralized
objective is J(u(30)) − J
?
= 1.36 · 10
−6
, thus achieving
numerical convergence.
We perform a parametric analysis, running Algorithm 1 for
different values of λ and c. In Table I the number of iterations
needed to achieve a relative error between the decentralized
and the centralized objective value
J(u(k))−J
?
J
?
< 10
−6
is
reported. Note that if we choose a value of c that does
not satisfy (9), Algorithm 1 does not always converge (see
first row of Table I, for λ = 0.01, 0.1). As c and λ
increase, numerical convergence requires more iterations,