1
Regularized Jacobi Iteration for Decentralized Convex Quadratic
Optimization with Separable Constraints
Luca Deori, Kostas Margellos and Maria Prandini
Abstract—We consider multi-agent, convex quadratic opti-
mization 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 focus on a regularized
variant of the so called Jacobi algorithm for decentralized
computation in such problems. We provide a fixed-point theoretic
analysis showing that the algorithm converges to a minimizer of
the centralized problem under more relaxed conditions on the
regularization coefficient from those available in the literature,
and in particular with respect to scaled projected gradient
algorithms. The efficacy of the proposed algorithm is illustrated
by applying it to the problem of optimal charging of electric
vehicles.
Index Terms—Decentralized optimization, Jacobi algorithm,
iterative methods, optimal charging control, electric vehicles.
I. INTRODUCTION
O
PTIMIZATION 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 [1], [2], mobility [3], [4], [5], robotic systems
[6], etc. We focus on multi-agent optimization programs that
are convex and are subject to constraints that are separable.
The agents’ decisions are, however, coupled by means of
a common objective function, which is considered to be
quadratic. The considered structure, although specific, captures
a wide class of problems, like the electric vehicle charging
problem studied in this paper, and is amenable to efficient
numerical solvers tailored for quadratic optimization [7].
Solving such problems in a centralized fashion would re-
quire agents to share their local constraint functions, while
even if this was possible it would unnecessarily increase
the computational burden. To alleviate these issues we adopt
an iterative, decentralized perspective, where agents perform
local computations in parallel, and then exchange with each
other their new solutions, or broadcast them to some central
authority that sends an update to each agent. Admittedly,
distributed optimization offers a more general communication
setup, however, the fact that agents decision vectors are
coupled via the objective function poses additional difficulties,
preventing the use of distributed algorithms [8], [9]. Even
upon an epigraphic reformulation, the resulting problem will
Research was supported by the European Commission, H2020, under the
project UnCoVerCPS, grant number 643921, and by EPSRC UK under
the grant EP/P03277X/1. The authors would like to thank one anonymous
reviewer for suggesting a refinement in the calculations of Section III-C.
L. Deori and M. Prandini are with the Dipartimento di Elettronica
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, Oxford, OX1 3PJ, United Kingdom, e-mail:
kostas.margellos@eng.ox.ac.uk
not exhibit the structure typically encountered in distributed
optimization, with the resulting coupling constraint not nec-
essarily being of “budget” form [10]. A distributed gossip
based gradient algorithm has been proposed in [11], but with
reference to a noncooperative counterpart of the problem under
study here. As such, it does not lead to a social welfare
solution. Moreover, it requires an iteration varying step-size
as opposed to the constant step-size considered in this paper.
A. Related work
From a cooperative optimization point of view, algorithms
for decentralized solutions to convex optimization problems
with separable constraints can be found in [12], [13], 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 [14]–[16]. 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 prob-
lem, for an appropriately chosen gradient step-size.
The second direction for decentralized optimization involves
mainly the so called Jacobi algorithm, which serves as a
proximal based alternative to gradient algorithms. The Gauss-
Seidel algorithm exhibits similarities with the Jacobi one,
but is not of parallelizable nature [17], unless a colouring
scheme is adopted (see Section 1.2.4 in [12]). Under the Jacobi
algorithmic setup, at every iteration, instead of performing a
gradient step, each agent minimizes the common objective
function subject to its local constraints in a best-response
fashion, while keeping the decisions of all other agents fixed
to their values at the previous iteration. In [12], it is shown that
the Jacobi algorithm converges under certain contractiveness
requirements, which are typically satisfied only under strong
(or strict in case of quadratic objective functions) convexity as-
sumptions that are, however, not imposed in the current work.
In [18], [19], a regularized version of the Jacobi algorithm is
proposed, however, an explicit condition on the regularization
coefficient for convergence is not provided. A similar paral-
lelizable, albeit different scheme, has been presented in [4],
[20], [21], without employing regularization, while [22], [23]
follow a randomized block coordinate descent approach and
provide convergence results concerning the expected value of
the objective functions. The results most closely related to our
work appear in [24], [25]. In all aforementioned references,
however, unlike our paper, convergence is limited to the
optimal value and not in iterates.
From a non-cooperative perspective there has recently been
a notable research activity using tools from mean-field and
2
aggregative game theory. Under a deterministic, discrete-time
setting, [3], [5], [26] deal with the non-cooperative counterpart
of our work, showing convergence not to a minimizer, but
to an approximate Nash equilibrium of a related game, and
to an exact Nash equilibrium in the limiting case where the
number of agents tends to infinity. Using an approach similar
to the regularized Jacobi algorithm it is shown in [27] that
convergence to an exact Nash equilibrium for a finite number
of agents can be achieved. A similar result, using a gradient
based variant is recently provided in [28].
B. Contributions of this work and organization of the paper
We adopt a cooperative point of view, and consider a
regularized Jacobi algorithm similar to the one in [18], [19],
[24]. Our contributions extend these results as follows:
1. Focusing on the case where the objective function is
quadratic, we show that the iterates generated by the regu-
larized Jacobi algorithm converge to an optimal solution of
the centralized problem counterpart, as opposed to the weaker
statement that the iterates sequence achieves the optimal value
allowing, however, an oscillatory behaviour (i.e., all limit
points are optimal solutions) [24]. To achieve this, we follow
a fundamentally different analysis from [24], relying on an
operator theoretic approach. Our result serves as the Jacobi
counterpart of gradient methods, thus complementing the work
of [12], [29], [30]. The recent paper [31] shows convergence to
an optimal solution of the centralized problem counterpart as
well. However, the converge proof in [31, Theorem 1] strongly
depends on results of this paper and relies on the agents’
constraint sets to be convex polyhedra while our result requires
these sets to be only compact and convex.
2. As opposed to [18], [19], we provide an explicit calculation
of the regularization coefficient that ensures convergence, and
show that the condition of Theorem 1 constitutes a relaxed
version of that of [24] (see Theorem 3 and discussion on
constant step-sizes therein), as well as of that of unscaled
projected gradient methods (see Proposition 3.3 in Chapter
3 of [12] for convergence in value, and Theorem 4.1 in [29]
or Theorem 2 in [30] for convergence in iterates) that ends up
being the same with that of [24]. We also show that the main
Jacobi iteration can be written as a scaled projected gradient
step and derive an improved convergence condition (however,
concerning convergence in optimal value not in iterates) under
a particular choice of the scaling matrix and projection norm.
Notably, the condition of Theorem 1 is less conservative.
This improvement can affect significantly how well-behaved
numerically the underlying optimization programs are.
3. From an application point of view, we extend the results of
[4] on electric vehicle charging, achieving convergence to an
optimal charging solution as opposed to convergence in value.
The results obtained here extend significantly our earlier
work in [32], where no formal comparison with the gradient
methods and [24] was provided. It should be noted that [12],
[24], provide algorithms that are limited to convergence in
optimal value under more restrictive choices on the step-size,
however, are applicable to convex function and not necessarily
quadratic as the focus of this paper. Our results can be
extended to the non-quadratic case (the proof is similar to
[24]), one can show convergence as far as the optimal value is
concerned using a less restrictive step-size condition. We refer
to this condition in Remark 3, while the reader is referred to
the technical memorandum [33] for more details and proofs.
Section II introduces the problem under study and states the
proposed algorithm. In Section III we provide the main conver-
gence result and a comparison with scaled projected gradient
methods and the algorithm of [24]. Section IV provides an
extensive simulation study for the electric vehicle charging
control case study, while Section V concludes the paper and
outlines some directions for future research.
II. DECENTRALIZED PROBLEM FORMULATION
A. Motivating example: 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 . Following [3], [5], [26], the PEV charging problem
is given by the following optimization problem.
min
{x
i
(t)}
m
i=1
T
t=0
1
m
T
X
t=0
p(t)
d(t) +
m
X
i=1
x
i
(t)
2
(1)
subject to
T
X
t=0
x
i
(t) = γ
i
, for all i = 1, . . . , m
x
i
(t) ≤ x
i
(t) ≤ x
i
(t), for all 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, x
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 x
i
(t), x
i
(t) ∈ R
are bounds on the minimum and maximum value of x
i
(t),
respectively. The objective function in (1) 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 (1). 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
introduction in [2] for further elaboration on price anticipating
agents). Problem (1) can be rewritten as
min
x∈R
m(T +1)
(d + Ax)
>
P (d + Ax) (2)
subject to: x
i
∈ X
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(T +1)
, where ⊗ denotes
the Kronecker product, and I ∈ R
(T +1)×(T +1)
the identity
matrix. Moreover, d = (d(0), . . . , d(T )) ∈ R
T +1
, x =
(x
1
, . . . , x
m
) ∈ R
m(T +1)
, x
i
= (x
i
(0), . . . , x
i
(T )) ∈ R
T +1
,
and X
i
is the constraint set of vehicle i, i = 1, . . . , m, in (1).
3
Algorithm 1 Decentralized algorithm
1: Initialization
2: k = 0.
3: Consider x
i
0
∈ X
i
, for all i = 1, . . . , m.
4: For i = 1, . . . , m repeat until convergence
5: Agent i receives x
−i
k
from central authority.
6: x
i
k+1
= arg min
z
i
∈X
i
f(z
i
, x
−i
k
) + ckz
i
− x
i
k
k
2
.
7: k ← k + 1.
B. Problem statement
Motivated by the electric vehicle charging control problem
in (2), we consider the following class of programs:
P : min
{x
i
∈R
n
i
}
m
i=1
f(x
1
, . . . , x
m
) (3)
subject to: x
i
∈ X
i
, for all i = 1, . . . , m, (4)
where each agent i, i = 1, 2, . . . , m, has a local decision vector
x
i
∈ R
n
i
and a local constraint set X
i
⊆ R
n
i
, and cooperates
to determine a minimizer of f : R
n
1
× . . . × R
n
m
→ R, which
couples its decision vector with those of the other agents.
Assumption 1. The objective function f : R
n
1
×. . .×R
n
m
→
R is given by f(x
1
, . . . , x
m
) = x
>
Qx + q
>
x, where x =
[(x
1
)
>
, . . . , (x
m
)
>
]
>
∈ R
n
with n =
P
m
i=1
n
i
, Q ∈ R
n×n
is symmetric and positive semi-definite (Q = Q
>
0) and
q ∈ R
n
. Moreover, the sets X
i
⊆ R
n
i
, i = 1, . . . , m, are
non-empty, compact and convex.
Note that Q is assumed to be symmetric without loss of
generality; in the opposite case it could be split in a symmetric
and an antisymmetric part, with the latter giving rise to terms
that simplify each other.
Remark 1 (Problem generalization). We also allow for objec-
tive functions of the form f(x
1
, . . . , x
m
) = x
>
Qx + q
>
x +
P
m
i=0
g
i
(x
i
), where the g
i
(x
i
),i = 1, . . . , m, are convex
functions that could encode a utility function for each agent.
In this case an epigraphic reformulation can be exploited to
bring the cost back to be quadratic. Letting y
i
= [x
i,>
h
i
]
>
be
the decision vector of agent i, the local constraint set can be
then defined as Y
i
= X
i
∩ {g
i
(x
i
) ≤ h
i
}, while the objective
function can be rewritten as x
>
Qx + q
>
x +
P
m
i=0
h
i
, which
is quadratic in y = [y
1,>
. . . y
m,>
]
>
.
Under Assumption 1, the function f is convex and hence
continuous, while the constraint set X = X
1
× · · · × X
m
is
non-empty and compact, as result of Weierstrass’ theorem [12,
Proposition A8, p. 625], P admits at least one optimal solution.
However, P does not necessarily admit a unique minimizer.
With a slight abuse of notation, for each i, i = 1, . . . , m,
let f(·, x
−i
) : R
n
i
→ R be the objective function in (3) as a
function of the decision vector x
i
of agent i, when the decision
vectors of all other agents are fixed to x
−i
∈ R
n−n
i
. We will
occasionally also write f (x) instead of f (x
1
, . . . , x
m
).
C. Regularized Jacobi algorithm
Solving problem P in a centralized fashion is not always
possible since agents may not be willing to share X
i
, i =
1, . . . , m. Moreover, even if this was the case, solving P in
one shot might be computationally challenging. To overcome
this and account for information sharing issues, motivated by
the separable structure of P we follow a decentralized, iterative
approach as described in Algorithm 1.
Initially, each agent i, i = 1, . . . , m, starts with some
value x
i
0
∈ X
i
, such that
x
1
0
, . . . , x
m
0
is feasible (step 3,
Algorithm 1). At iteration k + 1, each agent i receives x
−i
k
(step 5, Algorithm 1) from the central authority, and updates
its estimate for x
i
by solving a local minimization problem
(step 6, Algorithm 1). The performance criterion in this local
problem is a linear combination of the objective f(z
i
, x
−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
regularization term, penalizing the difference between z
i
and
the value of agent’s i own variable at iteration k, i.e., x
i
k
.
The relative importance of these two terms is dictated by the
regularization coefficient c ∈ R
+
, which plays a key role
in determining the convergence properties of Algorithm 1.
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.
Remark 2 (Information exchange). To implement Algorithm
1, at iteration k + 1, it is needed that some central authority
collects and broadcasts the current solution of each agent
to all others, so that each of them can compute f(·, x
−i
k
).
However, in the case where the coupling in the objective
function is only through the average of some agents’ variables
as in the example of Section II-A, at every iteration k the
central authority needs to broadcast only the average of the
agents’ decisions, or in other words the cumulative charging
d+Ax
k
with reference to the electric vehicle case study. Each
agent will then be able to compute f(·, x
−i
k
) by subtracting
from the average the value its local decision vector x
i
k
.
III. MAIN CONVERGENCE RESULT
We start defining some matrices that will be used in the
following: for all i = 1, . . . , m, let Q
i,i
denote the i-th block
of Q, with row and column indices corresponding to x
i
, where
x = [x
1,>
. . . x
m,>
]
>
. Denote then by Q
d
a block diagonal
matrix whose i-th block is Q
i,i
, and let Q
z
= Q − Q
d
denote
the off (block) diagonal part of Q. Since Q is assumed to be
symmetric, Q
z
is symmetric as well and its eigenvalues are all
real. Since Q
z
has zero trace, at least one of its eigenvalues
will be non-negative. As a result, λ
max
Q
z
≥ 0, where λ
max
Q
z
denotes the maximum eigenvalue of Q
z
.
Theorem 1. Under Assumption 1, if c > λ
max
Q
z
, then Algorithm
1 converges to a minimizer of P.
Theorem 1 provides an explicit bound on c that ensures
convergence. Such a bound is derived by a fixed-point theoretic
approach. Note that if the objective function f was strictly
convex with respect to x, then the standard Jacobi iteration
of [12] can be adopted instead of the regularized version. In
that case, geometric convergence to some minimizer of P is
guaranteed by means of Proposition 3.5 in [12].
4
c>λ
max
Q
z
c>λ
max
Q
− λ
min
Q
d
c>λ
max
Q
c>
m − 1
2m − 1
2λ
max
Q
z
Fig. 1: Bar plot for the bound on c (from bottom to top): Algo-
rithm of [33] (convergence in value); Theorem 1 (convergence
in iterates); scaled projected gradient algorithm (convergence
in value); Algorithm of [24] (convergence in value) and un-
scaled projected gradient algorithm (convergence in iterates).
Remark 3 (Connection with [24] and extensions). For the
more general case of a convex objective function, by [24,
Theorem 3] (see constant step-size condition), it can be shown
that Algorithm 1 converges to the optimal value of P for c
greater than one half of the Lipschitz constant of the objective
function gradient, which for the case of quadratic objective
functions is 2λ
max
Q
, thus leading to c > λ
max
Q
. The sequence of
iterates, however, may not converge and exhibit an oscillatory
behaviour. Under the same condition on c it is shown unscaled
projected gradient algorithms with step-size 1/c (i.e., two over
the Lipschitz constant of the gradient) can converge not only
in value, but also in iterates (see Proposition 3.3 in Chapter
3 of [12] for convergence in value, and Theorem 4.1 in [29]
or Theorem 2 in [30] for convergence in iterates).
In Section III-C we write the Jacobi iteration as a scaled
projected gradient step and show that it converges in value
but not in iterates if c > λ
max
Q
−λ
min
Q
d
, which is less restrictive
than the aforementioned conditions. This result is strengthened
even further if c > λ
max
Q
z
according to Theorem 1.
By Theorem 3 of [33] it can be shown that, as far as the
optimal value is concerned, Algorithm 1 converges for c >
m−1
2m−1
2λ
max
Q
z
. The latter is a relaxed version for the condition
c > λ
max
Q
z
of Theorem 1, since
1
2
>
m−1
2m−1
, for all m. However,
Theorem 1 ensures convergence to some minimizer and not
just convergence in value. This result is shown in [33] using
an analysis similar to the proof of Theorem 3 in [24], that is
based on Proposition 1 and sequence convergence properties
(see Exercise 1.19 in [34] (p. 18)). The relationship between
the various conditions on c is pictorially shown in Figure 1.
A. Preliminary results
The results of this section hold under Assumption 2.
Assumption 2. The function f : R
n
1
× . . . × R
n
m
→ R is
continuously differentiable, and jointly convex with respect to
all arguments, i.e., convex with respect to x. The sets X
i
⊆
R
n
i
, i = 1, . . . , m, are non-empty, compact and convex.
1) Minimizers and fixed-points definitions: By (3)-(4), the
set of minimizers of P is given by
M = arg min
{z
i
∈X
i
}
m
i=1
f(z
1
, . . . , z
m
) ⊆ X. (5)
Following the discussion below Assumption 1, M is non-
empty. Note that M is not necessarily a singleton; this is the
case if f is jointly strictly convex with respect to its arguments.
For each i, i = 1, . . . , m, consider the mappings T
i
:
X → X
i
and
e
T
i
: X → X
i
, defined such that, for any
x = (x
1
, . . . , x
m
) ∈ X,
T
i
(x) = arg min
z
i
∈X
i
kz
i
− x
i
k
2
(6)
subject to: f(z
i
, x
−i
) ≤ min
ζ
i
∈X
i
f(ζ
i
, x
−i
),
e
T
i
(x) = arg min
z
i
∈X
i
f(z
i
, x
−i
) + ckz
i
− x
i
k
2
. (7)
The mapping in (6) serves as a tie-break rule to select, in
case f(·, x
−i
) admits multiple minimizers over X
i
, the one
closer to x
i
with respect to the Euclidean norm. Note that
in (6) and (7) we use equality instead of inclusion since the
corresponding minimizers T
i
(x) and
e
T
i
(x), respectively, are
unique. Note also that with x
k
in place of x, (7) implies
that the update step 6 in Algorithm 1 can be equivalently
represented by x
i
k+1
=
e
T
i
(x
k
).
Define also the mappings T : X → X and
e
T : X → X,
such that their components are given by T
i
and
e
T
i
,
T (x) = arg min
z∈X
m
X
i=1
kz
i
− x
i
k
2
(8)
subject to: f(z
i
, x
−i
) ≤ min
ζ
i
∈X
i
f(ζ
i
, x
−i
), ∀i = 1, . . . , m,
e
T (x) = arg min
z∈X
m
X
i=1
f(z
i
, x
−i
) + ckz
i
− x
i
k
2
, (9)
where the terms inside the summation in (8) and (9) are
decoupled. The set of fixed-points of T and
e
T is, respectively,
are given by
F
T
=
x ∈ X : x
i
= T
i
(x), for all i = 1, . . . , m
, (10)
F
e
T
=
x ∈ X : x
i
=
e
T
i
(x), for all i = 1, . . . , m
. (11)
2) Connections between minimizers and fixed-points: We
report here a fundamental optimality result.
Proposition 1 ( [12, Proposition 3.1]). Assume that f is a
continuously differentiable function and X is a non-empty,
closed and convex set. We then have that,
1) if x ∈ X minimizes f over X, then (z − x)
>
∇f(x) ≥ 0,
for all z ∈ X.
2) if f is also convex on X, then the condition of the previous
part is also sufficient for x ∈ arg min
z∈X
f(z).
We show that the set of minimizers M of P in (5) and the
set of fixed-points F
T
of the mapping T in (8) coincide.
Proposition 2. Under Assumption 2, M = F
T
.
Proof. 1) M ⊆ F
T
: Fix any x ∈ M . For i = 1, . . . , m,
denote x by (x
i
, x
−i
). The fact that x ∈ M implies that
f(x
i
, x
−i
) will be no greater than f(ζ
i
, x
−i
), for all ζ
i
∈ X
i
,
i.e., f(x
i
, x
−i
) ≤ min
ζ
i
∈X
i
f(ζ
i
, x
−i
), which means that x
satisfies the inequality in (8). Moreover x is also optimal
for the objective function in (8), since it results in zero cost.
Hence, by (8), x is a fixed-point of T , i.e., x ∈ F
T
.
2) F
T
⊆ M: Fix any x ∈ F
T
. By the definition of F
T
we
have f(x
i
, x
−i
) ≤ min
ζ
i
∈X
i
f(ζ
i
, x
−i
), for all i = 1, . . . , m.
The last statement implies that x
i
is the minimizer of f(·, x
−i
)
5
over X
i
. For all i = 1, . . . , m, by the first part of Proposition
1 (with f(·, x
−i
) in place of f) we then have that
(z
i
− x
i
)
>
∇
i
f(x
i
, x
−i
) ≥ 0, for all z
i
∈ X
i
, (12)
where ∇
i
f(x
i
, x
−i
) is the i-th component of the gradient
∇f(·, x
−i
) of f(·, x
−i
), evaluated at x
i
. By (12), we then
have that
P
m
i=1
(z
i
− x
i
)
>
∇
i
f(x
i
, x
−i
) ≥ 0 for all z
i
∈ X
i
,
i = 1, . . . , m, which, by setting x = (x
1
, . . . , x
m
), z =
(z
1
, . . . , z
m
), can be written as (z − x)
>
∇f(x) ≥ 0, for
all z ∈ X. By the second part of Proposition 1, and since
f is jointly convex with respect to all elements of x, the last
statement implies that x minimizes f over X, i.e., x ∈ M.
The connection between minimizers, fixed-points and vari-
ational inequalities similar to (12), has been also investigated
in [35], in the context of non-cooperative games.
Proposition 3. Under Assumption 2, F
T
= F
e
T
.
Proof. 1) F
T
⊆ F
e
T
: Fix any x ∈ F
T
. By (10), this is
equivalent to the fact that x
i
= T
i
(x), for all i = 1, . . . , m,
which, due to the definition of T implies that, for all i =
1, . . . , m, f (x
i
, x
−i
) ≤ min
ζ
i
∈X
i
f(ζ
i
, x
−i
). This implies
that x
i
minimizes f(·, x
−i
) over X
i
, hence, by the first part
of Proposition 1 (with f(·, x
−i
) in place of f) we have
that (z
i
− x
i
)
>
∇
i
f(x
i
, x
−i
) ≥ 0, for all z
i
∈ X
i
. Let
f
c
(z
i
, x) = f(z
i
, x
−i
) + ckz
i
− x
i
k
2
, for all z
i
, i = 1, . . . , m,
and notice that ∇f
c
(x
i
, x) = ∇f(x
i
, x
−i
), since the gradient
of the quadratic penalty term vanishes at x
i
. We then have
that, for all i = 1, . . . , m,
(z
i
− x
i
)
>
∇
i
f
c
(x
i
, x) ≥ 0, for all z
i
∈ X
i
. (13)
Since f
c
(·, x) is strictly convex with respect to its first argu-
ment, by the second part of Proposition 1 (with f
c
(·, x) in
place of f), (13) implies that, for all i = 1, . . . , m, x
i
is the
unique minimizer of f
c
(·, x) over X
i
, i.e.,
x
i
= arg min
z
i
∈X
i
f(z
i
, x
−i
) + ckz
i
− x
i
k
2
. (14)
By (7), (14) is equivalent to x
i
=
e
T
i
(x), for all i = 1, . . . , m.
2) F
e
T
⊆ F
T
: Fix any x ∈ F
e
T
. By (11) this is equivalent to
the fact that x
i
=
e
T
i
(x), for all i = 1, . . . , m, which, by the
definition of
e
T
i
in (7), implies that, for all i = 1, . . . , m,
x
i
= arg min
z
i
∈X
i
f(z
i
, x
−i
) + ckz
i
− x
i
k
2
. (15)
Let again f
c
(z
i
, x) = f(z
i
, x
−i
) + ckz
i
− x
i
k
2
. Equation (15)
implies then that, for all i = 1, . . . , m, x
i
minimizes f
c
(·, x)
over X
i
, and by the first part of Proposition 1 (with f
c
(·, x) in
place of f) leads to (z
i
− x
i
)
>
∇
i
f
c
(x
i
, x) ≥ 0, for all z
i
∈
X
i
. Notice that ∇f
c
(x
i
, x) = ∇f(x
i
, x
−i
), since the gradient
of ckz
i
− x
i
k
2
with respect to z
i
vanishes at x
i
. Therefore,
for all i = 1, . . . , m, we have that
(z
i
− x
i
)
>
∇
i
f(x
i
, x
−i
) ≥ 0, for all z
i
∈ X
i
. (16)
Since f (·, x
−i
) is convex with respect to its first argu-
ment, by the second part of Proposition 1, (16) implies
that x
i
minimizes f (·, x
−i
) over X
i
. In other words,
x
i
∈ arg min
z
i
∈X
i
f(z
i
, x
−i
), for all i = 1, . . . , m.
This in turn implies that, for all i = 1, . . . , m,
f(x
i
, x
−i
) ≤ f (z
i
, x
−i
), for all z
i
∈ X
i
, i.e., f(x
i
, x
−i
) ≤
min
z
i
∈X
i
f(z
i
, x
−i
). The last inequality shows that x satisfies
the inequality in (8). Moreover, it minimizes the objective
function in (8), since it results in zero cost, so x = T(x).
By Propositions 2 and 3 we have that the set of minimizers
M of P coincides with the fixed-points of the mapping
e
T .
Corollary 1. Under Assumption 2, M = F
e
T
.
B. Proof of Theorem 1
Step 6 of Algorithm 1 can be equivalently written as
x
i
k+1
=
e
T
i
(x
k
), which entails that x
k+1
=
e
T (x
k
), i.e., a
Picard-Banach iteration of
e
T (see [36] (Chapter 1.2) for a
definition). Since
e
T is non-empty (it coincides with M by
Corollary 1), we only need to prove that
e
T is firmly non-
expansive (see [37] (Section 1) for a definition in general
Hilbert spaces). If that is the case, then, by [37], [38], we
have that the Picard-Banach iteration converges to a fixed-
point of
e
T , for any initial condition x
0
. By Corollary 1 this
fixed-point will also be a minimizer of P. We next show
that if c > λ
max
Q
z
, then
e
T (·) is indeed firmly non-expansive
with respect to k · k
Q
d
+I
c
−Q
(I
c
is the identity matrix I of
appropriate dimensions weighted by c), i.e.,
k
e
T (x) −
e
T (y)k
2
Q
d
+I
c
−Q
≤ (x − y)
>
(Q
d
+ I
c
− Q)(
e
T (x) −
e
T (y)), (17)
thus establishing Theorem 1. To this end, by Assumption 1,
e
T (x) = arg min
z∈X
m
X
i=1
f(z
i
, x
−i
) + ckz
i
− x
i
k
2
= arg min
z∈X
m
X
i=1
(z
i
)
>
(Q
i,i
+ I
c
)z
i
+ (2(x
−i
)
>
Q
−i,i
− 2(x
i
)
>
I
c
+ q
>
i
)z
i
= arg min
z∈X
z
>
(Q
d
+ I
c
)z + (2x
>
Q
z
− 2x
>
I
c
+ q
>
)z. (18)
Notice the slight abuse of notation in (18), where I
c
in the
second and the third equality are not of the same dimension.
Let ξ(x) = (Q
d
+ I
c
)
−1
(I
c
x − Q
z
x − q/2) denote the
unconstrained minimizer of (18). We then have that
e
T (x) = arg min
z∈X
(z − ξ(x))
>
(Q
d
+ I
c
)(z − ξ(x))
= [ξ(x)]
X
Q
d
+I
c
, (19)
where [ξ(x)]
X
Q
d
+I
c
denotes the projection, with respect to || ·
||
Q
d
+I
c
, of ξ(x) on X. Note that Q
d
+ I
c
is positive definite
for c ∈ R
+
, so its inverse exists and the projection is well
defined. We then have that
k
e
T (x) −
e
T (y)k
2
Q
d
+I
c
= k [ξ(x)]
X
Q
d
+I
c
− [ξ(y)]
X
Q
d
+I
c
k
2
Q
d
+I
c
≤ (ξ(x) − ξ(y))
>
(Q
d
+ I
c
)([ξ(x)]
X
Q
d
+I
c
− [ξ(y)]
X
Q
d
+I
c
)
= (x − y)
>
(I − Q(Q
d
+ I
c
)
−1
)(Q
d
+ I
c
)
× ([ξ(x)]
X
Q
d
+I
c
− [ξ(y)]
X
Q
d
+I
c
)
= (x − y)
>
(Q
d
+ I
c
− Q)([ξ(x)]
X
Q
d
+I
c
− [ξ(y)]
X
Q
d
+I
c
),
(20)
