Signal Processing - January 2016 - 71

COUpling COnStraintS
So far in this article, we have assumed that the constraints in the
optimization problem is separable and convex. In other words, the
constraint set X in (1) is of the form X = X 1 # fX n with each
X i being convex. A natural extension of the BSUM framework is
to modify it to deal with coupling and nonconvex constraints.
LINEAR COuPLING
Consider the following convex problem with linear coupling
constraints
min

(x 1, f, x n)

s.t.

f (x 1, f, x n)
n

/ Ai xi = b

i=1

(29)

x i ! X i, 6i = 1, 2, f, n,

where x i ! R mi, A i ! R k # mi, b ! R k and f ($) is a convex function. As seen in Example 4, the direct extension of the BCD/
BSUM approach does not work for this type problem. A popular
approach for solving the above optimization problem is the
alternating direction method of multipliers (ADMM) [88], [89].
This approach is based on finding a saddle point of the augmented Lagrangian function
L (x 1, f, x n; m) = f (x 1, f, x n)
n

+ m, / A i x i - b +
i=1

t

2

n

/ Ai xi - b

2

,

i=1

where m ! R k is the Lagrange multiplier corresponding to the
linear constraint; t 2 0 is the augmented Lagrangian coefficient;
and G ·, · H denotes the inner product operator.
At each iteration of the ADMM method, either a primal block
variable x i is updated according to

(BSUMM) [44], is guaranteed to converge to the global optimal
of (29) under some regularity assumptions [44]. For extensions
to nonconvex problems, see the recent work [90] and [91].
There are a few other interesting techniques that deal with linearly coupling constraint. For example, [92] and [93] propose to
randomly pick two blocks of variables to update at each iteration, and [94] proposes new algorithms based on minimizing
the augmented Lagrangian function. We refer the readers to
these papers for more related works in this direction.
Let us illustrate an application of BSUMM to a multicommodity
routing problem, which arises in the design of next-generation
cloud-based communication networks [95]. Consider a connected
wireline network N = (V, L) that is controlled by K + 1 network controllers (NCs) as illustrated in Figure 8. Let V denote the
set of network nodes, which is partitioned into K subsets, i.e.,
V = , iK= 1 V i, V i + V j = Q, 6i ! j. The set of directed links that
connect nodes of V is denoted as L _ {l = (s l, d l) 6s l, d l ! V},
where l = (s l, d l) denotes the directed link from node s l to node
d l . Each NC i controls V i and the links connecting these nodes,
i.e., L i _ {l = (s l, d l) ! L 6s l, d l ! V i}, (cf. Figure 8). The network N i _ (V i, Li) is called the subnetwork i.
Our objective is to transport M data flows over the network,
with each flow m being routed from a source node s m ! V to
the sink node d m ! V. We use rm $ 0 to denote flow m 's rate,
and use fl, m $ 0 to denote its rate on link l ! L. We also
assume that a master node exists, which controls the data flow
rates {rm} mM = 1 . The central NC 0 controls the subnetwork N 0,
consisting of the master node and the links connecting different
subnetworks, i.e., L 0 = , i ! j L 0ij .
We consider two types of network constraints:
1) Link capacity constraints. Assume each link l ! L has a
fixed capacity denoted as C l . The total flow rate on link l is
constrained by
1 T fl # C l, 6l ! L,

x ir + 1 ! arg min L (x i, x -r i; m r),
xi ! Xi

(30)

where 1 is the all-one vector and fl _ [fl,1, f, fl, M ] T .

or the dual Lagrange multiplier m is updated according to the
gradient ascent rule
mr + 1 ! mr + ar c

n

/ A i x ir - b m,

N = (V, L)

i=1

where a r is the dual step size at iteration r. The update orders
for the primal and dual variables could be either cyclic or
randomized.
Similar to the BSUM framework, one can replace the augmented Lagrangian function L (·, x -r i; m r) with its tight upper
u i (·, x r; m r) at iteration r + 1, where
bound L
Lu i (x i, x r, m r) = u i (x i, x r)
+ m r, A i x i + / A j x rj - b +
j!i

t

2

A i x i + / A j x rj - b

2

N1 = (V 1, L1)

NC 0

N 4 = (V 4, L4)
[NC4]

j!i

with u i (·, x r) being a locally tight approximation of the function f (·, x -r i) around the point x ir satisfying Assumption A. The
resulting algorithm, named the BSUM method of multipliers

[NC1]

[NC5]

N5 = (V 5, L5)

N 3 = (V 3, L3)

[NC3]

N 2 = (V 2, L2) [NC2]

[FIg8] a wireline network consists of five subnetworks. each of
them is controlled by an nc, and these ncs are coordinated
globally by a central nc 0 [95].

IEEE SIGNAL PROCESSING MAGAZINE [71] jANuARy 2016



Table of Contents for the Digital Edition of Signal Processing - January 2016

Signal Processing - January 2016 - Cover1
Signal Processing - January 2016 - Cover2
Signal Processing - January 2016 - 1
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Signal Processing - January 2016 - Cover3
Signal Processing - January 2016 - Cover4
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