Signal Processing - January 2016 - 66

min
h (x) +
x

,

/ (| A j x + b j | (2,

s.t. x ! R m,

(16)

j= 1

where A j ! R k i # m, b j ! R ki, and h (x) is some convex function
not necessarily smooth. For a set of applications for this model,
see [41, Sec. 4]. Consider the following smooth approximation
of (16):
,

min
h (x) + g (x): = h (x) + /
x

(17)

where h is some small constant and g (x) denotes the smooth
part of the objective. The IRLS algorithm solves (17) by performing the following iteration:
A jx + b j

,

x r + 1 = arg minm ) h (x) + 1
2
x!R

/

2

2

A j xr + b j

j=1

+ h2
+ h2

3.

Define the following function for g (x):
,

xk = vk sk

A j x + b j 2 + h 2 , s.t. x ! R m ,

j=1

u (x, x r) = 1 e /
2 j=1

where H kj ! C N # M is the channel from transmitter j to
receiver k and n k ! C N denotes the additive white Gaussian
noise at the receiver k with distribution CN (0, v 2 I).
When linear beamformers are employed at the transmitters
and receivers, the transmitted signal and the estimated received
data stream can be respectively written as

2

A jx + b j
r

A jx + b j

+ h2
2

+h

2

+

A j xr + b j

2

+ h 2 o.

and
ts k = u kH y k,
where v k ! C M and u k ! C N are, respectively, the transmit and
receive beamformers. Here the transmitted data stream and the
estimated data stream at the receiver are denoted by s k ! C
and ts k ! C, respectively.
A crucial task in modern wireless networks is to design
the transmit and receive beamformers v k and u k to maximize a given utility of the system. Here, for simplicity of presentation, we consider the sum rate utility function as our
objective. Therefore, our goal is to solve the following optimization problem:

(18)
max
u,v

It is clear that g (x r) = u (x r, x r), so Assumption (A1) is satisfied.
To verify Assumption (A2), we apply the arithmetic-geometric
inequality, and have
,

u (x, x ) = 1 e /
2 j=1
r

,

$

/

A jx + b j
A j xr + b j
A jx + b j

2

2

+ h2
2

+ h2

r

A jx + b j

+

2

2

+h o
2

m

+ h = g (x), 6 x ! R .

j=1

Then according to Example 1, Assumption (A3) is automatically
true, therefore we have verified that u (x, x r) defined in (18) is
indeed an upper-bound function for the smooth function g (x).
It follows that the IRLS algorithm corresponds to a single-block
BSUM algorithm. Notice that using the BSUM framework we
can easily generalize the IRLS to the multiblock scenario.
aPPlIcaTIonS oF The BSuM FraMework
Here we briefly review a few applications of the BSUM framework in wireless communication, bioinformatics, signal processing, and machine learning.
wireleSS COmmUniCatiOn and tranSCeiver deSign
Consider a multiple-input, multiple-output interference channel with K transmitter-receiver pairs. Let M (resp. N) be the
number of antennas at each transmitter (resp. receiver) and
each transmitter k, k = 1, 2, f, K, is interested in transmitting one data stream to its own receiver. Let x k ! C M be the
transmitted signal of user k; assuming linear channel model,
the received signal of user k can be written as
y k = H kk x k +
<
desired signal

/ H kj x j

+ nk ,
5

j!k

1442443

noise

multiuser interference

s.t.

K

/ R k (u, v)

k=1

vk

2

# Pk, 6k = 1, 2, f, K,

(19)

where Pk is the total power budget of user k and R k (u, v),
which is the communication rate of user k, is given by
R k (u, v) = log f 1 +

v

2

| u kH H kk v k | 2
.
u k 2 + / | u kH H kj v j | 2 p
j!k

Problem (19) is nonconvex and known to be NP-hard [58].
Using the well-known relation between the signal-to-interference-plus-noise ratio (SINR) and the mean square error (MSE)
value, one can rewrite (19) as [12], [59]:
min
v,u
s.t.

K

/ log ^e k (u, v)h

k=1

vk

2

# Pk, 6k = 1, 2, f, K,

(20)

where e k (u, v) is the MSE value and is given by
e k (u, v) = | u kH H kk v k - 1 | 2 +

/ | u kH H kj v j | 2 + v 2 .

j!k

Since the log (:) function is concave, it is upper bounded by its
first-order approximation (i.e., the linear upper bound in Table 3).
Therefore, we can define the function
u (x , x r ) =

K

/ ^log ^e k (x r)h + ^e k (x r)h-1 ^e k (x) - e k (x r)hh,

(21)

k=1

where x _ (u, v) is the optimization variable and x r _ (u r, v r)
denotes the beamformer at iteration r. It is not hard to see that
the approximation function in (21) is a valid upper bound in the

IEEE SIGNAL PROCESSING MAGAZINE [66] jANuARy 2016
Table of Contents for the Digital Edition of Signal Processing - January 2016
