1310 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 7, JULY 2006
Dual Methods for Nonconvex Spectrum
Optimization of Multicarrier Systems
Wei Yu, Member, IEEE, and Raymond Lui
Abstract—The design and optimization of multicarrier com-
munications systems often involve a maximization of the total
throughput subject to system resource constraints. The optimiza-
tion problem is numerically difficult to solve when the problem
does not have a convexity structure. This paper makes progress
toward solving optimization problems of this type by showing that
under a certain condition called the time-sharing condition, the
duality gap of the optimization problem is always zero, regardless
of the convexity of the objective function. Further, we show that
the time-sharing condition is satisfied for practical multiuser
spectrum optimization problems in multicarrier systems in the
limit as the number of carriers goes to infinity. This result leads to
efficient numerical algorithms that solve the nonconvex problem
in the dual domain. We show that the recently proposed optimal
spectrum balancing algorithm for digital subscriber lines can be
interpreted as a dual algorithm. This new interpretation gives
rise to more efficient dual update methods. It also suggests ways
in which the dual objective may be evaluated approximately,
further improving the numerical efficiency of the algorithm. We
propose a low-complexity iterative spectrum balancing algorithm
based on these ideas, and show that the new algorithm achieves
near-optimal performance in many practical situations.
Index Terms—Digital subscriber lines (DSLs), discrete multitone
(DMT), duality theory, dynamic spectrum management (DSM),
iterative spectrum balancing (ISB), nonconvex optimization, op-
timal spectrum balancing (OSB), orthogonal frequency-division
multiplex (OFDM).
I. INTRODUCTION
T
HE design of communication systems often involves the
optimization of a design objective subject to various re-
source constraints. The optimization problem becomes numeri-
cally difficult to solve when either the objective function or the
constraint lacks a convexity structure. This paper deals with effi-
cient numerical solutions for nonconvex optimization problems
for a particular class of optimization problems that often arise in
multicarrier communication systems. In a multicarrier system,
the transmission frequency spectrum is partitioned into a large
Paper approved by G.-H. Im, the Editor for Transmission Systems of the IEEE
Communications Society. Manuscript received July 14, 2005; revised December
12, 2005. This work was supported in part by Bell Canada University Laborato-
ries, in part by Communications and Information Technology Ontario (CITO),
in part by the Natural Sciences and Engineering Council (NSERC) of Canada,
and in part by the Canada Research Chairs program. This paper was presented
in part at the IEEE Global Telecommunications Conference, Dallas, TX, De-
cember 2004, and in part at the IEEE International Conference on Communica-
tions, Paris, France, May 2005.
The authors are with the Edward S. Rogers Sr. Electrical and Computer En-
gineering Department, University of Toronto, Toronto, ON M5S 3G4, Canada
(e-mail: weiyu@comm.utoronto.ca; rwmlui@gmail.com).
Digital Object Identifier 10.1109/TCOMM.2006.877962
number of frequency bins on which parallel data transmissions
take place. The most common examples of multicarrier sys-
tems include wireless orthogonal frequency-division multiplex
(OFDM) systems, such as the 802.11 and digital audio broad-
casting systems, and wireline discrete multitone (DMT) sys-
tems, such as the digital subscriber line (DSL) systems. In both
DMT and OFDM systems, a pair of discrete Fourier transform
(DFT) and inverse discrete Fourier transform (IDFT) is used
to partition the frequency band into independent subchannels.
Adaptive spectrum shaping and bit allocation may be easily im-
plemented on a carrier-by-carrier basis.
A central issue in the design of adaptive multicarrier sys-
tems is that of optimal spectrum and bit allocation across the
frequency domain. The issue is well understood for single-user
systems in which the optimal solution resembles an informa-
tion-theoretically optimal “waterfilling” solution. However, the
problem is nontrivial when multiple users are present at the same
time. In the latter case, the design objective function and the
constraints are often nonconvex, and the optimization problem
becomes computationally difficult to solve.
This paper makes progress toward numerical solution of
nonconvex optimization problems for multicarrier systems
by studying their fundamental properties. In particular, we
focus on the characterization of the Lagrangian dual of these
nonconvex problems. Our main result is that a nonconvex
optimization problem in multicarrier systems has a
zero duality
gap whenever a so-called “time-sharing” condition is satisfied.
Further, the time-sharing condition is always satisfied for
practical multiuser spectrum optimization problems in multi-
carrier systems when the number of frequency carriers goes to
infinity. This result is surprising at a first glance, as nonconvex
optimization problems generally have a nonzero duality gap.
Yet, the result is very useful, as it opens up the possibilities
of rigorously solving for the global optimum of nonconvex
problems in the dual domain.
Lagrangian duality theory for general convex optimization
problems is well known [1], [2]. For nonconvex multiuser
spectrum optimization problems, existing approaches in the
literature typically focus on either the convex relaxation of
the problem [3]–[5] or heuristic methods that approximate the
global solution of the problem [6]–[11]. In both cases, the global
optimality of the solution is difficult to prove. Recently, an op-
timal spectrum balancing (OSB) method for the DSL multiuser
spectrum optimization problem is proposed in [12], where a
first proof of global optimality of a bit-loading algorithm for
a nonconvex problem is provided. This paper generalizes the
result of [12] and reinterprets the OSB algorithm in a dual
optimization framework. Our main contribution is a theoretical
0090-6778/$20.00 © 2006 IEEE
YU AND LUI: DUAL METHODS FOR NONCONVEX SPECTRUM OPTIMIZATION OF MULTICARRIER SYSTEMS 1311
Fig. 1. Multiuser DSLs.
treatment of nonconvex optimization problems for multicarrier
systems, and a precise condition for zero duality gap. Our
general theory is inspired by the earlier work of Aubin and
Ekeland [13] and Bertsekas
et al. [14], [15], where an integer
programming problem is considered in a similar context.
The general theory presented in this paper also gives rise to
several algorithmic improvements to the OSB algorithm. First,
by reinterpreting the OSB algorithm as a dual algorithm, we
show that the dual updates may be done efficiently using sub-
gradient and ellipsoid algorithms. This significantly speeds up
the convergence of the implementation. Second, we propose a
low-complexity method to approximate the evaluation of the La-
grangian dual objective, which allows the computational com-
plexity to be further reduced at a small cost to performance. Both
are significant steps toward making the algorithm practical. This
paper aims to provide an optimization-theoretical viewpoint to
the multiuser spectrum optimization problem. Theoretical ap-
proaches to this problem have been proved to be fruitful in the
past. For example, in a recent work [16], a nonlinear comple-
mentarity approach to the spectrum optimization problem is
used to provide very interesting insights to the DSL problem.
The rest of the paper is organized as follows. Section II
contains the system model and the problem formulation.
In Section III, a general theory of nonconvex optimization
problems for multicarrier systems is presented and the main
duality gap result is proved. Section IV provides efficient
ways to update dual variables for OSB. In Section V, a new
low-complexity algorithm is proposed for the evaluation of
dual objective, which further reduces the computational com-
plexity of the algorithm. Section VI contains simulation results.
Concluding remarks are made in Section VII.
II. S
PECTRUM OPTIMIZATION PROBLEMS
This paper is primarily motivated by the recent surge of in-
terest in dynamic spectrum management (DSM) for digital sub-
scriber lines. In a DSL system, multiple copper twisted pairs
are bundled together. The electromagnetic coupling between
the copper pairs causes crosstalk interference, which has long
been identified as the primary source of line impairment in DSL
deployments. Current DSL systems use a static spectrum man-
agement (SSM) approach where a fixed transmit power spec-
tral density (PSD) is applied to each line regardless of the loop
topology or user service requirements. The performance pro-
jection under SSM is based on the levels of worst-case crosstalk
interference.
Future generations of DSL services are envisioned to imple-
ment DSM [17], [18], in which, each line is given an ability to
adapt to its loop environment and service requirements individ-
ually across the spectrum. DSM is facilitated by the adoption
of multicarrier modulation by DSL standardization bodies. The
ability to set the PSD level of each frequency carrier individu-
ally gives DSM techniques the potential to greatly improve the
achievable rates and service ranges of current DSL systems. On
the other hand, the large number of design variables also present
a research challenge from an optimization point of view. DSM
is an active area of research, both within the research commu-
nity and within the standardization bodies [19].
Under the SSM scheme, where the line interference is
assumed to be fixed (or assumed to have a worst-case PSD
level), the spectrum optimization problem simplifies to the
following. The design objective is to maximize the overall
system throughput, which is the sum of individual rates in each
frequency carrier. The design constraint is a power constraint
coupled across all the carriers. Let
denote the transmit PSD
at the
th carrier. The optimization problem is
maximize
(1)
subject to
where is the combined noise and interference PSD at the
th carrier normalized by . Here, is the gap to ca-
pacity,
is the channel transfer function in the th carrier,
is the total power constraint, and is the number of frequency
carriers. The above problem has a well-known waterfilling so-
lution. Efficient solution readily exists in this case, because the
objective function is concave in the optimizing variable
.
The spectrum optimization problem becomes much more
challenging when the PSDs of multiple users need to be opti-
mized at the same time. The need for such a joint optimization
is most clearly illustrated in the situation depicted in Fig. 1,
where the channel transfer functions are heavily unbalanced.
As shown in the figure, when an optical network unit (ONU)
is deployed remotely, it may emit excessive interference to a
neighboring customer-premise modem served from the central
office. DSM enables the joint optimization of the transmit
1312 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 7, JULY 2006
PSDs by both the central office modem and the ONU modem,
allowing both to operate at the same time.
Mathematically, the multiuser spectrum optimization
problem may be formulated as follows:
maximize
(2)
subject to
where the effective noise PSD for the th user at the th car-
rier
is again normalized by , and the effective in-
terference coefficient
is defined as . Here,
is the number of users, is the number of frequency car-
riers,
is the channel transfer function from the th trans-
mitter to the
th receiver in carrier is the relative weight
given to the
th user in the optimization problem, is the op-
timization variable denoting the power allocation for user
in
the
th carrier, and is the total power constraint for the user
. Throughout the paper, the sidelobe effect between adjacent
carriers is neglected.
1
This is realistic for frame-synchronous
DSL systems implementing a zipper-like modulation [20], or
where a sufficient amount of transmit windowing is included.
By solving the above optimization problem with varying
,
the entire achievable rate region can be generated. Because the
objective function is not concave in
, numerical optimization
is difficult. Clearly, an exhaustive search is not feasible, as the
complexity would be exponential in the total number of vari-
ables, which is
, where can be as large as 4096.
Iterative waterfilling (IWF) [7] is one of the early multiuser
spectrum optimization techniques that take advantage of the
ability for DSL modems to perform spectral shaping. In this al-
gorithm, each user iteratively maximizes its own achievable rate
by performing a single-user waterfilling with the crosstalk inter-
ference from all other users treated as noise. However, the IWF
process does not seek to find the global optimum for the entire
DSL bundle. Instead, each user participates in a noncoopera-
tive game, and the convergence point of the IWF process corre-
sponds to a competitive equilibrium. Although not optimum, the
IWF algorithm has been shown to outperform SSM schemes.
Recently, an exact OSB algorithm to solve this problem glob-
ally and optimally was proposed in [12]. The basic strategy is to
transform the spectrum optimization problem (2) into the dual
domain by forming its Lagrangian dual
maximize
(3)
subject to
1
When the desired DMT symbol and the interfering DMT symbols are not
frame-synchronized, the frequency tones of the desired and interfering signals
are not orthogonal to each other, creating sidelobes for each tone. In this case,
S
may interfere not only with
S
, but also with
S
,
S
, etc.
The idea is to solve the Lagrangian for each set of nonneg-
ative and fixed
. Then the solution to the orig-
inal problem may be found by a nested bisection search in the
-space. It can be shown that the OSB algorithm has a com-
putational complexity that is linear in the number of frequency
carriers
. As illustrated in [12], the OSB algorithm can pro-
vide a significant performance improvement, as compared with
IWF.
However, the computational complexity of the OSB algo-
rithm, although linear in
, is still exponential in the number
of users
. This is so for two reasons. First, with users,
nested loops of bisections are needed, one for each . Thus,
the
search is exponentially complex. Second, the maximiza-
tion of the Lagrangian for a fixed set of
involves
an exhaustive search over
in each tone , which
has a computational complexity that is also exponential in
.
When the number of users is large, the complexity of OSB be-
comes prohibitive.
The purpose of this paper is to refine the OSB algorithm with
an aim of eliminating its exponential complexity. Toward this
end, we establish a general theory of dual optimization for mul-
ticarrier systems, and show that contrary to general nonconvex
problems, the duality gap for multiuser spectrum optimization
always tends to zero as the number of frequency carriers goes
to infinity, regardless of whether the optimization problem is
convex. This key observation leads to efficient
search methods
that optimize the dual objective function directly.
Second, to overcome the exponential complexity of an ex-
haustive search over
, we propose iterative and
low-complexity ways to evaluate the dual objective approxi-
mately. The resulting algorithm is a middle ground between
IWF and OSB. We show by simulation that such an iterative
spectrum balancing (ISB) technique achieves most of the gain
of OSB in many cases of practical importance, while having a
much lower computational complexity.
The computational methods proposed in this paper have a
wider implication beyond that of DSL applications. The DSL
spectrum balancing problem is very similar to the optimal power
allocation and bit-loading problem for OFDM systems in wire-
less applications [3], [5], [21], [22]. A low-complexity solution
to the DSL problem is likely to be applicable to wireless sys-
tems, as well.
III. D
UALITY GAP OF
NONCONVEX OPTIMIZATION
In this section, we present a general duality theory for non-
convex optimization problems in multicarrier systems. In a mul-
ticarrier system, the optimization objective and the constraints
typically consist of a large number of individual functions, each
corresponding to one of the
frequency carriers. So, the opti-
mization problem has the following general form:
maximize
(4)
subject to
YU AND LUI: DUAL METHODS FOR NONCONVEX SPECTRUM OPTIMIZATION OF MULTICARRIER SYSTEMS 1313
where are vectors of optimization variables,
are functions that are not necessarily
concave, and
are functions that are
not necessarily convex. Power constraints are denoted by
an
-vector . Here, “ ” is used to denote a compo-
nent-wise inequality. For the multiuser spectrum optimiza-
tion considered before,
,
and
.
The idea of dual optimization is to solve (4) by forming its
Lagrangian dual
(5)
where
is a vector of Lagrangian dual variables. Define the
dual objective
as an unconstrained maximization of the
Lagrangian
(6)
The dual optimization problem is
minimize
subject to (7)
When ’s are concave and ’s are convex, standard
convex optimization results guarantee that the primal problem
(4) and the dual problem (7) have the same solution. When
convexity does not hold, the dual problem provides a solution,
which is an upper bound to the solution of (4). The upper bound
is not always tight, and the difference between the upper bound
and the true optimum is called the “duality gap.”
The main objective of this section is to characterize a con-
dition under which the duality gap is zero even when the opti-
mization problem is not convex. Toward this end, we define the
following
time-sharing condition.
Definition 1: Let
and be optimal solutions to the op-
timization problem (4) with
and , respec-
tively. An optimization problem of the form (4) is said to sat-
isfy the time-sharing condition if for any
and for any
, there always exists a feasible solution , such
that
, and
.
The time-sharing condition has the following intuitive in-
terpretation. Consider the maximum value of the optimization
problem (4) as a function of the constraint
. Clearly, a larger
implies a more relaxed constraint. So, roughly speaking,
the maximum value is an increasing function of
. The
time-sharing condition implies that the maximum value of the
optimization problem is a concave function of
.
Note that if
’s are concave and ’s are convex, then the
time-sharing condition is always satisfied. This can be easily
verified by setting
, in which case
the concavity of
implies
, and the convexity of implies
Fig. 2. Time-sharing property implies zero duality gap.
. However,
the converse is not necessarily true. As is shown later in the
paper, for many multicarrier systems of practical interest, the
time-sharing condition holds even when
’s are not concave
and
’s are not convex.
The main result of this section is that the time-sharing prop-
erty implies zero duality gap. Further, for many practical opti-
mization problems in the multicarrier context, the time-sharing
condition is satisfied.
Theorem 1: Consider an optimization problem of the form
(4). If the optimization problem satisfies the time-sharing prop-
erty, then it has a zero duality gap, i.e., the primal problem (4)
and the dual problem (7) have the same optimal value.
Proof: The theorem is a standard result in convex opti-
mization if
’s are concave and ’s are convex functions. In
this case, the optimization problem (4) is a convex program-
ming problem, which has a zero duality gap under general con-
straint qualification conditions. The main novelty of the the-
orem resides in cases where (4) is not convex, but for which the
time-sharing condition nevertheless holds. The proof is divided
into two parts.
Let
and be vectors of power constraints with
for some . Let , ,
and
be the optimal solutions to the optimization problem (4)
with constraints
and , respectively. In the first part of
the proof, we show that time sharing implies that
is a concave function of . The concavity follows from the
definition of the time-sharing property. Since
, the time-sharing property implies that there ex-
ists a
such that and
. Since is
a feasible solution for the optimization problem, this means that
,
thus proving the concavity.
Second, we show that the concavity of the optimal
in
implies zero duality gap. Fig. 2 is a graphical illustration of
the proof. Consider a sequence of the optimization problem pa-
rameterized by the constraint
. The solid line in Fig. 2 is a plot
of optimal
as the constraint varies. The
curve is plotted with
on the -axis. The -axis is lo-
cated at the point where
. Thus, the intersection
of the curve with the
-axis is exactly the primal optimal solu-
tion to (4), which is denoted as
on the plot.
1314 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 7, JULY 2006
Fig. 3. Duality gap is nonzero without the time-sharing condition.
Now, consider the dual objective function for a fixed
(8)
Let
be the optimal solution to the above optimization
problem. The value of
can be graphically obtained by
drawing a tangent line to the
curve
through the point
. By the definition of
, it is not difficult to see that the tangent line has a slope
. Further, the intersection of the tangent line with the -axis
is
, which is exactly the
value of
, as illustrated in Fig. 2. This allows the minimiza-
tion of
to be visualized. The dual optimum, denoted as ,
is the minimum
over all nonnegative ’s. Clearly, when
the optimal
curve is concave, among
all tangent lines with various slopes
, the that minimizes
the
-axis intersection is precisely the one that intersects the
-axis at . Thus, and the duality gap is zero.
To illustrate the importance of the time-sharing condition,
Fig. 3 depicts a situation in which time sharing does not hold.
In this case, the curve
is not concave,
and the minimum
is strictly larger than the maximum
.
It turns out that the time-sharing condition is always satisfied
for practical spectrum optimization problems in multicarrier ap-
plications in the limit, as the number of carriers
goes to in-
finity. The intuitive reason is as follows. The time-sharing con-
dition is clearly satisfied if an actual time-division multiplexing
may be implemented. Let
and be two spectrum alloca-
tions. In this case, the entire frequency band can be assigned to
for percentage of the time, and for percentage of
the time. The overall
then becomes a linear combination
. The constraint becomes a linear
combination also, thus satisfying the time-sharing condition.
In practical multicarrier systems with a large number of
frequency carriers, channel conditions in adjacent carriers are
often similar. Then, time sharing may be approximately im-
plemented with frequency sharing. By interleaving
and
in the frequency domain with a proportionality , the overall
becomes approximately ,
and the same applies to the constraints. The approximation is
exact when
.
To make the intuition precise, consider the spectrum opti-
mization problem (2) with continuous frequency variables
maximize
subject to
(9)
where the effective noise PSD
is again normalized by
, and the effective interference coefficient
is defined as . Again, is the
channel transfer function from the
th transmitter to the th
receiver.
Theorem 2: Consider an optimization problem (9) in which
and are both continuous functions of . Then,
the time-sharing condition is always satisfied. In addition, its
discretized version (2) also satisfies the time-sharing condition
in the limit as
.
Proof: Let
and be the optimal solutions to
the spectrum optimization problem (9) with power constraints
and , respec-
tively. Let
and be their respective optimal values. To
prove the time-sharing property, we need to construct an
such that it satisfies a power constraint and
achieves a rate equal to or higher than
for all
between zero and one.
We first prove the theorem for the case in which
and
are constant functions of for all and . First, observe
that the optimal solution to (9) is always a frequency-division
multiplex (FDM) of at most
frequency bands, where each
frequency band corresponds to a transmission strategy for which
a subset of
users transmit. Further, within each frequency
band, the optimal
and must be constant. This
is because within each band, the same Karush–Kuhn–Tucker
(KKT) condition (which is a necessary condition even for
nonconvex problems) must be satisfied for each frequency
,
and the optimal set of spectra is the KKT point that maximizes
the weighted sum rate. As the same condition applies to all
frequencies, the power allocation within each band must be flat.
Now, let
be the optimal solution of
(9) with a power constraint
, and be
the optimal solution of (9) with a power constraint
. Let the
achievable rate in the two cases be
and , respectively.
To prove the time-sharing property, we need to construct an
that achieves at least ,
with a power that is at most
for all between
zero and one. Such a
may be constructed by taking the
union of the two frequency partitions corresponding to
and
, then further subdividing each frequency band in the union
into two,
proportion of which has , and
proportion of which has . Clearly,
the resulting
satisfies the power constraint
. Further, it also achieves a rate
