AKL
et al.: MULTICELL CDMA NETWORK DESIGN
717
By relaxing the integer variables
,
, to contin-
uous variables, the optimizations in (31)–(34) are solved using a
sequential quadratic programming method [27]. In this method,
a quadratic programming subproblem is solved at each itera-
tion. An estimate of the Hessian of the Lagrangian is updated
at each iteration using the Broyden–Fletcher–Goldfarb–Shanno
formula [28]. A line search is performed using a merit function
[29]. The quadratic programming subproblem is solved using
an active set strategy [30].
In Section VI, we provide results for the optimization prob-
lems in (31)–(34) (with the continuous relaxation assumption),
which give an upper bound on the optimal value. What is pre-
sented is the rounded-down version of the solution. We also
solve the MIP problems and provide the number of branches
required to arrive at the solution.
We note that unlike (9), (31)–(34) are not convex optimization
problems, and so it may be possible for the approaches described
above not to converge to a global optimal solution. To ensure
that this did not occur, we verified the results of the optimization
using simulated annealing (SA) [31], which has many attractive
features. In particular, SA can statistically guarantee finding a
global optimal solution [32]. On the other hand, it can be quite
time consuming to use SA to find an optimal solution, and it
is difficult to fine tune. We use an adaptive simulated annealing
(ASA) algorithm, which is based on an associated proof that the
parameter space can be sampled much more efficiently than by
other previous SA algorithms.
1
Our initial capacity vector used
in ASA is the solution returned by the optimization problems
(31)–(34). The generating probability density function, the ac-
ceptance probability density function, and the cooling temper-
ature schedule used were the default values provided by the al-
gorithm itself.
VI. N
UMERICAL
R
ESULTS
We assume the following for the analysis. The COST-231
propagation model with a carrier frequency of 1800 MHz, av-
erage base-station height of 30 m, and average mobile height of
1.5 m is used to determine the coverage region. The path-loss
coefficient
is four. The shadow-fading standard deviation
is 6 dB. The processing gain
is 21.1 dB. (This corresponds
to the processing gain in IS-95.) The bit energy to interference
ratio threshold
is 9.2 dB. The interference to background noise
ratio
is 10 dB. The voice activity factor
is 0.375. The
whole area is divided into small grids of size 150 by 150 m .
In what follows, we will study an example with a uniform
and nonuniform user distribution in detail and show how the
optimization techniques described in this paper, i.e., (9) and
(31)–(34), maximize the capacity profile of such a network. The
following results have been obtained for the 27-cell CDMA net-
work shown in Fig. 2. The base stations are located at the centers
of a hexagonal grid whose radius is 1732 m. Base station 1 is
located at the origin. The base stations are numbered consecu-
tively in a spiral pattern. The pilot-signal power of every base
station is 1 W.
1
The ASA code and ample documentation are publicly available at
http://www.ingber.com/
Fig. 2.
Capacity in a 27-cell CDMA network with uniform user distribution.
