In this section, we propose a BCD-based algorithm to solve the formulated non-convex optimization problem. Specifically, to solve problem , we decouple the original problem into three sub-problems, i.e., the BD matching problem, the UAV trajectory optimization problem, and the reflection coefficient optimization problem, then respectively apply the game-based matching algorithm, SCA algorithm, and relaxation algorithm to solve these sub-problems. Afterwards, problem can be solved by iteratively addressing the sub-problems until reaching convergence.
3.1. BD Matching Optimization
With the given fixed UAV trajectory and the reflection coefficients of the BDs, the BD matching problem is a nonlinear mixed-integer programming problem and is difficult to transform into a convex problem. Therefore, we propose a game-based matching algorithm to schedule the BDs effectively according to the locations of the CEs and BDs and the trajectory of the UAV. The game-based matching algorithm can be divided into four steps: (1) CE Scheduling; (2) BD Scheduling; (3) Slot Selection; and (4) Optional Set Updating.
(1) CE Scheduling
In this step, each slot is associated with a CE. Then, for each CE, an optional slot set is obtained by collecting its associated slots. For the
n-th slot, we define the active CE as
, which is obtained by
We define the optional slot set for the m-th CE as . Subsequently, we initialize the set of optional slots for the k-th BD as , which indicates that the k-th BD is allocated slots to reflect the signal in the initialization of the proposed iterative algorithm.
(2) BD Scheduling
In this step, each BD selects one slot from its optional slot set based on transmission path loss. The selected slot for the
k-th BD is expressed as
We define the BDs that choose the n-th slot as the collected set . Note that more than one BD may choose the same slot. Hence, in the next step, an optimal BD is choesn for the n-th slot to reflect the signal from the set .
(3) Slot Selection
In this step, the slots selected by BDs in the previous step are used to select the most suitable BDs. For the
n-th slot, the most suitable BD is selected from the set
, which is defined as
and obtained by
where
is the already-allocated number of slots for the
k-th BD and
is the competition factor. Until this point, parts of BDs have been matched with the most appropriate slots; next, we update
with
if the
k-th BD is chosen by one slot in this step. Otherwise,
remains unchanged.
(4) Optional Set Updating
After the previous two steps, we update the optional slot set belonging to each BD. When updating the optional slot set belonging to each BD, there are two principles to follow. First, the BDs and slots that already have a matching relationship cannot be matched again. Second, the number of BDs that can be matched per slot depends on . To update , the slots are removed from the set if they satisfy any of the above principles.
Steps 2–4 are iterated until each slot matches
BDs, at which point the matching scheme
of each BD at slot
n is obtained. The detailed game-based matching algorithm is shown in Algorithm 1. The matching scheme
is updated after Algorithm 1 if it increases the objective function (
10a); otherwise, the previous
are utilized in solving the trajectory optimization problem and the reflection coefficient optimization problem.
Algorithm 1: Framework of the game-based matching algorithm. |
|
3.2. Trajectory Optimization
With the fixed BD matching scheme and the reflection coefficients of the BDs, the trajectory optimization problem can be expressed as
with the constraints (10b), (10f) and (10g). The constraint (14b) is equivalent to (10c).
Representing
as a variable
, the objective function (
14a) and left-hand side (LHS) of inequalities (14b) and (10b) are all non-convex with respect to
. Because (
14a) and the LHS of (14b) and (10b) have similar structures, we take (
14a) as an example and transform it into a convex problem with respect to
.
We first decompose the non-convex term in the logarithmic function of (
14a), which is shown as
where
Note that and are neither convex nor concave with respect to . However, we find that is a concave function with respect to and that is a convex function with respect to . Therefore, in the following we focus on obtaining the concave approximations of and with respect to based on .
For
, we can obtain the linear lower bound of
using the SCA. For more detail, in the
r-th iteration the first-order Taylor expansion is applied at the given local point
, then
is transformed into
where
e is the Euler constant and
satisfies
where
are the slack variables and satisfy
. According to (
3)–(
7), we can now obtain
Thus far, we have converted
into a concave form with respect to
. However, the new constraints (
20) on
remain non-convex. Next, we transform (
20) into a convex form. For analytical convenience, we introduce new auxiliary variables
, which satisfy
and (
20) can be expressed as
For the right half of the constraints in (
22),
is concave with respect to
and
is concave with respect to
, as
is always equal to or greater than
. Next, we obtain the upper bound of
and
in order to transform (
22) into a convex expression.
First, we define
as the upper bound of
through first-order Taylor expansion at a local point
, which is provided by
where
are the slack variables and satisfy
Then, we define
as the upper bound of
, through first-order Taylor expansion at the given local point
, which can be expressed as
where
are the slack variables and satisfy
With (
23)–(
26), we can transform (
22) into
Until now, we have made the constraints (
22) a convex expression. However, in the process of transforming (
27), we have introduced the constraints (
26), which are non-convex. For the in constraints (
26), similar to the treatment of the inverse trigonometric function in [
33], we have the upper bound of
through the first-order Taylor expansion at the local point
, and transform (
26) into
where
are the new slack variables and satisfy
Thus far, we have obtained the concave approximation of . Next, we focus on .
is a convex function about
. However, we need find a concave form about
to solve
. With the given local point
in the
r-th iteration, we obtain the upper bound of
through the first-order Taylor expansion, which is provided by
where
are auxiliary variables with the constraints
. Similar to (
22), we use the logarithmic form to represent the constraints, which can be expressed as
Up to now, we have converted
to a convex form with respect to
. However, the new constraints (
31) on
are non-convex. Thus, we need to transform (
31) into a convex form. In (
31), we find that
is convex with respect to
and
is convex with respect to
. Then, we apply the first-order Taylor expansion to
based on the local point
and introduce the new variables
to represent the lower bound of
. Finding that
is convex with respect to
, we introduce the slack variables
and define
by applying the first-order Taylor expansion to
based on the local point
, which can be obtained by
with the constraints
According to the above treatment, (
31) can be transformed into
which are convex with respect to
and
. Now, we have changed
into a concave form, and the constraints on the slack variables we added in the process of transformation are convex as well. After the above process, problem
can be expressed in the following form:
with the constraints (10f), (10g), (
24), (
27)–(
29), (
33) and (
34).
Note that problem is a convex optimization problem, which can be solved by standard convex optimization solvers such as CVX.
3.3. Reflection Coefficient Optimization
With the given trajectory of the UAV and BD matching scheme, the optimization problem for the reflection coefficients of the BDs without the discrete constraint is provided by
with the constraints (10b) and (14b).
It can be noted that the objective function in (36a) and the constraints (10b) and (14b) are non-convex for
. Similar to the processing of (
15),
in (36a) can be expressed as the sum of two terms, i.e.,
and
. It is clear that
is a concave function about
, while
is a convex function about
. Thus, our goal is to transform
into a concave form with respect to
. Here, we use the first-order Taylor expansion to deal with
at the given local point
at the
r-th iteration and introduce the new variables
; then, we can obtain the lower bound of
as
Therefore, we can reformulate problem
as
It can be noticed that problem
is a convex problem and can be solved directly. After continuous optimization of the backscattering coefficients, the discrete coefficients are obtained using downward rounding based on the discrete set
, where
, which can be expressed as