Proposed Algorithm for Problem P0

Table of Links

Abstract and I. Introduction

II. System Model

III. Problem Formulation

IV. Proposed Algorithm for Problem P0

V. Numerical Results

VI. Conclusion

APPENDIX A: PROOF OF LEMMA 1 and References

IV. PROPOSED ALGORITHM FOR PROBLEM P0

To solve P0, we utilize the BCD technology to decompose the original problem into multiple subproblems. Specifically, for the given other variables, A, P, H, and Q are optimized in each subproblem respectively. In addition, the SCA technology is utilized to transform the non-convex constraints into convex constraints.

A. Subproblem 1: Optimizing User Scheduling Variable

B. Subproblem 2: Optimizing Transmit Power of B

C. Subproblem 3: Optimizing Horizontal Trajectory and Velocity of B

In this subsection, the horizontal trajectory and velocity of B is optimized for provided {A,P,H}. The original optimization problem is rewritten as

To address the non-convexity in (19a), Lemma 1 is introduced.

D. Subproblem 4: Optimizing Horizontal Trajectory and Velocity of B

In this subsection, for given {A,P,Q}, the vertical trajectory H of B is optimized. The optimization problem is expressed as

With the same method as (13b), (23b) is reformulated as (19a)-(19d) and (1a) and (1b) are reformulated as (16c), (16e), and (19c). With the same method in Subproblem 3, (9) in this subsection is reformulated as (16a)-(16f) wherein (16b) and (16d) are reformulated as (18a) and (18b), respectively.

P4.2 is a convex optimization problem that can be solved using existing optimization tools such as CVX.

E. Convergence Analysis of Algorithm 1

The obtained suboptimal solution of the transformed subproblem is also the suboptimal solution of the original nonconvex subproblem, and each subproblem is solved using SCA convex transformation iteration. Finally, all suboptimal solutions of the subproblems that satisfy the threshold ε constitute the suboptimal solution of the original problem. Therefore, our algorithm is to alternately solve the subproblem P1.1, P2.1, P3.2 and P4.2 to obtain the suboptimal solution of the original problem until a solution that satisfies the threshold ε is obtained.

It is worth noting that in the classic BCD, to ensure the convergence of the algorithm, it is necessary to accurately solve and update the subproblems of each variable block with optimality in each iteration. But when we solve P3.1 and P4.1 , we can only optimally solve their approximation problem P3.2 and P4.2. Therefore, we cannot directly apply the convergence analysis of the classical BCD, and further proof of the convergence of Algorithm 1 is needed, as shown below.

(30) This is similar to the representation in (29), and from (27) to (30), we obtain

1
. (31) The above analysis indicates that the target value of P0 does not decrease after each iteration of Algorithm 1. Due to the objective value of P0 is a finite upper bound, therefore the proposed Algorithm 1 ensures convergence. The simulation results in the next section indicate that the proposed BCDbased method converges rapidly for the setting we are considering. In addition, since only convex optimization problems need to be solved in each iteration of Algorithm 1, which have polynomial complexity, Algorithm 1 can actually converge

quickly for wireless networks with a moderate number of users.