For any given
\(\{{\textbf{P}},\varvec{\rho },{\textbf{G}}\}\), the UAV trajectory of problem (
P2) can be optimized by solving the following problem:
$$\begin{aligned} {(P5):}&\mathop {\max }_{{\textbf{q}},\alpha }\alpha \nonumber \\ {\rm s.t.} \quad&(6),(7),(11),(13),(15a). \end{aligned}$$
Problem (
\(P5\)) is still non-convex due to the non-convex constraints (
11), (
13) and (
15a). As discussed in Sect.
3.1, we utilize SCA for trajectory optimization. To this end,
\(R_k [n]\) in constraint (
11) can be written as follows:
$$\begin{aligned} R_k [n]=d_m W_0 \left[ {\hat{R}}_k [n]-\log _2 \left( \sum \limits _{i=a+1}^{d}\frac{P_{g_{m,i}[n]}[n]\beta _0}{\Vert {\textbf{q}}[n]-{\textbf{w}}_{g_{m,i}[n]}\Vert ^2 +H^2}+N_0 d_m W_0\right) \right] , \end{aligned}$$
(19)
where
\({\hat{R}}_k [n]=\log _2( \frac{P_k [n]\beta _{0}}{\Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2 +H^2}+\sum \limits _{i=a+1}^{d}\frac{P_{g_{m,i}[n]}[n]\beta _0}{\Vert {\textbf{q}}[n]-{\textbf{w}}_{g_{m,i}[n]}\Vert ^2 +H^2} +N_0 d_m W_0)\).
By introducing slack variables
\({\textbf{S}}\triangleq \{S_k [n]=\Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2,\forall k,\forall n\}\), constraint (
11) can be written as follows:
$$\begin{aligned} \sum \limits _{n=1}^N d_m W_0[ {\hat{R}}_k[n] -\log _2 \left( \sum \limits _{i=a+1}^{d}\frac{P_{g_{m,i}[n]}[n]\beta _0}{S_{g_{m,i}[n]}[n]+H^2}+N_0 d_m W_0)] (1-\rho [n] \right) \delta \ge C, \end{aligned}$$
(20)
, and problem (
\(P5\)) can be formulated as follows:
$$\begin{aligned} {(P6):}&\mathop {\max }_{{\textbf{q}},\alpha ,{\textbf{S}}}\alpha \nonumber \\ {\rm s.t.} \quad&S_k [n]\le \Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2,~~\forall k,n,\nonumber \\&(6),(7),(13),(15a),(20). \end{aligned}$$
(21)
To tackle the non-convexity of constraints (
13), (
15a), (
20) and (
21), we approximate them with convex functions in each iteration. We define
\({\textbf{q}}^r\triangleq \{{\textbf{q}}^r[n],\forall n\}\) as the given UAV trajectory in the r-th iteration. Although
\(h_k [n]\) is not convex with resect to
\({\textbf{q}}[n]\), it is covex with respect to
\(\Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2\). Recall that any convex function is globally lower-bounded by its first-order Taylor expansion at any point. Therefore, we have the following convex lower bound at the given point
\({\textbf{q}}^r\)$$\begin{aligned}&h_k[n]\ge \frac{\beta _0}{\Vert {\textbf{q}}^r[n]-{\textbf{w}}_k\Vert ^2 +H^2}\\&\quad -\frac{\beta _0(\Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2-\Vert {\textbf{q}}^r[n] -{\textbf{w}}_k\Vert ^2)}{(\Vert {\textbf{q}}^r[n]-{\textbf{w}}_k\Vert ^2 +H^2)^2} \triangleq {\hat{h}}_k^{lb}[n]. \end{aligned}$$
(22)
By substituting
\({\hat{h}}_k^{lb}[n]\) for
\(h_k[n]\) in (
13), we can approximate (
13) as the following concave constraint:
$$\begin{aligned}&E_k^{init}+\sum \limits _{i=1}^n\eta P_0 h_k^{lb}[i]\rho [i]\delta -\sum \limits _{i=1}^{n-1}P_k[i](1-\rho [i])\delta \\&\quad \ge P_k[n](1-\rho [n])\delta ,~~\forall n,k. \end{aligned}$$
(23)
Also, by substituting
\({\hat{h}}_k^{lb}[n]\) for
\(h_k[n]\) in (
15a), we can approximate (
15a) as the following concave constraint
$$\begin{aligned} E_k^{init}+\sum \limits _{n=1}^N\eta P_0 h_k^{lb}&[n]\rho [n]\delta -\sum \limits _{n=1}^{N}P_k[n](1-\rho [n])\delta \ge \alpha ,~~\forall k. \end{aligned}$$
(24)
\({\hat{R}}_k[n]\) in constraint (
20) is convex with respect to
\(\Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2\). Therefore, we have the following convex lower bound at the given point
\({\textbf{q}}^r\)$$\begin{aligned} {\hat{R}}_k[n]\ge&\sum \limits _{i=a}^d A_{k,i}^r[n]( \Vert {\textbf{q}}[n]-{\textbf{w}}_{g_{m,i}[n]}\Vert ^2 -\Vert {\textbf{q}}^r[n]-{\textbf{w}}_{g_{m,i}[n]}\Vert ^2)\\&+B_{k,i}^r [n] \triangleq {\hat{R}}_k^{lb}[n], \end{aligned}$$
(25)
where
\(A_{k,i}^r[n]\) and
\(B_{k,i}^r [n]\) are constants that are given by
$$\begin{aligned} A_{k,i}^r[n]&=\frac{ -\frac{P_{g_{m,i}[n]}[n]\beta _0}{(\Vert {\textbf{q}}^r[n]-{\textbf{w}}_{g_{m,i}[n]}\Vert ^2+H^2)^2}}{\sum _{j=a}^{d}\frac{P_{g_{m,i}[n]}[n]\beta _0}{(\Vert {\textbf{q}}^r[n]-{\textbf{w}}_{g_{m,i}[n]}\Vert ^2+H^2)}+N_0 d_m W_0}, \end{aligned}$$
(26)
$$\begin{aligned} B_{k,i}^r[n]&=\log _2 \left( \sum \limits _{i=a}^d\frac{P_{g_{m,i}[n]}[n]\beta _0}{\Vert {\textbf{q}}^r[n] -{\textbf{w}}_{g_{m,i}[n]}\Vert ^2+H^2}+N_0 d_m W_0\right) . \end{aligned}$$
(27)
By substituting
\({\hat{R}}_k^{lb}[n]\) for
\({\hat{R}}_k[n]\) in (
20), we can approximate (
20) as the following jointly concave constraint with respect to
\({\textbf{q}}\) and
\({\textbf{S}}\)$$\begin{aligned} \sum \limits _{n=1}^{N}d_m W_0 [{\hat{R}}_k^{lb}[n]&-\log _2 \bigg (\sum \limits _{i=a+1}^d\frac{P_{g_{m,i}[n]}[n]\beta _{0}}{S_{g_{m,i}[n]}[n]+H^2}\\&+N_0 d_m W_0)](1-\rho [n] \bigg )\delta \ge C. \end{aligned}$$
(28)
In constraint (
21),
\(\Vert {\textbf{q}}[n]-{\textbf{w}}_k\Vert ^2\) is convex with respect to
\({\textbf{q}}[n]\), so constraint (
21) can be approximated as
$$\begin{aligned} S_k[n]\le \Vert {\textbf{q}}^r[n]-&{\textbf{w}}_k\Vert ^2+2({\textbf{q}}^r[n]-{\textbf{w}}_k)^T\\&({\textbf{q}}[n]-{\textbf{q}}^r[n]),~~\forall k,n. \end{aligned}$$
(29)
Based on these, with any given local point
\({\textbf{q}}^r\), problem (
\(P5\)) can be approximated as the following convex problem:
$$\begin{aligned} {(P6):}&\mathop {\max }_{{\textbf{q}},\alpha ,{\textbf{S}}}\alpha \nonumber \\ {\rm s.t.} \quad&(6),(7),(23),(24),(28),(29). \end{aligned}$$
, and problem (
\(P6\)) can be solved by existing convex optimization technologies.