Distributed ADMM-based approach for total harvested power maximization in non-linear SWIPT system

Pham Viet Tuan; Insoo Koo

doi:10.1007/s11276-019-02188-z

Wireless Networks

pp 1–15 | Cite as

Distributed ADMM-based approach for total harvested power maximization in non-linear SWIPT system

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Pham Viet Tuan
Insoo Koo

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First Online: 21 November 2019

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Abstract

In this paper, the distributed alternating direction method of multipliers (ADMM)-based approach is investigated for total harvested power maximization (THPM) in multiple-input single-output transceiver pairs with simultaneous wireless information and power transfer. The power-splitting architecture and practical non-linear energy harvesting are utilized at each receiver. The highly non-convex THPM problem is formulated under requirements for the data rate, the amount of harvested power, and with limited power at the transmitters. First, semidefinite relaxation (SDR) and sequential parametric convex approximation are exploited to convert the non-convex THPM to a series of convex subproblems. Then, slack variables replacing interference are introduced to allow decomposing into distributed subproblems via ADMM. The interesting point is that optimal precoding matrices of the local subproblems satisfy the rank-1 constraints of SDR. Finally, the numerical experiments present the convergence of the proposed algorithm, obtaining results similar to the centralized approach, as well as comparing it with some baseline schemes.

Keywords

Simultaneous wireless information and power transfer (SWIPT) Semidefinite relaxation (SDR) Non-linear energy harvesting model Sequential parametric convex approximation (SPCA) Alternating direction method of multipliers (ADMM)

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Notes

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant through the Korean Government (MSIT) under Grant NRF-2018R1A2B6001714.

Appendix 1

Proof of Lemma 1

We consider the local problem as follows:

$$\begin{aligned}&\min \,\, - {t_l} + \frac{\rho }{2}{\left\| {{\mathbf{{f}}_l} - {\mathbf{{A}}_l}{\mathbf{{f}}^{\left( m \right) }}} \right\| ^2} + \mathbf{{\lambda }}_l^{\left( m \right) T}\left( {{\mathbf{{f}}_l} - {\mathbf{{A}}_l}{\mathbf{{f}}^{\left( m \right) }}} \right) \,\,\,\,\text {s.t.} \end{aligned}$$

(21a)

$$\begin{aligned}&0 \ge \left( {{d_l}{t_l} + {c_l}} \right) - \left( {\frac{{{M_l}}}{{1 + x_l^{\left( n \right) }}} - \frac{{{M_l}\left( {{x_l} - x_l^{\left( n \right) }} \right) }}{{{{\left( {1 + x_l^{\left( n \right) }} \right) }^2}}}} \right) , \end{aligned}$$

(21b)

$$\begin{aligned}&0 \ge - \ln {x_l} - {a_l}\left( {2y_l^{\left( n \right) }{y_l} - y_l^{\left( n \right) 2}} \right) + {a_l}{b_l}, \end{aligned}$$

(21c)

$$\begin{aligned}&0 \ge \frac{{y_l^2}}{{\left( {1 - {\theta _l}} \right) }} - \left( {\mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) + {{\tilde{P}}_l} + \sigma _l^2} \right) , \end{aligned}$$

(21d)

$$\begin{aligned}&0 \ge \frac{{{\mu _l}}}{{\left( {1 - {\theta _l}} \right) }} - \left( {\mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) + {{\tilde{P}}_l} + \sigma _l^2} \right) , \end{aligned}$$

(21e)

$$\begin{aligned}&0 \ge \frac{{\delta _l^2}}{{{\theta _l}}} + \left( {{{\tilde{Q}}_l} + \sigma _l^2} \right) - \frac{1}{{{\gamma _l}}}\mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) , \end{aligned}$$

(21f)

$$\begin{aligned}&0 \ge {\tilde{p}_{lk}} - \mathrm{{Tr}}\left( {{\mathbf{{G}}_{lk}}{\mathbf{{B}}_l}} \right) ,\forall k \ne l, \end{aligned}$$

(21g)

$$\begin{aligned}&0 \ge \mathrm{{Tr}}\left( {{\mathbf{{G}}_{lk}}{\mathbf{{B}}_l}} \right) - {\tilde{q}_{lk}},\forall k \ne l, \end{aligned}$$

(21h)

$$\begin{aligned}&\mathrm{{0}} \ge \mathrm{{Tr}}\left( {{\mathbf{{B}}_l}} \right) - E_l^{\max }, \end{aligned}$$

(21i)

$$\begin{aligned}&{\mathbf{{B}}_l}\succeq 0, \end{aligned}$$

(21j)

$$\begin{aligned}&\begin{array}{l} 0< {\theta _l} < 1,\,{t_l} \ge 0, {x_l} \ge 0,{y_l} \ge 0,\,\\ {{\tilde{p}}_{kl}} \ge 0,\,{{\tilde{q}}_{kl}} \ge 0,{{\tilde{P}}_l} \ge 0,{{\tilde{Q}}_l} \ge 0,\forall k \ne l, \end{array} \end{aligned}$$

(21k)

where the variables are ${{{\mathbf{B}}_l},{\theta _l},{t_l},{x_l},{y_l},{\mathbf{f}}_l}$. The Lagrangian function is expressed as follows:

$$\begin{aligned}&\mathcal{L}\left( {{\mathbf{{B}}_l},{\theta _l},{t_l},{x_l},{y_l},{\mathbf{{f}}_l},{\alpha _l},{\beta _l},{\pi _l},{{\left\{ {{\varphi _k}} \right\} }_{k \ne l}},{{\left\{ {{\xi _k}} \right\} }_{k \ne l}},\eta ,{\mathbf{{Z}}_l}} \right) \\&\quad = - {t_l} + \frac{\rho }{2}{\left\| {{\mathbf{{f}}_l} - {\mathbf{{A}}_l}{\mathbf{{f}}^{\left( m \right) }}} \right\| ^2} + \mathbf{{\lambda }}_l^{\left( m \right) T}\left( {{\mathbf{{f}}_l} - {\mathbf{{A}}_l}{\mathbf{{f}}^{\left( m \right) }}} \right) \\&\qquad + {\alpha _l}\left( {\frac{{y_l^2}}{{\left( {1 - {\theta _l}} \right) }} - \left( {\mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) + {{\tilde{P}}_l} + \sigma _l^2} \right) } \right) \\&\qquad + \,{\beta _l}\left( {\frac{{{\mu _l}}}{{\left( {1 - {\theta _l}} \right) }} - \left( {\mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) + {{\tilde{P}}_l} + \sigma _l^2} \right) } \right) \\&\qquad + \,{\pi _l}\left( {\frac{{\delta _l^2}}{{{\theta _l}}} + \left( {{{\tilde{Q}}_l} + \sigma _l^2} \right) - \frac{1}{{{\gamma _l}}}\mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) } \right) \\&\qquad + \sum \limits _{k \ne l}^L {{\varphi _k}\left( {{{\tilde{p}}_{lk}} - \mathrm{{Tr}}\left( {{\mathbf{{G}}_{lk}}{\mathbf{{B}}_l}} \right) } \right) } \\&\qquad + \sum \limits _{k \ne l}^L {{\xi _k}\left( {\mathrm{{Tr}}\left( {{\mathbf{{G}}_{lk}}{\mathbf{{B}}_l}} \right) - {{\tilde{q}}_{lk}}} \right) } \\&\qquad + \eta \left( {\mathrm{{Tr}}\left( {{\mathbf{{B}}_l}} \right) - E_l^{\max }} \right) \\&\qquad - {\mathrm{{Tr}}\left( {{\mathbf{{Z}}_l}{\mathbf{{B}}_l}} \right) } \\&\qquad + \varUpsilon , \end{aligned}$$

(22)

where ${\alpha _l},{\beta _l},{\pi _l},{\varphi _k},{\xi _k},\eta \ge 0$, ${\mathbf{{Z}}_l}\succeq 0$ are the dual variables according to constraints (21d), $\ldots $ , (21j), and $\varUpsilon $ term is not relative to ${{\mathbf{B}}_l}$. To easily see the relationship of variable ${{\mathbf{B}}_l}$, we rewrite the Lagrangian function as follows:

$$\begin{aligned}&\mathcal{L}\left( {{\mathbf{{B}}_l},{\theta _l},{t_l},{x_l},{y_l},{\mathbf{{f}}_l},{\alpha _l},{\beta _l},{\pi _l},{{\left\{ {{\varphi _k}} \right\} }_{k \ne l}},{{\left\{ {{\xi _k}} \right\} }_{k \ne l}},\eta ,{\mathbf{{Z}}_l}} \right) \\&\quad = \mathrm{{Tr}}\left( {\left( {\eta \mathbf{{I}} + \sum \limits _{k \ne l}^L {\left( {{\xi _k} - {\varphi _k}} \right) {\mathbf{{G}}_{lk}}} } \right) {\mathbf{{B}}_l}} \right) \\&\qquad - \left( {{\alpha _l} + {\beta _l} + \frac{{{\pi _l}}}{{{\gamma _l}}}} \right) \mathrm{{Tr}}\left( {{\mathbf{{G}}_{ll}}{\mathbf{{B}}_l}} \right) - \mathrm{{Tr}}\left( {{\mathbf{{Z}}_l}{\mathbf{{B}}_l}} \right) + \varXi . \end{aligned}$$

(23)

From Karush–Kuhn–Tucker (KKT) optimal conditions, we have ${\nabla _{{{\mathbf{B}}_l}}}\mathcal{L} = 0$ and ${\mathrm{Tr}}\left( {\mathbf{{Z}}_l^ * \mathbf{{B}}_l^ * } \right) = 0$. Therefore, we obtain:

$$\begin{aligned}&\mathbf{{D}}_l^ * - \left( {\alpha _l^ * + \beta _l^ * + \frac{{\pi _l^ * }}{{{\gamma _l}}}} \right) {{\mathbf{G}}_{ll}} - \mathbf{{Z}}_l^ * = 0, \end{aligned}$$

(24)

$$\begin{aligned}&\mathbf{{Z}}_l^ * \mathbf{{B}}_l^ * \, = \,0, \end{aligned}$$

(25)

$$\begin{aligned}&\alpha _l^ * ,\,\,\beta _l^ * ,\pi _l^ * ,\varphi _k^ * ,\xi _k^ * \ge 0; \mathbf{{Z}}_l^ * ,\mathbf{{B}}_l^ * \succeq 0, \forall k \ne l, \end{aligned}$$

(26)

where $\mathbf{{D}}_l^ * = \left( {{\eta ^ * }{} \mathbf{{I}} + \sum \limits _{k \ne l}^L {\left( {\xi _k^ * - \varphi _k^ * } \right) {{\mathbf{G}}_{lk}}} } \right) $. The Eq. (25) is derived from Appendix in [9]. Moreover, the notation ‘*’ indicates the optimal variables that satisfy the KKT conditions. From (24), we derive:

$$\begin{aligned} \mathbf{{D}}_l^ * = \mathbf{{Z}}_l^ * + \left( {\alpha _l^ * + \beta _l^ * + \frac{{\lambda _l^ * }}{{{\gamma _l}}}} \right) {{\mathbf{G}}_{ll}}\succeq 0. \end{aligned}$$

Then, we will prove that $\mathbf{{D}}_l^ * $ is positive definite by contradiction. Assume that the contradiction is right, and so, there exists a zero eigenvalue according to eigenvector ${\mathbf{{v}}_l} \ne 0$. We consider ${{\mathbf{B}}_l} = {z_l}{\mathbf{{v}}_l}{} \mathbf{{v}}_l^H$ with ${z_l} > 0$ and $\mathbf{{D}}_l^ * {\mathbf{{v}}_l} = 0$ with ${{\mathbf{G}}_{lk}}\, = \,{\mathbf{{g}}_{lk}}{} \mathbf{{g}}_{lk}^H$. We observe that matrix $\mathbf{{D}}_l^ * \in {\mathbb {C}^{N \times N}}$ has $rank\left( {\mathbf{{D}}_l^ * } \right) \le N - 1$ and the rows of $\mathbf{{D}}_l^ * $ depend on ${\mathbf{{g}}_{lk}},\,\forall k \ne l$. Because random channels ${\mathbf{{g}}_{lk}},\,\forall k \ne l$ and ${\mathbf{{g}}_{ll}}$ are independent, then $\mathbf{{g}}_{ll}^H{\mathbf{{v}}_l} \ne 0$ with a probability of 1 [7].

In addition, we derive $( {\alpha _l^ * + \beta _l^ * + \frac{{\pi _l^ * }}{{{\gamma _l}}}}) > 0$ because the required SINR or EH constraints hold with equality under optimal conditions. Therefore,

$$\begin{aligned}&\mathcal{L}\left( {{{\mathbf{B}}_l} = {z_l}{\mathbf{{v}}_l}{} \mathbf{{v}}_l^H,{\varTheta ^ * }} \right) \, = {\mathrm{Tr}}\left( {\mathbf{{D}}_l^ * {z_l}{\mathbf{{v}}_l}{} \mathbf{{v}}_l^H} \right) \\&\quad - \left( {\alpha _l^ * + \beta _l^ * + \frac{{\pi _l^ * }}{{{\gamma _l}}}} \right) {\mathrm{Tr}}\left( {{{\mathbf{G}}_{ll}}{z_l}{\mathbf{{v}}_l}{} \mathbf{{v}}_l^H} \right) \\&\quad - {\mathrm{Tr}}\left( {\mathbf{{Z}}_l^ * {z_l}{\mathbf{{v}}_l}{} \mathbf{{v}}_l^H} \right) + \varDelta , \end{aligned}$$

where ${\varTheta ^ * }$ represents other optimal variables, and is unbounded below when ${z_l} \rightarrow + \infty $, and this contradicts the optimality. Therefore, $\mathbf{{D}}_l^ * = \mathbf{{Z}}_l^ * + ( {\alpha _l^ * + \beta _l^ * + \frac{{\pi _l^ * }}{{{\gamma _l}}}} ){{\mathbf{G}}_{ll}} \succ 0$. From a basic rank inequality property [30, 31], we obtain $rank\left( {\mathbf{{D}}_l^ * } \right) = N$ and

$$\begin{aligned} rank\left( {\mathbf{{Z}}_l^ * } \right)= & {} rank\left( {\mathbf{{D}}_l^ * - \left( {\alpha _l^ * + \beta _l^ * + \frac{{\pi _l^ * }}{{{\gamma _l}}}} \right) {{\mathbf{G}}_{ll}}} \right) \\\ge & {} rank\left( {\mathbf{{D}}_l^ * } \right) - rank\left( {{{\mathbf{G}}_{ll}}} \right) = N - 1. \end{aligned}$$

Based on (25), $\mathbf{{Z}}_l^ * \mathbf{{B}}_l^ * \, = \,0$, and we have $rank\left( {{\mathbf{B}}_l^ * } \right) = N - rank\left( {\mathbf{{Z}}_l^ * } \right) \le 1$. From SINR constraint (21f), $rank\left( {{\mathbf{B}}_l^ * } \right) = 1$ is derived. It means Lemma 1 is proven.

Solution of the linear EH model scheme

We formulate the harvested energy maximization for linear EH model as follows:

$$\begin{aligned}&\mathop {\mathrm{{maximize}}}\limits _{\left\{ {{{\mathbf{b}}_l},{\theta _l}} \right\} \,} \,\,\sum \limits _{l = 1}^L {\left( {1 - {\theta _l}} \right) \left( {\sum \limits _{k = 1}^L {{{\left| {\mathbf{{g}}_{kl}^H{{\mathbf{b}}_k}} \right| }^2}\,} + \,\sigma _l^2} \right) \,} \,\,\,\,\,\text {s.t.} \end{aligned}$$

(27a)

$$\begin{aligned}&\left( {1 - {\theta _l}} \right) \left( {\sum \limits _{k = 1}^L {{{\left| {\mathbf{{g}}_{kl}^H{{\mathbf{b}}_k}} \right| }^2}} + \sigma _l^2} \right) \ge {\mu _l},\,\forall l, \end{aligned}$$

(27b)

$$\begin{aligned}&\frac{{{\theta _l}{{\left| {\mathbf{{g}}_{ll}^H{{\mathbf{b}}_l}} \right| }^2}}}{{{\theta _l}\left( {\sum \limits _{k = 1,k \ne l}^L {{{\left| {\mathbf{{g}}_{kl}^H{{\mathbf{b}}_k}} \right| }^2}} + \sigma _l^2} \right) + \delta _l^2}}\, \ge {\gamma _l},\,\forall l, \end{aligned}$$

(27c)

$$\begin{aligned}&{\left\| {{{\mathbf{b}}_l}} \right\| ^2} \le E_l^{\max },\,\forall l, \end{aligned}$$

(27d)

$$\begin{aligned}&0< {\theta _l} < 1,\,\forall l. \end{aligned}$$

(27e)

Using SDR technique and introducing the auxiliary variables $\left\{ {{w_l}} \right\} $, we reformulate the optimization problem as follows:

$$\begin{aligned}&\mathop {\mathrm{{maximize}}}\limits _{\left\{ {{{\mathbf{B}}_l},{\theta _l},{w_l}} \right\} \,} \,\,\sum \limits _{l = 1}^L {w_l^2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text {s.t.} \end{aligned}$$

(28a)

$$\begin{aligned}&0 \ge \frac{{w_l^2}}{{\left( {1 - {\theta _l}} \right) }} - \left( {\sum \limits _{k = 1}^L {{\mathrm{Tr}}\left( {{{\mathbf{G}}_{kl}}{{\mathbf{B}}_k}} \right) \,\, + \sigma _l^2} } \right) ,\,\,\forall l, \end{aligned}$$

(28b)

$$\begin{aligned}&0 \ge \frac{{{\mu _l}}}{{\left( {1 - {\theta _l}} \right) }} - \left( {\sum \limits _{k = 1}^L {{\mathrm{Tr}}\left( {{{\mathbf{G}}_{kl}}{{\mathbf{B}}_k}} \right) + \sigma _l^2} } \right) ,\,\forall l, \end{aligned}$$

(28c)

$$\begin{aligned}&0 \ge \frac{{\delta _l^2}}{{{\theta _l}}} + \left( {\sum \limits _{k \ne l}^L {{\mathrm{Tr}}\left( {{{\mathbf{G}}_{kl}}{{\mathbf{B}}_k}} \right) + \sigma _l^2} } \right) - \frac{1}{{{\gamma _l}}}{\mathrm{Tr}}\left( {{{\mathbf{G}}_{ll}}{{\mathbf{B}}_l}} \right) ,\,\forall l, \end{aligned}$$

(28d)

$$\begin{aligned}&{\mathrm{0}} \ge {\mathrm{Tr}}\left( {{{\mathbf{B}}_l}} \right) - E_l^{\max },\,\forall l, \end{aligned}$$

(28e)

$$\begin{aligned}&{{\mathbf{B}}_l}\succeq 0,\,\forall l, \end{aligned}$$

(28f)

$$\begin{aligned}&0< {\theta _l} < 1,\,{w_l} \ge 0,\,\forall l, \end{aligned}$$

(28g)

$$\begin{aligned}&\mathrm{{rank}}\left( {{{\mathbf{B}}_l}} \right) = 1,\,\forall l. \end{aligned}$$

(28h)

We observed that the objective function is convex, so we use the first-order Taylor:

$$\begin{aligned} w_l^2 \ge 2w_l^{\left( n \right) }{w_l} - w_l^{\left( n \right) 2},\,\forall l. \end{aligned}$$

Next, we solve the $n\hbox {-th}$ convex subproblem by CVX without rank-1 constraints (28h):

$$\begin{aligned}&\mathop {{\mathrm{maximize}}}\limits _{\left\{ {{{\mathbf{B}}_l},{\theta _l},{w_l}} \right\} \,} \,\,\left( {2w_l^{\left( n \right) }{w_l} - w_l^{\left( n \right) 2}} \right) \end{aligned}$$

(29a)

$$\begin{aligned}&\text {subject to}: (28b), \ldots , (28g). \end{aligned}$$

(29b)

We also achieve the optimal solutions that satisfy the rank-1 constraints with probability one. This result is similarly proven in Lemma 1. When the value of iterative algorithm is convergent, we obtain the optimal value of the linear EH model scheme.

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Authors and Affiliations

Pham Viet Tuan
- 1
- 2
Email authorView author's OrcID profile
Insoo Koo
- 3

1.Division of Computational Mechatronics, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
2.Faculty of Electrical and Electronics EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam
3.School of Electrical and Computer EngineeringUniversity of UlsanUlsanKorea

Distributed ADMM-based approach for total harvested power maximization in non-linear SWIPT system

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