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A DCA-Based Approach for Outage Constrained Robust Secure Power-Splitting SWIPT MISO System

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Optimization of Complex Systems: Theory, Models, Algorithms and Applications (WCGO 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 991))

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Abstract

This paper studies the worst-case secrecy rate maximization problem under the total transmit power, the energy harvesting and the outage probability requirements. The problem is nonconvex, thus, hard to solve. Exploiting the special structure of the problem, we first reformulate as a DC (Difference of Convex functions) program. Then, we develop an efficient approach based on DCA (DC Algorithm) and alternating method for solving the problem. The computational results confirm the efficiency of the proposed approach.

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Notes

  1. 1.

    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.0. Online: http://cvxr.com/cvx, (2012).

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Authors and Affiliations

  1. LGIPM, University of Lorraine, 57073, Metz, France

    Phuong Anh Nguyen & Hoai An Le Thi

Authors
  1. Phuong Anh Nguyen
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  2. Hoai An Le Thi
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Corresponding author

Correspondence to Phuong Anh Nguyen .

Editor information

Editors and Affiliations

  1. Computer science and Applications Department, LGIPM, University of Lorraine, Metz Cedex 03, France

    Hoai An Le Thi

  2. Computer Science and Applications Department, LGIPM, University of Lorraine, Metz Cedex 03, France

    Hoai Minh Le

  3. Laboratory of Mathematics, National Institute for Applied Sciences (INSA)-Rouen Normadie, Saint-Étienne-du-Rouvray Cedex, France

    Tao Pham Dinh

Appendix A

Appendix A

First, we transform the EH constraint. According to [4], we rewrite \(\vert \mathbf h _{jk}^H\mathbf w _j\vert ^2 = \mathbf w _j^H (\overline{\mathbf{H }}_{jk} + \mathbf d _{jk})\mathbf w _j,\) where \(\overline{\mathbf{H }}_{jk} = \overline{\mathbf{h }}_{jk}\overline{\mathbf{h }}_{jk}^H, \ \mathbf d _{jk} = \overline{\mathbf{h }}_{jk}\varDelta \mathbf h _{jk}^H + \varDelta \mathbf h _{jk} \overline{\mathbf{h }}_{jk}^H + \varDelta \mathbf h _{jk}\varDelta \mathbf h _{jk}^H \). By applying the triangle inequality and the Cauchy-Schwarz inequality, we have

$$\begin{aligned} \Vert \mathbf d _{jk} \Vert \le \Vert \overline{\mathbf{h }}_{jk}\Vert \Vert \varDelta \mathbf h _{jk}^H\Vert +\Vert \varDelta \mathbf h _{jk}\Vert \Vert \overline{\mathbf{h }}_{jk}^H\Vert +\Vert \varDelta \mathbf h _{jk}\Vert \Vert \varDelta \mathbf h _{jk}^H\Vert \\ \Vert \mathbf d _{jk} \Vert \le \Vert \mathbf Q _{jk} \Vert ^{-1}+ 2 \Vert \overline{\mathbf{h }}_{jk} \Vert \sqrt{\Vert \mathbf Q _{jk} \Vert ^{-1}} = \epsilon _{jk} \Rightarrow - \epsilon _{jk} {{\,\mathrm{{\mathbf I}}\,}}_N \le \mathbf D _{jk} \le \epsilon _{jk} {{\,\mathrm{{\mathbf I}}\,}}_N, \\ {{\,\mathrm{thus}\,}}, \ {{\,\mathrm{Tr}\,}}\left[ (\overline{\mathbf{H }}_{jk} - \epsilon _{jk} {{\,\mathrm{{\mathbf I}}\,}}_N) \mathbf W _j\right] \le \vert \mathbf h _{jk}^H\mathbf w _j\vert ^2 = {{\,\mathrm{Tr}\,}}\left[ (\overline{\mathbf{H }}_{jk} + \mathbf d _{jk}) \mathbf W _j\right] . \end{aligned}$$

Therefore, the EH constraint is recast as \( \sum _{j=1}^K {{\,\mathrm{Tr}\,}}(\overline{\mathbf{H }}_{jk} \mathbf W _j - \epsilon _{jk} \mathbf W _j) \ge \frac{E_k}{(1-\rho _k)}. \)

$$\begin{aligned} {{\,\mathrm{Next,}\,}} \ R_{I,k} \le \log \Big (1+\frac{\rho _k \mathbf h _{kk}^H\mathbf W _k\mathbf h _{kk}}{\rho _k\sum _{j\ne k}{} \mathbf h _{jk}^H\mathbf W _j\mathbf h _{jk}+\rho _k\sigma _{1k}^2+\sigma _{2k}^2}\Big ) \end{aligned}$$
(22)

is reformulated similarly to [1] as

$$\begin{aligned} \left\{ \begin{matrix} \beta _k \ge \dfrac{1}{\rho _k}, \ \sum _{j=1}^K f_{jk} +\sigma _{1k}^2+ \beta _k\sigma _{2k}^2 \ge a_k, \ \sum _{j\ne k}f_{jk} +\sigma _{1k}^2+ \beta _k\sigma _{2k}^2 \le b_k, \\ 0 < u \le a_k, \ v \ge b_k > 0, \ \lambda _{jk} \ge 0, \ x \ge R_{I,k}, \ 2^x v - u \le 0,\\ \mathbf h _{kk}^H\mathbf W _k\mathbf h _{kk} \ge f_{kk} \Leftrightarrow \begin{bmatrix}\lambda _{kk} \mathbf Q +\mathbf W _k &{} \mathbf W _k\overline{\mathbf{h }}_{kk} \\ \overline{\mathbf{h }}_{kk}^H\mathbf W _k &{}\overline{\mathbf{h }}_{kk}^H\mathbf W _k\overline{\mathbf{h }}_{kk} - f_{kk} - \lambda _{kk} \end{bmatrix} \succeq \mathbf 0 , \\ \mathbf h _{jk}^H\mathbf W _k\mathbf h _{jk} \le f_{jk} \Leftrightarrow \begin{bmatrix}\lambda _{jk} \mathbf Q -\mathbf W _j &{} -\mathbf W _j\overline{\mathbf{h }}_{jk} \\ -\overline{\mathbf{h }}_{jk}^H\mathbf W _j &{}-\overline{\mathbf{h }}_{jk}^H\mathbf W _j\overline{\mathbf{h }}_{jk} + f_{jk} - \lambda _{jk} \end{bmatrix} \succeq \mathbf 0 , \ j\ne k. \end{matrix}\right. \end{aligned}$$

The relaxed constraints hold with equalities at the optimal solution [1].

By using slack variables \(\xi _k = \frac{2^{R_{I,k} - R_k} - 1}{{{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k)} \), the outage constraint is transformed into

$$\begin{aligned} \left\{ \begin{matrix} -\ln p_k - \xi _k \sigma _{3k}^2 - \sum _{j\ne k} \ln ( 1 + \xi _k{{\,\mathrm{Tr}\,}}(\mathbf G _{jk}{} \mathbf W _j) \le 0 , \\ \ {{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k) \le s_k, \ 0< s_k, \ 0 <\xi _k, \\ \xi _k \le \frac{2^{R_{I,k} - R_k} - 1}{s_k} \Leftrightarrow R_k - R_{I,k}+\log _{2}(1+\xi _k s_k) \le 0. \end{matrix}\right. \end{aligned}$$

At the optimum, \(\xi _k = \frac{2^{R_{I,k} - R_k} - 1}{{{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k)} \) at optimum, if not, we can increase \(R_k\). In addition, \({{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k) = s_k\) at the optimum, otherwise, we can decrease s leading to increase \(R_k\) due to \(\xi _k = \frac{2^{R_{I,k} - R_k} - 1}{{{\,\mathrm{Tr}\,}}(\mathbf G _{kk}{} \mathbf W _k)} \).

Such that, if the relaxed constraints do not hold with equalities at the optimum, the objective function can be further increased.

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Nguyen, P.A., Le Thi, H.A. (2020). A DCA-Based Approach for Outage Constrained Robust Secure Power-Splitting SWIPT MISO System. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_30

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