### Sum data rate maximization

By simple transformation, the problem (20) can be rewritten as

$$ \underset{\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{s_{j}}} \right\}_{j = 1}^{2},\alpha}{\max} {R_{\text{sum}}} $$

(22a)

$$ \begin{array}{l} ~~~~~~~{s.t.}~~~~~~~\min \left\{{\frac{{{\phi_{2}}\left(\theta\right){{\left| {\mathbf{h}_{R1}^{H}{\mathbf{s}_{1}}} \right|}^{2}}}}{{\sigma_{1}^{2}}},\frac{{{\phi_{1}}\left(\theta\right){{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{1}}} \right|}^{2}}}}{{{\varphi_{1}}\left({{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right)}}} \right\} \ge \bar \gamma, \end{array} $$

(22b)

$$ \begin{array}{l} \min \left\{{\frac{{{\phi_{1}}\left(\theta\right){{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{2}}} \right|}^{2}}}}{{{\varphi_{2}}\left({{\mathbf{w}_{1}},{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right)}},\frac{{{\phi_{2}}\left(\theta \right){{\left| {\mathbf{h}_{R1}^{H}{\mathbf{s}_{2}}} \right|}^{2}}}}{{{\varphi_{3}}\left({{\mathbf{s}_{1}}} \right)}},\frac{{{\phi_{3}}\left(\theta\right){{\left| {\mathbf{h}_{R2}^{H}{\mathbf{s}_{2}}} \right|}^{2}}}}{{{\varphi_{4}}\left({{\mathbf{s}_{1}}} \right)}}} \right\} \ge \bar \gamma, \end{array} $$

(22c)

$$ \frac{{{{\left\| {{\mathbf{s}_{1}}} \right\|}^{2}} + {{\left\| {{\mathbf{s}_{2}}} \right\|}^{2}}}}{\alpha} \le \frac{{{\eta_{1}}{\phi_{1}}\left(\theta \right)\sum\limits_{i = 1}^{2} {{{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right|}^{2}}} }}{{1 - \alpha }}{\mathrm{ + }}\frac{{{\eta_{2}}\sum\limits_{j = 1}^{2} {{{\left| {\mathbf{h}_{rr}^{H}{\mathbf{s}_{j}}} \right|}^{2}}} }}{\alpha }, $$

(22d)

$$ \begin{array}{l} (20c), (20e) \end{array} $$

(22e)

where \({\varphi _{2}}\left ({{\mathbf {w}_{1}},{\mathbf {s}_{1}},{\mathbf {s}_{2}}} \right) = {\phi _{1}}\left (\theta \right){\left | {\mathbf {h}_{SR}^{H}{\mathbf {w}_{1}}} \right |^{2}} + \rho {P_{r}}\left ({{{\left | {\mathbf {h}_{rr}^{H}{\mathbf {s}_{1}}} \right |}^{2}} + {{\left | {\mathbf {h}_{rr}^{H}{\mathbf {s}_{2}}} \right |}^{2}}} \right) + \sigma _{r}^{2}, {\varphi _{1}}\left ({{\mathbf {s}_{1}},{\mathbf {s}_{2}}} \right) = \rho {P_{r}}\left ({{{\left | {\mathbf {h}_{rr}^{H}{\mathbf {s}_{1}}} \right |}^{2}} + {{\left | {\mathbf {h}_{rr}^{H}{\mathbf {s}_{2}}} \right |}^{2}}} \right) + \sigma _{r}^{2}, {\varphi _{3}}\left ({{\mathbf {s}_{1}}} \right){\mathrm { = }}{\phi _{2}}\left (\theta \right){\left | {\mathbf {h}_{R1}^{H}{\mathbf {s}_{1}}} \right |^{2}} + \sigma _{1}^{2}, {\varphi _{4}}\left ({{\mathbf {s}_{1}}} \right) = {\phi _{3}}\left (\theta \right){\left | {\mathbf {h}_{R2}^{H}{\mathbf {s}_{1}}} \right |^{2}} + \sigma _{2}^{2}\) and \(\bar \gamma {\mathrm { = }}{{\mathrm {e}}^{\bar R}}{\mathrm { - }}1\).

Apparently, the constraint (22e) is convex. However, the formulated problem is highly non-convex due to the non-convex objective (22a) and constraints (22b)–(22d). Next, let us cope with non-convex terms by using inner approximation method.

According to the inequality (45), five non-convex terms of the objective and two non-convex terms of the right side of (22b) can be bounded respectively around \(\left ({\left \{{\mathbf {w}_{i}^{k}}\right \}_{i = 1}^{2},\left \{{s_{j}^{k}}\right \}_{j = 1}^{2},{\alpha ^{k}}} \right)\) by

$$ {\begin{aligned} \frac{{{\phi_{2}}\left(\theta\right){{\left| {\mathbf{h}_{R1}^{H}{\mathbf{s}_{1}}} \right|}^{2}}}}{{\sigma_{1}^{2}}}& \ge \frac{{2{\phi_{2}}\left(\theta\right)R\left\{ {{{\left({\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right)}^{*}}\mathbf{h}_{R1}^{H}{\mathbf{s}_{1}}} \right\}}}{{\sigma_{1}^{2}}} \\ &- \frac{{{\phi_{2}}\left(\theta\right){{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right|}^{2}}}}{{\sigma_{1}^{2}}}{\mathrm{ = }}{\lambda_{1}}\left({{\mathbf{s}_{1}}} \right), \end{aligned}} $$

(23)

$$ {\begin{aligned} \frac{{{\phi_{1}}\left(\theta\right){{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{1}}} \right|}^{2}}}}{{{\varphi_{1}}\left({{\mathbf{w}_{1}},{\mathbf{s}_{1}},{\mathbf{s}_{2}}}\right)}} &\ge \frac{{2{\phi_{1}}\left(\theta \right)R\left\{{{{\left({\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right)}^{*}}\mathbf{h}_{SR}^{H}{\mathbf{w}_{2}}} \right\}}}{{{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}} \\ &- \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{2}^{k}} \right|}^{2}}}}{{{{\left({{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)} \right)}^{2}}}}{\varphi_{1}}\left({{\mathbf{w}_{1}},{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right)\\ &={\lambda_{2}}\left({\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{{{s_{j}}} \right\}_{j = 1}^{2}} \right), \end{aligned}} $$

(24)

$$ {\begin{aligned} \frac{{{\phi_{1}}\left(\theta\right){{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{1}}} \right|}^{2}}}}{{\sigma_{1}^{2}}} &\ge \frac{{2{\phi_{1}}\left(\theta\right)R\left\{{{{\left({\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right)}^{*}}\mathbf{h}_{SR}^{H}{\mathbf{w}_{1}}} \right\}}}{{\sigma_{1}^{2}}}\\ & - \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}}\right|}^{2}}}}{{\sigma_{1}^{2}}}{\mathrm{ = }}{\lambda_{3}}\left({{\mathbf{w}_{1}}}\right) \end{aligned}} $$

(25)

$$ {\begin{aligned} \frac{{{\phi_{2}}\left(\theta\right){{\left| {\mathbf{h}_{R1}^{H}{\mathbf{s}_{2}}} \right|}^{2}}}}{{{\varphi_{2}}\left({{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right)}} &\ge \frac{{2{\phi_{2}}\left(\theta \right)R\left\{{{{\left({\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right)}^{*}}\mathbf{h}_{R1}^{H}{\mathbf{s}_{2}}} \right\}}}{{{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}} \\ &- \frac{{{{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}}}{{{{\left({{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}\right)}^{2}}}}{\varphi_{2}}\left({{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right) ={\chi_{4}}\left({\left\{ {\mathbf{s}_{j}^{k}} \right\}_{j = 1}^{2}} \right) \end{aligned}} $$

(26)

$$ \begin{array}{l} \frac{{{\phi_{3}}\left(\theta\right){{\left| {\mathbf{h}_{R2}^{H}{\mathbf{s}_{2}}} \right|}^{2}}}}{{{\varphi_{3}}\left({{\mathbf{s}_{1}}} \right)}} \ge \frac{{2{\phi_{3}}\left(\theta\right)R\left\{ {{{\left({\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right)}^{*}}\mathbf{h}_{R2}^{H}{\mathbf{s}_{2}}} \right\}}}{{{\varphi_{3}}\left({\mathbf{s}_{1}^{k}}\right)}} - \frac{{{\phi_{3}}\left(\theta\right){{\left| {\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}}}{{{{\left({{\varphi_{3}}\left({\mathbf{s}_{1}^{k}} \right)} \right)}^{2}}}}{\varphi_{3}}\left({{\mathbf{s}_{1}}} \right)\\ ~~~~~~~~~~~~{\mathrm{ = }}{\chi_{5}}\left({\left\{{{\mathbf{s}_{j}}} \right\}_{j = 1}^{2}} \right) \end{array} $$

(27)

$$ {\begin{array}{l} \frac{{{\eta_{1}}{\phi_{1}}\left(\theta \right)\sum\limits_{i = 1}^{2} {{{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right|}^{2}}} }}{{1 - \alpha }} \ge {\eta_{1}}{\phi_{1}}\left(\theta \right)\sum\limits_{i = 1}^{2} {\left({\frac{{2R\left\{ {{{\left({\mathbf{h}_{SR}^{H}\mathbf{w}_{i}^{k}} \right)}^{*}}\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right\}}}{{1 - {\alpha^{k}}}} - \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{i}^{k}} \right|}^{2}}\left({1 - \alpha} \right)}}{{{{\left({1 - {\alpha^{k}}} \right)}^{2}}}}} \right)} \\ ~~~~~~~~~~~~~~~~~~~~~~{\mathrm{ = }}{\chi_{6}}\left({\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\alpha} \right) \end{array}} $$

(28)

Thus, the constraints (22b)–(22d) can be innerly approximated by these convex ones, which are represented by

$$ \frac{{{{\left\| {{\mathbf{s}_{1}}} \right\|}^{2}} + {{\left\| {{\mathbf{s}_{2}}} \right\|}^{2}}}}{\alpha} \le {\chi_{6}}\left({\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\alpha} \right){\mathrm{ + }}{\chi_{7}}\left({\left\{ {{\mathbf{s}_{i}}} \right\}_{i = 1}^{2},\alpha} \right) $$

(29)

$$ \min \left\{{{\lambda_{1}}\left({{\mathbf{s}_{1}}}\right),{\lambda_{3}} \left({{\mathbf{w}_{1}}} \right)} \right\} \ge \bar \gamma $$

(30)

$$ {\begin{aligned} \min \left\{ {{\lambda_{2}}\left({\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{s_{j}}} \right\}_{j = 1}^{2}} \right),{\chi_{4}}\left({\left\{{\mathbf{s}_{j}^{k}} \right\}_{j = 1}^{2}} \right),{\chi_{5}}\left({\left\{ {{\mathbf{s}_{j}}} \right\}_{j = 1}^{2}} \right)} \right\} \ge \bar \gamma \end{aligned}} $$

(31)

Then, using the inequality (47), \(\ln \left ({1 + \gamma _{1}^{\mathrm {R}}}\right), \ln \left ({1 + \gamma _{1}^{{\mathrm {U1}}}}\right),\ln \left ({1 + \gamma _{2}^{\mathrm {R}}} \right),\ln \left ({1 + \gamma _{2}^{{\mathrm {U1}}}}\right), \ln \left ({1 + \gamma _{2}^{{\mathrm {U2}}}}\right)\) of the objective can be innerly approximated by

$$ {\begin{aligned} \ln \left({1 + \gamma_{1}^{\mathrm{R}}} \right) &\ge \ln \left({1{\mathrm{ + }}\frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right|}^{2}}}}{{\sigma_{1}^{2}}}} \right) - \frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right|}^{2}}}}{{\sigma_{1}^{2}}}\\ &+ \frac{{2{\phi_{1}}\left(\theta \right)R\left\{ {{{\left({\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right)}^{*}}\mathbf{h}_{SR}^{H}{\mathbf{w}_{1}}} \right\}}}{{\sigma_{1}^{2}}}\\ & - \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right|}^{2}}\left({{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right|}^{2}} + \sigma_{1}^{2}} \right)}}{{\sigma_{1}^{2}\left({\sigma_{1}^{2} + {{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{1}^{k}} \right|}^{2}}} \right)}}{\mathrm{ = }}{\Omega_{1}}\left({{\mathbf{w}_{1}}} \right) \end{aligned}} $$

(32)

$$ {\begin{aligned} \ln \left({1 + \gamma_{1}^{{\mathrm{U1}}}} \right) &\ge \ln \left({1{\mathrm{ + }}\frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right|}^{2}}}}{{\sigma_{1}^{2}}}} \right) - \frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right|}^{2}}}}{{\sigma_{1}^{2}}}\\ &+ \frac{{2{\phi_{1}}\left(\theta \right)R\left\{ {{{\left({\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right)}^{*}}\mathbf{h}_{R1}^{H}{\mathbf{s}_{1}}} \right\}}}{{\sigma_{1}^{2}}}\\ &- \frac{{{{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right|}^{2}}\left({{{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right|}^{2}} + \sigma_{1}^{2}} \right)}}{{\sigma_{1}^{2}\left({\sigma_{1}^{2} + {{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{1}^{k}} \right|}^{2}}} \right)}}{\mathrm{ = }}{\Omega_{2}}\left({{\mathbf{s}_{1}}} \right) \end{aligned}} $$

(33)

$$ {\begin{aligned} \ln \left({1 + \gamma_{2}^{\mathrm{R}}} \right) &\ge \ln \left({1 + \frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{2}^{k}} \right|}^{2}}}}{{{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}}} \right) - \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{2}^{k}} \right|}^{2}}}}{{{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}}\\ & + \frac{{2{\phi_{1}}\left(\theta \right)R\left\{ {{{\left({\mathbf{h}_{SR}^{H}\mathbf{w}_{2}^{k}} \right)}^{*}}\mathbf{h}_{SR}^{H}{\mathbf{w}_{2}}} \right\}}}{{{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}}\\ &- \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{2}^{k}} \right|}^{2}}{\varphi_{1}}\left({{\mathbf{w}_{1}},{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right)}}{{{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)\left({{\varphi_{1}}\left({\mathbf{w}_{1}^{k},\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right) + {{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{2}^{k}} \right|}^{2}}} \right)}}\\ &={\Omega_{3}}\left({\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{\mathbf{s}_{j}}} \right\}_{j = 1}^{2}} \right) \end{aligned}} $$

(34)

$$ {\begin{aligned} \ln \left({1 + \gamma_{2}^{{\mathrm{U1}}}} \right) &\ge \ln \left({1 + \frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}}}{{{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}}} \right) - \frac{{{{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}}}{{{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}} \\ &+ \frac{{2{\phi_{2}}\left(\theta \right)R\left\{ {{{\left({\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right)}^{*}}\mathbf{h}_{R1}^{H}{\mathbf{s}_{2}}} \right\}}}{{{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)}}\\ &- \frac{{{{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}{\varphi_{2}}\left({{\mathbf{s}_{1}},{\mathbf{s}_{2}}} \right)}}{{{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right)\left({{\varphi_{2}}\left({\mathbf{s}_{1}^{k},\mathbf{s}_{2}^{k}} \right) + {{\left| {\mathbf{h}_{R1}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}} \right)}}\\ &={\Omega_{4}}\left({\left\{ {{\mathbf{s}_{j}}} \right\}_{j = 1}^{2}} \right) \end{aligned}} $$

(35)

$$ {\begin{aligned} \ln \left({1 + \gamma_{2}^{{\mathrm{U2}}}} \right) &\ge \ln \left({1 + \frac{{{\phi_{1}}\left(\theta \right){{\left| {\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}}}{{{\varphi_{4}}\left({\mathbf{s}_{1}^{k}} \right)}}} \right) - \frac{{{{\left| {\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}}}{{{\varphi_{4}}\left({\mathbf{s}_{1}^{k}} \right)}} \\ &+ \frac{{2{\phi_{2}}\left(\theta \right)R\left\{{{{\left({\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right)}^{*}}\mathbf{h}_{R2}^{H}{\mathbf{s}_{2}}} \right\}}}{{{\varphi_{4}}\left({\mathbf{s}_{1}^{k}} \right)}}\\ &- \frac{{{{\left| {\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}{\varphi_{4}}\left({{\mathbf{s}_{1}}} \right)}}{{{\varphi_{4}}\left({\mathbf{s}_{1}^{k}} \right)\left({{\varphi_{4}}\left({\mathbf{s}_{1}^{k}}\right) + {{\left| {\mathbf{h}_{R2}^{H}\mathbf{s}_{2}^{k}} \right|}^{2}}} \right)}}={\Omega_{5}}\left({\left\{ {{\mathbf{s}_{j}}} \right\}_{j = 1}^{2}} \right) \end{aligned}} $$

(36)

For *k*th iteration, the feasible points of the original problem can be generated by solving the convex problem

$$ \mathop {\max }\limits_{\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{s_{j}}} \right\}_{j = 1}^{2},\alpha} {f_{k}}\left({\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{s_{j}}} \right\}_{j = 1}^{2},\alpha} \right) $$

(37a)

$$ ~~~~~~~{s.t.}~~~~~~~ (22e),(29)-(31) $$

(37b)

where \({f_{k}}\left ({\left \{ {{\mathbf {w}_{i}}} \right \}_{i = 1}^{2}\!,\!\left \{ {{\mathbf {s}_{j}}} \right \}_{j = 1}^{2}\!,\!\alpha } \right){\mathrm { = }}{\Omega _{1}}\left ({{\mathbf {w}_{1}}} \right){\mathrm {\! +\! }}{\Omega _{2}}\left ({{\mathbf {s}_{1}}} \right){\mathrm {\! +\! }}{\Omega _{3}} \left ({\left \{ {{\mathbf {w}_{i}}} \right \}_{i = 1}^{2}\!,\!\left \{ {{\mathbf {s}_{j}}} \right \}_{j = 1}^{2}} \right){\mathrm {\! +\! }}{\Omega _{4}}\left ({\left \{ {{\mathbf {s}_{j}}} \right \}_{j = 1}^{2}} \right) {\mathrm { + }}{\Omega _{5}}\left ({\left \{ {{\mathbf {s}_{j}}} \right \}_{j = 1}^{2}} \right)\) is the approximation of the objective of original problem.

**Convergence analysis:** The convergence performance of the proposed scheme can be presented in the following proposition.

###
**Proposition 1**

Algorithm *1* can generate the feasible points by iteration to make the objective value of (22) become bigger and finally converge to the Karush-Kuhn-Tucker point of (22) after finitely many iterations.

###
*Proof*

Refer to Appendix 5 for the detailed proof. □

**Complexity analysis:** The computational cost of problem (22) at each iteration is *O*(*m*^{2}*n*^{2.5}+*n*^{3.5}), where the optimization problem (22) involves *m*=2*M*+2*N*+1 the scalar real variables and *n*=4 quadratic and linear constraints [47].

Given the initial point \(\left ({\left \{ {\mathbf {w}_{i}^{0}} \right \}_{i = 1}^{2},\left \{ {s_{j}^{0}} \right \}_{j = 1}^{2},{\alpha ^{0}}} \right)\), we can achieve the feasible point of the problem (22) by iterating the following problem

$$ {\begin{aligned} &\underset{\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{{{s_{j}}}\right\}_{j = 1}^{2}}{\min} \\ &\left\{ {\frac{{\min \left\{{{\lambda_{1}}\left({{\mathbf{s}_{1}}} \right),{\lambda_{3}}\left({{\mathbf{w}_{1}}} \right)} \right\}}}{{\bar \gamma}},\frac{{\min \left\{ {{\lambda_{2}}\left({\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{{{s_{j}}} \right\}_{j = 1}^{2}}\right),{\chi_{4}}\left({\left\{ {\mathbf{s}_{j}^{k}} \right\}_{j = 1}^{2}} \right),{\chi_{5}}\left({\left\{{{\mathbf{s}_{j}}} \right\}_{j = 1}^{2}}\right)} \right\}}}{{\bar \gamma }}} \right\} \\ &~~~~~~s.t. ~~~~~~(22e),(29) \end{aligned}} $$

(38)

until the objective is greater than or equals 1.

### Harvested energy maximization

For convenient treatment, the problem can be rewritten as

$$ \underset{\left\{ {{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{s_{j}}} \right\}_{j = 1}^{2},\alpha}{\max} {\eta_{1}}{f_{1}}\left(\theta \right)\sum\limits_{i = 1}^{2} {{{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right|}^{2}}} \alpha {\mathrm{ + }}{\eta_{2}}{f_{1}}\left(\theta \right)\sum\limits_{j = 1}^{2} {{{\left| {\mathbf{h}_{rr}^{H}{\mathbf{s}_{j}}} \right|}^{2}}} \left({1{\mathrm{ - }}\alpha} \right) $$

(39a)

$$ \begin{array}{l} ~~~~~~~{s.t.}~~~~~~~ (22b)-(22e), \end{array} $$

(39b)

By introducing auxiliary variable *λ*_{1},*λ*_{2}, we have

$$ \alpha {\lambda_{1}} \le 1{\mathrm{ ~~~~and ~~~~ }}\left({1 - \alpha} \right){\lambda_{2}} \le 1 $$

(40)

Hence, the problem (39) can be expressed equivalently as

$$ \underset{\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{{{s_{j}}}\right\}_{j = 1}^{2},\alpha,{\lambda_{1}},{\lambda_{2}}}{\max} \frac{{{\eta_{1}}{f_{1}}\left(\theta\right)\sum\limits_{i = 1}^{2} {{{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right|}^{2}}\alpha}}}{{{\lambda_{1}}}}{\mathrm{ + }}\frac{{{\eta_{1}}{f_{1}}\left(\theta \right)\sum\limits_{j = 1}^{2} {{{\left| {\mathbf{h}_{rr}^{H}{\mathbf{s}_{j}}} \right|}^{2}}} \left({1{\mathrm{ - }}\alpha} \right)}}{{{\lambda_{2}}}} $$

(41a)

$$ \begin{array}{l} ~~~~~~~{s.t.}~~~~~~~ (22b)-(22d), \end{array} $$

(41b)

$$ \begin{array}{l} ~~~~~~~~~~~~~~~~~~~(22e),(40). \end{array} $$

(41c)

It is easily found that the constraints (41c) are convex. As for non-convex constraints (41b), we can approximate these into convex ones by referring to sum throughput maximization part, which can be expressed as (29)–(31).

Next, let us cope with the non-convex objective using (41a). Two terms of the objective can be bounded respectively by

$$ {\begin{aligned} \frac{{{\eta_{1}}{f_{1}}\left(\theta\right)\sum\limits_{i = 1}^{2} {{{\left| {\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right|}^{2}}}}}{\lambda_{1}}&\ge {\eta_{1}}{f_{1}}\left(\theta\right)\sum\limits_{i = 1}^{2} {\left({\frac{{2R\left\{{{{\left({\mathbf{h}_{SR}^{H}\mathbf{w}_{i}^{k}}\right)}^{*}}\mathbf{h}_{SR}^{H}{\mathbf{w}_{i}}} \right\}}}{{\lambda_{1}^{k}}} - \frac{{{{\left| {\mathbf{h}_{SR}^{H}\mathbf{w}_{i}^{k}} \right|}^{2}}{\lambda_{1}}}}{{{\left({\lambda_{1}^{k}}\right)}^{2}}}}\right)}\\ &= {{\mathrm{g}}_{1}}\left({\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},{\lambda_{1}}} \right) \end{aligned}} $$

(42)

$$ \begin{array}{l} \frac{{\eta_{2}}\sum\limits_{j = 1}^{2} {{{\left| {\mathbf{h}_{rr}^{H}{\mathbf{s}_{j}}} \right|}^{2}}}}{\lambda_{2}} \ge {\eta_{2}}\sum\limits_{j = 1}^{2} {\left({\frac{{2R\left\{{{{\left({\mathbf{h}_{rr}^{H}\mathbf{s}_{j}^{k}} \right)}^{*}}\mathbf{h}_{rr}^{H}{\mathbf{s}_{j}}} \right\}}}{{\lambda_{2}^{k}}} - \frac{{{{\left| {\mathbf{h}_{rr}^{H}\mathbf{s}_{j}^{k}}\right|}^{2}}{\lambda_{2}}}}{{{{\left({\lambda_{2}^{k}} \right)}^{2}}}}} \right)} \\ {\mathrm{ ~~~~~~~~~~~~~~~~ = }}{{\mathrm{g}}_{2}}\left({\left\{ {{\mathbf{s}_{j}}} \right\}_{j = 1}^{2},{\lambda_{2}}} \right) \end{array} $$

(43)

In result, at *k*th iteration, the feasible point of (41) can be generated by solving the following problem:

$$ \underset{\left\{{{\mathbf{w}_{i}}}\right\}_{i = 1}^{2},\left\{{{s_{j}}} \right\}_{j = 1}^{2},\alpha,{\lambda_{1}},{\lambda_{2}}}{\max} g\left({\left\{{{\mathbf{w}_{i}}} \right\}_{i = 1}^{2},\left\{ {{s_{j}}} \right\}_{j = 1}^{2},\alpha,{\lambda_{1}},{\lambda_{2}}} \right) $$

(44a)

$$ \begin{array}{l} ~~~~~~~{s.t.}~~~~~~~ (29)-(31), (41c), (42)-(43). \end{array} $$

(44b)

where \(g\left ({\left \{{{\mathbf {w}_{i}}} \right \}_{i = 1}^{2},\left \{ {{s_{j}}} \right \}_{j = 1}^{2},\alpha,{\lambda _{1}},{\lambda _{2}}} \right){\mathrm { = }}{{\mathrm {g}}_{1}}\left ({\left \{ {{\mathbf {w}_{i}}} \right \}_{i = 1}^{2},{\lambda _{1}}} \right)+{{\mathrm {g}}_{2}}\left ({\left \{ {{\mathbf {s}_{j}}} \right \}_{j = 1}^{2},{\lambda _{2}}} \right)\).

**Convergence analysis:** Algorithm 2 produces non-decreasing sequence and finally converges to the KKT point of (39) after finitely many iterations, whose proof can refer to Appendix 5.

**Complexity analysis:** The computational cost of problem (22) at each iteration is *O*(*m*^{2}*n*^{2.5}+*n*^{3.5}), where the optimization problem (22) involves *m*=2*M*+2*N*+1 the scalar real variables and *n*=6 quadratic and linear constraints.

Given the *λ*_{1} and *λ*_{2}, the initial feasible point of (41) can be generated by iterating optimization problem (38).