Table 1 Algorithm of the proposed iterative scheme for the optimization of the proposed system

1) set initial point

If \({P_{max}} - \frac {{\sigma _{SI}^{2}\left ({{N_{0}} + \sigma _{RSI}^{2}} \right)}}{2} \ge 0\)

Let \(\textbf {P}_{1}^{\left (1 \right)} = {\left [ {\begin {array}{*{20}{c}} {0,}&{{P_{\max }},}&0&{\left ({{2^{\frac {{{C_{2}}}}{{{1-\tau _{min}}}}}} - 1} \right)\frac {{{N_{0}+\sigma _{RSI}^{2}}}}{{{{\left | {{h_{21}}} \right |}^{2}}}}} \end {array}} \right ]^{T}}\)

\(\textbf {P}_{2}^{\left (1 \right)} \,=\, {\left [ {\begin {array}{*{20}{c}}{0,}&{\frac {{\eta {{\left | {{h_{21}}} \right |}^{2}}}}{{1 - \eta \sigma _{SI}^{2}}}{\alpha _{2}^{\left (1 \right)}} \,-\, \frac {{{N_{0} + \sigma _{RSI}^{2}}}}{{{{\left | {{h_{12}}} \right |}^{2}}}},}&{0,}{\eta {{\left | {{h_{21}}} \right |}^{2}}{\alpha _{2}^{\left (1 \right)}} \,-\, \frac {{{N_{0} + \sigma _{RSI}^{2}}}}{{{{\left | {{h_{12}}} \right |}^{2}}}}}\end {array}} \right ]^{T}}\)

\({{\boldsymbol {\tau }}^{\left (1 \right)}} = {\left [ {\begin {array}{*{20}{c}}{0,}&{{\tau _{\min }},}&{0,}&{1 - {\tau _{\min }}}\end {array}} \right ]^{T}}\)

\({\alpha _{1}^{\left (1 \right)}} = \frac {{{2^{\frac {{{C_{2}}}}{{{\tau _{3}} + {\tau _{4}}}}}}{{\left \{ {\left ({1 - \eta \sigma _{SI}^{2}} \right){N_{0}}} \right \}}^{\frac {{{\tau _{3}}}}{{{\tau _{3}} + {\tau _{4}}}}}}{{\left ({{N_{0}} + \sigma _{RSI}^{2}} \right)}^{\frac {{{\tau _{4}}}}{{{\tau _{3}} + {\tau _{4}}}}}}}}{{{{\left | {{h_{21}}} \right |}^{2}}}}\)

\(\alpha _{2}^{\left (1 \right)} = \left ({{P_{\max }} - \frac {{\sigma _{SI}^{2}\left ({{N_{0}} + \sigma _{RSI}^{2}} \right)}}{{{{\left | {{h_{21}}} \right |}^{2}}{{\left | {{h_{12}}} \right |}^{2}}}}} \right){\tau _{\min }} + \frac {{{ {{N_{0}} + \sigma _{RSI}^{2}} }}}{{\eta {{\left | {{h_{21}}} \right |}^{2}}{{\left | {{h_{12}}} \right |}^{2}}}}\)

\({\tau _{\min }} = \frac {{AW\left ({\left ({\log 2} \right){e^{y\log 2}}} \right) - BC\log 2}}{{AC\log 2}}\)

else

Let \(\textbf {P}_{1}^{\left (1 \right)} = {\left [ {\begin {array}{*{20}{c}}{0,} & {{P_{\max }},} & {0,} & {\left ({{2^{\frac {{{C_{2}}}}{{{1-\tau _{min}}}}}} - 1} \right)\frac {{{N_{0} + \sigma _{RSI}^{2}}}}{{{{\left | {{h_{21}}} \right |}^{2}}}}}\end {array}} \right ]^{T}}\)

\(\textbf {P}_{2}^{\left (1 \right)} = {\left [ {\begin {array}{*{20}{c}}{0,}&{\frac {{\eta {{\left | {{h_{21}}} \right |}^{2}}}}{{1 - \eta \sigma _{SI}^{2}}}{P_{\max }},}&{0,}&0\end {array}} \right ]^{T}}\)

\({{\boldsymbol {\tau }}^{\left (1 \right)}} = {\left [ {\begin {array}{*{20}{c}}{0,} & {{\tau _{\min }},} &{0,} & {1 - {\tau _{\min }}}\end {array}} \right ]^{T}}\)

\({\alpha _{1}^{\left (1 \right)}} = \frac {{{2^{\frac {{{C_{2}}}}{{{\tau _{3}} + {\tau _{4}}}}}}{{\left \{ {\left ({1 - \eta \sigma _{SI}^{2}} \right){N_{0}}} \right \}}^{\frac {{{\tau _{3}}}}{{{\tau _{3}} + {\tau _{4}}}}}}{{\left ({{N_{0}} + \sigma _{RSI}^{2}} \right)}^{\frac {{{\tau _{4}}}}{{{\tau _{3}} + {\tau _{4}}}}}}}}{{{{\left | {{h_{21}}} \right |}^{2}}}}\)

\(\alpha _{2}^{\left (1 \right)} = {P_{\max }} + \frac {{\left ({N_{0} + \sigma _{RSI}^{2}} \right)\left ({1 - \eta \sigma _{SI}^{2}} \right)}}{{\eta {{\left | {{h_{21}}} \right |}^{2}}{{\left | {{h_{12}}} \right |}^{2}}}} \)

\({\tau _{\min }} = \frac {{{C_{1}}}}{{{{\log }_{2}}\left ({1 + \frac {{{{\left | {{h_{12}}} \right |}^{2}}}}{{{N_{0} + \sigma _{RSI}^{2}}}}{P_{2,2}}} \right)}}\)

2) for k=1:k_max

i) Find ∇E_T(τ^(k)), p_k, and β_k

ii) Let τ^(k+1)=τ^(k)+β_kp_k

iii) If |τ^(k+1)−τ^(k)|<ε

Algorithm terminated. The power and time are set to \(\textbf {P}_{1}^{\left ({k} \right)}\), \(\textbf {P}_{2}^{\left ({k} \right)}\),

and τ.

else

Find \(\textbf {P}_{1}^{\left ({k + 1} \right)}\) and \(\textbf {P}_{2}^{\left ({k + 1} \right)}\)