$$
\begin{array}{l} + 2 \left\langle q _ {k + 1} ^ {i} - q _ {k} ^ {i}, R _ {i} \pi_ {k} ^ {- i} - R _ {i} \pi_ {k + 1} ^ {- i} \right\rangle + 2 \left\langle R _ {i} \pi_ {k} ^ {- i} - R _ {i} \pi_ {k + 1} ^ {- i}, q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle \\ = \alpha_ {k} ^ {2} \| F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) \| _ {2} ^ {2} + \| q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \| _ {2} ^ {2} + \beta_ {k} ^ {2} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} ^ {2} \\ + 2 \alpha_ {k} \left\langle \bar {F} _ {\pi_ {k}} ^ {i} \left(q _ {k} ^ {i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle + 2 \alpha_ {k} \left\langle F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) - \bar {F} _ {\pi_ {k}} ^ {i} \left(q _ {k} ^ {i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle \\ - 2 \alpha_ {k} \beta_ {k} \langle F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right), R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \rangle - 2 \beta_ {k} \langle R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \rangle \tag {Algorithm 1 Lines 3 and 5} \\ \leq \alpha_ {k} ^ {2} \| F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) \| _ {2} ^ {2} + \| q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \| _ {2} ^ {2} + \beta_ {k} ^ {2} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} ^ {2} \\ + 2 \alpha_ {k} \left\langle \bar {F} _ {\pi_ {k}} ^ {i} \left(q _ {k} ^ {i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle + 2 \alpha_ {k} \left\langle F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) - \bar {F} _ {\pi_ {k}} ^ {i} \left(q _ {k} ^ {i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle \\ + 2 \alpha_ {k} \beta_ {k} \| F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) \| _ {2} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} \\ + 2 \beta_ {k} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} \| q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \| _ {2} \quad (\text {C a u c h y - S c h w a r z i n e q u a l i t y}) \\ \leq \alpha_ {k} ^ {2} \| F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) \| _ {2} ^ {2} + \| q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \| _ {2} ^ {2} + \beta_ {k} ^ {2} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} ^ {2} \\ + 2 \alpha_ {k} \left\langle \bar {F} _ {\pi_ {k}} ^ {i} \left(q _ {k} ^ {i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle + 2 \alpha_ {k} \left\langle F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) - \bar {F} _ {\pi_ {k}} ^ {i} \left(q _ {k} ^ {i}\right), q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \right\rangle \\ + \frac {\alpha_ {k} \beta_ {k}}{c _ {1}} \| F ^ {i} \left(q _ {k} ^ {i}, A _ {k} ^ {i}, A _ {k} ^ {- i}\right) \| _ {2} ^ {2} + \alpha_ {k} \beta_ {k} c _ {1} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} ^ {2} \\ + \frac {\beta_ {k}}{c _ {2}} \| R _ {i} \left(\sigma_ {\tau} \left(q _ {k} ^ {- i}\right) - \pi_ {k} ^ {- i}\right) \| _ {2} ^ {2} + c _ {2} \beta_ {k} \| q _ {k} ^ {i} - R _ {i} \pi_ {k} ^ {- i} \| _ {2} ^ {2} \\ \end{array}
$$