Wireless Resource Management for Green Communications

Abstract

The time-frequency spatial distribution of multi-network radiant energy ${\mathbf{e}}(t,f,{\mathbf{s}})$ depends on the topology of the network, the location of the base station, the transmit power of the antenna, and the physical environment of the wireless communication.

Appendices

Appendix 1

Let $x_{1}$ and $x_{2}$ represent the optimal solution under $\mu_{1}$ and $\mu_{2}$, i.e., $x_{1} = x*\left( {\mu_{1} ;\mu_{1} } \right)$, $x_{2} = x*\left( {\mu_{2} ;\mu_{1} } \right)$. Thus the following inequality is obtained

$$ \begin{array}{*{20}l} {\left\| {\nabla_{u} \phi } \right.(u_{1} ;\mu_{1} ) - \left. {\nabla_{u} \phi (u_{2} ;\mu_{1} )} \right\|} \hfill \\ { = \left\| {\left. {\sum\limits_{i = 1}^{N} {\{ \nabla_{u} \phi_{i} (u_{ 1} ;\mu_{1} ){ - }\nabla_{u} \phi_{i} (u_{2} ;\mu_{1} )\} } } \right\|} \right.} \hfill \\ { = \left\| {\left. {\sum\limits_{i = 1}^{N} {\{ g_{ij} (x_{1i} ) - g_{ij} (x_{2i} )\} } } \right\|} \right.} \hfill \\ { \le \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {\left\| {\left. {g_{ij} (x_{1i} ) - g_{ij} (x_{2i} )} \right\|} \right.} } } \hfill \\ \end{array} $$

(5.61)

Based on the assumption in Sect. 5.4.1, the gradient norm is bounded by a scalar $M_{ij} (g)$. Thus, according to the mean value theorem, we have $\left\| {\left. {g_{i} (x_{1i} ) - g_{i} (x_{2i} )} \right\|} \right. \le M_{ij} (g)\left\| {\left. {x_{1i} - x_{2i} } \right\|} \right.$. By combining with (5.61), the following inequality can be obtained

$$ \begin{array}{*{20}l} {\left\| {\nabla_{u} \phi } \right.(u_{1} ;\mu_{1} ) - \left. {\nabla_{u} \phi (u_{2} ;\mu_{1} )} \right\|} \hfill \\ { \le \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {M_{ij} (g)\left\| {\left. {x_{1i} - x_{2i} } \right\|} \right.} } } \hfill \\ \end{array} . $$

(5.62)

For the first order optimality conditions of (5.10), we obtain

$$ \left\{ {\begin{array}{*{20}c} {\left\langle {\left. {\nabla f_{i} (x_{1i} ) + \sum\limits_{j = 1}^{m} {u_{1j} \nabla g_{ij} (x_{1i} ) + \mu_{1} \nabla d_{i} (x_{1i} ),x_{2i} - x_{1i} } } \right\rangle \ge 0} \right.} \\ {\left\langle {\left. {\nabla f_{i} (x_{2i} ) + \sum\limits_{j = 1}^{m} {u_{2j} \nabla g_{ij} (x_{2i} ) + \mu_{1} \nabla d_{i} (x_{2i} ),x_{1i} - x_{2i} } } \right\rangle \ge 0} \right.} \\ \end{array} } \right.. $$

(5.63)

By adding these inequalities of (5.63), underlying inequalities can be obtained

$$ \begin{array}{*{20}l} {\left\langle {\left. {\sum\limits_{j = 1}^{m} {u_{2j} \nabla g_{ij} (x_{2i} ) - \sum\limits_{j = 1}^{m} {\mu_{1j} } \nabla g_{ij} (x_{1i} ),x_{1i} - x_{2i} } } \right\rangle } \right.} \hfill \\ { \ge \left\langle {\nabla f_{i} (x_{1i} ) - \nabla f_{i} (x_{ 2i} ),x_{1i} - x_{2i} } \right\rangle } \hfill \\ { + \mu_{1} \left\langle {\nabla d_{i} (x_{1i} ) - \nabla d_{i} (x_{ 2i} ),x_{1i} - x_{2i} } \right\rangle } \hfill \\ { \ge \mu_{1} \sigma_{i} \left\| {x_{1i} - x_{2i} } \right\|^{2} } \hfill \\ \end{array} , $$

(5.64)

where we use the convexity of $f_{i} (\cdot)$ and strong convexity of $d_{i} (\cdot)$ in the second inequality. Next, the following inequality is considered

$$ \begin{array}{*{20}l} {\left\langle {\left. {\sum\limits_{j}^{m} {u_{2j} \nabla g_{ij} (x_{2i} ) + \mu_{1} \nabla d_{i} (x_{2i} ),x_{1i} - x_{2i} } } \right\rangle } \right.} \hfill \\ { - \sum\limits_{j = 1}^{m} {[g_{ij} (x_{1i} ) - g_{ij} (x_{2i} )]} (u_{2j} - u_{1j} )} \hfill \\ { = \sum\limits_{j = 1}^{m} {u_{1j} [g_{ij} (x_{1i} ) - g_{ij} (x_{2i} )]} + \left\langle {\nabla g_{ij} (x_{1i} ),x_{2i} - x_{1i} } \right\rangle } \hfill \\ { + \sum\limits_{j = 1}^{m} {u_{2j} [g_{ij} (x_{2i} ) - g_{ij} (x_{1i} )]} + \left\langle {\nabla g_{ij} (x_{2i} ),x_{1i} - x_{2i} } \right\rangle } \hfill \\ { \le 0} \hfill \\ \end{array} . $$

(5.65)

The inequality of (5.65) is derived from the convexity of $g_{ij} (\cdot)$, and it indicates a relationship as follows

$$ \begin{array}{*{20}l} {\left\langle {\left. {\sum\limits_{j = 1}^{m} {u_{2j} \nabla g_{ij} (x_{2i} ) + \sum\limits_{j = 1}^{m} {u_{1j} } \nabla g_{ij} (x_{1i} ),x_{1i} - x_{2i} } } \right\rangle } \right.} \hfill \\ { \le \sum\limits_{j = 1}^{m} {\left[ {g_{{i{\text{j}}}} (x_{1i} ) - g_{ij} (x_{2i} )} \right]} (u_{2j} - u_{1j} )} \hfill \\ \end{array} . $$

(5.66)

The right hand of (5.66) suggests

$$ \begin{array}{*{20}l} {\sum\limits_{j = 1}^{m} {\left[ {g_{{i{\text{j}}}} (x_{1i} ) - g_{ij} (x_{2i} )} \right]} (u_{2j} - u_{1j} )} \hfill \\ { \le \sqrt {\sum\limits_{j = 1}^{m} {[g_{ij} (x_{1j} ) - g_{ij} (x_{2i} )]^{2} } } \left\| {u_{2} - u_{ 1} } \right\|} \hfill \\ { \le \sqrt {\sum\limits_{j = 1}^{m} {M_{ij}^{2} (g)} } \left\| {x_{1i} - x_{2i} } \right\|\left\| {u_{2} - u_{ 1} } \right\|,} \hfill \\ \end{array} , $$

(5.67)

where the first inequality adopts the Hölder’s inequality. By applying (5.66), (5.67) to (5.64), we have

$$ \left\| {x_{1i} - x_{2i} } \right\| \le \frac{1}{{\mu_{1} \sigma_{i} }}\sqrt {\sum\limits_{j = 1}^{m} {M_{ij}^{2} (g)} } \left\| {u_{2} - u_{ 1} } \right\|. $$

(5.68)

And after substituting (5.68) into (5.61), we obtain

$$ \begin{array}{*{20}l} {\left\| {\nabla_{u} \phi } \right.(u_{1} ;\mu_{1} ) - \left. {\nabla_{u} \phi (u_{2} ;\mu_{1} )} \right\|} \hfill \\ { \le \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {\frac{{{\text{M}}_{ij} (g)}}{{\mu_{1} \sigma_{i} }}} } \sqrt {\sum\limits_{j = 1}^{m} {M_{ij}^{2} (g)} } \left\| {u_{2} - u_{ 1} } \right\|} \hfill \\ \end{array} . $$

(5.69)

Then, the Lipschitz constant $L_{i}^{d} \left( {\mu_{1} } \right)$ can be obtained from (5.69).

Appendix 2

First, we recall the projection inequality [27], i.e., $\left\langle {v - P_{\Omega } (v),y - P_{\Omega } (v)} \right\rangle \le 0,\forall y \in\Omega$. Then, we consider $\Phi (\overline{u} ;\mu_{1} )$ in light of (5.21)

$$ \begin{array}{*{20}l} {\phi (\overline{u} ;\mu_{1} ) = \sum\limits_{i = 1}^{N} {\mathop {\text{min} }\limits_{{x_{i} \in X_{i} }} \{ f_{i} (x_{i} ) + \left\langle {\overline{u} ,g_{i} (x_{i} )} \right\rangle + \mu_{1} d_{i} (x_{i} )\} } } \hfill \\ { \le \sum\limits_{i = 1}^{N} {\mathop {\text{min} }\limits_{{x_{i} \in X_{i} }} \{ f_{i} (x_{i} ) + \left\langle {\overline{u} ,g_{i} (x_{i} )} \right\rangle \} + } \mu_{1} \sum\limits_{i = 1}^{N} {D_{i} } } \hfill \\ { = \phi (\overline{u} ) + \mu_{1} \sum\limits_{i = 1}^{N} {D_{i,} } } \hfill \\ \end{array} $$

(5.70)

where the inequality follows the definition of $D_{i}$. After that, we consider the function

$$ \begin{array}{*{20}l} {f(\overline{x} ;\mu_{2} )} \hfill \\ { = f(\overline{x} ) + \text{max} \{ \left\langle {u,\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right\rangle - \frac{1}{2}\mu_{2} \left\| u \right\|^{2} \} } \hfill \\ { = f(\overline{x} ) + \mathop {\text{max} }\limits_{{\theta \in \left[ {0,\pi } \right],\left\| u \right\| \le M_{u} }} \{ \cos \theta \left\| u \right\|\left\| {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right\| - \frac{1}{2}\mu_{2} \left\| u \right\|^{2} \} } \hfill \\ { = f(\overline{x} ) + \mathop {\text{max} }\limits_{{\left\| u \right\| \le M_{u} }} \{ \left\| u \right\|\left\| {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right\| - \frac{1}{2}\mu_{2} \left\| u \right\|^{2} \} } \hfill \\ { \ge f(\overline{x} ) + \mathop {\text{max} }\limits_{{\left\| u \right\| \le M_{u} }} \{ \left\| u \right\|\left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\| - \frac{1}{2}\mu_{2} \left\| u \right\|^{2} \} }, \hfill \\ \end{array} $$

(5.71)

where we use the geometric definition of dot product in the second equality, and the inequality is based on the non-expansive property of $[\cdot]^{ + }$. The last term $\{ \cdot\}_{1}$ in (5.71) is nonnegative

$$ \left\{ \cdot \right\}_{1} = \left\{ {\begin{array}{*{20}c} {M_{u} \left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\| - \frac{1}{2}\mu_{2} M_{u}^{2} ,\left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\| \ge \mu_{2} M_{u} } \\ {\left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\|^{2} /2\mu_{2} ,\left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\| \le \mu_{2} M_{u} }. \\ \end{array} } \right. $$

(5.72)

Through combining (5.70) and (5.71) together, we obtain the right hand of (5.72). Then, we consider

$$ \begin{array}{*{20}l} {\phi (\overline{u} ) \le \phi (u^{ * } )} \hfill \\ { = \mathop {\text{min} }\limits_{x \in X} \left\{ {\sum\limits_{i = 1}^{N} {f_{i} (x_{i} )} + \left\langle {u^{ * } ,\sum\limits_{i = 1}^{N} {g_{i} (x_{i} )} } \right\rangle } \right\}} \hfill \\ { \le f(\overline{x} ) + \left\langle {u^{ * } ,\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right\rangle } \hfill \\ { \le f(\overline{x} ) + \left\langle {u^{ * } ,\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\rangle + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} - \left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } ,\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\rangle } \hfill \\ { \le f(\overline{x} ) + \left\| {u^{ * } } \right\|\left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]} \right\|} \hfill \\ \end{array} , $$

(5.73)

where the third inequality adopts the projection inequality with $\Omega = [ \cdot ]^{ + }$, and the fourth inequality follows the fact that the term $\left\langle {\sum\nolimits_{i = 1}^{N} {g_{i} (\overline{x}_{i} ) - } \left[ {\sum\nolimits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } ,\left[ {\sum\nolimits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\rangle$ negative. Thus the left hand of (5.22) is proved. Furthermore, the following inequality can be obtained by taking (5.70), (5.71), (5.72) into consideration

$$ - \left\| {u^{ * } } \right\|\left\| {\left[ {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right]^{ + } } \right\| \le \sigma + \mu_{1} \sum\limits_{i = 1}^{N} {D_{i} - \{ \cdot \}_{1} } . $$

(5.74)

The inequality (5.23) follows from (5.73) and the specific expression of $\{ \cdot \}_{1}$ after a few simple calculations.

Appendix 3

Because the Bregman projection is taken into consideration, the proof is different from the corresponding contents in [12] that a special mapping is considered. For brevity, we denote $x_{i}^{ * } = x_{i}^{ * } (0^{m} ;\mu_{1} )$ and $\overline{x}_{i}^{0} = \overline{x}_{i}^{ * } (0^{m} ;\mu_{1} )$. After that, the following equalities are defined

$$ \begin{array}{*{20}l} {\tilde{\phi }(0^{m} ;\mu_{1} ) = \sum\limits_{i = 1}^{N} {f_{i} (\overline{x}_{i}^{0} )} + \left\langle {0^{m} ,\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} )} } \right\rangle + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\overline{x}_{i}^{0} )} ,} \hfill \\ {\phi (0^{m} ;\mu_{1} ) = \sum\limits_{i = 1}^{N} {f_{i} (x_{i}^{ * } )} + \left\langle {0^{m} ,\sum\limits_{i = 1}^{N} {g_{i} (x_{i}^{ * } )} } \right\rangle + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (x_{i}^{ * } )} .} \hfill \\ \end{array} $$

(5.75)

By considering the inexactness inequality in (5.24), the following equalities can be obtained

$$ \tilde{\phi }(0^{m} ;\mu_{1} ) - \phi (0^{m} ;\mu_{1} ) \le \frac{{\mu_{1} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } . $$

(5.76)

Now we consider the following inequalities

$$ \begin{array}{*{20}l} {\phi (\overline{u}^{0} ;\mu_{1} )} \hfill \\ { \ge \phi (0^{m} ;\mu_{1} ) + \left\langle {\nabla \phi (0^{m} ;\mu_{1} ),\overline{u}^{0} - 0^{m} } \right\rangle - \frac{1}{2}L^{d} (\mu_{1} )\left\| {\overline{u}^{0} - 0^{m} } \right\|^{2} } \hfill \\ { \ge \tilde{\phi }(0^{m} ;\mu_{1} ) + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (x_{i}^{ * } )} ,\overline{u}^{0} } \right\rangle - \frac{1}{2}L^{d} (\mu_{1} )\left\| {\overline{u}^{0} } \right\|^{2} - \frac{{\mu_{2} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } } \hfill \\ { = \left[ {\tilde{\phi }(0^{m} ;\mu_{1} ) + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} )} ,\overline{u}^{0} } \right\rangle - \frac{1}{2}L^{d} (\mu_{1} )\left\| {\overline{u}^{0} } \right\|^{2} } \right]_{1} } \hfill \\ { + \left[ {\left\langle {\sum\limits_{i = 1}^{N} {g_{i} (x_{i}^{ * } )} - \sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} )} ,\overline{u}^{0} } \right\rangle - \frac{{\mu_{2} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } } \right]_{2} ,} \hfill \\ \end{array} $$

(5.77)

where we use the concavity of $\varphi (\cdot;\mu_{1} )$ in the first inequality, and the second inequality follows (5.74). First, the term $[\cdot]_{1}$ of (5.75) is considered

$$ \begin{array}{*{20}l} {\left[ \cdot \right]_{1} = \tilde{\phi }(0^{m} ;\mu_{1} ) - \frac{1}{\gamma }\left[ {\left\langle { - \gamma \sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} ),\overline{u}^{0} } } \right\rangle + \frac{1}{2}\left\| {\overline{u}^{0} } \right\|^{2} } \right]} \hfill \\ { \ge \tilde{\phi }(0^{m} ;\mu_{1} ) - \frac{1}{\gamma }\left[ {\left\langle { - \gamma \sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} ),\overline{u}^{0} } } \right\rangle + B_{u} (0^{m} ;\overline{u}^{0} )} \right]} \hfill \\ { = \tilde{\phi }(0^{m} ;\mu_{1} ) - \frac{1}{\gamma }\mathop {\text{min} }\limits_{u \in U} \left[ { - \left\langle {\gamma \sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} ),u} } \right\rangle + B_{u} (0^{m} ;\overline{u}^{0} )} \right]} \hfill \\ { \ge \mathop {\text{max} }\limits_{u \in U} \left[ {\tilde{\phi }(0^{m} ;\mu_{1} ) - \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} ),u - 0^{m} } } \right\rangle - \frac{1}{2\gamma }\left\| u \right\|^{2} } \right]} \hfill \\ { = \mathop {\text{max} }\limits_{u \in U} \left[ {\sum\limits_{i = 1}^{N} {f_{i} (\overline{x}_{i}^{0} )} + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} ),u} } \right\rangle - \frac{1}{2\gamma }\left\| u \right\|^{2} } \right] + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\overline{x}_{i}^{0} )} } \hfill \\ { \ge f(\overline{x}^{0} ;\mu_{2} ) + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\overline{x}_{i}^{0} )} ,} \hfill \\ \end{array} $$

(5.78)

where the first equality uses the fact that $\gamma = 1/L^{d} (\mu_{1} )$, the first inequality follows (5.28), the second equality is based on the definition of Bregman projection (5.29), the second inequality follows (5.27) and $u^{c} = 0^{m}$, and the third inequality uses $\mu_{2} \ge L^{d} (\mu_{1} )$ in (5.31).

Furthermore, $[ \cdot ]_{2}$ is taken into account

$$ \begin{array}{*{20}l} {[ \cdot ]_{2} \ge - \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {\left\| {g_{ij} (x_{i}^{ * } ) - g_{ij} (\overline{x}^{0} )} \right\|} \left\| {\overline{u}^{0} } \right\|} - \frac{{\mu_{1} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } } \hfill \\ { \ge - \left\| {\overline{u}^{0} } \right\|\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {M_{ij} (g)\left\| {x_{i}^{ * } - \overline{x}^{0} } \right\|} } - \frac{{\mu_{1} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } } \hfill \\ { \ge - \left\| {\overline{u}^{0} } \right\|\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {M_{ij} (g)\varepsilon_{i} } } - \frac{{\mu_{1} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } } \hfill \\ { \ge - \left\| {\overline{u}^{0} } \right\|M\varepsilon_{\left[ 1 \right]} - \frac{{\mu_{1} }}{2}\varepsilon_{\left[ \sigma \right]}^{2} ,} \hfill \\ \end{array} $$

(5.79)

where the second inequality uses the boundness of the gradient, the third inequality follows the fact $\left\| {x_{i}^{ * } - \overline{x}^{0} } \right\| \le \varepsilon_{i}$, and the last inequality adopts Hölder’s inequality. Now we consider the following inequalities about $\left\| {\overline{u}^{0} } \right\|$

$$ \begin{array}{*{20}l} {\left\| {\overline{u}^{0} } \right\| = \left\| {P_{U} \left[ {\frac{{\mu_{1} }}{{L^{d} }}\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} )} } \right]} \right\|} \hfill \\ { \le \left\| {\frac{{\mu_{1} }}{{L^{d} }}\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} )} } \right\|} \hfill \\ { \le \frac{{\mu_{1} }}{{L^{d} }}\left[ {\left\| {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{0} ) - \sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{c} )} } } \right\| + \left\| {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{c} )} } \right\|} \right]} \hfill \\ { \le \frac{{\mu_{1} }}{{L^{d} }}\left[ {MD_{\sigma } + \left\| {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{c} )} } \right\|} \right],} \hfill \\ \end{array} , $$

(5.80)

where the first equality uses the definition of Bregman projection (5.21), the second inequality follows the non-expansive property of the norm operator, and the third inequality adopts the boundness of the gradient and Hölder’s inequality. By combining (5.76), (5.77) and (5.78), we complete the proof

$$ \begin{array}{*{20}l} {\phi (\overline{u}^{0} ;\mu_{1} )} \hfill \\ { \ge f(\overline{u}^{0} ;\mu_{2} ) + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\overline{x}_{i}^{0} )} } \hfill \\ { - \frac{{\mu_{1} }}{{L^{d} }}\left[ {M^{2} D_{\sigma } + M\left\| {\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i}^{c} )} } \right\|} \right]\varepsilon_{[1]} - \frac{{\mu_{1} }}{2}\varepsilon_{[\sigma ]}^{2} } \hfill \\ { \ge f(\overline{x}^{0} ;\mu_{2} ) - \sigma_{0.} } \hfill \\ \end{array} $$

(5.81)

Appendix 4

For the clarity of the exposition, the following equalities are denoted, $u_{2} = u^{ * } (\overline{x} ;\mu_{2} )$, $\tilde{x}_{i} = \tilde{x}_{i}^{ * } (\widehat{u};\mu_{1} )$, $\widehat{x}_{i} = x_{i}^{ * } (\widehat{x};\mu_{1} )$. According to the definition of $f( \cdot ;\mu_{1} )$ and $\mu_{ 2}^{ + } { = (1 - }\tau )\mu_{2}$, we have

$$ \begin{array}{*{20}l} {f(\bar{x}^{ + } ;\mu _{2}^{ + } )} \hfill \\ { = \mathop {{\text{max}}}\limits_{{u \in U}} \left\{ {\sum\limits_{{i = 1}}^{N} {f_{i} (\bar{x}^{ + } ) + \left\langle {u,\sum\limits_{i}^{N} {g_{i} (\bar{x}^{ + } )} } \right\rangle - \frac{1}{2}\mu _{2}^{ + } \left\| u \right\|^{2} } } \right\}} \hfill \\ { = \mathop {{\text{max}}}\limits_{{u \in U}} \left\{ {\sum\limits_{{i = 1}}^{N} {f_{i} [(1 - \tau )\bar{x}_{i} + \tau \tilde{x}_{i} ] + \left\langle {u,\sum\limits_{i}^{N} {f_{i} [(1 - \tau )\bar{x}_{i} + \tau \tilde{x}_{i} ]} } \right\rangle - \frac{{(1 - \tau )\mu _{2} }}{2}\left\| u \right\|^{2} } } \right\}} \hfill \\ \begin{gathered} \le \mathop {{\text{max}}}\limits_{{u \in U}} \left\{ {(1 - \tau )\left[ {\sum\limits_{{i = 1}}^{N} {f_{i} (\bar{x}_{i} ) + \left\langle {u,\sum\limits_{i}^{N} {g_{i} (\bar{x}_{i} )} } \right\rangle - \frac{{\mu _{2} }}{2}\left\| u \right\|^{2} } } \right]_{1} } \right. \hfill \\ \left. \quad + \, {\tau \left[ {\sum\limits_{{i = 1}}^{N} {f_{i} (\tilde{x}_{i} )} + \left\langle {u,\sum\limits_{i}^{N} {g_{i} (\tilde{x}_{i} )} } \right\rangle } \right]_{2} } \right\}, \hfill \\ \end{gathered} \hfill \\ \end{array} $$

(5.82)

where the first inequality follows the convexities of $f_{i} ( \cdot )$ and ${\text{g}}_{ij} ( \cdot )$. The first term $[ \cdot ]_{1}$ of (5.82) can be denoted as

$$ [ \cdot ]_{1} { = } - \mu_{ 2} \left[ {B(u_{2} ,u) + \frac{1}{2}\left\| {u_{2} } \right\|^{2} + \left\langle {u_{2} ,u - u_{2} } \right\rangle } \right] + \sum\limits_{i = 1}^{N} {f_{i} (\overline{x}_{i} )} { + }\left\langle {u,\sum\limits_{i = 1}^{N} {g_{i} (\overline{x}_{i} )} } \right\rangle . $$

(5.83)

The first order optimality condition is

$$ \left\langle {\sum\limits_{i}^{N} {g_{i} (\overline{x}_{i} )} - \mu_{2} u_{2} ,u - u_{2} } \right\rangle \le 0,\quad \forall {\text{u}} \in U. $$

(5.84)

Therefore, the term $[ \cdot ]_{1}$ can be further estimated as

$$ \begin{array}{*{20}l} {[ \cdot ]_{1} \le - \mu _{2} B(u_{2} ,u) + \sum\limits_{{i = 1}}^{N} {f_{i} (\bar{x}_{i} )} + \left\langle {u_{2} ,\sum\limits_{{i = 1}}^{N} {g_{i} (\bar{x}_{i} )} } \right\rangle - \frac{1}{2}\mu _{2} \left\| {u_{2} } \right\|^{2} } \hfill \\ { = - \mu _{2} B(u_{2} ,u) + f(\bar{x};u_{2} )} \hfill \\ { \le - \mu _{2} B(u_{2} ,u) + \phi (\bar{u};\mu _{1} ) + \sigma } \hfill \\ { \le - \mu _{2} B(u_{2} ,u) + \phi (\hat{u};\mu _{1} ) + \left\langle {\nabla \phi (\hat{u};\mu _{1} ),\bar{u} - \hat{u}} \right\rangle + \sigma } \hfill \\ \begin{gathered} = - \mu _{2} B(u_{2} ,u) + \phi (\hat{u};\mu _{1} ) + \left\langle {\sum\limits_{{i = 1}}^{N} {g_{i} (\tilde{x}_{i} )} ,\bar{u} - \hat{u}} \right\rangle \hfill \\ \quad + \, \left\langle {\sum\limits_{{i = 1}}^{N} {g_{i} (\hat{x}_{i} ) - \sum\limits_{{i = 1}}^{N} {g_{i} (\tilde{x}_{i} )} ,\bar{u} - \hat{u}} } \right\rangle + \sigma , \hfill \\ \end{gathered} \hfill \\ \end{array} $$

(5.85)

where we use the definition of δ-excessive gap condition in the second inequality, the third inequality is based on the concavity of $\phi ( \cdot \mu_{1} )$, and the second equality follows the definition of $\tilde{\nabla }\phi (u;\mu_{1} )$. Meanwhile, the estimation about the term $[ \cdot ]_{ 2}$ of (5.82) may lead to

$$ \begin{aligned} [ \cdot ]_{ 2} \, & { = }\sum\limits_{i = 1}^{N} {f_{i} (\tilde{x}_{i} )} + \left\langle {\hat{u},\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} } \right\rangle + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\tilde{x}_{i} )} + \left\langle {u - \hat{u},\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} } \right\rangle \\ & \quad - \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\tilde{x}_{i} )} \le \sum\limits_{i = 1}^{N} {f_{i} (\hat{x}_{i} )} + \left\langle {\hat{u},\sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} } \right\rangle + \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\hat{x}_{i} )} \\ & \quad + \frac{{\mu_{2} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } + \left\langle {u - \hat{u},\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} } \right\rangle - \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\tilde{x}_{i} )} \\ & = \phi (\hat{u};\mu_{1} ) + \left\langle {u - \hat{u},\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} } \right\rangle - \mu_{1} \sum\limits_{i = 1}^{N} {d_{i} (\tilde{x}_{i} )} + \frac{{\mu_{2} }}{2}\sum\limits_{i = 1}^{N} {\sigma_{i} \varepsilon_{i}^{2} } . \\ \end{aligned} $$

(5.86)

Through substituting $[ \cdot ]_{ 1}$ and $[ \cdot ]_{ 2}$ into (5.82), we can demonstrate

$$ \begin{aligned} & f(\bar{x}^{ + } ;\mu _{2}^{ + } ) \\ & \le \mathop {{\text{max}}}\limits_{{u \in U}} \left[ { - \mu _{2} (1 - \tau )B(u_{2} ,u) + \phi (\hat{u};\mu _{1} ) + \left\langle {\sum\limits_{{i = 1}}^{N} {g_{i} (\tilde{x}_{i} )} ,(1 - \tau )(\bar{u} - \hat{u}) + \tau (\bar{u} - \hat{u})} \right\rangle } \right]_{3} \\ & + \left[ {(1 - \tau )\sigma + (1 - \tau )\left\langle {\sum\limits_{{i = 1}}^{N} {g_{i} (\hat{x}_{i} )} - \sum\limits_{{i = 1}}^{N} {g_{i} (\tilde{x}_{i} )} ,\bar{u} - \hat{u}} \right\rangle - \mu _{1} \tau \sum\limits_{{i = 1}}^{N} {d_{i} (\tilde{x}_{i} )} + \frac{{\mu _{1} \tau }}{2}\sum\limits_{{i = 1}}^{N} {\sigma _{i} \varepsilon _{i}^{2} } } \right] \\ \end{aligned} $$

(5.87)

Furthermore, in order to explore (5.87), the term $[ \cdot ]_{ 3}$ is analyzed as follows

$$ \begin{array}{*{20}l} {[ \cdot ]_{ 3} = \mathop { \text{max} }\limits_{u \in U} \left[ { - \mu_{2} (1 - \tau )B(u_{2} ,u){ + }\phi (\hat{u};\mu_{1} ) + \tau \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,u - u_{2} } \right\rangle } \right]} \hfill \\ { = \phi (\hat{u};\mu_{1} ) - \mu_{2} (1 - \tau )\mathop {\text{min} }\limits_{u \in U} \left[ { - \frac{\tau }{{\mu_{2} (1 - \tau )}}\left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,u - u_{2} } \right\rangle + B(u_{2} ,u)} \right]} \hfill \\ {{ = }\phi (\hat{u};\mu_{1} ) + \tau \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,u - u_{2} } \right\rangle - \mu_{2} (1 - \tau )B(u_{2} ,\mathop u\limits^{\sim} )} \hfill \\ { \le \phi (\hat{u};\mu_{1} ) + \tau \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,\tilde{u} - u_{2} } \right\rangle - \frac{{\mu_{2} (1 - \tau )}}{ 2}\left\| {u_{2} - \tilde{u}} \right\|^{ 2} } \hfill \\ { \le \phi (\hat{u};\mu_{1} ) + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,\overline{u}^{ + } - \hat{u}} \right\rangle - \frac{1}{2}L^{d} (\mu_{1} )\left\| {\overline{u}^{ + } - \hat{u}} \right\|^{ 2} } \hfill \\ { = \phi (\hat{u};\mu_{1} ) + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,\overline{u}^{ + } - \hat{u}} \right\rangle - \frac{1}{2}L^{d} (\mu_{1} )\left\| {\overline{u}^{ + } - \hat{u}} \right\|^{ 2} } \hfill \\ { + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} ) - \sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} } ,\overline{u}^{ + } - \hat{u}} \right\rangle } \hfill \\ { \le \phi (\overline{u}^{ + } ;\mu_{1} ) + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} ) - \sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} ),\overline{u}^{ + } - \hat{u}} } } \right\rangle } \hfill \\ { \le \phi (\overline{u}^{ + } ;\mu_{1}^{ + } ) + (\mu_{1} - \mu_{1}^{ + } )\sum\limits_{i = 1}^{N} {d_{i} (x_{i}^{ * } (\overline{u}^{ + } ;\mu_{1}^{ + } ))} } \hfill \\ { + \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} ) - \sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} ),\overline{u}^{ + } - \hat{u}} } } \right\rangle }, \hfill \\ \end{array} $$

(5.88)

where the second inequality follows the fact $\overline{u}^{ + } - \hat{u}{ = }\tau (\tilde{u} - u_{2} )$, the third inequality derives from the concavity of $\phi ( \cdot ;\mu_{1} )$, and the fourth inequality uses the concavity of $\phi (u; \cdot )$. In order to analyze the term $[ \cdot ]_{ 3}$ + $[ \cdot ]_{ 4}$, we need to take underlying inequalities into consideration respectively (5.89)–(5.92).

$$ \begin{array}{*{20}l} {\left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} - \sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} ,\overline{u}^{ + } - \hat{u}} \right\rangle + (1 - \tau )\left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} - \sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} ,\overline{u} - \hat{u}} \right\rangle } \hfill \\ { = \tau \left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} - \sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} ,\tilde{u} - \hat{u}} \right\rangle } \hfill \\ { \le \tau \left\| {\tilde{u} - \hat{u}} \right\|\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {M_{ij} (g)\left\| {\tilde{x}_{i} - \hat{x}_{i} } \right\|} } } \hfill \\ { \le \tau \left\| {\tilde{u} - \hat{u}} \right\|\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{m} {M_{ij} (g)\varepsilon_{i} } } } \hfill \\ { \le \tau \left\| {\tilde{u} - \hat{u}} \right\|M\varepsilon_{[1]} }, \hfill \\ \end{array} $$

(5.89)

where the first equality follows line 1 and line 4 in the recursion, the first inequality is based on the boundness of the gradient, and the second inequality follows $\left\| {\tilde{x}_{i}^{*} (\hat{u};\mu_{1} ) - x_{i}^{*} (\hat{u};\mu_{1} )} \right\| \le \varepsilon_{i}$. Actually, the term $\tau \left\| {\tilde{u} - \hat{u}} \right\|$ is bounded by

$$ \begin{array}{*{20}l} {\tau \left\| {\tilde{u} - \hat{u}} \right\|} \hfill \\ {{ = }\tau \left\| {Pu\left( {u_{2} + \frac{\tau }{{\mu_{2} (1 - \tau )}}\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} } \right) - (1 - \tau )\bar{u} - \tau u_{2} } \right\|} \hfill \\ { \le \tau \left\| {\frac{\tau }{{\mu_{2} (1 - \tau )}}\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} - (1 - \tau )\bar{u} - (1 - \tau )u_{2} } \right\|} \hfill \\ { \le \frac{{\tau^{2} }}{{\mu_{2} (1 - \tau )}}\left\| {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} } \right\| + \tau (1 - \tau )\left\| {\frac{1}{{\mu_{2} }}\sum\limits_{i = 1}^{N} {g_{i} (\bar{x}_{i} )} - \bar{u}} \right\|} \hfill \\ { \le \frac{1}{{L^{d} (\mu_{1} )}}\left( {\left\| {\sum\limits_{i = 1}^{N} {[g_{i} (\tilde{x}_{i} ) - g_{i} (x_{i}^{c} )]} } \right\| + \sum\limits_{i = 1}^{N} {g_{i} (x_{i}^{c} )} } \right)} \hfill \\ { + \frac{\tau (1 - \tau )}{{\mu_{2} }}\left( {\left\| {\sum\limits_{i = 1}^{N} {[g_{i} (\bar{x}_{i} ) - g_{i} (x_{i}^{c} )]} } \right\| + \sum\limits_{i = 1}^{N} {g_{i} (x_{i}^{c} )} } \right)} \hfill \\ { + \tau (1 - \tau )\left\| {\overline{u} } \right\|} \hfill \\ { \le \left( {\frac{1}{{L^{d} (\mu_{1} )}} + \frac{\tau (1 - \tau )}{{\mu_{2} }}} \right)\left( {D_{\sigma } M + \left\| {\sum\limits_{i = 1}^{N} {g_{i} (x_{i}^{c} )} } \right\|} \right)} \hfill \\ { + \tau (1 - \tau )\left\| {\bar{u}} \right\|}, \hfill \\ \end{array} $$

(5.90)

where the first equality is based on the definition of Bregman projection, the first and the second inequality follow the projection inequality and the nonexpansive property of norm, the third inequality uses, and the fourth inequality adopts the boundness of the gradient and Hölder’s inequality. We combine (5.89) and (5.90) to obtain

$$ \begin{array}{*{20}l} {\left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} { - }\sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} ,\overline{u}^{ + } - \hat{u}} \right\rangle } \hfill \\ { + (1 - \tau )\left\langle {\sum\limits_{i = 1}^{N} {g_{i} (\hat{x}_{i} )} { - }\sum\limits_{i = 1}^{N} {g_{i} (\tilde{x}_{i} )} ,\overline{u} - \hat{u}} \right\rangle } \hfill \\ { \le \left[ {\frac{{\mu_{1} }}{{L^{d} }}C_{d} + \tau (1 - \tau )\left( {\frac{{C_{d} }}{{L^{d} }} + M\left\| {\overline{u} } \right\|} \right)} \right]\varepsilon_{[1]} } \hfill \\ \end{array} . $$

(5.91)

With the definition of $\alpha$, the following inequality is available

$$ \begin{array}{*{20}l} {\left( {\mu_{1} - \mu_{1}^{ + } } \right)\sum\limits_{i = 1}^{N} {d_{i} (x_{i}^{*} (\overline{u}^{ + } ;\mu_{1}^{ + } ))} - \mu_{1} \tau \sum\limits_{i = 1}^{N} {d_{i} (\tilde{x}_{i} )} } \hfill \\ { \le \mu_{1} \tau \left( {\alpha \sum\limits_{i = 1}^{N} {D_{i} } - \sum\limits_{i = 1}^{N} {d_{i} (\tilde{x}_{i} )} } \right)} \hfill \\ { = 0}. \hfill \\ \end{array} $$

(5.92)

Therefore, the conclusion of Lemma 4 can be elucidated by combining (5.87)–(5.92) together

$$ \begin{array}{*{20}l} {f(\overline{x}^{ + } ;\mu_{2}^{ + } )} \hfill \\ { \le \phi (\overline{u}^{ + } ;\mu_{1}^{ + } ) + (1 - \tau )\sigma } \hfill \\ { + \left[ {\frac{{\mu_{1} }}{{L^{d} }}C_{d} + \tau (1 - \tau )\left( {\frac{{C_{d} }}{{L^{d} }} + M\left\| {\overline{u} } \right\|} \right)\varepsilon_{[1]} + \frac{{\mu_{1} \tau }}{2}\varepsilon_{[\sigma ]}^{2} } \right]} \hfill \\ { = \phi (\overline{u}^{ + } ;\mu_{1}^{ + } ) + (1 - \tau )\sigma + \eta (\tau ,\mu_{1} ,\mu_{2} ,\overline{u} ,\varepsilon )} \hfill \\ { = \phi (\overline{u}^{ + } ;\mu_{1}^{ + } ) + \sigma_{ + } .} \hfill \\ \end{array} . $$

(5.93)