#  Channel Memory

This example involves a two-operator inclusion problem. Memory effects are present in the communication link to and from evaluation of $F_1$. 
The intensity of the memory effects are represented by a scalar forgetting factor $\alpha > 0$. 
The network effects for all $k \in \N$ are
```{math}
\begin{align*}
z^1_k &= u_k^1 - \alpha z^1_{k-1},  & y_k^1 &= w_k^1 - \alpha y_{k-1}^1, \\
z^2_k &= u_k^2, & y_k^2 &= w_k^2. 
\end{align*}
```

The degenerate case of $\alpha=0$ is no network dynamics.
A state-space realization of these memory effects is
```{math}
 \text{Network}: \qquad  \mat{c}{x_{k+1}^N \hl z_k \hdl y_k} =  \mat{cc|cc:cc}{-\alpha I & 0 & \frac{1}{2} I & 0 & 0 & 0 \\
 0 & -\alpha I & 0 & 0 & \frac{1}{2}I & 0  \hl
 0 & -2\alpha I & 0 & 0 & I & 0  \\
 0 & 0 & 0 & 0 & 0 & I  \hdl
 -2 \alpha I & 0 & I & 0 & 0 & 0 \\
 0 & 0 & 0 & I & 0 & 0}  \mat{c}{x_{k}^N \hl w_k \hdl u_k}.
```



The controller structure with parameters $(\gamma, \lambda) \geq 0$ used to solve the two-operator inclusion problem is
```{math}
 \mat{c}{x_{k+1}^c \hl u_k^1 \\ u_k^2} =  \mat{c|cc}{I & -\gamma \lambda & \frac{ -\gamma \lambda}{\alpha+1} \hl
 (1+\alpha) I & 0 & 0 \\
 I & -\gamma I & -\frac{ -\gamma }{\alpha+1} I }  \mat{c}{x_{k}^c \hl y_k^1 \\ y_k^2}.
```

This controller structure is parameterized by $\alpha$. If $\alpha = 0$ and $\lambda = 1$, then this controller is the same as Projected Gradient Descent.
The controller structure is chosen to ensure that the {doc}`Regulator Equation <../how_it_works/index_how_it_works>` condition for algorithm convergence is satisfied for all values $(\gamma, \lambda)$. Projected Gradient Descent fails the regulator equation requirement of convergence when $\alpha > 0$. 

We use this algorithm to solve a composite optimization problem 
```{math}
\beta^* \in \argmin_{\norm{\beta}_1 \leq 100 } f(\beta)
```
where $f$ is a convex quadratic with $m=1$, $L=5$. 

Figure [1](#sym-trace) plots a trace of algorithm execution starting from $x_0 = 0$, highlighting the states of the network and controller.

:::{figure} _static/sim_channel_sym_dark.png
:align: center
:class: only-dark
:name: sym-trace
*Figure 1:* Trace of execution and convergence
:::

:::{figure} _static/sim_channel_sym_light.png
:align: center
:class: only-light
:name: sym-trace
*Figure 1:* Trace of execution and convergence
:::

The Regulator Equations are used to  establish tracking properties of solution trajectories. The solution to the Regulator Equations for this network and controller are
```{math}
\begin{align*}
\Pi &= \mat{cc}{0 & \frac{1}{2(\alpha+1)}I \\ -\frac{1}{2} I & 0 }, &  \Gamma &= \mat{cc}{-(\alpha+1) I & 0 \\
-I & 0 }, \\
\Phi &= \mat{cc}{0 & \frac{1}{2(\alpha+1)}I \\ 0  & -I }, &  \Theta &= \mat{c}{-I & 0}.
\end{align*}
```

The pair $(\beta^*, w^*)$ solving the composite optimization problem is unique, because  $f$ is strongly convex with a nonempty constraint set is nonempty, and $f$ is smooth.
Algorithm convergence implies tracking of the signals $(x^N, x^c, y, u)$ with
```{math}
\begin{align*}
    \lim_{k \rightarrow \infty}
    \mat{c}{x_k^N \\ x^c_k \hl  y_k \\u_k} & = \mat{c}{\Pi \\ \Theta \hl \Phi \\ \Gamma} \mat{c}{-\beta^* \\ \nabla f(\beta^*)}.
\end{align*}
```


Figure [2](#sym-track) plots the tracking of these signals over time

:::{figure} _static/sim_channel_sym_track_dark.png
:align: center
:class: only-dark
:name: sym-track
*Figure 2:* Tracking errors
:::

:::{figure} _static/sim_channel_sym_track_light.png
:align: center
:class: only-light
:name: sym-track
*Figure 2:* Tracking errors
:::

Figure [3](#sym-trace-sq) plots the squared norm of the tracking error


:::{figure} _static/sim_channel_sym_track_sq_dark.png
:align: center
:class: only-dark
:name: sym-track-sq
*Figure 3:* Tracking residuals
:::

:::{figure} _static/sim_channel_sym_track_sq_light.png
:align: center
:class: only-light
:name: sym-track-sq
*Figure 3:* Tracking residuals
:::

```{literalinclude} ../../../examples/simulation/dr_example/sim_channel_symmetric.m
:linenos: true
:caption: Code for optimization with symmetric channel memory effects
:language: matlab
```

:::{seealso}
{doc}`Analysis <../examples_analysis/ana_channel_memory>` of the channel-memory  algorithm.
:::