What is the optimal forecast for the following moving average model? Suppose that we have an MA(1) model where $y_t=\epsilon_t-\frac 1 2 \epsilon_{t-1}$ and $\epsilon_t$ are i.i.d.
I want to find an optimal forecast of $y_{T+1}$ in the form of $Cy_T$ where C is a constant. What is the best choice of C?
I have computed that 

$\begin{aligned}
&amp;y_{T+1}=E(y_{T+1}|I_T)=E(\epsilon_{T+1}-\frac 1 2 \epsilon_T|I_T)\\
&amp;=E(\epsilon_{T+1}|I_T)-\frac 1 2E(\epsilon_T|I_T)\\
&amp;=0-\frac 1 2\epsilon_T=-\frac 1 2 \epsilon_T\\
&amp;\epsilon_T=y_T+\frac 1 2y_{T-1}+\frac 1 4 y_{T-2}+......+{1\over 2^{t-1}}y_1+0\\
&amp;y_{T+1}=-\frac 12 \epsilon_T=-\frac 12y_T-\frac 1 4y_{T-1}-\frac 1 8 y_{T-2}-......-{1\over 2^{t}}y_1
\end{aligned}$
Your equation $y_{T+1} = E(y_{T+1}|I_T)$ clearly does not make sense.

In any case,
$$
E(y_{T+1}|I_T)
$$ is the best one step ahead forecast conditional on $Y_1, \cdots, Y_T$, whereas you're asked to find the best linear one step ahead forecast given $Y_T$. They are in general not the same.
The best linear forecast for $Y_{T+1}$ given $Y_T$ is
$$
\mu + \frac{\gamma(1)}{\gamma(0)} (Y_T - \mu),
$$
where $\mu$ is the unconditional mean of the series and $\gamma$ is the autocovariance function. In your case, $\mu = 0$ and $\gamma$ is that of a MA(1) series ($\frac{\gamma(1)}{\gamma(0)} = \frac{\theta}{1+\theta^2}, \; \theta = \frac12$). 

In general, the best linear forecast for covariance-stationary time series can be obtained by solving the Yule-Walker equations.

For best linear forecast, the assumption that innovations are i.i.d. is extraneous.