I'm working on an optimization problem where I'm trying to recover a real vector from a subset of that vector's elements that are observed, subject to a regularizing function and some constraints.
$$ \min_{\hat{f}}||(f-\hat{f})\circ[1\ 0\ 1\ 1\ 0\ \dots]^\intercal||_2^2 + r(\hat{f}) \text{ s.t. } A\hat{f}=c $$
Where $f$ is the partial observed vector, and $\hat{f}$ is the estimator. The vector of $1$'s and $0$'s corresponds to which elements are observed and which aren't.
That is, I would like the norm term to only penalize error in the observed elements. Instinctively, I would like to use the Lagrange multiplier for this, but the partial observation bit is throwing me off. Is there a particular approach that tends to work well for these sorts of problems?
Thanks