I am given a vector x. My objective is to find an optimal y (minimize $||y-x||_2^2$). With the constraint $y(c) = a$ (a and c are known scalars).
$$\text{minimize}_y ||y-x||_2^2 \\ \text{subject to}\ \ y(c) = a $$
Further, I am confused about how to enforce a smoothness constraint. I am aware that in theory we add a regularization term to the objective function $ ||y-x|| + \lambda \bigtriangledown y $
I am trying to use CVX to achieve this. More info on CVX is Matlab Software for Convex Optimization (http://cvxr.com/cvx/) How is it possible to have such a regularization?
The optimization variable is $y \in \mathbb R^N$. Without a regularization term, the solution to the optimization problem written is the vector that agrees with $x$ except in the $c$th component, which is equal to $a$. You can add a regularization term involving a (discrete) gradient of $y$ as you have mentioned. One popular choice for a regularization term is \begin{equation*} r(y) = \sum_{i=1}^{N-1} |y_{i+1} - y_i|. \end{equation*} With this regularization term, you are solving a "total variation denoising" problem.
The optimization problem including this regularization term is \begin{align} \text{minimize} & \quad \|y - x\|_2^2 + \lambda \sum_{i=1}^{N-1} |y_{i+1} - y_i| \\ \text{subject to}& \quad y_c = a, \end{align} with variable $y \in \mathbb R^N$.