\begin{equation}\label \arg\min\limits_{\mathbf{c}} \mathbf{c}^TA\mathbf{c}-\mathbf{b}^T\mathbf{c}\\ s.t. \quad c_i = k_i, \quad i\in I \end{equation}
where $A\in R^{M\times M}$ is a large sparse semi-positive matrix, $\mathbf{c} \in R^{M \times 1}$, and $k_i$ are given constants, $I\subseteq \{1,2,\ldots,M\}$.
How to solve this quadratic constrained problem using MATLAB or CVX software effectively?
If you simply fix the variables which you now have written as equality constraints, you can reduce the objective in the remaining free variables $\tilde{c}$ to $\tilde{c}^T\tilde{A}\tilde{c} + \tilde{b}^T\tilde{c} + f$, and the optimal solution is given by the solution to the stationarity condition $2\tilde{A}\tilde{c} + \tilde{b} = 0$, i.e., a linear system of equations. No reason to use a general solver here.