I aim to find $x \in \mathbb{R}^n$ that $\min_x |D \cdot F \cdot x|^2$ subject to $x_i =  X_i$ and $x_j \geq X_j$ , $i \in I, j \in J$ and I and J partition ${1\cdots N}$ into two sets.it is easy to do $D \cdot F \cdot x$ because D is diagonal and F is the discrete Fourier transform. But actually storing and computing on F is not practical.MATLAB's lsqlin works for small cases. It looks like it uses a direct method, qpsub, if the problem has a mix of equality and inequality constraints.Are there any solvers or quick implementations you could refer me to?
indirect method for least-squares with inequality constraints
optimization;linear solver;sparse;least squares