I have the following optimization problem: Find $\mathbf{w}$ such that the following error measure is minimised:
- $E_u = \dfrac{1}{N_u}\sum_{i=0}^{N_u-1}\lVert \mathbf{w}^Tx(t_{i+1})-\mathbf{F}(\{\mathbf{w}^Tx(t_j)_{j=0,i}\})\rVert$,
- $t_i \text{ being the i-th timestamp and } \lVert \cdot \rVert \text{ the } L_2 \text{ norm}$
- $\mathbf{F}$ is something of the form $\sum_{j=0}^{i}\alpha_j\mathbf{w}^Tx(t_j)$ with $\sum_{j=0}^i \alpha_j = 1$.
It's important to note the $\alpha$'s are fixed (because they are from a subsystem).
With the constraints that: $\mathbf{w}>\mathbf{0}$ and $\mathbf{w}<\mathbf{L}$. Both $\mathbf{0}$ and $\mathbf{L}$ are vectors in $\mathbb{R}^6$, $\mathbf{L}$ being a vector of positive arbitrary limits I set.
Unfortunately, this doesn't look like the standard least-squares problem, due to that pesky $\mathbf{w}$ term that pops in both places (this is fixed in a certain epoch). Essentially, is like least squares but the target $\mathbf{y}$ is the value of the series at the next timestamp.
Is this another class of problems? Unfortunately, I don't have enough background on this area, but I've read something about Recursive Least Squares and Kalman filters - is this something that could be solved with this?