# Optimization with probability constrain

Is there a clever way (analitically or nummerically) to minimize following objective function

$$L(W, W_{in}, W_1, W_2) = \sum_{t=0}^T (W + W_{in}W_{out})\vec{x}(t) \cdot (W_1+W_2 W_{out})\vec{x}(t)$$ with $W \in R^{1 \times N},W_{in} \in R, W_{out} \in R^{1 \times N}, W_1 \in R^{1 \times N},W_2 \in R$ and $x(t) \in R^N$ is an arbitrary continious function.

The matrices should all have the constrain though, that it "looks" like they were drawn from a uniform (or gaussian) distribution. (I dont know how to formalize it mathematically correctly, maybe with something like: When $N \rightarrow \inf$, the values of the entries of the matrices "fill" the intervall $[-1,1]$ ) Otherwise they can be arbitrary, even in $N$.

• I can't follow what you are trying to do. You need to elaborate what your constraints are. – copper.hat Jun 4 '17 at 17:23
• I want to change the entries in the 4 matrices that way, that they minimize L for a given x(t), but the entries of the matrices should no all be simply 0, but if you consider them as measurements, it should look like they come from a uniform or gaussian distribution. (Sorry, I cant really formulate it mathematically better) – Luca Thiede Jun 4 '17 at 17:30
• Sorry, it is not clear to me what you want. – copper.hat Jun 4 '17 at 17:31
• Maybe something like add a second term to L, which is the Kullback Leibler divergence between a unifrom/gaussian and the the probability distribution of the entries. If it is still uncomprehensable I delete the question. – Luca Thiede Jun 4 '17 at 17:35
• Well, don't be hasty, I don't understand, but someone else may be able to help. – copper.hat Jun 4 '17 at 17:36