Hi,
Sorry my question is a little long. I have a optimization problem like this:
A and D are M\times M matrix, x,y are M\times 1 vector. C is
covariance of x, \lambda is a regularization parameter which I can
specify.
$$ \begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& &(Ax-y)^TC^{-1}(Ax-y)+\lambda |Dx|2^2 \
& \text{subject to}
& & x_i\geq 0, \quad i=1,\cdots, M \
&&& \sum{i=1}^M{x_i}=N\
&&& \sum_{i=1}^M{i (x_i/N)}= \mu\
&&& \sum_{i=1}^M{(i-\mu)^2 (x_i/N)=\sigma^2}
\end{aligned}
\end{equation*}
* First, how to write WLS in DCP?
I tried
L=chol(C,'lower');
invL=inv(L);
invC=inv(C);
minimize(sum_square(invL*A*x-invL*y)+lambda*sum_square(D*x))
and
minimize (quad_form(A*x-y,invC)+lambda*sum_square(D*x));
I found the latter is much more efficient than the former. Another
concerned is that since the covariance matrix I have is ill conditioned,
so getting an inverse of it directly through inv(.) may not be a good
idea. So I was wondering if there is any way I can put C instead
of inv(C) directly in the objective function to avoid crude inverse of C, and still comply with the rules of
cvx? I just want the ill condition of C to impact the stability of the
result as less as possible.
* Second, in the setting of the problem, $x_i/N$ can be considered as
frequency of a discrete distribution. The third and fourth constraints
are the constraints of the mean and variance. So I can also write these
two constraints to
$$\sum_{i=1}^M{i x_i}= \mu*N$$
and
$$ \sum_{i=1}^M{i^2 x_i}=\text{2nd raw moment}$$
I found that also the feasible set should be the same but the result
from CVX a is different. Moving $N$ to the right hand side of the constraints makes a great difference in the solution, especially when $\lambda$ is small.
Which way do you think I can get a more accurate solution?
* In addition, I want to do the optimization many times, each time with a different
$\lambda$ or $y$. So far I have been using for loop. Do you know any
equivalent formulation that cvx can process more efficiently?
Thank you very much!
YZ