I have the following constrained optimization problem
\begin{align} \mathop{\text{min}}_\mathbf{Kp,Kd} && (\mathbf{Kd} - \mathbf{Kd_d})^2 + (\mathbf{Kp} - \mathbf{Kp_d})^2 \\ \text{subject to:} && \lambda_m\{\mathbf{Kp}\} > \mathbf{v1}\\ && \lambda_m\{\mathbf{Kd}\} > \mathbf{v2}\\ && \lambda_m\{\mathbf{Kd}\}*\lambda_m\{\mathbf{Kp}\} + \mathbf{n}*\lambda_m\{\mathbf{Kd}\} > \mathbf{v3}\\ \end{align}
Assume that the only unknown variables are the squared matrices $\mathbf{Kp,Kd}$, whereas $\mathbf{Kp_d,Kd_d}$ are known squared full rank matrices. For simplicity, we can assume that all the matrices are diagonal.
$\lambda_m$ denotes the smallest eigenvalue of its argument.
$\mathbf{v1},\mathbf{v2},\mathbf{v3},\mathbf{n}$ are known scalar.
Which approach can I use to solve such a problem? I am open to numerical methods as well.
EDIT: Since someone made me notice that dealing with matrices makes everything harder, I will take advantage of the fact that the matrices are diagonal to turn the problem into a scalar one.
Assume that $\mathbf{Kd},\mathbf{Kp},\mathbf{Kd_d},\mathbf{Kp_d}$ are diagonal $NxN$ positive definite matrices.
Let $\mathbf{kd_i}$ be the element in position $(i,i)$ of $\mathbf{Kd}$.
In the same way, $\mathbf{kp_i}$ is the element in position $(i,i)$ of $\mathbf{Kp}$.
Let $\mathbf{S_d}$ be the set containing all the elements in the diagonal of $\mathbf{Kd}$ and $\mathbf{S_p}$ be the set containing all the elements in the diagonal of $\mathbf{Kp}$
I can rewrite the problem as:
\begin{align} \mathop{\text{min}} && \sum_{i=0}^N[(\mathbf{kd_i} - \mathbf{kd_{di}})^2 + (\mathbf{kp_i} - \mathbf{kp_{di}})^2] \\ \text{subject to:} && min(\mathbf{S_p}) > \mathbf{v1}\\ && min(\mathbf{S_d}) > \mathbf{v2}\\ && min(\mathbf{S_d})*min(\mathbf{S_p}) + \mathbf{n}*min(\mathbf{S_d}) > \mathbf{v3}\\ \end{align}
I didn't quite understand the question. It seems you are trying to get the diagonal values of the resulting matrix as low as possible? Lets look just at Kd- Kdd. each eigenvalue in Kd is larger than the minimum and positive as you said in the comments. then you just need to optimize |Kd_i - Kdd_i| , _i being the signifier for eigenvalue i.
Obviously you want to take Kd_i = Kdd_i, but that's not necessarily possible. If you can't take it so since Kdd_i < min(Kd eigenvalues) then you want to take Kd_i = min(Kd eigenvalues).
Same for Kp. Now you only have to solve the constraints at the bottom. Am I missing something?