I want to solve the following optimization problem, in which I have two cost functions $J_1 = \mathbf{w}^T R_1 \mathbf{w}$ and $J_2 = \mathbf{w}^T R_2 \mathbf{w}$, where $R_1$ and $R_2$ are covariance matrices (symmetric PSD).
The problem is to find $\mathbf{w}$ such that $J_1$ is minimized and $J_2$ s maximized constrained on $\mathbf{w}^T \mathbf{w} = 1$.
NB:- This problem arises in signal processing algorithm development and hence posted in this forum.
Thanks for any help.
Purna.
Ideas:
1) Find the minimum eigen vector of $R_1$ and assign this to $w$. This will minimize $ J_1$ , doesn't maximize $J_2$. But when $R_1$ and $R_2$ are positive definite or full rank matrices and computation is an issue, this is a decent solution.
2) Form a new objective $J_1 - J_2$ and minimize this with the given constraint
3) Try and formulate as max SINR problem with $J_1$ as the interference term and $J_2$ as the signal term and the given constraint, you should be able to apply the relaxation $w^Tw < 1$ to convert this to a convex problem. This is readily solved in literature ( simply google (Max SINR) optimization.
If you want to maximize $J_2$ and minimize $J_1$, another way to state it is that you want to maximize the difference $(J_2-J_2)$. You can define a new cost function:
$$ \begin{split} J_3 &=(J_2-J_1)\\ &= \mathbf{w}^T\big(\mathbf{R}_2-\mathbf{R}_1\big)\mathbf{w} \\ &= \mathbf{w}^T\mathbf{R}_3\mathbf{w}\end{split} $$
At this point, you have reframed your problem into a quadratic optimization problem with a constraint, to which you can apply a wide variety of methods: https://en.wikipedia.org/wiki/Quadratic_programming.
