Simultaneously minimize and maximize two cost functions 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 $w^Tw &lt; 1$ to convert this to a convex problem. This is readily solved in literature ( simply google (Max SINR) optimization.