Simultaneously minimize and maximize two cost functions
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.