# 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 $$w^Tw < 1$$ 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.