If you want the lowest correlation then just short your portfolio. Correlation -1, now you have zero exposure. But I don't think that's really your objective function.
For a two asset portfolio ...
$$
\sigma =\sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2+2w_1w_2Cov_1,_2}
$$
And your real objective function incorporates your return expectation.
Bear with me now, because we're just stating the obvious so far. And I haven't been thinking through the generalization to more assets lately nearly as much as calling library functions to do this and as sunny as it is outside, I'm in no mood to do matrixes in markdown or stare at the equation until I'm sure I generalize right. I'm going to suggest you go have a look at this explanation with a spreadsheet.
On to some of the assumptions about where to look for your non-correlated positive return expectation.
You can't assume that inverting your signal generation will give you a set of low correlation signals in the form of shorts. It might look that way on daily returns, but my experience with long/short systems of this nature is that the long and short can be highly correlated on time frames longer than one day. Furthermore, you have different market structure issues with shorts. I see some feasibility issues in terms of share availability, costs issues (there's a highly profitable sub-industry thriving off of lending shares to folks like you and me), and stocks just behave differently going down than going up.
You have to diversify in terms of signal generation methods entirely. This might apply more to some strategies and less to others. You should certainly test for it in your strategies. Even hope you will find it if you must. But run the numbers and don't assume.
Keep diversifying. And I certainly don't mean to suggest you shouldn't trade some inverted signals, because most people I know do that. Just don't expect them to be negative correlations or even the best diversification for your longs. Uncorrelated alpha is the holy grail. Happy questing!
