Using Black two-fund theorem and Corner Portfolios in practice...

I’m trying to wrap my head around these concepts, and ideally in real-life scenarios to help me understand.

  1. One can only use the Black two-fund theorem in an Unconstrained optimization?? They teach us that once you’ve found 2 portfolio’s on the efficient frontier, you can use Black theorem to find any others. But how did you find those two? (I assume some optimization software) - would it then be that much more rigorous for the software to build the entire efficient frontier? - ie. what is the benefit of the Black two fund theorem when I assume you had to use computer software to find the first 2 portfolio’s (inputs)

  2. As far as the corner portfolios, I assume again, you can specify the software to find these corner portfolios in a Sign-constrained optimization? Like the software/optimization will pump out the corner portfolios, ie. every portfolio where there’s an asset-class with 0% weight, and the minimum variance portfolio. Again, would it be that much additional computer power for it to build out entire efficient frontier? I’m just struggling to gain a practical understanding of these two theories (Black-fund and Corner Portfolio).

Bonus question: I assume that neither the Black fund, nor the Corner Portfolio theories could be used in the opposite MVO (Sign-constrained vs. Unconstrained)…

bump

anyone?

3rd time’s a charm…

still looking for any additional clarity here