Why must the matrices be positive semidefinite? What is the input authority cost? What is the purpose of multiplying the transpose then the positive semidefinite matrix then the matrix itself?

Why must the matrices be positive semidefinite?

If we only consider real numbers, the definition of a PSD matrix $$A\in\mathbb{R}^{n\times n}$$ is $$z^\top A z \ge 0$$ for $$z \in \mathbb{R}^n$$.

By restricting ourselves to PSD matrices, we know that the loss $$J$$ must always be bounded below by 0 because the sum of non-negative numbers must be non-negative. In particular, this problem is minimizing a strongly convex quadratic so there must be a unique minimum. That's nice!

Now consider a matrix $$B$$ that does not have the PSD property. The quantity $$z^\top B z$$ could be positive, negative, or neither.

Your optimization procedure is minimizing the loss $$J$$. If your matrix is, for example negative definite, then you could always improve the loss by making the sums of these quadratic forms arbitrarily negative, ever smaller. This is akin to minimizing a line with nonzero slope: there's no minimum to find!

What is the input authority cost?

No idea. You'll have to read the slides, or the cited works, or contact the author.

What is the purpose of multiplying the transpose then the positive semidefinite matrix then the matrix itself?

This is called a quadratic form, and it shows up all over the place in math because of its role in defining PD and PSD matrices. What it means in the specific terms of this optimization depends on the context of the problem: where do these matrices come from, and what do they mean?