Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly now the new set, $\mathbb{R}^n$, is convex and optimizing here is equivalent to optimizing on the sphere (minus one point of course, which will be assumed to be a bad point, the point at infinity). So all the grand theory of convex optimization may be applied.
Now suppose, instead, that your constrained space is the following:
Given a matrix $A \in M(n,n)$, there are $n$ constraints,
$\sum_{j=1}^n A_{i,j}^2=1$ for every $i\in \{ {1,\cdots, n}\}$
Is there a 'neat' way to turn this constrained set into a convex one, without losing much information much akin to the previous example.
We may think of this matrix $A$ as lying in the vector space $\mathbb{R}^{n^2}$ where the representation is given by a vector $v=(A_{1,*},\cdots, A_{n,*})$; where $A_{i,*}$ is the $i$th row of $A%$.