I have a problem that looks very much like a norm-constrained linear program, but with an extra constraint that is unusual for me. The problem is the following. Given a matrix $A$ and a vector $w$,
$$ \min_{x \in \mathbb{R}^n} w^\top x $$
subject to
$$ \|x\|_2 = 1,$$
$$x_i \ge 0 ~\forall~ i,$$
$$x = Az.$$
Note that $A$ and $w$ are fixed, and $z$ is a free variable. It almost looks like a standard QCLP, but with the extra requirement that $x$ be in the column space of $A$. We cannot assume that $A$ is invertible, but $z$ can take on any value in $\mathbb{R}^n$.
Of course we can substitute $Az$ for $x$ to get an equivalent problem with fewer constraints,
$$\min_{z \in \mathbb{R}^n} w^\top A z$$
subject to
$$\|Az\|_2 = 1,$$
$$Az \ge 0$$
(where the last inequality is element-wise), but I still don't know of methods that are designed to solve this problem, or even better if there is a closed-form solution.
Are problems like this well-studied in some corner of constrained optimization? I have seen similar ones, but not this one exactly.
Best Answer
The projection to the column space of $A$ is done by $A A^\dagger x$ (where $A^\dagger$ is the pseudoinverse). If $x$ is in the column space, $A A^\dagger x=x$. Hence, the above constraint can be written as: $$(A A^\dagger -I)x=0$$.