I am currently researching some robust control problems, and I ended up with the following optimization problem:
\begin{align} \max_{x,w} ~&~ f(x,w) \\ {\rm s.t.} ~&~ g(x,w) \le 0 \\ ~&~ \|w\|\le \|x\| \\ ~&~ x,w\in\mathbb{R}^n \end{align}
The functions $f,g$ are linear, e.g. $f(x,w) = c^\top x + d^\top w$ and $g(x,w) = Ax + Bw - g$ for matrices $A,B$ and vectors $c,d,g$.
If $n=1$, the problem can be recast as an SOCP as $\|w\| \le \|x\|$ is equivalent to $w^\top w \le x^2$, which can be recast as the pair of contstraints $w^\top w \le xv$ and $x=v$, and the first can be rewritten as $\left\|\begin{bmatrix} w \\ x-v\end{bmatrix}\right\| \le x+v$.
Can this program be recasted as an SOCP problem for $n\ge 2$?
Edit: The problem might be recasted as a convex program even if the constraint itself is non-convex. For example, if $g = g(x)$ is independent of $w$, and $f(x,w)=c^\top x + d^\top w$, then for any fixed $x$, we have:
$$ \max_{w: \|w\|\le \|x\|} f(x,w) = c^\top x + \|d\|\|x\| $$
Indeed, the inequality $\le$ follows from Hoelder's inequality, and $\ge$ follows from choosing $w = \frac{\|x\|}{\|d\|}d$. Thus, the problem is equivalent to: \begin{align} \max_{x} ~&~ c^\top x + \|d\|\|x\| \\ {\rm s.t.} ~&~ g(x) \le 0 \\ ~&~ x\in\mathbb{R}^n \end{align} which is convex. I wonder if "tricks" like that can be used for the general problem.
You can omit $x$ from the objective since you can always perform a reformulation via the epigraph. For fixed $w$, the problem is equivalent to deciding if there is a $x$ such that $||x|| \geq ||w||$ and $Ax \leq g - Bw$.
This is NP-hard since it a generalization of maximizing a quadratic function over the unit box $\{ x : ||x||_1 \leq 1\}$, which is known to be NP-complete (see Some optimal inapproximability results by Håstad).