I need your expertise in solving the following problem:
Given a convex set $C \subseteq \mathbb{R}^n$, and a positive (or semi-positive) definite matrix $A \in \mathbb{R}^{n \times n}$, we wish to solve the following maximization problem: $$ \max_{x \in C} \left\| Ax \right\|_2^2$$
This function is convex, hence by Rockafellar's Convex Analysis (specifically Theorem 32.2), the maximum can be achieved at $x^\prime \in \partial(C)$, i.e. a boundary point of $C$.
Is there a way to convert the maximization problem to minimization problem while:
- $C$ is still the set of instances.
- The new function we wish to minimize is convex.
This is speaking only regarding the function: $$ \left\| Ax\right\|_2^2 $$
If this is not possible, is there a way to convert this function to (I don't that this is possible): $$ c^Ty,$$ such that $y$ has been constructed from $x \in C$, i.e. we wish to have $$ c^Ty = \left\| Ax \right\|_2^2.$$