  Given an $n\times n$ matrix $A_n$ and $1\leq r, p \leq\infty$, consider the
following quadratic optimization problem referred to as the
$\ell_r$-Grothendieck problem:
\begin{align}M_r(A_n)\coloneqq\max_{\boldsymbol{x}\in\mathbb{R}^n:\|\boldsymbol{x}\|_r\leq1}\boldsymbol{x}^{\top}
A_n \boldsymbol{x},\end{align} as well as the $r\rightarrow p$ operator norm of
the matrix $A_n$, defined as \begin{align}\|A_n\|_{r \rightarrow p}\coloneqq
\sup _{\boldsymbol{x}\in\mathbb{R}^n:\|\boldsymbol{x}\|_r \leq 1}\|A_n
\boldsymbol{x}\|_p,\end{align} where $\|\boldsymbol{x}\|_r$ denotes the
$\ell_r$-norm of the vector $\boldsymbol{x}$. This work analyzes
high-dimensional asymptotics of these quantities when $A_n$ are symmetric
random matrices with independent and identically distributed heavy-tailed
upper-triangular entries with index $\alpha$. When $1\leq r\leq 2$
(respectively, $1\leq r\leq p$) and $\alpha\in(0,2)$, suitably scaled versions
of $M_r(A_n)$ and $\|A_n\|_{r\rightarrow p}$ are shown to converge to a
Fr\'echet distribution as $n\rightarrow\infty$. In contrast, when $2< r<\infty$
(respectively, $1\leq p< r$), it is shown that there exists $\alpha_*\in(1,2)$
such that for every $\alpha\in(0,\alpha_*)$, suitably scaled versions of
$M_r(A_n)$ and $\|A_n\|_{r\rightarrow p}$ converge to the power of a stable
distribution. Furthermore, it is shown that there exists
$\bar\alpha_*>\alpha_*$ such that when $\alpha\in(\alpha_*,\bar\alpha_*)$, the
latter convergence result holds only when the matrix entries are centered; when
the entries have non-zero mean, a different limit arises after additional
centering and scaling. As a corollary, these results yield a characterization
of the limiting ground state of the Levy spin glass when $\alpha \in (0,1)$.
The analysis uses a combination of tools from the theory of heavy-tailed
distributions, the nonlinear power method and concentration inequalities.
