  We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in
S^{n-1}} ( x^T W x/2 + h^T x )\,, $$ with $W$ a (random) matrix and $h$ a
random external field. We study the probabilities of large deviation of
$F_n^{W,h}$ for $h$ a centered Gaussian vector with i.i.d. entries, both
conditioned on $W$ (a general Wigner matrix), and unconditioned when $W$ is a
GOE matrix. Our results validate (in a certain region) and correct (in another
region), the prediction obtained by the mathematically non-rigorous replica
method in Y. V. Fyodorov, P. Le Doussal, J. Stat. phys. 154 (2014).
