  We analyze statistical features of the ``optimization landscape'' in a random
version of one of the simplest constrained optimization problems of the
least-square type: finding the best approximation for the solution of an
overcomplete system of $M>N$ linear equations $({\bf a}_k,{\bf x})=b_k, \,
k=1,\ldots,M$ on the $N-$sphere ${\bf x}^2=N$. We treat both the $N-$component
vectors ${\bf a}_k$ and parameters $b_k$ as independent mean zero real Gaussian
random variables. First, we derive the exact expressions for the mean number of
stationary points of the least-square loss function in the framework of the
Kac-Rice approach combined with the Random Matrix Theory for Wishart Ensemble,
and then perform its asymptotic analysis as $N\to \infty$ at a fixed
$\alpha=M/N>1$ in various regimes. In particular, this analysis allows to
extract the Large Deviation Function for the density of the smallest Lagrange
multiplier $\lambda_{min}$ associated with the problem, and in this way to find
its most probable value. This can be further used to predict the asymptotic
minimal value ${\cal E}_{min}$ of the loss function as $N\to \infty$. Finally,
we develop an alternative approach based on the replica trick to conjecture the
form of the Large Deviation function for the density of ${\cal E}_{min}$ at
$N\gg 1$. As a by-product, we find the value of the {\it compatibility
threshold} $\alpha_c$ which is the minimal value of the asymptotic ratio $M/N$
such that the random linear system on the $N-$sphere is typically compatible.
