  Finding the global minimum of a cost function given by the sum of a quadratic
and a linear form in N real variables over (N-1)- dimensional sphere is one of
the simplest, yet paradigmatic problems in Optimization Theory known as the
"trust region subproblem" or "constraint least square problem". When both terms
in the cost function are random this amounts to studying the ground state
energy of the simplest spherical spin glass in a random magnetic field. We
first identify and study two distinct large-N scaling regimes in which the
linear term (magnetic field) leads to a gradual topology trivialization, i.e.
reduction in the total number N_{tot} of critical (stationary) points in the
cost function landscape. In the first regime N_{tot} remains of the order $N$
and the cost function (energy) has generically two almost degenerate minima
with the Tracy-Widom (TW) statistics. In the second regime the number of
critical points is of the order of unity with a finite probability for a single
minimum. In that case the mean total number of extrema (minima and maxima) of
the cost function is given by the Laplace transform of the TW density, and the
distribution of the global minimum energy is expected to take a universal
scaling form generalizing the TW law. Though the full form of that distribution
is not yet known to us, one of its far tails can be inferred from the large
deviation theory for the global minimum. In the rest of the paper we show how
to use the replica method to obtain the probability density of the minimum
energy in the large-deviation approximation by finding both the rate function
and the leading pre-exponential factor.
