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

Topology trivialization and large deviations for the minimum in the simplest random optimization