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This paper deals with the problem of finding the globally optimal subset of helements from a larger set of n elements in d space dimensions so as tominimize a quadratic criterion, with an special emphasis on applications tocomputing the Least Trimmed Squares Estimator (LTSE) for robust regression. Thecomputation of the LTSE is a challenging subset selection problem involving anonlinear program with continuous and binary variables, linked in a highlynonlinear fashion. The selection of a globally optimal subset using the branchand bound (BB) algorithm is limited to problems in very low dimension,tipically d<5, as the complexity of the problem increases exponentially with d.We introduce a bold pruning strategy in the BB algorithm that results in asignificant reduction in computing time, at the price of a negligeable accuracylost. The novelty of our algorithm is that the bounds at nodes of the BB treecome from pseudo-convexifications derived using a linearization technique withapproximate bounds for the nonlinear terms. The approximate bounds are computedsolving an auxiliary semidefinite optimization problem. We show through acomputational study that our algorithm performs well in a wide set of the mostdifficult instances of the LTSE problem.

SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression