  For linearly constrained least-squares problems that depend on a vector of
parameters, this paper proposes techniques for reducing the number of involved
optimization variables. After first eliminating equality constraints in a
numerically robust way by QR factorization, we propose a technique based on
singular value decomposition (SVD) and unsupervised learning, that we call
$K$-SVD, and neural classifiers to automatically partition the set of parameter
vectors in $K$ nonlinear regions in which the original problem is approximated
by using a smaller set of variables. For the special case of parametric
constrained least-squares problems that arise from model predictive control
(MPC) formulations, we propose a novel and very efficient QR factorization
method for equality constraint elimination. Together with SVD or $K$-SVD, the
method provides a numerically robust alternative to standard condensing and
move blocking, and to other complexity reduction methods for MPC based on basis
functions. We show the good performance of the proposed techniques in numerical
tests and in a linearized MPC problem of a nonlinear benchmark process.
