Reduction of the Number of Variables in Parametric Constrained Least-Squares Problems
This work provides a numerically robust and efficient method for reducing computational complexity in parametric constrained least-squares problems, particularly beneficial for real-time applications like Model Predictive Control.
This paper addresses the problem of reducing the number of optimization variables in parametric constrained least-squares problems. It proposes a K-SVD technique combined with neural classifiers to partition parameter vectors into regions, allowing approximation with fewer variables, and for MPC problems, a novel QR factorization method for efficient equality constraint elimination. The techniques demonstrate good performance in numerical tests and a linearized MPC problem.
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.