A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables

TLDR

The method targets general indefinite quadratic functions with bound constraints, especially for large‑scale problems. The paper proposes a reflective Newton algorithm to minimize high‑dimensional quadratic functions with bound constraints. The algorithm employs reflective Newton steps and is evaluated on moderately large sparse problems using sparse Cholesky and preconditioned conjugate gradient solvers. The method demonstrates strong global and second‑order convergence, generates strictly feasible points, and shows significant practical potential.

Abstract

We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper arid lower bounds on some of the variables. The method applies to a general (indefinite) quadratic function for which a local minimizes subject to bounds is required and is particularly suitable for the large-scale problem. Our new method exhibits strong convergence properties and global and second-order convergence and appears to have significant practical potential. Strictly feasible points are generated. We provide experimental results on moderately large and sparse problems based on both sparse Cholesky and preconditioned conjugate gradient linear solvers.