An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds

TLDR

The authors propose a new trust‑region method for minimizing nonlinear functions with simple bounds. The method avoids solving a quadratic programming subproblem by instead solving a trust‑region subproblem that minimizes a quadratic function under an ellipsoidal constraint, yielding strictly feasible iterates and reducing to the standard trust‑region approach when no bounds are present. The approach achieves global and local quadratic convergence, and preliminary experiments demonstrate its practical viability.

Abstract

We propose a new trust region approach for minimizing a nonlinear function subject to simple bounds. Unlike most existing methods, our proposed method does not require that a quadratic programming subproblem, with inequality constraints, be solved in each iteration. Instead, a solution to a trust region subproblem is defined by minimizing a quadratic function subject only to an ellipsoidal constraint. The iterates generated are strictly feasible. Our proposed method reduces to a standard trust region approach for the unconstrained problem when there are no upper or lower bounds on the variables. Global and local quadratic convergence is established. Preliminary numerical experiments are reported indicating the practical viability of this approach.