On global minimizers of quadratic functions with cubic regularization

TL;DR

This paper investigates the properties of quadratic functions with cubic regularization, showing how to find global minimizers and approximate stationary points, with implications for optimization algorithms.

Contribution

It provides theoretical insights into the structure of minimizers and stationary points for cubic-regularized quadratic functions, including closed-form solutions and finite-step methods.

Findings

Any non-global stationary point can be improved in closed form.

A global minimizer can be reached after finitely many stationary points.

Results extend to approximate stationary conditions, aiding algorithm design.

Abstract

In this paper, we analyze some theoretical properties of the problem of minimizing a quadratic function with a cubic regularization term, arising in many methods for unconstrained and constrained optimization that have been proposed in the last years. First we show that, given any stationary point that is not a global solution, it is possible to compute, in closed form, a new point with a smaller objective function value. Then, we prove that a global minimizer can be obtained by computing a finite number of stationary points. Finally, we extend these results to the case where stationary conditions are approximately satisfied, discussing some possible algorithmic applications.