Controlled Optimization of Quadratic Functions in $\mathbb{R}^n$

TL;DR

This paper introduces a novel controlled dynamical system approach for optimizing quadratic functions in finite-dimensional spaces, including new discretization methods for gradient-based minimization.

Contribution

It presents the controlled quadratic gradient flow and its discretizations, advancing the controllability and optimization of quadratic functions.

Findings

Controlled quadratic gradient flow effectively minimizes quadratic functions.

Discretization methods improve convergence control.

Framework enhances understanding of controllability in optimization trajectories.

Abstract

In this work, we introduce and study the controllability of the trajectories of a linear dynamical system, which can be used to solve the minimization of a quadratic function in finite dimension. We named this dynamical system the controlled quadratic gradient flow. Finally, we introduce what we call the controlled quadratic gradient descent and the controlled proximity operator which are respectively the Euler explicit and implicit discretization of the controlled gradient flow.