Projection onto quadratic hypersurfaces

Abstract

We address the problem of projecting a point onto a quadratic hypersurface, more specifically a central quadric. We show how this problem reduces to finding a given root of a scalar-valued nonlinear function. We completely characterize one of the optimal solutions of the projection as either the unique root of this nonlinear function on a given interval, or as a point that belongs to a finite set of computable solutions. We then leverage this projection and the recent advancements in splitting methods to compute the projection onto the intersection of a box and a quadratic hypersurface with alternating projections and Douglas-Rachford splitting methods. We test these methods on a practical problem from the power systems literature, and show that they outperform IPOPT and Gurobi in terms of objective, execution time and feasibility of the solution.

BibTex

@misc{projection_vh_21,
title = {Projection onto quadratic hypersurfaces},
year = {2021},
note = {submitted},
author = {Loïc {Van Hoorebeeck} and P.-A. Absil and Anthony Papavasiliou},
}

Reference

• (2021) L. Van Hoorebeeck, P.-A. Absil and A. Papavasiliou, “Projection onto quadratic hypersurfaces”, submitted. (preprint, abstract/BibTex)