Projection onto quadratic hypersurfaces

TL;DR

This paper presents a novel approach for projecting points onto quadratic hypersurfaces, reducing the problem to root-finding and applying splitting methods for intersection problems, with demonstrated efficiency in power systems applications.

Contribution

It introduces a complete characterization of the projection problem onto quadratic hypersurfaces and develops efficient splitting algorithms for intersection problems involving these surfaces.

Findings

Outperforms IPOPT and Gurobi in speed and feasibility

Provides a complete characterization of projection solutions

Demonstrates effectiveness on power systems problem

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.