The Geometry of SDP-Exactness in Quadratic Optimization

TL;DR

This paper investigates the geometric conditions under which quadratic optimization problems are exactly solvable via semidefinite relaxation, characterizing the boundary and degree of the solution region.

Contribution

It provides a geometric and algebraic characterization of the region where semidefinite relaxation yields exact solutions for quadratic problems.

Findings

Characterizes the algebraic boundary of the SDP-exactness region.

Derives a formula for the degree of this boundary.

Describes the structure of spectrahedral shadows surrounding the variety.

Abstract

Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree.