# The Geometry of SDP-Exactness in Quadratic Optimization

arXiv: https://arxiv.org/abs/1804.01796 · HTML: https://mlsanity.com/paper/1804.01796/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2018.

## Built on by (most cited first, 3 of 3)

- [Advances in Inference and Representation for Simultaneous Localization and Mapping](https://mlsanity.com/paper/2103.05041/) (2021)
- [On the Tightness of Semidefinite Relaxations for Rotation Estimation](https://mlsanity.com/paper/2101.02099/) (2021)
- [A Thorough View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy](https://mlsanity.com/paper/2102.08019/) (2021)

## Abstract (arXiv)

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.