The geometry of SDP-exactness in quadratic optimization

• Published in: Mathematical programming Vol. 182; no. 1-2; pp. 399 - 428
• Main Authors: Cifuentes, Diego, Harris, Corey, Sturmfels, Bernd
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020; Springer Nature B.V
• Subjects: Algebra; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Euclidean geometry; Full Length Paper; Geometry; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Quadratic equations; Semidefinite programming; Theoretical; Semidefinite programming; Quadratic optimization; 68W30 (secondary); 90C22 (primary); 14P10; Convex relaxation; Algebraic degree; 14C17

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...