The geometry of SDP-exactness in quadratic optimization

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...

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Bibliographic Details
Published in:Mathematical programming 2020-07, Vol.182 (1-2), p.399-428
Main Authors: Cifuentes, Diego, Harris, Corey, Sturmfels, Bernd
Format: Article
Language:English
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
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Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01399-8