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 inMathematical programming Vol. 182; no. 1-2; pp. 399 - 428
Main Authors Cifuentes, Diego, Harris, Corey, Sturmfels, Bernd
Format Journal Article
LanguageEnglish
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
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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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01399-8