On the tightness of SDP relaxations of QCQPs
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework fo...
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| Published in | Mathematical programming Vol. 193; no. 1; pp. 33 - 73 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.05.2022
Springer Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0025-5610 1436-4646 |
| DOI | 10.1007/s10107-020-01589-9 |
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| Abstract | Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs. |
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| AbstractList | Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs. |
| Audience | Academic |
| Author | Wang, Alex L. Kılınç-Karzan, Fatma |
| Author_xml | – sequence: 1 givenname: Alex L. orcidid: 0000-0002-4293-0359 surname: Wang fullname: Wang, Alex L. organization: Carnegie Mellon University – sequence: 2 givenname: Fatma orcidid: 0000-0001-5939-4575 surname: Kılınç-Karzan fullname: Kılınç-Karzan, Fatma email: fkilinc@andrew.cmu.edu organization: Carnegie Mellon University |
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| Keywords | Relaxation Semidefinite program Convex hull 90C22 (Semidefinite programming) 90C26 (Nonconvex programming, global optimization) 90C20 (Quadratic programming) Quadratically constrained quadratic programming Lagrange function |
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| Snippet | Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we... |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Convexity Eigenvalues Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Theoretical Tightness |
| Title | On the tightness of SDP relaxations of QCQPs |
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