Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

We study a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs , which contain no off-diagonal terms x j x k for j ≠ k , and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complem...

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Vydáno v:Mathematical programming Ročník 181; číslo 1; s. 1 - 17
Hlavní autoři: Burer, Samuel, Ye, Yinyu
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2020
Springer Nature B.V
Témata:
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Constraints
Formulations
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Polynomials
Theoretical
Variables
90C20
Low-rank solutions
90C26
Semidefinite relaxation
Quadratically constrained quadratic programming
90C22
ISSN:0025-5610, 1436-4646
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Popis
Shrnutí:We study a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs , which contain no off-diagonal terms x j x k for j ≠ k , and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-time-checkable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.
Bibliografie:ObjectType-Article-1
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01367-2