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Exact SDP relaxations of quadratically constrained quadratic programs with forest structures
- Title
- Exact SDP relaxations of quadratically constrained quadratic programs with forest structures
- Authors
- Azuma G.; Fukuda M.; Kim S.; Yamashita M.
- Ewha Authors
- 김선영
- SCOPUS Author ID
- 김선영
- Issue Date
- 2022
- Journal Title
- Journal of Global Optimization
- ISSN
- 0925-5001
- Citation
- Journal of Global Optimization vol. 82, no. 2, pp. 243 - 262
- Keywords
- Exact semidefinite relaxations; Forest graph; Quadratically constrained quadratic programs; The rank of aggregated sparsity matrix
- Publisher
- Springer
- Indexed
- SCIE; SCOPUS
- Document Type
- Article
- Abstract
- We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than n- 1 and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- DOI
- 10.1007/s10898-021-01071-6
- Appears in Collections:
- 자연과학대학 > 수학전공 > Journal papers
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