Exact SDP relaxations of quadratically constrained quadratic programs with forest structures
- 주제(키워드) Exact semidefinite relaxations , Forest graph , Quadratically constrained quadratic programs , The rank of aggregated sparsity matrix
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 Springer
- 발행년도 2022
- 총서유형 Journal
- URI http://www.dcollection.net/handler/ewha/000000190363
- 본문언어 영어
- Published As https://doi.org/10.1007/s10898-021-01071-6
초록/요약
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.
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