Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exact SDP relaxations of quadratically constrained quadratic programs
  with forest structures   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.

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The categories of this paper are: hep-th
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga