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Semidefinite Programming Relaxation for Nonconvex Quadratic Programs

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Tetsuya Fujie and Masakazu Kojima

Any quadratic inequality in the n-dimensional
Euclidean space can be relaxed into a linear matrix inequality in
(1+n) times (1+n) symmetric matrices. Based on this principle,
we extend the Lovasz-Schrijver SDP (semidefinite programming)
relaxation developed for a 0-1 integer program to a general
nonconvex QP (quadratic program), and present some fundamental
characterization of the SDP relaxation including its equivalence
to a relaxation using convex-quadratic valid inequalities for
the feasible region of the QP.
Report B-298,
Dept. of Mathematical and Computing Sciences, Tokyo Institute
of Technology, Tokyo, May 1995; revised August 1996. To
appear in Journal of Global Optimization.