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Linear and parabolic relaxations for quadratic constraints. (English) Zbl 1351.90131
Summary: This paper presents new techniques for filtering boxes in the presence of an additional quadratic constraint, a problem relevant for branch and bound methods for global optimization and constraint satisfaction. This is done by generating powerful linear and parabolic relaxations from a quadratic constraint and bound constraints, which are then subject to standard constraint propagation techniques. The techniques are often applicable even if the original box is unbounded in some but not all variables. As an auxiliary tool – needed to make our theoretical results implementable in floating-point arithmetic without sacrificing mathematical rigor – we extend the directed Cholesky factorization from F. Domes and A. Neumaier [SIAM J. Matrix Anal. Appl. 32, No. 1, 262–285 (2011; Zbl 1242.90152)] to a partial directed Cholesky factorization with pivoting. If the quadratic constraint is convex and the initial bounds are sufficiently wide, the final relaxation and the enclosure are optimal up to rounding errors. Numerical tests show the usefulness of the new factorization methods in the context of filtering.

MSC:
90C26 Nonconvex programming, global optimization
90C20 Quadratic programming
65F30 Other matrix algorithms (MSC2010)
65G20 Algorithms with automatic result verification
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