Linear and parabolic relaxations for quadratic constraints

• Published in: Journal of global optimization Vol. 65; no. 3; pp. 457 - 486
• Main Authors: Domes, Ferenc, Neumaier, Arnold
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2016; Springer; Springer Nature B.V
• Subjects: Algorithms; Boxes; Cholesky factorization; Computer Science; Factorization; Filtering; Filtration; Floating point arithmetic; Interval arithmetic; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Propagation; Quadratic programming; Real Functions; Studies; 90C26 nonconvex programming, global optimization; Ellipsoid relaxations; Interval hull; 90C20 quadratic programming; 65F30 other matrix algorithms; Rounding error control; Constraint satisfaction problems; Verified computing; Linear relaxations; 65G20 algorithms with automatic result verification; Non-convex constraints; Interval analysis; Directed modified Cholesky factorization; Parabolic relaxations; Quadratic constraints
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-015-0381-5

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 Domes and Neumaier (SIAM J Matrix Anal Appl 32:262–285, 2011 ) 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.