Solving algebraic problems with high accuracy
Proc. of the symposium on A new approach to scientific computation
SIAM Review
Convex quadratic and semidefinite programming relaxations in scheduling
Journal of the ACM (JACM)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Convexification and Global Optimization in Continuous And
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Super-fast validated solution of linear systems
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
SCAN '06 Proceedings of the 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics
A scaling algorithm for polynomial constraint satisfaction problems
Journal of Global Optimization
Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations
Mathematical Programming: Series A and B
Optimization Methods & Software - GLOBAL OPTIMIZATION
Optimization Methods & Software - GLOBAL OPTIMIZATION
GloMIQO: Global mixed-integer quadratic optimizer
Journal of Global Optimization
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This paper discusses the rigorous enclosure of an ellipsoid by a rectangular box, its interval hull, providing a convenient preprocessing step for constrained optimization problems. A quadratic inequality constraint with a strictly convex Hessian matrix defines an ellipsoid. The Cholesky factorization can be used to transform a strictly convex quadratic constraint into a norm inequality, for which the interval hull is easy to compute analytically. In exact arithmetic, the Cholesky factorization of a nonsingular symmetric matrix exists iff the matrix is positive definite. However, to cope efficiently with rounding errors in inexact arithmetic is nontrivial. Numerical tests show that even nearly singular problems can be handled successfully by our techniques. To rigorously account for the rounding errors involved in the computation of the interval hull and to handle quadratic inequality constraints having uncertain coefficients, we define the concept of a directed Cholesky factorization and give two algorithms for computing one. We also discuss how a directed Cholesky factorization can be used for testing positive definiteness. Some numerical tests are given in order to exploit the features and boundaries of the directed Cholesky factorization methods.