Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

摘要

We consider a symmetric matrix, the entries of which depend linearly on someparameters. The domains of the parameters are compact real intervals. Weinvestigate the problem of checking whether for each (or some) setting of theparameters, the matrix is positive definite (or positive semidefinite). Westate a characterization in the form of equivalent conditions, and also proposesome computationally cheap sufficient\,/\,necessary conditions. Our resultsextend the classical results on positive (semi-)definiteness of intervalmatrices. They may be useful for checking convexity or non-convexity in globaloptimization methods based on branch and bound framework and using intervaltechniques.