Tolerances, robustness and parametrization of matrix properties related to optimization problems

Milan Hladík

When we speak about parametric programming, sensitivity analysis, or related topics, we usually mean the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied. In this presentation, we turn to another question. Suppose that A is a matrix having a specific property P. What are the maximal allowable variations of the data such that the property still remains valid for the matrix? We study two basic forms of perturbations. The first is a perturbation in a given direction, which is closely related to parametric programming. The second type consists of all possible data variations in a neighbourhood specified by a certain matrix norm; this is related to the tolerance approach to sensitivity analysis, or to stability. The matrix properties that will be discussed are positive definiteness; P-matrix, H-matrix and P-matrix property; total positivity and inverse non-negativity.