Interval convex quadratic programming problems in a general form

Abstract

This paper addresses the problem of computing the minimal and the maximal optimal value of a convex quadratic programming (CQP) problem when the coefficients are subject to perturbations in given intervals. Contrary to the previous results concerning on some special forms of CQP only, we present a unified method to deal with interval CQP problems. The problem can be formulated by using equation, inequalities or both, and by using sign-restricted variables or sign-unrestricted variables or both. We propose simple formulas for calculating the minimal and maximal optimal values. Due to NP-hardness of the problem, the formulas are exponential with respect to some characteristics. On the other hand, there are large sub-classes of problems that are polynomially solvable. For the general intractable case we propose an approximation algorithm. We illustrate our approach by a geometric problem of determining the distance of uncertain polytopes. Eventually, we extend our results to quadratically constrained CQP, and state some open problems.