For Interval Computations, if Absolute-Accuracy Optimization is NP-Hard, then so is Relative-Accuracy Optimization

One of the basic problems of interval computations is to compute a range of a given function f(x1,...,xn) over a given box (i.e., to compute the maximum and the minimum of the function on the box). For many classes of functions (e.g., for quadratic functions) this problem is NP-hard; it is even NP-hard if instead of computing the minimum and maximum exactly, we want to compute them with a given (absolute) accuracy. In practical situations, it is more realistic to ask for a relative accuracy; are the corresponding problems still NP-hard? We show that under some reasonable conditions, NP-hardness of absolute-accuracy optimization implies that relative-accuracy optimization is NP-hard as well.