A revised monotonicity-based method for computing tight image enclosures of functions

The computation of tight interval image enclosures of functions over bounded variable domains is in the heart of interval-based branch and bound optimization (and constraint satisfaction) solvers. Interval arithmetic extends arithmetic operators, such as $$+$$ + , −, $$*$$ ∗ , $$\backslash $$ \ , $$\sin $$ sin , $$\cos $$ cos , etc., to intervals. In this way, the operators can be used directly for computing image enclosures of real functions over bounded domains (i.e., natural interval evaluations). Importantly, it is widely recognized that when a function f is monotonic w.r.t. some variable(s) in a given domain, we can compute tighter images of f on this domain than by using natural interval evaluations. This work presents a more general monotonicity-based method that may be applied even if the function is non-monotonic w.r.t. its variables. The method combines basic interval-based filtering techniques with a straightforward analysis of function derivatives. First, filtering based on partial derivatives detects sub-intervals in the domain where the function certainly increase or decrease. Then, we can determine in which subdomains within the interval the value should be maximizing (or minimizing) the function. Finally, we use the natural interval evaluation on the subdomains where f is maximized to compute an upper bound of the enclosure. We show that this method is equivalent to computing an enclosure by using the traditional method when f is monotonic. However, it may be more effective when f is not.