An Interval Constraint Propagation Algorithm Exploiting Monotonicity

When a function f is monotonic w.r.t. a variable x in a given box, it is well-known that the monotonicity-based interval extension of f computes a sharper image than the natural interval extension does. Indeed, the overestimation due to the variable x with multiple occurrences in f disappears. However, monotonicity has not been exploited in interval filtering/contraction algorithms for solving systems of nonlinear constraints over the reals. We propose in this paper a new interval constraint propagation algorithm, called MOnotonic Hull Consistency (Mohc), that exploits monotonicity of functions. The propagation is standard, but the Mohc-Revise procedure, used to filter/contract the variable domains involved in an individual constraint, is novel. This revise procedure uses two main bricks for narrowing intervals of the variables involved in f . One procedure is a monotonic version of the well-known HC4-Revise. A second procedure performs a dichotomic process calling interval Newton iterations, close to (while less costly than) the procedure BoxNarrow used in the Box contraction algorithm. When f is monotonic w.r.t. every variable with multiple occurrences, Mohc is proven to compute the optimal/sharpest box enclosing all the solutions of the constraint (hull consistency). Experiments show that Mohc is a relevant approach to handle constraints having several variables with multiple occurrences, contrarily to HC4 and Box.