Resolution of nonlinear interval problems using symbolic interval arithmetic

An interval problem is a problem where the unknown variables take interval values. Such a problem can be defined by interval constraints, such as “the interval [ a , b ] ⊂ [ a , b ] 2 ”. Interval problems often appear when we want to analyze the behavior of an interval solver. To solve interval problems, we propose to transform the constraints on intervals into constraints on their bounds. For instance, the previous interval constraint [ a , b ] ⊂ [ a , b ] 2 can be transformed into the following bound constraints “ a ≥ min ( a 2 , ab , b 2 ) and b ≤ max ( a 2 , ab , b 2 ) ”. Classical interval solvers can then be used to solve the resulting bound constraints. The procedure which transforms interval constraints into equivalent bound constraints can be facilitated by using symbolic interval arithmetic. While classical intervals can be defined as a pair of two real numbers, symbolic intervals can be defined as a pair of two symbolic expressions. An arithmetic similar to classical interval arithmetic can be defined for symbolic intervals. The approach will be illustrated on several applications.