QUAD-CLP(R): Adding the Power of Quadratic Constraints

We report on a new way of handling non-linear arithmetic constraints and its implementation into the QUAD-CLP(R) language. Important properties of the problem at hand are a discretization through geometric equivalence classes and decomposition into convex pieces. A case analysis of those equivalence classes leads to a relaxation (and sometimes recasting) of the original constraints into linear constraints, much easier to handle. Complementing earlier expositions in 18] and 19], the present focus is on applications upholding its worth. 1. Motivation. This paper presents the constraint programming language QUAD-CLP(R) which ooers a powerful novel solving strategy for non-linear arithmetic constraints under the computing paradigm of logic programming. Emphasis will be given here to the techniques involved in the constraint solver for quadratic constraints over R and to applications making use of this added power. Despite the enormous potential of non-linear arithmetic constraints in several spheres of scientiic activity, typical eeorts to provide for them amidst constraint languages have brought mostly disappointments as the resulting solvers either lacked eeectiveness or scalability. The delay strategy implemented in languages such as CLP(R) 10] and PRO-LOG III 1] yields an incomplete solver which will be eeective only if the problem under attack is such that reasoning about linear constraints ultimately becomes suu-cient. Unfortunately, this is seldom the case for interesting problems, even very simple ones. One classic example is the multiplication of complex numbers, which can be expressed as cmult((R1,I1),(R2,I2),(R1*R2-I1*I2,R1*I2+R2*I1)) in predicate calculus. On the other hand, languages like CAL 20] and RISC-CLP(Real) 7] bear witness that the price to pay to achieve a complete solver seems to be the use of costly computational algebra techniques which connne their usefulness to very small (albeit interesting) problems. Our approach, introduced in 18], takes advantage of the ease with which quadratic constraints can be replaced or approximated by linear constraints. It is therefore especially well-suited to problems involving quadratic and linear constraints. There