Combining Interval Constraint Solving Methods

2. Interval constraint solving In the following, let R be the set of real numbers. In this section, the notions of interval arithmetic, constraint (nonlinear equality/inequality over the real numbers) and interval-based constraint solving are introduced. 2.1. Interval arithmetic Let R∞ be the set of reals compactified with the infinities {−∞, +∞} in the obvious way, and F ⊂ R a finite subset of reals corresponding to binary floating-point numbers in a given format [15]. Let F∞ be F ∪ {−∞, +∞}. For every r ∈ R, let r↑ be the smallest element in F∞ greater or equal than r , and r↓ the greatest element in F∞ smaller or equal than r . Given a ∈ F ∪ {−∞} and b ∈ F ∪ {+∞} such that a b, let [a, b] denote the set {r ∈ R | a r b} called interval. The left bound of an interval I is denoted I , the right bound I . An interval is said
c 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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2 first order Taylor expansion and preconditioning, in order to increase the precision of computed domains. The interaction of box consistency and interval Newton is a form of coarse-grained cooperation since both only alternate after the computation of a fixed-point. Our domain pruning algorithm combines the three aforementioned techniques under the guidance of heuristics/theoretic criteria in order to prevent slow convergence phenomena and useless redundant computations. These heuristics are based on some knowledge of the solvers’ behaviors (w.r.t. the constraints’ expressions, the size of domains, etc.), and mainly fix a partial ordering of application of the solvers and avoid the computation of fixed-points of each iterative algorithm. A set of experimental results on a set of benchmarks from interval analysis and homotopy continuation validate the algorithm, that is also compared with the constraint solving algorithm of Numerica [26] and PHC [28]. We believe that the contribution of this paper is more a methodology of combination of solvers than the originality of the proposed combination. The outline of this paper is the following: Section 2 introduces some materials from interval arithmetic and local consistency techniques. Section 3 presents the novel domain pruning algorithm along with a set of examples motivating the combination approach. Experimental results are discussed in Section 4. Directions for future research are sketched in Section 5.
Computing, 2000. Kluwer Academic Publishers.
Combining Interval Constraint Solving Methods
Laurent Granvilliers
IRIN, Universit´ e de Nantes B.P. 92208, F-44322 Nantes Cedex 3, France Laurent.Granvilliers@irin.univ-nantes.fr Abstract. This paper discusses the combination of interval methods for solving nonlinear systems, and focuses on interval-based local consistencies and the multidimensional interval Newton method. The existing algorithms are revised in order to introduce a more cooperative strategy of application of elementary solvers that may prevent slow convergence phenomena and unnecessary computations. A set of experiments for comparing several algorithms is reported. Keywords: Interval analysis, constraint solving, nonlinear systems, local consistency, interval Newton method.
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3 to be canonical if none of its bounds is an infinity and if it contains at most two elements of F. Given a set ρ ⊆ R, let ¾ρ denote the interval hull of ρ defined as the smallest interval enclosing ρ, that is [min(ρ)↓, max(ρ)↑]. Assuming that no confusion is possible, any tuple of intervals (I1 , . . . , Ik ) is identified with the Cartesian product I1 ×· · ·× Ik called a box. Interval arithmetic has been designed by R. E. Moore [20] for automatically computing roundoff error bounds of numerical computations. In this framework, a real quantity is represented by an enclosing interval, and an interval operation is a set theoretic extension of the corresponding real one, usually implemented by floating point computations operating on the bounds of the parameters. The main concept from this framework is the notion of interval extension (also called inclusion function) of a continuous function f over R, that is an interval function designed to compute outer approximations of the range of f over a domain. DEFINITION 1. (Interval extension). A function F : Ik → I is an interval extension of f : Rk → R if for every I ∈ Ik in the domain of f , the range of f over I is included in the interval F (I). The obvious interval extension of a function f , called natural interval extension, corresponds to a componentwise extension of a given expression of f , where a real r is replaced with (approximated by) ¾{r }, a variable with an interval variable, and an operation with the corresponding interval one. The k-order Taylor interval extension results from a k-order Taylor expansion of f around a point, where the derivative f (k)(ξ ) involved in the error term is bounded by the evaluation of an interval extension of f (k) over the whole domain. Some other interval extensions are presented in [23, 24]. 2.2. Constraints and constraint solving In the following, let F be a set of operation symbols, R = {=, , } a set of relation symbols, Σ = R, F , R a real-based structure, and X = {x1 , x2 , . . . , xn } a set of real-valued variables. Given a set S , a tuple s = (s1 , . . . , sn ) ∈ S n , and a set of variables v = {xi1 , . . . , xik } ⊆ X , let πv (s) denote the tuple (si1 , . . . , sik ) (the projection of s over v ). A term is a syntactic expression built from real numbers, operations and variables. A constraint is a conjunction of atomic formulas made from terms and relation symbols. Given a constraint c, let ρc denote the relation defined by c in the standard interpretation of Σ (usual interpretation over the set of real numbers). Two constraints c and c′ over the same set of variables are said to be equivalent if ρc = ρc′ .