Solving Polynomial Systems Using a Branch and Prune Approach

Abstract

This paper presents Newton, a branch and prune algorithm used to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists of enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Sengupta narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

New interval methodologies for reliable chemical process modeling

“…Ratscheck and Rokne, 1984;Makino and Berz, 1999;Jansson, 2000); 2. Techniques that involve changes to the methodology at the level of the nonlinear equation system (e.g., Alefeld et al, 1998;Granvilliers and Hains, 2000;van Hentenryck et al, 1997;Ratz, 1994;Herbort and Ratz, 1997;Yamamura et al, 1998;Yamamura and Nishizawa, 1999); 3. Methods that seek to make improvements in solving the linear interval system defined by Eq.…”

Hull Consistency under Monotonicity

“…Since [x] is convex, the corresponding point is inside [x] and its i th component is x * i − ε i , which contradicts the fact that x * i is the infimum among the solutions. Let us compare OCTUM with three other standard generic contractors, namely HC4 [2,9], BOX [4,18] and 3B [15]. We have implemented a very naive method for detecting monotonicity, using a simple interval evaluation of the gradient that is systematically computed for every box (a better method would be to manage flags w.r.t.…”

Novel Approaches to Numerical Software with Result Verification

“…These techniques have been further developed and implemented in Numerica [67]. Furthermore box-consistency has been shown to drastically improve hull-consistency for a large set of problems in [66].…”