This paper presents Newton, a branch & prune algorithm 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 in 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-Segupta's 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.

The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory are collaborating on the development of the software package MINPACK-2. As part of the MINPACK-2 project we are developing a collection of significant optimization problems to serve as test problems for the package. This report describes the problems in the preliminary version of this collection. 1 Introduction The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory have initiated a collaboration for the development of the software package MINPACK-2. As part of the MINPACK-2 project, we are developing a collection of significant optimization problems to serve as test problems for the package. This report describes some of the problems in the preliminary version of this collection. Optimization software has often bee...

Non-linear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifier-free equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of inner-approximation of real relations.

. A new approach is proposed for finding all ffl--feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem(P) whose multiple global minimum solutionswith a zero objectivevalue (if any)correspond to all solutionsof the initial constrainedsystem of equalities. All ffl--globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite ffl--convergence to each of the multiple global minima of (P) through the successive refinement of a convexrelaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products o...

Homotopy continuation methods have been proven to be reliable for computing numerically approximations to all isolated solutions of polynomial systems. The performance of...

This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analysis, such as the interval Newton method. Its key conceptual idea is to introduce the notion of box-consistency, which approximates arc-consistency, a notion well-known in articial intelligence. Box-consistency achieves an eective pruning at a reasonable computation cost and generalizes some traditional interval operators. Newton has been applied to numerous applications in science and engineering, including nonlinear equation-solving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equation-solving benchmarks and outperforms the interval-based methods we are aware of on optimization problems. Key words: Constraint Programming, Nonlinear Programming, Interval Reasoning 1 Introduction Many applications in science and engineering (e.g., chemistry, robotics, economics, mechanics) require nding all isolated solutions to a system of nonlinear real constraints or nding the minimum value of a nonlinear function subject to nonlinear constraints. These problems are dicult due to their inherent computational complexity (i.e., they are NP-hard) and due to the numerical issues involved to guarantee correctness (i.e., nding all solutions or the global optimum) and to ensure termination. Preprint submitted to Elsevier Preprint 11 June 2001 Newton is a constraint programming language designed to support this class of applications. It originates from an attempt to reconcile the declarative nature of CLP(Intervals) languag...

Abstract—This paper proposes a new perspective for solving systems of complex nonlinear equations by simply viewing them as a multiobjective optimization problem. Every equation in the system represents an objective function whose goal is to minimize the difference between the right and left terms of the corresponding equation. An evolutionary computation technique is applied to solve the problem obtained by transforming the system into a multiobjective optimization problem. The results obtained are compared with a very new technique that is considered as efficient and is also compared with some of the standard techniques that are used for solving nonlinear equations systems. Several well-known and difficult applications (such as interval arithmetic benchmark, kinematic application, neuropsychology application, combustion application, and chemical equilibrium application) are considered for testing the performance of the new approach. Empirical results reveal that the proposed approach is able to deal with highdimensional equations systems. Index Terms—Computational intelligence, evolutionary multiobjective optimization, metaheuristics, nonlinear equation systems. I.

This paper presents Newton, a branch & prune algorithm 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 in enforcing at each node of the search tree a unique local consistency condition, called boxconsistency, 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 Hansen-Segupta 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 continuatio...

This paper addresses the problem of nonlinear multivariate root finding. In an earlier paper we describe a system called Newton which finds roots of systems of nonlinear equations using refinements of interval methods. The refinements are inspired by AI constraint propagation techniques. Newton is competitive with continuation methods on most benchmarks and can handle a variety of cases that are infeasible for continuation methods. This paper presents three "cuts" which we believe capture the essential theoretical ideas behind the success of Newton. This paper describes the cuts in a concise and abstract manner which, we believe, makes the theoretical content of our work more apparent. Any implementation will need to adopt some heuristic control mechanism. Heuristic control of the cuts is only briefly discussed here. Copyright c fl Massachusetts Institute of Technology, 1995 This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of ...

This paper presents a branch & cut algorithm to find all isolated solutions of a system of polynomial constraints. Our findings show that fairly straightforward refinements of interval methods inspired by AI constraint propagation techniques result in a multivariate root finding algorithm that is competitive with continuation methods on most benchmarks and which can solve a variety of systems that are totally infeasible for continuation methods. For example, we can solve the Broyden Banded function benchmark for hundreds of variables and hundreds of cubic equations. 1 Introduction In this paper we consider the classical problem of finding solutions to systems of nonlinear polynomial equations in many variables. This is an old problem with many applications and a large literature. In engineering applications it is generally sufficient to find an assignment of floating point numbers to variables such that the given constraints are satisfied to within the uncertainty introduced by the q...