Results 1  10
of
14
CLP(Intervals) Revisited
, 1994
"... The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedu ..."
Abstract

Cited by 121 (18 self)
 Add to MetaCart
The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedures to be applied in the solver. This idea is embodied in a new CLP language Newton whose operational semantics is based on the notion of boxconsistency, an approximation of arcconsistency, and whose implementation uses Newton interval method. Experimental results indicate that Newton outperforms existing languages by an order of magnitude and is competitive with some stateoftheart tools on some standard benchmarks. Limitations of our current implementation and directions for further work are also identified.
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... 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 ..."
Abstract

Cited by 101 (7 self)
 Add to MetaCart
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 arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce the HansenSegupta'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 stateoftheart 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.
Interval constraint logic programming
 CONSTRAINT PROGRAMMING: BASICS AND TRENDS, VOLUME 910 OF LNCS
, 1995
"... Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate nary relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variabl ..."
Abstract

Cited by 47 (5 self)
 Add to MetaCart
Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate nary relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variables represent real values whose domains are intervals de ned in the same manner. Narrowing operators are de ned from approximations. These operators compute, from an interval and a relation, aset included in the initial interval. Sets of constraints are then processed thanks to a local consistency algorithm pruning at each stepvalues from initial intervals. This algorithm is shown to be correct and to terminate, on the basis of a certain number of properties of narrowing operators. We focus here on the description of the general framework based on approximations, on its application to interval constraint solving over continuous and discrete quantities, we establish a strong link between approximations and local consistency notions and show that arcconsistency is an instance of the approximation framework. We nally describe recentwork on di erent variants of the initial algorithm proposed by John Cleary and developed by W. Older and A. Vellino which havebeen proposed in this context. These variants address four particular points: generalization of the constraint language, improvement of domain reductions, e ciency of the computation and nally, cooperation with other solvers. Some open questions are also identi ed. 1
Fast And Parallel Interval Arithmetic
 BIT
"... . Inmumsupremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpointradius interval arithmetic compared to power set operations is uniformly bounded by a factor 1.5 in radius. This is true for the four bas ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
. Inmumsupremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpointradius interval arithmetic compared to power set operations is uniformly bounded by a factor 1.5 in radius. This is true for the four basic operations as well as for vector and matrix operations, over real and over complex numbers. Moreover, we describe an implementation of midpointradius interval arithmetic entirely using BLAS. Therefore, in particular, matrix operations are very fast on almost any computer, with minimal eort for the implementation. Especially, with the new denition it is seemingly the rst time that full advantage can be taken of the speed of vector and parallel architectures. The algorithms have been implemented in the Matlab interval toolbox INTLAB. Keywords. Interval arithmetic, parallel computer, BLAS, midpointradius, inmumsupremum, AMS subject classications. 65G10 1. Introduction and notati...
Newton: Constraint Programming over Nonlinear Constraints
 SCIENCE OF COMPUTER PROGRAMMING
, 1998
"... 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 analy ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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 boxconsistency, which approximates arcconsistency, a notion wellknown in articial intelligence. Boxconsistency 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 equationsolving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equationsolving benchmarks and outperforms the intervalbased 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 NPhard) 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...
Three Cuts for Accelerated Interval Propagation
 MIT, Artif. Intell. Lab
, 1995
"... 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 c ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
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 ...
Local Convergence of an InexactRestoration Method and Numerical Experiments 1
"... Communicated by C. T. Leondes 1This work was supported by PRONEXCNPq/FAPERJ Grant E26/171.164/2003 APQ1, FAPESP Grants 03/091696 and 01/045974, and CNPq. The authors are indebted to Juliano B. Francisco and Yalcin Kaya for their careful reading of the first draft of this paper. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Communicated by C. T. Leondes 1This work was supported by PRONEXCNPq/FAPERJ Grant E26/171.164/2003 APQ1, FAPESP Grants 03/091696 and 01/045974, and CNPq. The authors are indebted to Juliano B. Francisco and Yalcin Kaya for their careful reading of the first draft of this paper.
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... 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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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 arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce HansenSegupta 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 stateoftheart continuatio...
A Geometric Optimization Approach to Detecting and Intercepting Dynamic Targets
"... Abstract — A methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane. The sensingpursuit problem considered in this paper is analogous to the Marco Polo game, in which the pursuer must capture multiple mobile targets that are ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract — A methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane. The sensingpursuit problem considered in this paper is analogous to the Marco Polo game, in which the pursuer must capture multiple mobile targets that are sensed intermittently, and with very limited information. In this paper, the mobile sensor network consists of a set of robotic sensors that must track and capture mobile targets based on the information obtained through cooperative detections. Since the sensors are installed on robotic platforms and have limited range, the geometry of the platforms and of the sensors fieldofview play a key role in obstacle avoidance and target detection. Thus, a new cell decomposition approach is presented to formulate the probability of detection and the cost of operating the robots based on the geometric properties of the network. Numerical simulations verify the validity and flexibility of our methodology. I.
Interval Methods Revisited
, 1995
"... 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 co ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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...