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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 ..."
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Cited by 134 (19 self)
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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 ..."
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Cited by 110 (7 self)
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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.
A Review of Preconditioners for the Interval GaussSeidel Method
, 1991
"... . Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system ..."
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Cited by 59 (19 self)
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. Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finiteprecision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system 0 = F (M) +F 0 (X)( ~ X \Gamma M); if interval arithmetic is then used to bound the solutions of this system, the resulting box ~ X contains all roots of the nonlinear system. We may use the interval GaussSeidel method to find these solution bounds. In order to increase the overall efficiency of the interval Newton / generalized bisection algorithm, the linear interval system is multiplied by a preconditioner matrix Y before the interval GaussSeidel method is applied. Here, we review results we have obtained over the past few years concerning computation of such preconditioners. We emphasize importance and connecting relationships,...
Some tests of generalized bisection
 ACM Trans. Math. Software
, 1987
"... This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically co ..."
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Cited by 57 (5 self)
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This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasiNewton methods. A specific algorithm containing novel “expansion ” and “exclusion ” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, highdegree polynomial system that is appropriate for generalized bisection, but very difticult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.
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 ..."
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Cited by 47 (5 self)
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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
Contractor Programming
 Artificial Intelligence
"... Abstract. This paper describes a solver programming method, called contractor programming, that copes with two issues related to constraint processing over the reals. First, continuous constraints involve an inevitable step of solver design. Existing softwares provide an insufficient answer by restr ..."
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Cited by 36 (17 self)
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Abstract. This paper describes a solver programming method, called contractor programming, that copes with two issues related to constraint processing over the reals. First, continuous constraints involve an inevitable step of solver design. Existing softwares provide an insufficient answer by restricting users to choose among a list of fixed strategies. Our first contribution is to give more freedom in solver design by introducing programming concepts where only configuration parameters were previously available. Programming consists in applying operators (intersection, composition, etc.) on algorithms called contractors that are somehow similar to propagators. Second, many problems with real variables cannot be cast as the search for vectors simultaneously satisfying the set of constraints, but a large variety of different outputs may be demanded from a set of constraints (e.g., a paving with boxes inside and outside of the solution set). These outputs can actually be viewed as the result of different contractors working concurrently on the same search space, with a bisection procedure intervening in case of deadlock. Such algorithms (which are not strictly speaking solvers) will be made easy to build thanks to a new branch & prune system, called paver. Thus, this paper gives a way to deal harmoniously with a larger set of problems while giving a fine control on the solving mechanisms. The contractor formalism and the paver system are the two contributions. The approach is motivated and justified through different cases of study. An implementation of this framework named Quimper is also presented. 1
Motivations for an arbitrary precision interval arithmetic and the MPFI library
 Reliable Computing
, 2002
"... Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is t ..."
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Cited by 36 (7 self)
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Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is the dimension of the matrix and u = 1 + − 1, with 1 + the smallest floatingpoint larger than 1; this means that n must be less than 200,000, which is almost reached by modern simulations. The numerical quality of solvers is now an issue, and not only their mathematical quality. Let us cite studies performed by the CEA (French Nuclear Agency) on the simulation of nuclear plant accidents and also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic [21, 2, 18]. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but it
Robust Process Simulation Using Interval Methods
 Comput. Chem. Eng
, 1996
"... Ideally, for the needs of robust process simulation, one would like a nonlinear equation solving technique that can find any and all roots to a problem, and do so with mathematical certainty. In general, currently used techniques do not provide such rigorous guarantees. One approach to providing suc ..."
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Cited by 32 (19 self)
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Ideally, for the needs of robust process simulation, one would like a nonlinear equation solving technique that can find any and all roots to a problem, and do so with mathematical certainty. In general, currently used techniques do not provide such rigorous guarantees. One approach to providing such assurances can be found in the use of interval analysis, in particular the use of interval Newton methods combined with generalized bisection. However, these methods have generally been regarded as extremely inefficient. Motivated by recent progress in interval analysis, as well as continuing advances in computer speed and the availability of parallel computing, we consider here the feasibility of using an interval Newton/generalized bisection algorithm on process simulation problems. An algorithm designed for parallel computing on an MIMD machine is described, and results of tests on several problems are reported. Experiments indicate that the interval Newton/generalized bisection method works quite well on relatively small problems, providing a powerful method for finding all solutions to a problem. For larger problems, the method performs inconsistently with regard to efficiency, at least when reasonable initial bounds are not provided.