Interval solution of nonlinear equations using linear programming

Yamamura, Kenichiro

doi:10.1109/iscas.1997.621843

Cited by 27 publications

(40 citation statements)

References 33 publications

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“…Benchmarks eco6, katsura5, katsura6, katsura7, tangets2, ipp, assur44, cyclic5, tangents0, chemequ, noon5, geneig, kinema, reimer5, and camera1s were taken from Verschelde's web site, 10 kin2 from [60], didrit from [15], lee from [29], and finally yama194, yama195, and yama196 from [63]. The most challenging benchmark is stewgou40 [16].…”

Section: Resultsmentioning

confidence: 99%

“…the Quad algorithm where product terms are relaxed with the rAI heuristic. Note that the BP(box+simplex) solver implements a strategy that is slightly different from the approach of Yamamura, Kawata, and Tokue [63].…”

Section: Resultsmentioning

confidence: 99%

“…In the interval analysis community, Yamamura, Kawata, and Tokue [63] used a simple linear relaxation procedure where nonlinear terms are replaced by new variables to prove that some box does not contain solutions. No convex/concave outerestimations are proposed to obtain a better approximation of the nonlinear terms.…”

Section: Discussionmentioning

confidence: 99%

“…No convex/concave outerestimations are proposed to obtain a better approximation of the nonlinear terms. As pointed out by Yamamura, Kawata, and Tokue, this approach is well adapted to quasi-linear problems: "This test is much more powerful than the conventional test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms" [63].…”

Section: Discussionmentioning

confidence: 99%

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Efficient and Safe Global Constraints for Handling Numerical Constraint Systems

Lebbah

Michel

Rueher

et al. 2005

SIAM J. Numer. Anal.

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Abstract. Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.Key words. systems of equations and inequalities, constraint programming, reformulation linearization technique, global constraint, interval arithmetic, safe rounding AMS subject classifications. 65H10, 65G40, 65H20, 93B18, 65G20 DOI. 10.1137/S00361429034361741. Introduction. Many applications in engineering sciences require finding all isolated solutions to systems of constraints over real numbers. These systems may be nonpolynomial and are difficult to solve: the inherent computational complexity is NP-hard and numerical issues are critical in practice (e.g., it is far from being obvious to guarantee correctness and completeness as well as to ensure termination). These systems, called numerical CSP (constraint satisfaction problem) in the rest of this paper, have been approached in the past by different interesting methods:

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Efficient and Safe Global Constraints for Handling Numerical Constraint Systems

Lebbah

Michel

Rueher

et al. 2005

SIAM J. Numer. Anal.

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“…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.…”

Section: Interval-newton Methodsmentioning

confidence: 99%

New interval methodologies for reliable chemical process modeling

Gau

Stadtherr

2002

Computers & Chemical Engineering

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The use of interval methods, in particular interval-Newton/generalized-bisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method, is presented, as is a new scheme for selection of the real point used in formulating the interval-Newton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the intervalNewton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.

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