## A Constraint Satisfaction Approach to a Circuit Design Problem (1998)

Citations: | 21 - 1 self |

### BibTeX

@MISC{Puget98aconstraint,

author = {Jean-François Puget and Pascal Van Hentenryck},

title = {A Constraint Satisfaction Approach to a Circuit Design Problem},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

A classical circuit-design problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc-1 workstations) but they state that "at this time, we know no other method which has been applied to this circuit design problem and which has led to the same guaranteed result of locating exactly one solution in this huge domain, completed with a reliable error estimate." The present paper gives a novel branch-and-prune algorithm that obtains a unique safe box for the above system within reasonable computation times. The algorithm combines traditional interval techniques with an adaptation of discrete constraint-satisfaction techniques to continuous problems. Of particular interest is the simplicity o...

### Citations

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Citation Context ...VALUE INTERVAL EXTENSION The second interval extension is based on the Taylor expansion around a point. This extension is an example of centered forms that are interval extensions introduced by Moore =-=[17]-=- and have been studied by many authors, since they have important properties. The mean value interval extension of a function is parametrized by the intervals for the variables in the function. It als... |

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Citation Context ...m was applied to find all solutions of the transistor modelling problem. Branching was applied until a safe box or a box of width smaller than 10 −8 was obtained. The algorithm returned a unique box x=-=[1]-=- =0.8999999 +[0.48517e −7 , 0.566954e −7 ] x[2] =0.4499874 +[0.6902216e −7 , 0.7493801e −7 ] x[3] =1.00000648 +[0.60195e −9 , 0.43303e −8 ] x[4] =2.00006854 +[0.5787e −10 , 0.319179e −8 ] x[5] =7.9999... |

423 |
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Citation Context ...m in particular, better performance can be obtained by using a stronger local consistency condition that we call box(2)-consistency. Box(2)-consistency is in fact an approximation of path consistency =-=[16]-=- and is related to some consistency notions presented in L’Homme [14]. 5.3.1. Informal presentation Box(2)-consistency generalizes box(1)-consistency by projecting all but two variables. The original ... |

169 | Older: Applying Interval Arithmetic to Real, Integer and Bolean Constraints
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Citation Context ...CIRCUIT DESIGN PROBLEM 87 to solve discrete combinatorial problems [e.g., 16, 15]. They have been adapted to continuous problems [e.g., 5, 14] and used inside systems such as BNR- Prolog and CLP(BNR) =-=[3, 21]-=- and many systems since then. The techniques used in systems like BNR-Prolog and CLP(BNR) are weaker than box(1)-consistency, since they decompose all constraints into ternary constraints on distinct ... |

130 | ROKNE New Computer Methods for Global optimization - RATSCHEK, J - 1988 |

121 | Hentenryck. CLP(intervals) revisited
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Citation Context ... starts with an informal discussion, then specifies the pruning operator, and presents a simple implementation. ⋆ 5.2.1. Informal presentation The first fundamental idea underlying box(1)-consistency =-=[2]-=- is to project all variables but one or, more precisely, to replace all variables but one by their intervals. This produces a stronger pruning than box(0)-consistency but, of course, at a higher cost.... |

101 | Solving Polynomial Systems Using a Branch and Prune Approach
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- 1997
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Citation Context ...within reasonable computation times. ⋆ The main novelty of the procedure is in the way in which constraints are used to prune the search space. The pruning techniques, some of which were presented in =-=[24]-=- and some of which are novel, are based on constraint- satisfaction techniques from artificial intelligence and are particularly effective when far from a solution. These techniques are thus orthogona... |

80 |
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Citation Context ... branchings. The paper also indicated that box(2)-consistency may be too strong a local condition for many problems, since it is slower than box(1)-consistency on benchmarks from continuation methods =-=[25]-=-. An interesting avenue of research is to characterize more formally the class of applications for which box(1)- and box(2)-consistency are effective pruning techniques. jogo436.tex; 30/06/1998; 12:38... |

67 | Constraint arithmetic on real intervals
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Citation Context ...CIRCUIT DESIGN PROBLEM 87 to solve discrete combinatorial problems [e.g., 16, 15]. They have been adapted to continuous problems [e.g., 5, 14] and used inside systems such as BNR- Prolog and CLP(BNR) =-=[3, 21]-=- and many systems since then. The techniques used in systems like BNR-Prolog and CLP(BNR) are weaker than box(1)-consistency, since they decompose all constraints into ternary constraints on distinct ... |

57 |
Bounding Solutions of Systems of Equations Using Interval Analysis
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Citation Context ...on is a monotonic interval extension. It is interesting to note that box consistency on the mean value interval extension of a system of constraints is closely related to the Hansen-Sengupta operator =-=[8]-=-, which is an improvement over Krawczyk’s operator [13]. Hansen and Smith [9] also argue that these operators are more effective when the interval Jacobian of the system is diagonally dominant and the... |

52 |
Newton-algorithmen zur bestimmung von nullstellen mit fehlerschranken
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Citation Context ...g to note that box consistency on the mean value interval extension of a system of constraints is closely related to the Hansen-Sengupta operator [8], which is an improvement over Krawczyk’s operator =-=[13]-=-. Hansen and Smith [9] also argue that these operators are more effective when the interval Jacobian of the system is diagonally dominant and they suggest conditioning the system S.For the purpose of ... |

51 | Extending Prolog with Constraint Arithmetic on Real Intervals - Older, Vellino - 1990 |

50 | Preconditioners for the interval Gauss-Seidel method
- Kearfott
- 1990
(Show Context)
Citation Context ...st conditioning the system S.For the purpose of this paper, we simply assume that we have a conditioning operator cond(S, ⃗I) and use the notation τc(S, ⃗I) to denote τ(cond(S, ⃗I), ⃗I). See Kearfott =-=[11, 12]-=- for extensive coverage of conditioners. 6.3. THE BRANCH-AND-PRUNE ALGORITHM We are now in position to reconsider our branch-and-prune algorithm. The new version, given in Figure 4, differs in two way... |

37 |
An Interval Newton Method
- Hansen, Greenberg
- 1983
(Show Context)
Citation Context ... −7 ] x[3] =1.00000648 +[0.60195e −9 , 0.43303e −8 ] x[4] =2.00006854 +[0.5787e −10 , 0.319179e −8 ] x[5] =7.9999714 +[0.3767867e −7 , 0.4259589e −7 ] x[6] =7.99969268 +[0.14994e −8 , 0.692803e −8 ] x=-=[7]-=- =5.00003127 +[0.338646e −8 , 0.848255e −8 ] x[8] =0.99998772 +[0.69887e −9 , 0.621097e −8 ] x[9] =2.00005248 +[0.47411e −9 , 0.649037e −8 ] in the original range [0, 10] 9 , together with a proof tha... |

29 |
Starting regions by fixed points and tightening
- Hong, Stahl
- 1994
(Show Context)
Citation Context ...nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne (1993) in the box =-=[0, 10]-=- 9 . Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc-1 workstations) but they state that “at this time, we know no other method which has been applied to this ... |

28 |
Interval arithmetic in matrix computation
- Hansen
(Show Context)
Citation Context ...istency on the mean value interval extension of a system of constraints is closely related to the Hansen-Sengupta operator [8], which is an improvement over Krawczyk’s operator [13]. Hansen and Smith =-=[9]-=- also argue that these operators are more effective when the interval Jacobian of the system is diagonally dominant and they suggest conditioning the system S.For the purpose of this paper, we simply ... |

20 |
Safe starting regions for iterative methods
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- 1977
(Show Context)
Citation Context ... (⃗I)(Ij ⊖ mj ) ⊕ fi(m1,... ˆ ,mn) ⎦⎠ ∂xj j=1,j̸=1 where mi = center(Ii).If I ′ i ⊆Ii (16 i6 n) then there exists a zero in 〈I ′ i ,... ,I′ n 〉. A proof of this result can be found in Moore and Jones =-=[19]-=-. 7. Experimental results This section reports experimental results of the branch-and-prune algorithms. We compare the branch-and-prune algorithm with two instantiations of the pruning operator: Prune... |

16 |
Experiments using interval analysis for solving a circuit design problem
- Ratschek, Rokne
- 1993
(Show Context)
Citation Context ...-F. PUGET AND P. VAN HENTENRYCK The problem is very challenging both for local and global methods because small variations in the inputs produce large differences in the functions. Ratschek and Rokne =-=[23]-=- summarize various attempts to find a solution to this problem using local methods; these descriptions are not repeated here. It suffices to say that successful attempts require very elaborate procedu... |

8 | Mean Value Forms in Interval Analysis - Caprani, Mandsen - 1980 |

6 |
Large-scale behaviour of junction transistors
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- 1954
(Show Context)
Citation Context ...Key words: Global zero search, Electrical circuit, Transistor modelling, Interval methods, Branch and prune, Constraint satisfaction 1. Introduction The transistor modelling problem of Ebers and Moll =-=[6]-=- is the system of nonlinear equations ⎧ ⎪⎨ ⎪⎩ (1 − x1x2)x3[exp(x5(g1k − g3kx710 −3 − g5kx810 −3 )) − 1] −g5k + g4kx2 = 0 (1 6 k 6 4) (1 − x1x2)x3[exp(x6(g1k − g2k − g3kx710 −3 + g4kx910 −3 )) − 1] −g5... |

4 |
Mean value forms in interval analysis, Computing 25: 147
- Caprani, Madsen
- 1980
(Show Context)
Citation Context ... ⃗I) = ... ⎪⎩ 0=τ(fn, ⃗I)(X1,... ,Xn) Note that the mean value interval extensions is defined in terms of natural extensions. The proof of the following proposition can be found in Caprani and Madsen =-=[4]-=-. jogo436.tex; 30/06/1998; 12:38; p.14A CONSTRAINT SATISFACTION APPROACH TO A CIRCUIT DESIGN PROBLEM 89 function Search(S, ⃗I0) begin ⃗I := Prune( ˆS ∪ τc(S, ⃗I0), ⃗I0); if Empty( ⃗I)then return ∅ el... |

3 |
Logical arithmetic, Future Generation Computing Systems 2(2
- Cleary
- 1987
(Show Context)
Citation Context ...ue box x[1] =0.8999999 +[0.48517e −7 , 0.566954e −7 ] x[2] =0.4499874 +[0.6902216e −7 , 0.7493801e −7 ] x[3] =1.00000648 +[0.60195e −9 , 0.43303e −8 ] x[4] =2.00006854 +[0.5787e −10 , 0.319179e −8 ] x=-=[5]-=- =7.9999714 +[0.3767867e −7 , 0.4259589e −7 ] x[6] =7.99969268 +[0.14994e −8 , 0.692803e −8 ] x[7] =5.00003127 +[0.338646e −8 , 0.848255e −8 ] x[8] =0.99998772 +[0.69887e −9 , 0.621097e −8 ] x[9] =2.0... |

1 |
Consistency techniques for numerical constraint satisfaction problems
- L’Homme
- 1993
(Show Context)
Citation Context ...er local consistency condition that we call box(2)-consistency. Box(2)-consistency is in fact an approximation of path consistency [16] and is related to some consistency notions presented in L’Homme =-=[14]-=-. 5.3.1. Informal presentation Box(2)-consistency generalizes box(1)-consistency by projecting all but two variables. The original existence problem ∃X1 ⊆ I1,... ,∃Xn ⊆In :S(X1,... ,Xn) is thus approx... |