## Interval Methods Revisited (1995)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Hentenryck95intervalmethods,

author = {P. Van Hentenryck and D. Mcallester and D. Kapur},

title = {Interval Methods Revisited},

institution = {},

year = {1995}

}

### OpenURL

### Abstract

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...

### Citations

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1020 |
Interval Analysis
- Moore
- 1966
(Show Context)
Citation Context ...r describes a particular "branch and cut" search method based on interval techniques. Interval techniques manipulate upper and lower bounds on variables and have been explored by a variety o=-=f authors [12, 5, 4, 8]. The main-=- innovation presented here consist of two "local interval cuts" that are effective in regions of large uncertainty, i.e., where the bounds associated with variables remain quite loose. Our t... |

593 | Methods and Applications of Interval Analysis - Moore - 1979 |

506 |
Introduction to Interval Computation
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...to the above rules (NE stands for "natural evaluation"). For example if B(x) is [\Gamma1; 1] then NE(x+ x; B) is the interval [\Gamma2; 2]; NE(xsx; B) is the interval [\Gamma1; 1]; but NE(x =-=2 ; B) is [0; 1]-=-. For most nontrivial expressions the interval NE(e; B) is larger than the actual range of values of e over points in B. If B(x) is [\Gamma1; 1] we have that NE(x 2 \Gamma x; B) is [\Gamma1; 2] while ... |

177 | Applying interval arithmetic to real, integer and Boolean constraints
- Benhamou, Older
- 1997
(Show Context)
Citation Context ...cal in the sense that each cut examines only a single constraint. Our emphasis on local constraints is inspired by constraint propagation techniques developed for AI and logic programming communities =-=[18, 3]-=-. Our local interval cuts seem empirically to reduce the time required to solve certain benchmarks, such as the Broyden Banded function, from exponential to linear in the number of variables. Continua... |

173 | Design, implementation and evaluation of the constraint language cc(FD - Hentenryck, Saraswat, et al. - 1998 |

144 | New Computer Methods for Global Optimization - Ratschek, Rokne - 1988 |

130 | Hentenryck, P.: CLP(intervals) revisited - Benhamou, McAllister, et al. - 1994 |

80 |
Homotopies exploiting Newton polytopes for solving sparse polynomial systems
- Verschelde, Verlinden, et al.
- 1994
(Show Context)
Citation Context ... benchmarks, such as the Broyden Banded function, from exponential to linear in the number of variables. Continuation methods provide another approach to solving systems of nonlinear equations, e.g., =-=[17, 23]-=-. Continuation methods are quite different from the traditional branch and bound or branch and cut approaches to NP complete problems. Unlike standard branch and cut approaches, continuation methods h... |

66 |
Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken
- Krawczyk
- 1969
(Show Context)
Citation Context ...r describes a particular "branch and cut" search method based on interval techniques. Interval techniques manipulate upper and lower bounds on variables and have been explored by a variety o=-=f authors [12, 5, 4, 8]. The main-=- innovation presented here consist of two "local interval cuts" that are effective in regions of large uncertainty, i.e., where the bounds associated with variables remain quite loose. Our t... |

59 |
Bounding solutions of systems of equations using interval analysis
- Hansen, Sengupta
- 1981
(Show Context)
Citation Context ...r describes a particular "branch and cut" search method based on interval techniques. Interval techniques manipulate upper and lower bounds on variables and have been explored by a variety o=-=f authors [12, 5, 4, 8]. The main-=- innovation presented here consist of two "local interval cuts" that are effective in regions of large uncertainty, i.e., where the bounds associated with variables remain quite loose. Our t... |

54 | Extending prolog with constraint arithmetic on real intervals
- Older, Vellino
- 1990
(Show Context)
Citation Context ...cal in the sense that each cut examines only a single constraint. Our emphasis on local constraints is inspired by constraint propagation techniques developed for AI and logic programming communities =-=[18, 3]-=-. Our local interval cuts seem empirically to reduce the time required to solve certain benchmarks, such as the Broyden Banded function, from exponential to linear in the number of variables. Continua... |

38 |
Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems
- Morgan
- 1987
(Show Context)
Citation Context ... benchmarks, such as the Broyden Banded function, from exponential to linear in the number of variables. Continuation methods provide another approach to solving systems of nonlinear equations, e.g., =-=[17, 23]-=-. Continuation methods are quite different from the traditional branch and bound or branch and cut approaches to NP complete problems. Unlike standard branch and cut approaches, continuation methods h... |

34 |
Computing All Solutions to Polynomial Systems Using Homotopy Continuation
- Morgan, Sommese
- 1987
(Show Context)
Citation Context ...we expect the procedure described above to have essentially the same behavior. The benchmarks were taken from papers on numerical analysis [15], interval analysis [5, 7, 14], and continuation methods =-=[23, 17, 16, 11]-=-. Our system was run on a Sun Sparc 10 workstation to obtain all solutions. In our runs the final intervals must have widths smaller than 10 \Gamma8 . The timings for our system are listed in the colu... |

31 |
Safe start region by fixed points and tightening
- Hong, Stahl
- 1994
(Show Context)
Citation Context ... procedure described above, although we expect the procedure described above to have essentially the same behavior. The benchmarks were taken from papers on numerical analysis [15], interval analysis =-=[5, 7, 14]-=-, and continuation methods [23, 17, 16, 11]. Our system was run on a Sun Sparc 10 workstation to obtain all solutions. In our runs the final intervals must have widths smaller than 10 \Gamma8 . The ti... |

30 | Interval Arithmetic in Matrix Computations - Hansen - 1967 |

22 |
Numerical solution of non-linear equations
- MoreĢ, Cosnard
- 1979
(Show Context)
Citation Context ...at more complex than the procedure described above, although we expect the procedure described above to have essentially the same behavior. The benchmarks were taken from papers on numerical analysis =-=[15]-=-, interval analysis [5, 7, 14], and continuation methods [23, 17, 16, 11]. Our system was run on a Sun Sparc 10 workstation to obtain all solutions. In our runs the final intervals must have widths sm... |

21 |
Chemical Equilibrium Systems as Numerical Test Problems
- Meintjes, Morgan
- 1990
(Show Context)
Citation Context ...we expect the procedure described above to have essentially the same behavior. The benchmarks were taken from papers on numerical analysis [15], interval analysis [5, 7, 14], and continuation methods =-=[23, 17, 16, 11]-=-. Our system was run on a Sun Sparc 10 workstation to obtain all solutions. In our runs the final intervals must have widths smaller than 10 \Gamma8 . The timings for our system are listed in the colu... |

21 |
Safe starting regions for iterative methods
- Jones
- 1977
(Show Context)
Citation Context ... procedure described above, although we expect the procedure described above to have essentially the same behavior. The benchmarks were taken from papers on numerical analysis [15], interval analysis =-=[5, 7, 14]-=-, and continuation methods [23, 17, 16, 11]. Our system was run on a Sun Sparc 10 workstation to obtain all solutions. In our runs the final intervals must have widths smaller than 10 \Gamma8 . The ti... |

11 | A Logic Language for Combinatorial Optimization - Hentenryck - 1989 |

2 | A Class of interval Newton - Krawczyk - 1986 |