## Interval Analysis on Directed Acyclic Graphs for Global Optimization (2004)

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Venue: | J. Global Optimization |

Citations: | 41 - 8 self |

### BibTeX

@TECHREPORT{Schichl04intervalanalysis,

author = {Hermann Schichl and Arnold Neumaier},

title = {Interval Analysis on Directed Acyclic Graphs for Global Optimization},

institution = {J. Global Optimization},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the ow of the computation.

### Citations

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Interval Methods for Systems of Equations
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Citation Context ...cally Lipschitz. Using (5) wesnd an enclosure of the range of f over the box x by f(x) 2 f(z) + f [z; x](x z); for all x 2 x: This is a centered form and has the quadratic approximation property (cf. =-=[2-=-1]). The most general slope denition is the one with interval center f(x) f(z) + f [z; x](x z); and the special case x = z gives f [z; z] = f 0 (z) the interval derivative. Slopes can be calculated a... |

299 | Rigorous Global Search: Continuous Problems - Kearfott - 1996 |

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Citation Context ...(CP). This method for solving constraint satisfaction problems (CSPs) and global optimization problems (GLOPs) wassrst developed in the discrete case [15] and later transferred to the continuous case =-=[7, 10, 16, 28]-=-. The basics of constraint propagation on DAGs are outlined in Section 5. The results of constraint propagation, especially the ranges of the inner nodes, can be used to improve the ranges of the stan... |

171 | Applying interval arithmetic to real, integer, and boolean constraints
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Citation Context ...mplicit equation v1 =v2. 546 H. SCHICHL AND A. NEUMAIER Figure 1. DAG representation of Problem (2). Of course, variables usually appear more than once, and many algorithms for constraint propagation =-=[1, 3, 25]-=- use the principle that the variable nodes of identical variables can be identified, hereby reducing the size of the graph. However, this principle can be generalized. DEFINITION 3.1. Two vertices v1 ... |

114 |
Computationaly of global solutions to factorable nonconvex programs: Part I - convex underestimations problems
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Citation Context ..., constraint propagation, slope, interval analysis 2000 MSC Classication: primary 65G40, secondary 90C26 1 1 Introduction Deterministic algorithms for solving factorable global optimization problems [=-=11, 20]-=- usually use branch-and-bound like schemes [2, 12, 17, 24, 27]. The success of such a method heavily relies on the quality of the range estimates computed for the functions involved. This paper discus... |

75 |
A General Purpose Global Optimization Software Package
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Citation Context ...2000 MSC Classication: primary 65G40, secondary 90C26 1 1 Introduction Deterministic algorithms for solving factorable global optimization problems [11, 20] usually use branch-and-bound like schemes [=-=2, 12, 17, 24, 27]-=-. The success of such a method heavily relies on the quality of the range estimates computed for the functions involved. This paper discusses a new representation technique for global optimization pro... |

63 |
Introduction to Numerical Analysis
- Neumaier
- 2001
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Citation Context ...AG representation. Details will be presented elsewhere.) Finally, in Section 8 we will make some statements about implementation issues and performance. Our notation follows the notation suggested in =-=[22]-=-. In particular, inequalities between vectors are interpreted component-wise, I denotes the identity matrix, intervals and boxes are written in bold face, and rad x = 1 2 (x x) denotes the radius of a... |

58 | Consistency techniques for continuous constraints
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Citation Context ...int propagation (see Secton 5) the algorithms will not make use of the implicit equation v 1 = v 2 . Of course, variables usually appear more than once, and many algorithms for constraint propagation =-=[1, 3, 2-=-5] use the principle that the variable nodes of identical variables can be identied, hereby reducing the size of the graph. However, this principle can be generalized. 3.1 Denition. Two vertices v 1 a... |

51 |
Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models
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- 1998
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Citation Context ... • slopes over boxes with fixed center, • linear enclosures. To illustrate the techniques, we will use throughout the simple example min f(x1, x2, x3) = (4x1 − x2x3)(x1x2 + x3) s.t. x1 ∈ [1, 2], x2 ∈ =-=[3, 4]-=-, x3 ∈ [3, 4], whose DAG representation can be found in Figure 2. [1, 2] [3, 4] [3, 4] x1 x2 x3 4 −1 ∗ ∗ + + ∗ min Figure 2: Directed Acyclic Graph representation of (4) 7 (4)s4.1 Forward Evaluation S... |

43 |
Convexification and Global Optimization
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- 2003
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Citation Context ...2000 MSC Classication: primary 65G40, secondary 90C26 1 1 Introduction Deterministic algorithms for solving factorable global optimization problems [11, 20] usually use branch-and-bound like schemes [=-=2, 12, 17, 24, 27]-=-. The success of such a method heavily relies on the quality of the range estimates computed for the functions involved. This paper discusses a new representation technique for global optimization pro... |

39 |
GAMS: A User’s Guide, Release 2.25
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Citation Context ...G solver), the computational trees provided by the parsers of high-level programming language compilers (FORTRAN 90, C++) are used, in others the parsers of modelling languages like AMPL [13] or GAMS =-=[-=-8] provide the graph representation of the mathematical problem. In Sections 2 and 3 we will introduce the special DAGs used in problem representation and talk about dierent interpretations and simpli... |

38 | ProfileDriven instruction level parallel scheduling with applications to superblocks
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- 1996
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Citation Context ...tation technique for global optimization problems using directed acyclic graphs (DAGs). Traditionally, DAGs have been used in automatic differentiation [6, 14] and in the theory of parallel computing =-=[9]-=-. We will show that the DAG representation of a global 542 H. SCHICHL AND A. NEUMAIER optimization problem serves many purposes. In some global optimization algorithms [17] and constraint propagation ... |

36 |
eds), Automatic Differentiation of Algorithms
- Griewank, Corliss
- 1991
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Citation Context ...ns involved. This paper discusses a new representation technique for global optimization problems using directed acyclic graphs (DAGs). Traditionally, DAGs have been used in automatic differentiation =-=[6, 14]-=- and in the theory of parallel computing [9]. We will show that the DAG representation of a global optimization problem serves many purposes. In some global optimization algorithms [17] and constraint... |

31 |
Editors, Towards Global Optimization
- DIXON, SZEGO
- 1975
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Citation Context ..., constraint propagation, slope, interval analysis 2000 MSC Classication: primary 65G40, secondary 90C26 1 1 Introduction Deterministic algorithms for solving factorable global optimization problems [=-=11, 20]-=- usually use branch-and-bound like schemes [2, 12, 17, 24, 27]. The success of such a method heavily relies on the quality of the range estimates computed for the functions involved. This paper discus... |

25 |
Floudas. Deterministic global optimization: theory, methods and application
- A
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Citation Context ...2000 MSC Classication: primary 65G40, secondary 90C26 1 1 Introduction Deterministic algorithms for solving factorable global optimization problems [11, 20] usually use branch-and-bound like schemes [=-=2, 12, 17, 24, 27]-=-. The success of such a method heavily relies on the quality of the range estimates computed for the functions involved. This paper discusses a new representation technique for global optimization pro... |

25 |
forms - use and limits
- Taylor
- 2002
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Citation Context ... construct linear enclosures of the form f(x) 2 f + s(x z); for x 2 x; with thin slope s 2 R n and thick constant term. This approach corresponds tosrst order Taylor arithmetic as, e.g., presented in =-=[4, 5, 23]. Si-=-nce linear Taylor expression also obey a chain rule similar to slopes, these enclosures can be computed by backward evaluation with little eort quite similar to \thick" slopes. Kolev [19] showed ... |

23 |
αBB: a global optimization method for general constrained nonconvex problems
- Androulakis, Maranas, et al.
- 1995
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Citation Context ..., we first compute the values of the centers in forward mode, which is an ordinary (interval) function evaluation. Then we change the map dm to slm : E → IR 10szf ,sf 4[6,12]+ [3,4][−12,−1] =[−24,45] =-=[1, 2]-=- [3, 4] [3, 4] x1 [−72,−19] x2 [−60,−19] x3 4 [3,4] −1 [1,2] [3,4] [3,4] ∗ ∗ 4 [−12,−1] [−12,−6] 1 −1 + + [6,12] [−12,−1] [6,12] [−12,−1] [1,1] ∗ min Figure 6: Interval gradient evaluation for (4) f g... |

20 | Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems
- Kearfott
- 1991
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Citation Context ...(CP). This method for solving constraint satisfaction problems (CSPs) and global optimization problems (GLOPs) wassrst developed in the discrete case [15] and later transferred to the continuous case =-=[7, 10, 16, 28]-=-. The basics of constraint propagation on DAGs are outlined in Section 5. The results of constraint propagation, especially the ranges of the inner nodes, can be used to improve the ranges of the stan... |

16 | GLOPT - A Program for Constrained Global Optimization
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Citation Context ...(CP). This method for solving constraint satisfaction problems (CSPs) and global optimization problems (GLOPs) wassrst developed in the discrete case [15] and later transferred to the continuous case =-=[7, 10, 16, 28]-=-. The basics of constraint propagation on DAGs are outlined in Section 5. The results of constraint propagation, especially the ranges of the inner nodes, can be used to improve the ranges of the stan... |

16 |
Deterministic Global Optimization: Theory, Algorithms and Applications
- Floudas
- 1999
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Citation Context ...c graphs, global optimization, interval analysis, slope 1. Introduction Deterministic algorithms for solving factorable global optimization problems [11, 20] usually use branch-and-bound like schemes =-=[2, 12, 17, 24, 27]-=-. The success of such a method heavily relies on the quality of the range estimates computed for the functions involved. This paper discusses a new representation technique for global optimization pro... |

15 | Using directed acyclic graphs to coordinate propagation and search for numerical constraint satisfaction problems
- Vu, Schichl, et al.
- 2004
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Citation Context ...hing steps. In addition, the DAG representation also greatly improves the speed of constraint propagation when compared to traditional methods, often by one or more orders of magnitude, see Vu et al. =-=[29]-=-. Our notation follows the notation suggested in [22]. In particular, inequalities between vectors are interpreted component-wise, I denotes the identity matrix, intervals and boxes are written in bol... |

12 |
COSY INFINITY version 8 reference manual
- Berz
- 1997
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Citation Context ... construct linear enclosures of the form f(x) 2 f + s(x z); for x 2 x; with thin slope s 2 R n and thick constant term. This approach corresponds tosrst order Taylor arithmetic as, e.g., presented in =-=[4, 5, 23]. Si-=-nce linear Taylor expression also obey a chain rule similar to slopes, these enclosures can be computed by backward evaluation with little eort quite similar to \thick" slopes. Kolev [19] showed ... |

12 |
Use of interval slopes for the irrational part of factorable functions
- Kolev
- 1997
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Citation Context ...e narrower interval derivatives and slopes than those provided by using only interval automatic dierentiation preceded by constraint propagation. The implementation is based on earlier work by Kolev [=-=18]-=- on optimal slopes and by Bliek [6] on backward slope evaluation. Care is taken to ensure that rounding errors are treated correctly. Interval techniques are presented for computing from the DAG usefu... |

10 |
BB : A global optimization method for general constrained nonconvex problems
- Androulakis, Maranas, et al.
- 1995
(Show Context)
Citation Context ...neral slopes, wesrst compute the values of the centers in forward mode, which is an ordinary (interval) function evaluation. Then we change the map dm to slm : E ! IR 10 x 1 x 2 x 3 + + 4 1 min [1;=-= 2]-=- [3; 4] [3; 4] [6;12] [ 12; 1] 4 1 [3;4] [1;2] [3;4] [3;4] 1 1 =[ 24;45] [3;4][ 12; 1] 4[6;12]+ [ 72; 19] [ 60; 19] [ 12; 1] [ 12; 6] [6;12] [ 12; 1] [1;1] Figure 6: Interval gradient evaluation for (... |

10 |
Algorithms for Solving Nonlinear Constrained and Optimization Problems: The State of the Art. Report of the European Community funded project COCONUT, Mathematisches Institut der Universitat
- Bliek, Spellucci, et al.
- 2001
(Show Context)
Citation Context ...P). This method for solving constraint satisfaction problems (CSPs) and global optimization problems (GLOPs) was first developed in the discrete case [15] and later transferred to the continuous case =-=[7, 10, 16, 28]-=-. The basics of constraint propagation on DAGs are outlined in Section 5. The results of constraint propagation, especially the ranges of the inner nodes, can be used to improve the ranges of the stan... |

10 |
Automatic Di erentiation of Algorithms
- Greiwank, Corliss
- 1991
(Show Context)
Citation Context ...ons involved. This paper discusses a new representation technique for global optimization problems usingsdirected acyclic graphs (DAGs). Traditionally, DAGs have been used in automatic dierentiation [=-=6, 14]-=- and in the theory of parallel computing [9]. We will show that the DAG representation of a global optimization problem serves many purposes. In some global optimization algorithms [17] and constraint... |

9 |
Computer Methods for Design Automation
- Bliek
- 1992
(Show Context)
Citation Context ...slopes than those provided by using only interval automatic dierentiation preceded by constraint propagation. The implementation is based on earlier work by Kolev [18] on optimal slopes and by Bliek [=-=6]-=- on backward slope evaluation. Care is taken to ensure that rounding errors are treated correctly. Interval techniques are presented for computing from the DAG useful redundant constraints, in particu... |

8 |
An improved interval linearization for solving non-linear problems, Manuscript
- Kolev
- 2002
(Show Context)
Citation Context ...n [4, 5, 23]. Since linear Taylor expression also obey a chain rule similar to slopes, these enclosures can be computed by backward evaluation with little eort quite similar to \thick" slopes. Ko=-=lev [19-=-] showed that propagating them in 15 forward mode leads to better enclosures; however, the eort for computing in forward mode is n times higher. 8 Implementation Issues 8.1 Multiplication and Division... |

8 | Taylor forms – use and limits
- Neumaier
- 2002
(Show Context)
Citation Context ... construct linear enclosures of the form f (x)∈f + s(x − z), for x ∈x, with thin slope s ∈Rn and thick constant term. This approach corresponds to first order Taylor arithmetic as, e.g., presented in =-=[4, 5, 23]-=-. Since the linear Taylor expressions also obey a chain rule similar to slopes, these enclosures can be computed by backward evaluation with little effort quite similar to “thick” slopes. Kolev [19] s... |

7 |
The Krawczyk operator and Kantorovich’s theorem
- Shen, Neumaier
- 1990
(Show Context)
Citation Context ...or x = [1; 2] [3; 4] [3; 4]. A very useful tool for calculating enclosures of the range of f over a box is a slope. This is a linear approximation of the form f(x) = f(z) + f [z; x](x z); (5) see [2=-=6, -=-18]. In one dimension the slope is unique, if it is continuous, and we have f [z; x] = 8 : f(x) f(z) x z x 6= z f 0 (z) x = z: 9 x 1 x 2 x 3 + + 4 1 min [1; 2] [3; 4] [3; 4] 12 8 4 1 4 2 4 4 1 1 4 ... |

5 |
Veri integration of ODEs and using dierential algebraic methods on high-order Taylor models
- Berz, Makino
- 1998
(Show Context)
Citation Context ... boxes withsxed center, linear enclosures. To illustrate the techniques, we will use throughout the simple example min f(x 1 ; x 2 ; x 3 ) = (4x 1 x 2 x 3 )(x 1 x 2 + x 3 ) s.t. x 1 2 [1; 2]; x 2 2 [=-=3;-=- 4]; x 3 2 [3; 4]; (4) whose DAG representation can be found in Figure 2. x 1 x 2 x 3 + + 4 1 min [1; 2] [3; 4] [3; 4] [1; 2] [3; 4] [3; 4] Figure 2: Directed Acyclic Graph representation of (4) 7 ... |

1 |
Computer Methods for Design Automation
- Bliel
- 1992
(Show Context)
Citation Context ...ns involved. This paper discusses a new representation technique for global optimization problems using directed acyclic graphs (DAGs). Traditionally, DAGs have been used in automatic differentiation =-=[6, 14]-=- and in the theory of parallel computing [9]. We will show that the DAG representation of a global 542 H. SCHICHL AND A. NEUMAIER optimization problem serves many purposes. In some global optimization... |