#### DMCA

## Formal global optimisation with Taylor models (2006)

Venue: | Automated Reasoning, LNCS |

Citations: | 12 - 0 self |

### Citations

1264 |
Interval analysis,
- Moore
- 1966
(Show Context)
Citation Context ...h exponential complexity easily be generalised to the multi-dimensional case. Although important, this technique is not yet sufficient for most applications, which is why numerous other refinements =-=[11,19]-=- have been developed. In section 4 we are going to see how Taylor models address the dependency problem more directly and are able to overcome it in part. 2.3 The Set of Interval Bounds Traditionally ... |

576 |
Global Optimization Using Interval Analysis
- Hansen
- 1992
(Show Context)
Citation Context ...nd-bound algorithm to optimise a function over an interval [18]. This work has been the basis for many sophisticated refinements, making up the core of numerous current global optimisation algorithms =-=[11]-=-. From Formal Proof in Principle to Formal Proof in Fact In 1879 Frege was first to introduce the notion of formal proof. With his Begriffsschrift he gave a language to express propositions and rules ... |

202 | A proof of the Kepler conjecture
- Hales
- 2005
(Show Context)
Citation Context ...s [4], robotics [16]), over experimental physics (particle motion in accelerators [12]) to geometry. A prominent instance of the last class is the proof of the Kepler conjecture given by Thomas Hales =-=[10]-=-, in which some thousand lemmata asserting bounds on geometric functions occur. From Extremum Criteria to Global Optimisation Algorithms In 1755 Euler gave (based on previous work by Fermat) the well-... |

94 |
A computer-checked proof of the four colour theorem
- Gonthier
- 2004
(Show Context)
Citation Context ...e Bruijn's system. Solutions to other computationally difficult problems have already been formalised: a formal proof of the Four Colour Theorem has been given by Georges Gonthier and Benjamin Werner =-=[6]-=-. Also, a verification algorithm for Pocklington certificates of prime numbers has been proven correct [7]. In a slightly different setting, other computational parts of the Kepler conjecture proof ha... |

87 | A compiled implementation of strong reduction, in
- Grégoire, Leroy
- 2002
(Show Context)
Citation Context ...5764579] 3 1.5346721 · 10−4 [2.4236733; 2.5763267] 4 4.0386107 · 10−6 [2.4236875; 2.5763165] ∞ 0 [2.4263158; 2.5761905] All these results have been obtained in less than a second using Coq's compiler =-=[8]-=-. uunionsq Example 4. Lemma I_751442360 in Thomas Hales' proof of the Kepler conjecture [10] states that −x1x3 − x2x4 + x1x5 + x3x6 − x5x6 + x2(−x2 + x1 + x3 − x4 + x5 + x6)√√√√√√√4x2 x2x4(−x2 + ... |

63 |
Institutiones Calculi Differentialis cum eius usu in Analysi Finitorum ac Doctrina Serierum
- Euler
(Show Context)
Citation Context ...tions occur. From Extremum Criteria to Global Optimisation Algorithms In 1755 Euler gave (based on previous work by Fermat) the well-known necessary condition ∇fx = 0 for f to assume an extremum at x =-=[5]-=-. However, in most interesting cases effectively solving this equation is an equally difficult problem. During the following centuries a lot of more sophisticated criteria have been developed, but lik... |

52 |
Interval arithmetic and automatic error analysis in digital computing [Ph.D. thesis],
- Moore
- 1962
(Show Context)
Citation Context ... methods to treat this kind of problems. In 1962 Ramon E. Moore described the use of interval arithmetic on a computer, refined by a branch-and-bound algorithm to optimise a function over an interval =-=[18]-=-. This work has been the basis for many sophisticated refinements, making up the core of numerous current global optimisation algorithms [11]. From Formal Proof in Principle to Formal Proof in Fact In... |

41 |
Remainder differential algebras and their applications
- Makino, Berz
- 1996
(Show Context)
Citation Context .... 4.3 What Reference Point to Choose? With this procedure established, a point that merits some more study is the choice to be made for the reference point y0. As mentioned, the strategy described in =-=[13]-=- is equivalent to setting y0 = c. However Taylor's theorem can be applied with any y0 ∈ Y as reference point. A good choice is one that minimises the width of the resulting Taylor model's error interv... |

40 | Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis.
- Merlet
- 2004
(Show Context)
Citation Context ...; b1] × . . . × [an; bn]. f x ∈ [c; d], of course desiring [c; d] as narrow as possible. Problems of this kind arise in a wide spectrum of science, ranging from engineering (aeronautics [4], robotics =-=[16]-=-), over experimental physics (particle motion in accelerators [12]) to geometry. A prominent instance of the last class is the proof of the Kepler conjecture given by Thomas Hales [10], in which some ... |

34 |
Rigorous Analysis of Nonlinear Motion in Particle Accelerators
- Makino
- 1998
(Show Context)
Citation Context ...as narrow as possible. Problems of this kind arise in a wide spectrum of science, ranging from engineering (aeronautics [4], robotics [16]), over experimental physics (particle motion in accelerators =-=[12]-=-) to geometry. A prominent instance of the last class is the proof of the Kepler conjecture given by Thomas Hales [10], in which some thousand lemmata asserting bounds on geometric functions occur. Fr... |

32 |
Automatic error analysis in digital computation
- Moore
- 1959
(Show Context)
Citation Context ...s inefficient if implemented as suggested by the formula given in table 1, because all of ac, ad, bc, and bd are computed before determining their minimum and maximum. In fact this can be accelerated =-=[17]-=- by looking at the signs of a, b, c, and d, using the fact that a ≤ b and c ≤ d. Definition 3. An interval [a; b] has sign + if a > 0, sign − if b < 0 and sign ± if a ≤ 0 ≤ b. Given the signs of the t... |

22 | Models and Floating-Point Arithmetic: Proof that Arithmetic Operations are Validated
- Revol, Makino, et al.
- 2005
(Show Context)
Citation Context ...ts will only yield approximations without error bounds, which is unacceptable for formal proof. An explicit treatment of rounding errors is possible, as has been shown for addition and multiplication =-=[24]-=-. However this approach can be expected to be much more difficult for the Taylor development of general smooth functions. Intervals: Using intervals with rational or floating-point number bounds as ... |

21 |
Tactical conflict detection and resolution in a 3-D airspace
- Dowek, Geser, et al.
- 2001
(Show Context)
Citation Context ... that ∀x ∈ [a1; b1] × . . . × [an; bn]. f x ∈ [c; d], of course desiring [c; d] as narrow as possible. Problems of this kind arise in a wide spectrum of science, ranging from engineering (aeronautics =-=[4]-=-, robotics [16]), over experimental physics (particle motion in accelerators [12]) to geometry. A prominent instance of the last class is the proof of the Kepler conjecture given by Thomas Hales [10],... |

21 | A Computational Approach to Pocklington Certificates in Type Theory
- Grégoire, Théry, et al.
- 2006
(Show Context)
Citation Context ...ormal proof of the Four Colour Theorem has been given by Georges Gonthier and Benjamin Werner [6]. Also, a verification algorithm for Pocklington certificates of prime numbers has been proven correct =-=[7]-=-. In a slightly different setting, other computational parts of the Kepler conjecture proof have been formalised, namely a large graph enumeration problem [20] and linear programs [22]. Computational ... |

14 | Flyspeck I: Tame Graphs
- Nipkow, Bauer, et al.
- 2006
(Show Context)
Citation Context ... of prime numbers has been proven correct [7]. In a slightly different setting, other computational parts of the Kepler conjecture proof have been formalised, namely a large graph enumeration problem =-=[20]-=- and linear programs [22]. Computational proofs are supported by an important characteristic of type theory: The so-called conversion rule identifies terms modulo β-conversion (computation). In fact, ... |

12 | Proving bounds for real linear programs in isabelle/hol
- Obua
- 2005
(Show Context)
Citation Context ...n proven correct [7]. In a slightly different setting, other computational parts of the Kepler conjecture proof have been formalised, namely a large graph enumeration problem [20] and linear programs =-=[22]-=-. Computational proofs are supported by an important characteristic of type theory: The so-called conversion rule identifies terms modulo β-conversion (computation). In fact, functional programs writt... |

12 |
A monadic, functional implementation of real numbers
- O’Connor
(Show Context)
Citation Context ... (x2,m2) := (k 7→ min (x1 k) (x2 k), ε 7→ max (m1 ε) (m2 ε)) sgnε (x,m) := 1 if x ε < −ε−1 if ε < x ε0 otherwise More details on constructive real numbers and their implementations can be found in =-=[25,21,23,2,15]-=-. Irrational functions (square root, trigonometry etc.) can be computed without rounding. This means that in order to obtain an approximation for a larger formula we have to provide only one precision... |

11 |
Grégoire and Assia Mahboubi. Proving equalities in a commutative ring done right in coq
- Benjamin
- 2005
(Show Context)
Citation Context ... polynomial with n-variate polynomials as coefficients. This is justified by the canonical polynomial isomorphism: R[X1, . . . ,Xn+1] ' R[X1, . . . ,Xn][Xn+1] This can be translated to Coq as follows =-=[9]-=-: Fixpoint PolyN (n:nat) struct n : Type := match n with | O => C | S m => list (PolyN m) end. The coefficients of the Taylor models' polynomials are represented by constructive real numbers because t... |

11 |
exacte, conception, algorithmique et performances d’une implémentation informatique en précision arbitraire. Thèse, Université Paris 7
- Ménissier-Morain, Arithmétique
- 1994
(Show Context)
Citation Context ... (x2,m2) := (k 7→ min (x1 k) (x2 k), ε 7→ max (m1 ε) (m2 ε)) sgnε (x,m) := 1 if x ε < −ε−1 if ε < x ε0 otherwise More details on constructive real numbers and their implementations can be found in =-=[25,21,23,2,15]-=-. Irrational functions (square root, trigonometry etc.) can be computed without rounding. This means that in order to obtain an approximation for a larger formula we have to provide only one precision... |

8 |
models and other validated functional inclusion methods
- Taylor
(Show Context)
Citation Context ...difficulty for the working mathematician. However, finding an appropriate addition theorem for a given g requires a certain amount of creativity, which can only be provided by a human. This is why in =-=[14]-=- this strategy has been applied to many different functions manually: x 7→ 1x , sin, cos, arctan, log etc., so that they could be implemented. In contrast, our construction (2) can entirely be carried... |

7 |
Formalising Exact Arithmetic: Representations, Algorithms and Proofs
- Niqui
- 2004
(Show Context)
Citation Context ... (x2,m2) := (k 7→ min (x1 k) (x2 k), ε 7→ max (m1 ε) (m2 ε)) sgnε (x,m) := 1 if x ε < −ε−1 if ε < x ε0 otherwise More details on constructive real numbers and their implementations can be found in =-=[25,21,23,2,15]-=-. Irrational functions (square root, trigonometry etc.) can be computed without rounding. This means that in order to obtain an approximation for a larger formula we have to provide only one precision... |

5 | Constructive analysis with witnesses
- Schwichtenberg
- 2012
(Show Context)
Citation Context |

2 |
Calcul de formules affines et de séries entières en arithmétique exacte avec types co-inductifs
- Bertot
- 2005
(Show Context)
Citation Context ...r n. Use a more precise set of bounds B ⊂ R. In practice this can mean to increase some precision parameter. The first option would be a good choice for f x = x − x, the second one for f x = √ x on =-=[0; 2]-=-. However, it is not easy to say which choice is better in general. Making the wrong one will lead to unnecessary computations. 3 Constructive Real Numbers No solution has been given to the loss of sh... |

2 |
de Bruijn. The mathematical language AUTOMATH, its usage, and some of its extensions
- Nicolaas
- 1968
(Show Context)
Citation Context ...il and paper. Again, the arrival of computers changed this situation: In 1967, after mathematical logic had become a more thoroughly studied topic, Nicolaas G. de Bruijn developed the Automath system =-=[3]-=-, which could syntactically verify that a given proof indeed demonstrates a given theorem. The fact that formal proofs could now be constructed and checked on a machine made their development more pra... |

1 |
A076741
- Azarian
- 2002
(Show Context)
Citation Context ...− c]√ [a; b]B = [⌊√ a ⌋ ; ⌈√ b ⌉] Now consider the function f = x 7→ √x−√x. Its natural B-interval extension is f̂ = [a; b] 7→ [ b√ac − ⌈√ b ⌉ ; ⌈√ b ⌉ − b√ac ] . For its united extension we have f̂n =-=[0; 1]-=- = n⋃ i=1 [−δi,n; δi,n] = [−δ1,n; δ1,n] where δi,n = ⌈√ i n ⌉ − ⌊√ i− 1 n ⌋ Note that δ1,n = ⌈√ 1 n ⌉ ≥ 11000 for any n. Thus limn→∞ f̂n [0; 1] = [0; 11000 ] 6= [0; 0], which means that f̂n is not B-s... |