Adaptive Bisection of Numerical CSPs Laurent Granvilliers Univ. - - PowerPoint PPT Presentation

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Adaptive Bisection of Numerical CSPs Laurent Granvilliers Univ. - - PowerPoint PPT Presentation

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Adaptive Bisection of Numerical CSPs Laurent Granvilliers Univ. Nantes, Lab. Computer Science, France L. Granvilliers (Nantes) Adaptive Bisection CP 2012 1 / 10 Bisection Algorithm Goal : Solving numerical CSPs using interval computations. x

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SLIDE 1

Adaptive Bisection of Numerical CSPs

Laurent Granvilliers

  • Univ. Nantes, Lab. Computer Science, France
  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 1 / 10

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SLIDE 2

Bisection Algorithm

Goal : Solving numerical CSPs using interval computations. x1 x2 x = x1 × x2    x1(x2

1 + x2 2) = 6x2 2

(x1 − 0.25)2 + x2

2 = 4

(x1, x2) ∈ x1 × x2

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 2 / 10

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SLIDE 3

Bisection Algorithm

Goal : Solving numerical CSPs using interval computations. x1 x2 x′    x1(x2

1 + x2 2) = 6x2 2

(x1 − 0.25)2 + x2

2 = 4

(x1, x2) ∈ x1 × x2

  • Contract x → x′
  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 2 / 10

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SLIDE 4

Bisection Algorithm

Goal : Solving numerical CSPs using interval computations. x1 x2 x′    x1(x2

1 + x2 2) = 6x2 2

(x1 − 0.25)2 + x2

2 = 4

(x1, x2) ∈ x1 × x2

  • Contract x → x′
  • Select x2
  • Bisect x′ along x2
  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 2 / 10

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SLIDE 5

Bisection Algorithm

Goal : Solving numerical CSPs using interval computations. x1 x2    x1(x2

1 + x2 2) = 6x2 2

(x1 − 0.25)2 + x2

2 = 4

(x1, x2) ∈ x1 × x2

  • Contract x → x′
  • Select x2
  • Bisect x′ along x2
  • Iterate and stop on

ǫ-boxes

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 2 / 10

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SLIDE 6

MaxDom Strategy

Let f(x1, x2) = x1(x2

1 + x2 2) − 6x2 2 be the function defining the cissoid.

Let x = [2, 4] × [0, 1] and evaluate the natural extension of f : f(x) = [2, 4] ([2, 4]2 + [0, 1]2) − 6 [0, 1]2 = [2, 68]

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 3 / 10

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SLIDE 7

MaxDom Strategy

Let f(x1, x2) = x1(x2

1 + x2 2) − 6x2 2 be the function defining the cissoid.

Let x = [2, 4] × [0, 1] and evaluate the natural extension of f : f(x) = [2, 4] ([2, 4]2 + [0, 1]2) − 6 [0, 1]2 = [2, 68]

Bisect x1

f([2, 3] , x2) = [2, 30] f([3, 4] , x2) = [21, 68]

Bisect x2

f(x1, [0, 0.5]) = [6.5, 65] f(x1, [0.5, 1]) = [2.5, 66.5]

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 3 / 10

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SLIDE 8

MaxDom Strategy

Let f(x1, x2) = x1(x2

1 + x2 2) − 6x2 2 be the function defining the cissoid.

Let x = [2, 4] × [0, 1] and evaluate the natural extension of f : f(x) = [2, 4] ([2, 4]2 + [0, 1]2) − 6 [0, 1]2 = [2, 68]

Bisect x1

f([2, 3] , x2) = [2, 30] f([3, 4] , x2) = [21, 68]

Bisect x2

f(x1, [0, 0.5]) = [6.5, 65] f(x1, [0.5, 1]) = [2.5, 66.5] MaxDom : bisecting the largest domain reduces overestimation in interval computations = ⇒ more contraction using local consistency

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 3 / 10

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SLIDE 9

MaxSmear Strategy

Let x = [0.5, 1] × [0.5, 1] and evaluate the derivatives of f : ∇f(x) = ([1, 4] , [−11.5, −4])

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 4 / 10

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SLIDE 10

MaxSmear Strategy

Let x = [0.5, 1] × [0.5, 1] and evaluate the derivatives of f : ∇f(x) = ([1, 4] , [−11.5, −4]) Let c be the midpoint of x and evaluate the mean value extension of f : f(x, c) = f(c) + ∇f(x)(x − c) = [−6.5, 1.4]

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 4 / 10

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SLIDE 11

MaxSmear Strategy

Let x = [0.5, 1] × [0.5, 1] and evaluate the derivatives of f : ∇f(x) = ([1, 4] , [−11.5, −4]) Let c be the midpoint of x and evaluate the mean value extension of f : f(x, c) = f(c) + ∇f(x)(x − c) = [−6.5, 1.4]

Bisect x1

f([0.5, 0.75] , x2) = [−6.0, 0.5] f([0.75, 1.] , x2) = [−5.6, 1.1]

Bisect x2

f(x1, [0.75, 1]) = [−6.0, −1.2] f(x1, [0.5, 0.75]) = [−3.6, 0.4]

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 4 / 10

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SLIDE 12

MaxSmear Strategy

Let x = [0.5, 1] × [0.5, 1] and evaluate the derivatives of f : ∇f(x) = ([1, 4] , [−11.5, −4]) Let c be the midpoint of x and evaluate the mean value extension of f : f(x, c) = f(c) + ∇f(x)(x − c) = [−6.5, 1.4]

Bisect x1

f([0.5, 0.75] , x2) = [−6.0, 0.5] f([0.75, 1.] , x2) = [−5.6, 1.1]

Bisect x2

f(x1, [0.75, 1]) = [−6.0, −1.2] f(x1, [0.5, 0.75]) = [−3.6, 0.4] MaxSmear : selecting the variable having the maximum smear value produces tighter linear relaxations = ⇒ Newton operator stronger

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 4 / 10

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SLIDE 13

RoundRobin Strategy

  • Total order xi1 < xi2 < · · · < xin

– xi < xj if xi occurs more than xj – xi < xj if xi occurs in more constraints than xj – . . .

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 5 / 10

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SLIDE 14

RoundRobin Strategy

  • Total order xi1 < xi2 < · · · < xin

– xi < xj if xi occurs more than xj – xi < xj if xi occurs in more constraints than xj – . . .

  • RoundRobin : select xi1, xi2, . . . , xin, xi1, xi2, . . . xin, xi1, . . .

– every domain is regularly bisected – fair strategy

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 5 / 10

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SLIDE 15

New Adaptive Strategy

  • Motivation

– robust smear strategy – efficient fair strategy

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 6 / 10

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SLIDE 16

New Adaptive Strategy

  • Motivation

– robust smear strategy – efficient fair strategy

  • Adaptive bisection strategy

– diversification (≈ RoundRobin) in the early steps of the algorithm – intensification (≈ MaxSmear) in the vicinity of solutions

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 6 / 10

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SLIDE 17

New Adaptive Strategy

  • Motivation

– robust smear strategy – efficient fair strategy

  • Adaptive bisection strategy

– diversification (≈ RoundRobin) in the early steps of the algorithm – intensification (≈ MaxSmear) in the vicinity of solutions

  • GRASP (Feo & Resende 1995)

– adaptive search procedure for derivative-free optimization – greedy + randomized

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 6 / 10

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SLIDE 18

New Algorithm

  • Initialization : for every variable (i = 1, . . . , n)

– si ≥ 0 : smear value of xi in x – ni ≥ 0 : number of times xi has been selected – [smin, smax] : range of smear values

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 7 / 10

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SLIDE 19

New Algorithm

  • Initialization : for every variable (i = 1, . . . , n)

– si ≥ 0 : smear value of xi in x – ni ≥ 0 : number of times xi has been selected – [smin, smax] : range of smear values

  • Subset of the best variables : let S ∈ [smin, smax] and mark every

variable xj s.t. sj ≥ S ∧ width(xj) > ǫ

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 7 / 10

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SLIDE 20

New Algorithm

  • Initialization : for every variable (i = 1, . . . , n)

– si ≥ 0 : smear value of xi in x – ni ≥ 0 : number of times xi has been selected – [smin, smax] : range of smear values

  • Subset of the best variables : let S ∈ [smin, smax] and mark every

variable xj s.t. sj ≥ S ∧ width(xj) > ǫ

  • Fair choice : select a marked variable xk s.t.

nk = min{ni : 1 ≤ i ≤ n ∧ xi marked}

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 7 / 10

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SLIDE 21

Adaptive Behaviour

  • Threshold on smear values

S = smin + α(smax − smin), 0 ≤ α ≤ 1

– α = 1 = ⇒ greedy – α = 0 = ⇒ fair

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 8 / 10

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SLIDE 22

Adaptive Behaviour

  • Threshold on smear values

S = smin + α(smax − smin), 0 ≤ α ≤ 1

– α = 1 = ⇒ greedy – α = 0 = ⇒ fair

  • Adaptive behaviour

α = 1 1 + βσ , β > 0

– σ : standard deviation of the smear values – σ → 0 = ⇒ α → 1 – σ → ∞ = ⇒ α → 0

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 8 / 10

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SLIDE 23

Experimental Results

Problem n MaxDom RoundRobin MaxSmear Adaptive Celestial 3 3 361 2 599 4 258 3 074 Combu. 10 559 463 1 303 509 Neuro 6 1 288 779 6 367 7 223 7 215 Trigo1 10 1 258 1 244 2 086 2 071 Broyden3 20 24 723 23 23 Geineg 6 8 546 8 116 1 707 2 400 Kapur 5 2 791 1 651 231 323 Nbody 8 1 976 2 049 1 532 1 765 Brown 10 12 443 5 850 6 827 4 996 Eco 6 1 108 1 530 1 063 1 023 Nauheim 8 1 204 722 816 708 Transistor 9 112 795 121 765 83 711 42 085

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 9 / 10

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SLIDE 24

Experimental Results

Problem n MaxDom RoundRobin MaxSmear Adaptive Celestial 3 3 361 2 599 4 258 3 074 Combu. 10 559 463 1 303 509 Neuro 6 1 288 779 6 367 7 223 7 215 Trigo1 10 1 258 1 244 2 086 2 071 Broyden3 20 24 723 23 23 Geineg 6 8 546 8 116 1 707 2 400 Kapur 5 2 791 1 651 231 323 Nbody 8 1 976 2 049 1 532 1 765 Brown 10 12 443 5 850 6 827 4 996 Eco 6 1 108 1 530 1 063 1 023 Nauheim 8 1 204 722 816 708 Transistor 9 112 795 121 765 83 711 42 085

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 9 / 10

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SLIDE 25

Experimental Results

Problem n MaxDom RoundRobin MaxSmear Adaptive Celestial 3 3 361 2 599 4 258 3 074 Combu. 10 559 463 1 303 509 Neuro 6 1 288 779 6 367 7 223 7 215 Trigo1 10 1 258 1 244 2 086 2 071 Broyden3 20 24 723 23 23 Geineg 6 8 546 8 116 1 707 2 400 Kapur 5 2 791 1 651 231 323 Nbody 8 1 976 2 049 1 532 1 765 Brown 10 12 443 5 850 6 827 4 996 Eco 6 1 108 1 530 1 063 1 023 Nauheim 8 1 204 722 816 708 Transistor 9 112 795 121 765 83 711 42 085

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 9 / 10

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SLIDE 26

Conclusion

  • The adaptive strategy is more robust.
  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 10 / 10

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SLIDE 27

Conclusion

  • The adaptive strategy is more robust.
  • Other adaptive schemes could be designed.

– aggregation of si and ni : (maxj nj − ni + 1) × si – learning techniques

  • L. Granvilliers (Nantes)

Adaptive Bisection CP 2012 10 / 10