Solving and visualizing nonlinear constraint satisfaction problems - - PowerPoint PPT Presentation

Solving and visualizing nonlinear constraint satisfaction problems

Elif Garajova ´, Martin Mec ˇiar

Department of Applied Mathematics Faculty of Mathematics and Physics Charles University in Prague

SWIM, June 2015

Outline

1 Interval solver for nonlinear constraints 2 Application: Complex intervals 3 Visualization techniques

Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 2/11

Library of Interval MEthods

Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 3/11

Library of Interval MEthods

Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 3/11

Problem: Solving a nonlinear CSP

Find the set of all (x, y) ∈ [−3, 3] × [−3, 3] satisfying:

• x2 + y2 − 9

1

3x − y2

• ≥ 1

2 (y − 2)2 + (x − 1)2 ≥ 1 7

• How to describe the solution set?
• How to find all solutions?

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Describing the solution set

• visual representation of the set
• projection from higher dimensions
• basic information about the set

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Describing the solution set

• visual representation of the set
• projection from higher dimensions
• basic information about the set
• description using interval boxes
• outer and inner approximation

Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 5/11

Solution: SIVIA

inner approximation: S ⊂ X

• uter approximation:

X ⊂ (S ∪ E) boxes with no solutions: (N ∩ X) = ∅

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Interval solver for nonlinear constraints

• solver based on the SIVIA algorithm
• uses interval contractors to enhance its efficiency
• written in MATLAB (and C++) using the INTLAB toolbox
• the interval solver can:
• solve a nonlinear CSP using interval methods
• reduce the number of boxes on the output
• plot the solution set (or its projection) in 2D
• visualize complex interval arithmetic

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Complex intervals

c r θ ρ Re Im x x y y

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Complex interval arithmetic I

Exact operations (a, b) + (c, d) = (a + c, b + d) (a, b) − (c, d) = (a − c, b − d) Overestimated operations (a, b) · (c, d) = (ac − bd, ad + bc) (a, b) (c, d) =

ac + bd

c2 + d2 , bc − ad c2 + d2

• Elif Garajova

´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 9/11

Complex interval arithmetic II

Interval operation: (a, b) · (c, d) = (ac − bd, ad + bc) Exact operation: {(a + bi) · (c + di) | (a, b) ∈ (a, b), (c, d) ∈ (c, d)}

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Visualizing nonlinear CSPs

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