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: � � 1 � ≥ 1 x 2 + y 2 − 9 � 3 x − y 2 2 ( y − 2 ) 2 + ( x − 1 ) 2 ≥ 1 7 • How to describe the solution set? • How to find all solutions? Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 4/11

Describing the solution set • visual representation of the set • projection from higher dimensions • basic information about the set Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 5/11

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 X ⊂ ( S ∪ E ) outer approximation: boxes with no solutions: ( N ∩ X ) = ∅ Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 6/11

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 Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 7/11

Complex intervals r c ρ θ Im y y Re x x Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 8/11

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 ) c 2 + d 2 , bc − ad ( a , b ) � ac + bd � ( c , d ) = c 2 + d 2 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 ) } Elif Garajova ´, Martin Mec ˇiar Solving and visualizing nonlinear CSPs 10/11

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