Discontinuous Images in Constraint On Utilizing Disconnected Images Propagation within GlobSol’s Constraint Introduction Example Propagation Software The GlobSol Context Notation GlobSol’s overall algorithm Constraint propagation within GlobSol Implementation of disconnected Ralph Baker Kearfott intervals in GlobSol Experimental Results The test problems Department of Mathematics The results University of Louisiana at Lafayette Conclusions SCAN 2010 1 / 23
Outline Introduction Discontinuous Images in Constraint Example Propagation Introduction The GlobSol Context Example The GlobSol Context Notation Notation GlobSol’s overall algorithm GlobSol’s overall algorithm Constraint propagation within Constraint propagation within GlobSol GlobSol Implementation of disconnected intervals in GlobSol Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results Experimental Results Conclusions The test problems The results Conclusions 2 / 23
General Context – Constraint propagation Components of Images Discontinuous Images in Constraint ◮ Constraint propagation is commonly used as one Propagation of various tools to reduce the size of search Introduction Example regions in branch and bound-based software for The GlobSol Context global optimization. Notation GlobSol’s overall algorithm Constraint propagation within GlobSol Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results Conclusions 3 / 23
General Context – Constraint propagation Components of Images Discontinuous Images in Constraint ◮ Constraint propagation is commonly used as one Propagation of various tools to reduce the size of search Introduction Example regions in branch and bound-based software for The GlobSol Context global optimization. Notation GlobSol’s overall algorithm ◮ Constraint propagation greatly speeds the overall Constraint propagation within GlobSol solution process and makes some problems that Implementation of disconnected intervals in GlobSol otherwise may not be practical to solve. Experimental Results The test problems The results Conclusions 3 / 23
General Context – Constraint propagation Components of Images Discontinuous Images in Constraint ◮ Constraint propagation is commonly used as one Propagation of various tools to reduce the size of search Introduction Example regions in branch and bound-based software for The GlobSol Context global optimization. Notation GlobSol’s overall algorithm ◮ Constraint propagation greatly speeds the overall Constraint propagation within GlobSol solution process and makes some problems that Implementation of disconnected intervals in GlobSol otherwise may not be practical to solve. Experimental Results The test problems ◮ The volume of regions returned from constraint The results Conclusions propagation is reduced more if inverse images with multiple, disconnected components are taken, rather than by taking the interval hull of such components and continuing with a single interval. 3 / 23
Outline Introduction Discontinuous Images in Constraint Example Propagation Introduction The GlobSol Context Example The GlobSol Context Notation Notation GlobSol’s overall algorithm GlobSol’s overall algorithm Constraint propagation within Constraint propagation within GlobSol GlobSol Implementation of disconnected intervals in GlobSol Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results Experimental Results Conclusions The test problems The results Conclusions 4 / 23
Discontinuous Images An Example Take the constraint Discontinuous Images in Constraint Propagation 1 / x − y = 0 Introduction Example along with the bounds x ∈ [ − 3 , 3 ] and y ∈ [ − 1 , 1 ] , The GlobSol Context solving for x in terms of y gives Notation GlobSol’s overall algorithm Constraint propagation within GlobSol x ∈ � � ( −∞ , − 1 ] ∪ [ 1 , ∞ ) ∩ [ − 3 , 3 ] = [ − 3 , − 1 ] ∪ [ 1 , 3 ] . Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results ◮ Similar disconnected solution sets occur when: Conclusions 5 / 23
Discontinuous Images An Example Take the constraint Discontinuous Images in Constraint Propagation 1 / x − y = 0 Introduction Example along with the bounds x ∈ [ − 3 , 3 ] and y ∈ [ − 1 , 1 ] , The GlobSol Context solving for x in terms of y gives Notation GlobSol’s overall algorithm Constraint propagation within GlobSol x ∈ � � ( −∞ , − 1 ] ∪ [ 1 , ∞ ) ∩ [ − 3 , 3 ] = [ − 3 , − 1 ] ∪ [ 1 , 3 ] . Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results ◮ Similar disconnected solution sets occur when: Conclusions ◮ taking inverses of even-order integer powers, 5 / 23
Discontinuous Images An Example Take the constraint Discontinuous Images in Constraint Propagation 1 / x − y = 0 Introduction Example along with the bounds x ∈ [ − 3 , 3 ] and y ∈ [ − 1 , 1 ] , The GlobSol Context solving for x in terms of y gives Notation GlobSol’s overall algorithm Constraint propagation within GlobSol x ∈ � � ( −∞ , − 1 ] ∪ [ 1 , ∞ ) ∩ [ − 3 , 3 ] = [ − 3 , − 1 ] ∪ [ 1 , 3 ] . Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results ◮ Similar disconnected solution sets occur when: Conclusions ◮ taking inverses of even-order integer powers, ◮ taking inverses of trigonometric functions. 5 / 23
Discontinuous Images An Example Take the constraint Discontinuous Images in Constraint Propagation 1 / x − y = 0 Introduction Example along with the bounds x ∈ [ − 3 , 3 ] and y ∈ [ − 1 , 1 ] , The GlobSol Context solving for x in terms of y gives Notation GlobSol’s overall algorithm Constraint propagation within GlobSol x ∈ � � ( −∞ , − 1 ] ∪ [ 1 , ∞ ) ∩ [ − 3 , 3 ] = [ − 3 , − 1 ] ∪ [ 1 , 3 ] . Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results ◮ Similar disconnected solution sets occur when: Conclusions ◮ taking inverses of even-order integer powers, ◮ taking inverses of trigonometric functions. ◮ Should we consider [ − 3 , − 1 ] and [ 1 , 3 ] separately, or should we simply use [ − 3 , 3 ] in subsequent calculations? 5 / 23
Individual Image Components Disadvantages – Should they be used? Discontinuous Images ◮ A disadvantage of proceeding with each in Constraint Propagation component separately is that more boxes are produced. Introduction Example The GlobSol Context Notation GlobSol’s overall algorithm Constraint propagation within GlobSol Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results Conclusions 6 / 23
Individual Image Components Disadvantages – Should they be used? Discontinuous Images ◮ A disadvantage of proceeding with each in Constraint Propagation component separately is that more boxes are produced. Introduction Example ◮ The volume reduction obtained from considering The GlobSol Context each component separately might also be effected Notation GlobSol’s overall algorithm through other means. Constraint propagation within GlobSol Implementation of disconnected intervals in GlobSol Experimental Results The test problems The results Conclusions 6 / 23
Individual Image Components Disadvantages – Should they be used? Discontinuous Images ◮ A disadvantage of proceeding with each in Constraint Propagation component separately is that more boxes are produced. Introduction Example ◮ The volume reduction obtained from considering The GlobSol Context each component separately might also be effected Notation GlobSol’s overall algorithm through other means. Constraint propagation within GlobSol Implementation of disconnected ◮ Knowing whether or not to use separate intervals in GlobSol Experimental Results components can only be determined by The test problems experimentation within particular software. The results Conclusions 6 / 23
Individual Image Components Disadvantages – Should they be used? Discontinuous Images ◮ A disadvantage of proceeding with each in Constraint Propagation component separately is that more boxes are produced. Introduction Example ◮ The volume reduction obtained from considering The GlobSol Context each component separately might also be effected Notation GlobSol’s overall algorithm through other means. Constraint propagation within GlobSol Implementation of disconnected ◮ Knowing whether or not to use separate intervals in GlobSol Experimental Results components can only be determined by The test problems experimentation within particular software. The results Conclusions ◮ Here, we report on the effect of using disconnected components with our GlobSol global optimization software. 6 / 23
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