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Lecture 3 - Sets of Linear Constraints

# Lecture 3 - Sets of Linear Constraints - Recap Navigating...

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Lecture 3 Sets of Linear Constraints UCSD Math 171A: Numerical Optimization Philip E. Gill ( [email protected] ) Friday, January 9th, 2009 Recap: Navigating hyperspace We can define the linear path : x ( α ) = ¯ x + α p where p is a nonzero direction vector that emanates from ¯ x . ¯ x p x ( α ) = ¯ x + αp UCSD Center for Computational Mathematics Slide 2/35, Friday, January 9th, 2009 Example: Given ¯ x = 1 1 ! and the direction p = 0 1 ! the linear path is x ( α ) = 1 1 ! + α 0 1 ! = 1 1 + α ! for all α 0 UCSD Center for Computational Mathematics Slide 3/35, Friday, January 9th, 2009 x ( α ) = 1 1 + α ! for all α 0 ¯ x p x ( α ) = ¯ x + αp x 1 x 2 UCSD Center for Computational Mathematics Slide 4/35, Friday, January 9th, 2009

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Question How does the constraint value r x ) vary as we move from ¯ x along the linear path x ( α ) = ¯ x + α p ? ¯ x p x ( α ) = ¯ x + αp UCSD Center for Computational Mathematics Slide 5/35, Friday, January 9th, 2009 r x + α p ) = a T x + α p ) - b = a T ¯ x + α a T p - b = ( a T ¯ x - b ) + α a T p = r x ) + α a T p r x + α p ) is a linear function of α , with d dt r x + α p ) α =0 = a T p r changes at the constant rate a T p . UCSD Center for Computational Mathematics Slide 6/35, Friday, January 9th, 2009 r is decreasing if a T p < 0 constant if a T p = 0 increasing if a T p > 0 Since | r | reflects the distance to the constraint, the sign of a T p tells us if we are moving away from, or towards the constraint boundary. UCSD Center for Computational Mathematics Slide 7/35, Friday, January 9th, 2009 Suppose that ¯ x is feasible. Let p (1) , p (2) , and p (3) be directions such that a T p (1) < 0 a T p (2) = 0 a T p (3) > 0 Then: a T p (1) < 0 p (1) points towards the constraint boundary a T p (2) = 0 p (2) is parallel to the constraint boundary a T p (3) > 0 p (3) points away from the constraint boundary UCSD Center for Computational Mathematics Slide 8/35, Friday, January 9th, 2009
Summary Given a feasible ¯ x and any direction p 6 = 0, a step α > 0 along p moves towards the infeasible half space if a T p < 0 and moves away

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Lecture 3 - Sets of Linear Constraints - Recap Navigating...

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