Lecture 4 - Properties of the Objective Function

# Lecture 4 - Properties of the Objective Function - Recap...

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Lecture 4 Properties of the Objective Function UCSD Math 171A: Numerical Optimization Philip E. Gill ( [email protected] ) Monday, January 12th, 2009 Recap: Properties of linear constraints x 1 + x 2 1 x 1 0 - x 1 ≥ - 2 x 2 0 x 1 + 2 x 2 1 In matrix-vector form Ax b , with A = 1 1 1 0 - 1 0 0 1 1 2 b = 1 0 - 2 0 1 UCSD Center for Computational Mathematics Slide 2/59, Monday, January 12th, 2009 x 1 x 2 a 1 # 1 #2 #3 #4 a 4 #5 a 5 a 2 UCSD Center for Computational Mathematics Slide 3/59, Monday, January 12th, 2009 An important feature of the feasible region Result Every point on the boundary of the feasible region lies on intersection of a subset of the constraint hyperplanes UCSD Center for Computational Mathematics Slide 4/59, Monday, January 12th, 2009

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x 1 x 2 a 1 # 1 #2 #3 #4 a 4 #5 a 5 a 2 UCSD Center for Computational Mathematics Slide 5/59, Monday, January 12th, 2009 x 1 x 2 ¯ x = 0 1 x 1 + x 2 = 1 x 1 = 0 UCSD Center for Computational Mathematics Slide 6/59, Monday, January 12th, 2009 The boundary point ¯ x = 0 1 ! lies on 2 hyperplanes: H 1 = { x : x 1 + x 2 = 1 } H 2 = { x : x 1 = 0 } ¯ x satisﬁes the square nonsingular system of equations 1 1 1 0 ! x 1 x 2 ! = 1 0 ! UCSD Center for Computational Mathematics Slide 7/59, Monday, January 12th, 2009 x 1 x 2 a 1 # 1 #2 #3 #4 a 4 #5 a 5 a 2 UCSD Center for Computational Mathematics Slide 8/59, Monday, January 12th, 2009
x 1 x 2 x 1 + x 2 = 1 ¯ x = 1 2 1 2 UCSD Center for Computational Mathematics Slide 9/59, Monday, January 12th, 2009 The boundary point ¯ x = 1 2 1 2 ! lies on 1 hyperplane: H 1 = { x : x 1 + x 2 = 1 } ¯ x satisﬁes the underdetermined system of equations ± 1 1 ² x 1 x 2 ! = ± 1 ² UCSD Center for Computational Mathematics Slide 10/59, Monday, January 12th, 2009 x 1 x 2 a 1 # 1 #2 #3 #4 a 4 #5 a 5 a 2 UCSD Center for Computational Mathematics Slide 11/59, Monday, January 12th, 2009 x 1 x 2 x 1 + x 2 = 1 x 1 + 2 x 2 = 1 ¯ x = 1 0 x 2 = 0

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The boundary point ¯ x = 1 0 ! lies on 3 hyperplanes: H 1 = { x : x 1 + x 2 = 1 } H 2 = { x : x 1 + 2 x 2 = 1 } H 3 = { x : x 2 = 0 } ¯ x satisﬁes the overdetermined system of equations 1 1 1 2 0 1 x 1 x 2 ! = 1 1 0 UCSD Center for Computational Mathematics Slide 13/59, Monday, January 12th, 2009 study of the boundary points of IF ⇐⇒ study of linear systems Deﬁnition In R n , a corner point of IF = { x : Ax b } is deﬁned as a feasible point that lies on n hyperplanes. UCSD Center for Computational Mathematics
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## This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Lecture 4 - Properties of the Objective Function - Recap...

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