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Unformatted text preview: Lecture 11 Finding a Vertex UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Wednesday, February 4th, 2009 Recap: Vertices Definition (Vertex) Given a set of m linear constraints in n variables, a vertex is a feasible point at which there are at least n linearly independent constraints active. (Equivalently, A a has at least one subset of n linearly independent rows, which implies that A a has rank n .) UCSD Center for Computational Mathematics Slide 2/36, Wednesday, February 4th, 2009 Example: x 1 + x 2 ≥ 1 x 1 ≥ x 1 ≥  2 x 2 ≥ x 1 + 2 x 2 ≥ 1 In matrixvector form Ax ≥ b , with A = 1 1 1 1 1 1 2 b = 1 2 1 UCSD Center for Computational Mathematics Slide 3/36, Wednesday, February 4th, 2009 x 2 x 1 #2 #1 #4 #3 #5 UCSD Center for Computational Mathematics Slide 4/36, Wednesday, February 4th, 2009 x 2 x 1 #2 #1 #4 #3 #5 A a = 1 1 1 2 0 1 A a = 1 1 1 0 A a = 1 1 UCSD Center for Computational Mathematics Slide 5/36, Wednesday, February 4th, 2009 Definition (Nondegenerate Vertex) A vertex at which exactly n constraints are active is called a nondegenerate vertex . Result If ¯ x is a nondegenerate vertex, then A a is nonsingular. Definition (Degenerate Vertex) A vertex at which more that n constraints are active is called a degenerate vertex . UCSD Center for Computational Mathematics Slide 6/36, Wednesday, February 4th, 2009 x 2 x 1 #2 #1 #4 #3 #5 A a = 1 1 1 2 0 1 A a = 1 1 1 0 (nondegenerate vertex) (degenerate vertex) UCSD Center for Computational Mathematics Slide 7/36, Wednesday, February 4th, 2009 Result At a degenerate vertex ¯ x , any subset of n linearly independent constraints uniquely defines ¯ x . UCSD Center for Computational Mathematics Slide 8/36, Wednesday, February 4th, 2009 n × n nonsingular A a m a × n ¯ A a b a ¯ b a Nonsingular system ¯ A a ¯ x = ¯ b a UCSD Center for Computational Mathematics Slide 9/36, Wednesday, February 4th, 2009 When can we guarantee that Ax ≥ b has a vertex?...
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