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Unformatted text preview: satisFes ∇ f ( x * ) = 0 and ∇ 2 f ( x * ) positive deFnite, then x * is a local minimizer for f ( x ). (p. 298) 8. Let A be an m × n matrix with m < n of rank m . An ndimensional vector x can be written in the form x = p + q , where p ∈ N ( A ) and q ∈ R ( A T ). (p. 56) 9. Let Z be a basis null space matrix for an m × n matrix A of rank m . If Z T Zv = 0, then Zv = 0. (p. 432) 10. ±or the problem Min f ( x ) subject to Ax = b and Z the null space matrix of A , let φ ( v ) = f (¯ x + Zv ), then ∇ φ ( v ) = 0 if and only if Z T ∇ f ( x ) = 0 if and only if ∇ f ( x ) = A T λ for some vector λ . (p. 431) 1 . 2...
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 Spring '10
 Brown
 Linear Algebra, Operations Research, Linear Programming, Sets, Optimization, linear programming problem, ZV, null space matrix

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