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Unformatted text preview: m × n matrix A of rank m . If Z T Zv = 0, then Zv = 0. (p. 432) [* means omit] 10. ±or the problem Min f ( x ) subject to Ax = b and Z a basis null space matrix for A , let φ ( v ) = f (¯ x + Zv ), then ∇ φ ( v ) = Z T ∇ f ( x ) and ∇ 2 φ ( v ) = Z T ∇ 2 f ( x ) Z . (p. 429) 11. ±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) 12. ±or the problem Min f ( x ) subject to Ax ≥ b , suppose that x * is a stationary point of φ ( v ) = f (¯ x + Zv ) such that ∇ f ( x * ) = ˆ A T ˆ λ * for ˆ A the submatrix of A of active constraints for x * . If ˆ λ * 1 < 0, then x * is not a local minimizer. (p. 440) 1 . 2...
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 Spring '10
 Brown
 Linear Algebra, Linear Programming, Optimization, ZV, null space matrix

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