proofs_2

# proofs_2 - m × n matrix A of rank m If Z T Zv = 0 then Zv...

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Let S = { x : Ax = b,x 0 } so that standard form for linear programming is Min z = c T x subject to x S . 1. The set S is convex. 2. In a linear programming problem, a local minimizer is a global minimizer. (p. 22) *3. If x is a basic feasible solution, then x is an extreme point of S . (p. 80) [* means omit] 4. If a linear programming problem with S bounded has an optimal solution, then it has an optimal solution which is basic feasible. (p. 90) 5. Given feasible solutions x to Min z = c T x subject to Ax = b,x 0 and y to Max w = b T y subject to A T y c , then w = b T y c T x = z . (p. 150) 6. If x * is optimal for Min z = c T x subject to Ax = b,x 0 and y * is optimal for Max w = b T y subject to A T y c , then x T * ( c A T y * ) = 0. Conversely, if x and y are feasible solutions to these problems such that x T ( c A T y ) = 0, then they are optimal. (p. 154) 7. If x * 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 n -dimensional 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

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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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proofs_2 - m × n matrix A of rank m If Z T Zv = 0 then Zv...

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