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Unformatted text preview: x w = −tc − ∇φ(ˆ) x 0 Suppose λ(ˆ) ≤ 1 where λ(ˆ) is the Newton decrement at x. x x ˆ 83 . (a) Show that x + ∆x is primal feasible. ˆ (b) Show that y = −(1/t)w is dual feasible. (c) Let p⋆ be the optimal value of the LP. Show that √ n + λ(ˆ) n x c x−p ≤ ˆ . t T ⋆ 84 10 Mathematical background 10.1 Some famous inequalities. The Cauchy-Schwarz inequality states that | a T b| ≤ a b 2 2 for all vectors a, b ∈ Rn (see page 633 of the textbook). (a) Prove the Cauchy-Schwarz inequality. Hint. A simple proof is as follows. With a and b fixed, consider the function g (t) = a + tb 2 2 of the scalar variable t. This function is nonnegative for all t. Find an expression for inf t g (t) (the minimum value of g ), and show that the Cauchy-Schwarz inequality follows from the fact that inf t g (t) ≥ 0. (b) The 1-norm of a vector x is defined as x to show that x 1 1 ≤ n k=1 |xk |. = √ nx Use the Cauchy-Schwarz inequality 2 for all x. (c) The harmonic mean of a positive vector x ∈ Rn is defined as ++ 1n1 n k=1 xk −...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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