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for a quadratic minimization problem newtons method

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Unformatted text preview: l guess x0 Newton’s Method: 1. Find ∇f (x), ∇2 f (x) and ∇2 f (x)−1 2. xk +1 = xk − ∇2 f (xk )−1 ∇f (xk ) 3. If ￿xk +1 − xk ￿2 < ￿ return xk +1 . Otherwise, repeat step 2 38 / 45 Unconstrained Newton’s Method Example 2 2 Let f (x) = x1 + 2x2 + x1 x2 + 3x2 with x0 = (0, 0) and ￿ = .001. Minimixe f (x) Step 1: ￿ ∇ f ( x) = ∇ 2 f ( x) = ∇ 2 f ( x) − 1 2x1 + x2 4x 2 + x 1 + 3 ￿ 21 14 = 1 7 ￿ ￿ ￿ 4 −1 −1 2 ￿ 39 / 45 Unconstrained Newton’s Method Example 2 2 Minimize f (x) = x1 + 2x2 + x1 x2 + 3x2 First Iteration ￿ x0 = (0, 0) ￿ x1 ￿￿ ￿ ￿ 3 ￿ 4 −1 0 7 = (0, 0) − = −6 −1 2 3 7 Check for Convergence ￿ ￿￿￿ 23 − 6 0 1) = 7 7 ￿ ∇f ( x = −6 3 0 4 7 + 7 +3 ￿ Terminate algorithm! Converged with only 1 step 1 7 ￿ 40 / 45 Newton’s Method Convergence When Newton’s Method converges, it does so pretty quickly (quadratically or better.) ￿ For a quadratic minimization problem, Newton’s Method converges in just one step But Newton’s Met...
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This note was uploaded on 12/10/2013 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

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