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001 initial guess x0 newtons method 1 find g x g x

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Unformatted text preview: a global optimum. The convex hull of the level set X0 is the smallest convex set that contains all of the points in X0 35 / 45 Unconstrained Newton’s Method Assume we have a continuously twice-differentiable f : Rn → R. We want to solve minx∈Rn f (x) Recall: Newton’s Method iteratively uses a function’s first-order approximations to find its roots. g (x) ≈ g (x0 ) + ∇g (x0 )(x − x0 ) g (x0 ) + ∇g (x0 )(x − x0 ) = 0 To use Newton’s Method to solve minx∈Rn f (x) we will let g (x) = ∇f (x) and use the Hessian ∇2 f = ∇g (x) to find an x where ∇f = 0 36 / 45 Newton’s Method - General Algorithm Given: ￿ A continuously differentiable g : Rn → Rn for which we want to find x such that g (x) = 0 ￿ ￿ = .001 ￿ Initial guess x0 Newton’s Method: 1. Find g (x), ∇g (x) and ∇g (x)−1 2. xk +1 = xk − ∇g (xk )−1 g (xk ) 3. If ￿xk +1 − xk ￿2 < ￿ return xk +1 . Otherwise, repeat step 2 37 / 45 Newton’s Method - Uncontrained Minimization Algorithm Given: ￿ ￿ ￿ A continuously twice-differentiable f : Rn → R for which we want to solve minx∈Rn f (x) ￿ = .001 Initia...
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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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