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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.
- Fall '07