Unformatted text preview: that the iterates are
(k ) x1 = γ γ−1
γ+1 k , (k ) x2 = − γ−1
γ+1 k . Therefore x(k) converges to (0, 0). However, this is not the optimum, since f is unbounded
8.2 A characterization of the Newton decrement. Let f : Rn → R be convex and twice diﬀerentiable,
and let A be a p × n-matrix with rank p. Suppose x is feasible for the equality constrained problem
minimize f (x)
subject to Ax = b.
Recall that the Newton step ∆x at x can be computed from the linear equations
∇2 f (ˆ) AT
u = −∇f (ˆ)
0 , and that the Newton decrement λ(ˆ) is deﬁned as
λ(ˆ) = (−∇f (ˆ)T ∆x)1/2 = (∆xT ∇2 f (ˆ)∆x)1/2 .
Assume the coeﬃcient matrix in the linear equations above is nonsingular and that λ(ˆ) is positive.
Express the solution y of the optimization problem
minimize ∇f (ˆ)T y
subject to Ay = 0
y T ∇2 f ()y ≤ 1
in terms of Newton step ∆x and the Newton decrement λ(ˆ).
8.3 Suggestions for exercises 9.30 in Convex Optimization. We recommend the following to generate
a problem instance:
74 n = 100...
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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 Berkeley.
- Fall '13
- The Aeneid