Convex Optimization · Problem 9.10

Equality constrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · pure-newton-method

Problem

The pure Newton method. Newton's method with fixed step size

t = 1

can diverge if the initial point is not close to

x^\star

. In this problem we consider two examples. ((a))

f(x) = \log(e^x + e^{-x})

has a unique minimizer

x^\star = 0

. Run Newton's method with fixed step size

t = 1

, starting at

x^{(0)} = 1

and at

x^{(0)} = 1.1

. ((b))

f(x) = -\log x + x

has a unique minimizer

x^\star = 1

. Run Newton's method with fixed step size

t = 1

, starting at

x^{(0)} = 3

. Plot

f

and

f'

, and show the first few iterates.