Exercise · ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-010 · ★★☆ Intermediate

For \(f(x,y)=x^3-3x+y^2\), what happens if the Newton step is applied from \((0,1)\)? Use the Hessian \(\mathrm{diag}(6x,2)\).

Choose one option.

Show answer

B. The Hessian is singular at \((0,1)\), so the step is undefined

Explanation

At \((0,1)\) the Hessian \(\mathrm{diag}(0,2)\) is singular because \(6x=0\), so it cannot be inverted. The other options assume an invertible Hessian or a vanishing gradient that do not hold.

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Review the concept: Newton's method for optimization

Newton's method for optimization is an iterative algorithm that seeks a critical point of a differentiable function \(f:\mathbb{R}^n\to\mathbb{R}\) using the gradient \(\nabla f\) and the Hessian \(H_f\).

\[x_{k+1}=x_k-H_f(x_k)^{-1}\nabla f(x_k)\]

More practice problems: newton's method for optimization

ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-001 What is the iteration formula of Newton's method for minimizing f:ℝ^n→ℝ?
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-002 Which condition on the Hessian guarantees that the Newton step is a descent direction?
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-003 If the Hessian is the identity matrix everywhere, what does the Newton step reduce to?
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-004 What type of convergence does Newton's method have near a minimizer with invertible Hessian?
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-005 Which object must be computed at each iteration of Newton's method besides the gradient?
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-006 For f(x,y)=x^2+2y^2 at (1,1), compute the Newton step p=-H_f^-1∇ f.
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-007 For f(x,y)=x^4+y^2 at (1,2), what is the Newton step in the x coordinate?
ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-008 For f(x)=e^x-2x on ℝ, what is x_1 starting from x_0=0?