Exercise · ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-004 · ★☆☆ Basic
What type of convergence does Newton's method have near a minimizer with invertible Hessian?
Choose one option.
Show answer
- D. Quadratic
Explanation
Near the minimizer the error is squared at each iteration, which is quadratic convergence. Linear convergence is slower, and logarithmic convergence does not describe this case.
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-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?
- ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-009 For f(x,y)=x^2+xy+y^2-3x, compute the Newton step from (0,0) and verify that the resulting point is…
← ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-003 ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-005 →