Exercise · ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-002 · ★☆☆ Basic
Which condition on the Hessian guarantees that the Newton step is a descent direction?
Choose one option.
Show answer
- B. That the Hessian is positive definite
Explanation
With a positive definite Hessian, the Newton step makes an acute angle with the negative gradient. Symmetry with zero trace is not enough, and a negative determinant signals an indefinite matrix.
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-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?
- 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…
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