Exercise · ZW-NEWTON-S-METHOD-FOR-OPTIMIZATION-009 · ★★★ Advanced

For \(f(x,y)=x^2+xy+y^2-3x\), compute the Newton step from \((0,0)\) and verify that the resulting point is the minimizer.

Choose one option.

Show answer

C. \((2,-1)\), because \(p=H_f^{-1}(3,0)\)

Explanation

At \((0,0)\) the gradient is \((-3,0)\) and the Hessian \(\begin{pmatrix}2&1\\1&2\end{pmatrix}\) has determinant 3. Solving \(Hp=(3,0)\) gives \(p=(2,-1)\), and at \((2,-1)\) the gradient vanishes, so this is the minimizer.

Study Continuous optimization in ZWMath →

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?