The constrained optimization problem

Question

I would like to find the minimum value $F(x)=x^{T}Ax$ and $\|x\|_{2}=1$, where $A$ is symmetric and positive-definite. I know that the minimum value is the smallest eigenvalue problem of the matrix $A$. However, I want to know that how to use an optimization method to find it. The method should be based on an iterative method. Please give me the iterative method. I hope this iterative method is convergent to the exact solution as quickly as possible. Thank you!**

Answer 1

The set $\|x\|_2=1$ is a riemannian manifold, you can use riemannian gradient descent to solve this. Alternatively, if the matrix $A$ is invertible, you can use the power method on $A^{-1}$.