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Proof of convergence theory in optimization


Feb 3, 2013
The question is:

Suppose that lim $x_k=x_*$, where $x_*$ is a local minimizer of the nonlinear function $f$. Assume that $\triangledown^2 f(x_*)$ is symmetric positive definite. Prove that the sequence $\left \{ f(x_k)-f(x_*) \right \}$ converges linearly if and only if $\left \{ ||x_k-x_*|| \right \}$ converges linearly. Prove that the two sequences converge at the same rate, regardless of what the rate is. What is the relationship between the rate constant for the two sequences?

(I guess we may use the orthogonal diagonalization of a symmetric matrix and $f(x_k)-f(x_*)=\triangledown f(x_*)+\frac{1}{2}(x_k-x_*)^T\cdot\triangledown^2 f(\xi)(x_k-x_*)$ and $\triangledown f(x_*)=0$...... But I got stuck here. So what's your answer?)