# Speed of convergence for minimizing Rosenbrock's function

I am minimizing $$f(x_1,x_2) = 100(x_2-x_1^2)^2 + (1-x_1)^2$$, where I try many algorithms to compare with each other. All of the algorithms find the optimal solution $$(1,1)$$ quickly, so I am not bothered with asymptotics right now.

Let $$x = (x_1,x_2)^\top$$ and $$x^*$$ define the optimal solution. We know that $$\epsilon= \|x_k - x^*\|$$ is the error of iteration $$k$$ compared to the optimal solution, and in convergence we are usually interested in the relation between $$\epsilon_k, \epsilon_{k+1}$$.

My question is very simple. Which one is the best option here, to plot $$\dfrac{\epsilon_{k+1}}{\epsilon_{k}}$$ for each algorithm, or to plot $$\epsilon_{k+1} - \epsilon_{k}$$? The first one is usually studied in asymptotic convergence, so I'm not sure whether it really makes sense for my simple Rosenbrock minimization.

• Why not both? And I would also use several (hundreds) runs with each different starting point and then put the average or median values fur each algorithm. Jan 4, 2020 at 18:48
• Change the 100 to 1e6 to make it a little more fun. Jan 4, 2020 at 20:02