In data fitting, regularization is a technique to stabilize an ill-posed problem. Ridge Regression (aka Tikhonov regularizaton) adds the 2-norm of the coefficient vector to the least squares problem. The cost function looks like this:

J=1/2*norm(A*x-b,2)^2+norm(alpha*x,2)^2 (MATLAB equation if image doesn't display)

where x is the coefficient vector, b corresponds to observations, A comes from evaluating terms of an equation, alpha corresponds to a chosen scalar weight value, and J is the cost. The minimization of this cost function is ridge regression. When alpha is zero and the least squares problem is ill-conditioned, this problem is unstable meaning that small changes in b cause large changes in x. Choosing alpha appropriately for a given problem stabilizes the problem causing small changes in b causes a small change in x.

This Cody problem is about solving the Ridge Regression problem. Implement the solution to the ridge regression problem. This is best done by solving a dual problem. Minimizes a different cost function yields the solution that solves the Ridge Regression problem:

J = 1/2*norm([A;alpha*eye(n)]*x-[b;zeros(1,n)],2)^2 (MATLAB equation if image doesn't display)

Therefore, write a function that minimum of the dual problem for a given A, alpha, and b.