I have faced a problem of programming the computation of next function for each p:
$$f(p) = \min_\limits{\| b_{0} - A_{0} x \| = p} \| x \|$$
For now I work with Euclidean norm (or Frobenius for matrix). Also I know about this function is that is it monotonically decreasing on
$$R_{0} \le p \le R_{1}, \qquad R_{0} = \min \|b_{0} - A_{0}x\|, \qquad R_{1} = \|b_{0}\|$$
I heard that this is quadratic programming problem, but when I started to research I didn't found what I need. Can someone please help me with solving this problem?
Edit:
Partial derivatives by each $x_{k}$
Using that derivative of $|ax + by|^{2}~by~x~is~2a(ax+by)$
$$L_{x_{k}} = 2x_{k} - 2\lambda \sum \limits_{i=1}^{m}a_{ik}\sum \limits_{j = 1}^{n}(a_{ij}x_{j} - b_{i}) = 0$$
$$x_{k} - \lambda \sum \limits_{i=1}^{m}a_{ik}\sum \limits_{j = 1}^{n}a_{ij}x_{j} + \lambda \sum \limits_{i = 1}^{m} a_{ik} b_{i} * n = 0$$
$$x_{k} - \lambda \sum \limits_{i = 1}^n x_{i} \sum \limits_{j = 1}^{m} a_{jk}a_{ji} = -\lambda*n\sum \limits_{i = 1}^m a_{ik}b_{i}$$
$A^{k} - column~of~matrix~A,~A_{k} -~row$
$$\begin{pmatrix} 1 - \lambda ||A^{1}||^{2} & - \lambda (A^{1}, A^{2}) & \dots & - \lambda (A^{1}, A^{n}) \\ - \lambda (A^{2}, A^{1}) & 1 - \lambda ||A^{2}||^{2} & \dots & - \lambda (A^{2}, A^{n}) \\ \vdots & \vdots & \ddots & \vdots \\ - \lambda (A^{n}, A^{1}) & \dots & \dots & 1 - \lambda (|| A^{n}||^{2}) \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n}\end{pmatrix} = \begin{pmatrix} -\lambda*n*(A^{1}, b) \\ -\lambda*n*(A^{2}, b) \\ \vdots \\ -\lambda*n*(A^{n}, b) \end{pmatrix}$$
So now I have problem with solving this for $x(\lambda)$