I am trying to solve following minimization problem.
$$\,\min_{x \in R^n} \left\lVert Ax -b\right\rVert_2$$ $$s.t \left\lVert x\right\rVert_2 \le t, t\ge0$$
My procedure is as follows.
First I find it's Lagrangian
$L(x,v) = \left\lVert Ax -b\right\rVert_2 + v^T(\left\lVert x\right\rVert_2 -t)$
I can express this in $l2$ norm
$L(x,v) = \left\lVert Ax -b\right\rVert_2^2 + v^T(\left\lVert x\right\rVert_2^2 -t^2)$ (Is this correct?)
Then I write the Lagrangian dual
$g(v) = \,\inf_{x\in R^n} L(x,v)$
$g(v) = \,\inf_{x\in R^n} \left\lVert Ax -b\right\rVert_2^2 + v^T(\left\lVert x\right\rVert_2^2 -t^2)$
Considering the convexity of two terms I find the gradient of the Lagrangian and considering first order condition, find the x which minimize Lagrangian
$\nabla L(x,v) = 2A^T(Ax-b)+ 2vx^T = 0$
$x = b^TA/(v + AA^T)$
My plan is to plug this x to Lagrangian and obtain the dual of it and then maximize it. But I not sure how to plug it and proceed
- Am I on the correct path to solve this?
- Are there any mistakes in my procedure?