I want to solve the standard problem $\min_{x\in D} \frac{1}{2}||Ax-b ||$ where:
D = \begin{matrix} 0\leq x_1 \leq C \\ |x_2| \leq x_1 \hspace{2mm} \text{and} \hspace{2mm} |x_2| \leq C- x_1 \\ 0\leq x_3\leq D \end{matrix}
where $C,D>0$. I want to find a solution numerically.
So the idea is the following:
1) Solve the unconstrained problem in which there exist a closed form solution. If the solution is in the domain $D$ stop. Else: 2) Seek for a solution in the boundary using Lagrange multipliers.
Does anyone know how may I solve the problem with Lagrange-multipliers on the boundary (especially in the second inequalities)?
There is a scipy-python optimizer but the bounds of the x-vector that can be used as inputs do not depend on the x-vector components...Any ideas?
Thanks in advance