The linear programming can be described as
$\min\ {\bf c}^{T}{\bf x}$
$ s.t.\ {\bf Ax=b} $,
$\ \ \ \ \ \ \ \ \ \|{\bf x}\|^{2}_{2}\le U$,
$\ \ \ \ \ \ \ \ \ x_{i}\ge 0,\ i=0,1,...,N-1$,
where ${\bf x}\in\mathbb{R}^{N\times 1}$, ${\bf c}\in\mathbb{R}^{N\times 1}_{+}$, ${\bf A}\in\mathbb{R}^{M\times N}_{+}$, ${\bf b}\in\mathbb{R}^{M\times 1}_{+}$, $\|\cdot\|_{2}$ is the $\ell2$-norm and $U$ is a constant scalar.
The first question is:
How to handle the last constraint (the positivity constraint)?
I can solve the problem analytically by using the method of Lagrange multipliers after I relax the the constraint. But I don't know how to solve the entire problem.
And other question is:
Is there any algorithm that can solve the problem efficiently?
Numerical results can be solved by MATLAB using the SQP algorithm, but I think there must be some simple and efficient algorithms.
Thanks.
I would first try to throw this at a state-of-the-art (convex) QCP (Quadratically Constrained Program) solver such as Cplex, Gurobi or Mosek. They have specialized solvers for this type of problem that typically outperform a general purpose NLP solver (sometimes by a large margin).