Quadratic problem solving with absolute value constraint Hello,
I have been trying to solve a problem of the form : 
$\max_x\quad -\tfrac{1}{2}x^TAx + b^Tx - C\sum_i |x_i|$
without the C term it is a simple quadratic problem, 
but I haven't been able to find any reasearch paper related to this kind of problem solving.
If you could share your opinion on this problem it would be greatly appreciated
best regards
edit : A is a positive-definite matrix
 A: Well, you have a non-smooth unconstrained problem but there exists a standard reformulation of the absolute function into linear constraints.
$
\begin{align}
&\max_{x} -\frac{1}{2}x^T A x + b^T x - C \sum_{i} z_{i}\\
s.t.\quad
& z_{i} = s_{i}^{+} + s_{i}^{-}, \quad \forall i\\
& x_{i} = s_{i}^{+} - s_{i}^{-}, \quad \forall i\\
& s_{i}^{+}, s_{i}^{-} \geq 0, \quad \forall i\\
\end{align}
$
with $A \succ 0$.
Mind you, this formulation may not give you the correct results if you decide to add constraints on $x_{i}$; if you are constraining $x_{i}$, you may need to reformulate this into a Mixed Integer Quadratic Program (MIQP). 
A: After your edit, the problem becomes equivalent to the convex problem:
$$\min x^TAx - b^Tx + C\|x\|_1$$
This is a very-well studied problem, and here are the keywords that will help you find algorithms and papers that solve it:


*

*Iterative soft-thresholding

*L1-LS (L1 regularized least squares)

*Forward Backward Splitting

*LASSO

*Also see Mark Schmidt's webpage
Good luck.
