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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 have 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).

1

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 have constraints on $x_{i}$; if you are constraining $x_{i}$, you may need to reformulate this into a Mixed Integer Quadratic Program (MIQP).