# Reformulating a convex optimization problem with $x \mapsto \max(x,0)$ in the constraint

I am wondering if there is a well-known transformation allowing one to solve convex optimization problems of the form

$$\begin{array}{ll} \underset{x}{\text{maximize}} & r^T x\\ \text{subject to} & \mathbf{1}^T x + \displaystyle\sum_i f_i\left(x_i\right) \leq 0\end{array}$$

where the penalty consists of convex losses

$$f_i\left(x\right) := c_i \max\left(x, 0\right)$$

Many thanks for pointers!

Because you say the losses are convex, I will presume that all $$c_i \ge 0$$, which means that max is used in a convex fashion. Given that, this problem can be formulated as a Linear Programming problem (LP).
Define additional optimization variables, $$y_i$$. Replace $$f_i(x_i)$$ with $$c_iy_i$$, and add the constraints $$y_i \ge x_i, y_i \ge 0$$. The result is an LP.
$$\max_{x,y}\quad r^Tx\\\text{subject to }\mathbf{1}^T x + \sum_i c_iy_i \leq 0$$ $$y\ge x$$ $$y \ge 0$$ where the latter two inequalities are interpreted as applying to each element of the vector.