# Variant of the linear programming problem

Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem:

$$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$ $$s.a.\quad Ax=b,$$ $$~~~~~~~~\qquad x\geq 0,$$ where $$A$$ is a matrix $$m\times n$$ with rank $$m$$, $$x\in \mathbb{R}^n$$, $$b\in\mathbb{R}^m$$, and $$\begin{equation*} d_j(x_j) = \begin{cases} 0, & \text{if x_j= 0},\\ d_jx_j+t_j, & \text{if x_j> 0}, \end{cases} \end{equation*}$$ $$d_j,x_j$$ are constants.

If $$x_j \le u_j$$ for some constant $$u_j$$, you can introduce a binary variable $$y_j$$, nonnegative variable $$z_j$$, and linear constraints: \begin{align} x_j &\le u_j y_j \\ d_j x_j + t_j - z_j &\le t_j(1-y_j) \end{align} The objective is then to minimize $$\sum_{j=1}^n z_j$$.