Variant of the linear programming problem

Question

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.

Very grateful for your answers and references of the problem mentioned.

Answer 1

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 $\sum_{j=1}^n z_j$.