```markdown
### Equivalent Convex Problems

Two problems are (informally) equivalent if the solution of one is readily obtained from the solution of the other, and vice-versa.

#### Some common transformations that preserve convexity:

- **Eliminating equality constraints**

  Original problem:
  \[
  \begin{align*}
  \text{minimize} & \quad f_0(x) \\
  \text{subject to} & \quad f_i(x) \leq 0, \quad i = 1, \ldots, m \\
  & \quad Ax = b
  \end{align*}
  \]

  Equivalent problem:
  \[
  \begin{align*}
  \text{minimize (over } z) & \quad f_0(Fz + x_0) \\
  \text{subject to} & \quad f_i(Fz + x_0) \leq 0, \quad i = 1, \ldots, m
  \end{align*}
  \]

  Where \( F \) and \( x_0 \) are such that:
  \[
  Ax = b \iff x = Fz + x_0 \text{ for some } z
  \]
```
