Goal program

1. Add the variable, $LaTeX: v$, and the constraints, $LaTeX: G(x) - v \le 0$ and $LaTeX: v \ge 0$, to measure the level of violation. (If the goal were $LaTeX: H(x) = 0$, instead of $LaTeX: G(x) \le 0$, the added constraint would be $LaTeX: H(x) + u - v = 0$, where $LaTeX: u$ and $LaTeX: v$ are (non-negative) levels of violation below and above the goal, respectively.)
2. Add the penalty term to the objective: $LaTeX: P(v)$, where $LaTeX: P(0)=0$, and $LaTeX: P$ is strictly increasing -- i.e., $LaTeX: v' \ge v$ and $LaTeX: v'_i > v_i$ for some $LaTeX: i$ imply $LaTeX: P(v') > P(v)$.
The resulting mathematical program represents the goal program. If the goal is satisfied, $LaTeX: v=0$; otherwise, the penalty term, $LaTeX: P(v)$, reflects "adequate compensation" for the violation.