# Artificial variable

A variable, say $LaTeX: v$, added to an equation, $LaTeX: h(x) = 0$. The resulting system, $LaTeX: h(x) + v = 0$, is feasible upon letting $LaTeX: v = -h(x)$ for a chosen $LaTeX: x$. The objective function is modified to penalize nonzero values of $LaTeX: v$. Often, $LaTeX: v \ge 0$ is required, multiplying $LaTeX: h$ by -1 if necessary. This originated in linear programming, where a Phase I objective is used to find a solution with $LaTeX: v=0$ (or ascertain that the original system has no feasible solution) by minimizing $LaTeX: \textstyle \sum_i v_i$ (ignoring the original objective, $LaTeX: c^T x$).