Artificial variable

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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).

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