In Linear programming, when is it beneficial to use the Two-Phase Method? Why not just use the Simplex Method?
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The standard simplex method requires that you have a feasible solution to start with - often by setting all original variables to $0$ and finding the resulting values of the slack variables (i.e., letting the slack variables be basic).
The two-phase method is used when you don't have or can't easily find an initial feasible solution. The first phase finds a feasible solution, and the second phase works from that feasible solution to an optimal solution. (So the second phase is really equivalent to the standard simplex method.)