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Assuming that we are working with a minimization objective function and we have identified the best basic feasible solution, i.e. the basic feasible solution that yields the most minimal objective value, when is this best basic feasible solution NOT the optimal solution to the LP? Are there any conditions in which this wouldn't be?

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The best basic feasible is optimal in the sense that there are no other solutions with a better objective. (There may be other (non-basic) solutions with the same optimal objective value).

This is of course exploited in the Simplex method: we can limit the search to basic feasible solutions only.

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