Simplex optimality criterion ($z=z_0+{\sum }_j{\bar{c}}_j{x}_j$)

In a feasible minimization simplex dictionary, the nonbasic variables are nonnegative. If every reduced cost ${\bar{c}}_j\ge 0$, the displayed objective identity proves $z\ge z_0$ for every feasible point. Thus the current basic feasible solution is globally optimal. If all reduced costs are strictly positive, equality forces every nonbasic variable to vanish and gives a unique basic solution, provided the dictionary determines all basic variables.