If the KKT conditions are violated, a ''pivot operation'' consisting of introducing a column of <{{math>\boldsymbol{|'''''N'''''}}</math> into the basis at the expense of an existing column in <math>\boldsymbol{{math|'''''B'''''}}</math> is performed. In the absence of [[Degeneracy (mathematics)|degeneracy]], a pivot operation always results in a strict decrease in {{math|'''''c'''''<mathsup>\boldsymbol{c}^{\mathrm{T}} \boldsymbol{x}</mathsup>'''''x'''''}}. Therefore, if the problem is bounded, the revised simplex method must terminate at an optimal vertex after repeated pivot operations because there are only a finite number of vertices.{{sfn|Nocedal|Wright|2006|p=370|loc=Theorem 13.4}}
Select an index <{{math>|''m'' < ''q'' \le≤ ''n</math>''}} such that <{{math|''s<sub>s_qq</sub>'' < 0</math>}} as the ''entering index''. The corresponding column of <math>\boldsymbol{{math|'''''A'''''}}</math>, <math>\boldsymbol{{math|'''''A}_q'''<sub>q</mathsub>''}}, will be moved into the basis, and <{{math|''x<sub>x_qq</mathsub>''}} will be allowed to increase from zero. It can be shown that
i.e., every unit increase in <{{math|''x<sub>x_qq</mathsub>''}} will results in a decrease by <{{math|−''s<sub>-s_qq</mathsub>''}} in <math>\boldsymbol{c}^{\mathrm{math|'''''c'''''<sup>T}} \boldsymbol{x}</mathsup>'''''x'''''}}.{{sfn|Nocedal|Wright|2006|p=369|loc=Eq. 13.24}} Since
<{{math|'''''x<sub>\boldsymbol{x_B}B</mathsub>'''''}} must be correspondingly decreased by <{{math|Δ'''''x<sub>\DeltaB</sub>''''' \boldsymbol{x_B} {= \boldsymbol{B}^{-1} \boldsymbol{'''''B'''''<sup>−1</sup>'''''A}_q x_q'''<sub>q</mathsub>x<sub>q</sub>''}} subject to <{{math|'''''x<sub>\boldsymbol{x_B}B</sub>''''' -− \DeltaΔ'''''x<sub>B</sub>''''' \boldsymbol{x_B}≥ \ge \boldsymbol{'''0'''}}</math>. Let <math>\boldsymbol{{math|'''''d}''''' = \boldsymbol{B{=}^{-1} \boldsymbol{'''''B'''''<sup>−1</sup>'''''A}_q'''<sub>q</mathsub>''}}. If <math>\boldsymbol{{math|'''''d}''''' \le≤ \boldsymbol{'''0'''}}</math>, no matter how much <{{math|''x<sub>x_qq</mathsub>''}} is increased, <{{math|'''''x<sub>\boldsymbol{x_B}B</sub>''''' -− \Delta \boldsymbol{x_B}Δ'''''x<sub>B</mathsub>'''''}} will stay nonnegative. Hence, <math>\boldsymbol{c}^{\mathrm{math|'''''c'''''<sup>T}} \boldsymbol{x}</mathsup>'''''x'''''}} can be arbitrarily decreased, and thus the problem is unbounded. Otherwise, select an index <{{math>|''p'' = \operatorname{argmin}_{1 \le=}} argmin<sub>1≤''i \le ''≤''m}''</sub> \{x_i {(}}''x<sub>i</sub>''/''d<sub>i</sub>'' d_i \mathop{|{!}} d_i''d<sub>i</sub>'' > 0\{{)}}}}</math> as the ''leaving index''. This choice effectively increases <{{math|''x<sub>x_qq</mathsub>''}} from zero until <{{math|''x<sub>x_pp</mathsub>''}} is reduced to zero while maintaining feasibility. The pivot operation concludes with replacing <math>\boldsymbol{{math|'''''A}_p'''<sub>p</mathsub>''}} with <math>\boldsymbol{{math|'''''A}_q'''<sub>q</mathsub>''}} in the basis.
