The Application Method of Karush-Kuhn-Tucker Condition for Judging Boundary Extremum Points

The Karush-Kuhn-Tucker (KKT) condition is crucial in optimization algorithms as a basis for determining whether the optimum point is a boundary extremum point or not. This article proposes application methods for several situations encountered in numerical algorithms and provides the program flowchart. Before applying this condition, all of the redundant constraints among the working constraints must be removed. For the case where the number of effective constraints is greater than the dimension, all of the basic gradient groups must be checked. As long as the Lagrange multipliers of a group are all non negative, the optimum point satisfies the KKT condition. For the case where the number of effective constraints is less than the dimension number, it is necessary to take some equations to calculate the Lagrange multiplier, and then to take other equations to test the calculation result. The calculation steps of this application method are illustrated with examples. The solution idea to the optimization problems with equality constraints is proposed.