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This work contains the statement and proof of the Karush Kuhn Turker’s theorem as a characterization of the behavior of objective function and the constraint function at local optima of inequality constrained optimization problem together with the necessary and sufficient conditions for the Lagrangian method as prerequisite for the convergence of the Lagrangian iterative method. Hence, it’s presentation as a method better in solving the constrained optimization problem. To start, it is ensured that the non-negativity constraints , if any are included in the m constraints and if the unconstrained optimum of does not satisfy all constraints, the constrained optimum must occur at a boundary point of the solution space. This means that one constraint must be satisfied in equation form for the Kahn-Tucker approach to strictly follow before the necessary iteration of the Lagrangian method can be able to work in the maximization of concave function problem or that of the minimization of convex functions.
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