A generalized procedure for the constrained optimization problem

Abstract

Abstract The necessary and the sufficient conditions for the solution of the equality constrained optimization problem with N variables and (N−1) constraints are first derived and generalized to N variables and m constraints. Directional derivatives are used in the approach. The necessary and sufficient conditions for the problem with N−1 constraints are shown to be equivalent to the unconstrained one-variable problem, when the ordinary derivatives are replaced by the corresponding directional derivatives of the objective function in the direction tangent to the intersection of the constraints. The general equality constrained optimization problem of N variables and m constraints is then analysed using the directional derivative approach. Feasible direction vectors are defined and obtained in terms of first partial derivatives of the constraints. Necessary and sufficient conditions in terms of directional derivatives are derived and their equivalent with results in the literature. Sufficient conditions hi...