A conjugate points theory for a nonlinear programming problem

The conjugate point is an important global concept in the calculus of variations and optimal control. In these extremal problems, the variable is not a vector in Rⁿ but a function. So a simple and natural question arises. Is it possible to establish a conjugate points theory for a nonlinear programming problem, Min f(x) on x∈Rⁿ? This paper positively answers this question. We introduce the Jacobi equation and conjugate points for the nonlinear programming problem, and we describe necessary and sufficient optimality conditions in terms of conjugate points.