Nonlinear Analysis and Convex Analysis,

” Conjugate points ” is a global concept in calculus of variation, and plays an important role in discussing optimality. Though it has been a theme of differential geometry, math-ematical programming approach has been recently developed with several extensions of the conjugate points theory to optimal control problems and variational problems with state constraints. In these extremal problems, the variable is not a vector $x $ in $R^{n} $ but a function $x(t) $. So a simple and natural question arises. Is it possible to establish a conjugate points theory for a minimization problem: minimize $f(x) $ on $x\in R^{n}$? In [3], the author positively answered this question. He introduced ”the Jacobi equation” and ”conjugate points ” for it, and describe optimality conditions in terms of”conjugate points”. In this paper, we extend it to a constrained nonlinear programming. 1. Jacobi’s conjugate points theory In this section, we review the classical conjugate points theory for the simplest problem in calculus of variations: $(SP) $ Minimize $\int_{0}^{T}f(t, x(t),\dot{X}(t))dt$ subject to $x(\mathrm{O})=A, $ $x(T)=B $