  This expository paper contains a concise introduction to some significant
works concerning the Karush-Kuhn-Tucker condition, a necessary condition for a
solution in local optimality in problems with equality and inequality
constraints. The study of this optimality condition has a long history and
culminated in the appearance of subdifferentials. The 1970s and early 1980s
were important periods for new developments and various generalizations of
subdifferentials were introduced, including the Clarke subdifferential and
Demyanov-Rubinov quasidifferential. In this paper, we mainly present four
generalized Karush-Kuhn-Tucker conditions or Fritz John conditions in
variational analysis and set-valued analysis via Lagrange multiplier methods
besides Fr$\Acute{e}$chet differentiable situation, namely subdifferentials of
convex functions, generalized gradients of locally Lipschitz functions,
quasidifferentials of quasidifferentiable functions and contingent
epiderivatives of set-valued maps and discuss the limits of Lagrangian methods
slightly in the last chapter. These results represent remarkable developments
in the theory of generalized differentiation. The purpose of this paper is to
use Karush-Kuhn-Tucker condition as a guide to provide our readers with some
advanced topics in modern nonlinear analysis.
