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This expository paper contains a concise introduction to some significantworks concerning the Karush-Kuhn-Tucker condition, a necessary condition for asolution in local optimality in problems with equality and inequalityconstraints. The study of this optimality condition has a long history andculminated in the appearance of subdifferentials. The 1970s and early 1980swere important periods for new developments and various generalizations ofsubdifferentials were introduced, including the Clarke subdifferential andDemyanov-Rubinov quasidifferential. In this paper, we mainly present fourgeneralized Karush-Kuhn-Tucker conditions or Fritz John conditions invariational analysis and set-valued analysis via Lagrange multiplier methodsbesides Fr$\Acute{e}$chet differentiable situation, namely subdifferentials ofconvex functions, generalized gradients of locally Lipschitz functions,quasidifferentials of quasidifferentiable functions and contingentepiderivatives of set-valued maps and discuss the limits of Lagrangian methodsslightly in the last chapter. These results represent remarkable developmentsin the theory of generalized differentiation. The purpose of this paper is touse Karush-Kuhn-Tucker condition as a guide to provide our readers with someadvanced topics in modern nonlinear analysis.

Generalized Karush-Kuhn-Tucker Conditions in Variational and Set-Valued Analysis