In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.

在优化问题的研究中，优化条件起着重要作用。通过使用一些导数，许多研究人员已经引入了各种类型的必要和/或充分的最优条件。特别地，在凸编程中，关于次微分的必要和充分的最优条件已被广泛研究。最近，作者研究了拟凸编程的必要和充分的最优条件。但是，关于不可微拟凸凸规划的Karush–Kuhn–Tucker类型最优条件，没有太多结果。在本文中，我们根据格林伯格-皮尔斯卡亚次微分研究了拟凸编程的Karush-Kuhn-Tucker型最优条件。我们显示了Greenberg–Pierskalla次微分的一些封闭性。在Slater约束条件下，我们从Greenberg-Pierskalla次微分论证了本质上拟凸编程的一个充要条件。另外，我们介绍了最优条件的必要和充分的约束条件。作为推论，我们证明了在次微分条件下凸编程的充要条件。