Characterizations of the solution set for quasiconvex programming in terms of Greenberg-Pierskalla subdifferential

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In convex programming, characterizations of the solution set in terms of the subdifferential have been investigated by Mangasarian. An invariance property of the subdifferential of the objective function is studied, and as a consequence, characterizations of the solution set by any solution point and any point in the relative interior of the solution set are given. In quasiconvex programming, how-ever, characterizations of the solution set by any solution point and an invariance property of Greenberg-Pierskalla subdifferential, which is one of the well known subdifferential for quasiconvex functions, have not been studied yet as far as we know. In this paper, we study characterizations of the solution set for quasiconvex programming in terms of Greenberg-Pierskalla subdifferential. To the purpose, we show an invariance property of Greenberg-Pierskalla subdifferential, and we introduce a necessary and sufficient optimality condition by Greenberg-Pierskalla subdifferential. Also, we compare our results with previous ones. Especially, we prove some of Mangasarian's characterizations as corollaries of our results.