Geodesic
First-Order Conditions for Optimization Problems with Quasiconvex Inequality Constraints
Serdica Mathematical Journal, Tome 34 (2008) no. 3, pp. 607-618.

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The constrained optimization problem min f(x), gj(x) ≤ 0 (j = 1,…p) is considered, where f : X → R and gj : X → R are nonsmooth functions with domain X ⊂ Rn. First-order necessary and first-order sufficient optimality conditions are obtained when gj are quasiconvex functions. Two are the main features of the paper: to treat nonsmooth problems it makes use of Dini derivatives; to obtain more sensitive conditions, it admits directionally dependent multipliers. The two cases, where the Lagrange function satisfies a non-strict and a strict inequality, are considered. In the case of a non-strict inequality pseudoconvex functions are involved and in their terms some properties of the convex programming problems are generalized. The efficiency of the obtained conditions is illustrated on examples.
Keywords: Nonsmooth Optimization, Dini Directional Derivatives, Quasiconvex Functions, Pseudoconvex Functions, Quasiconvex Programming, Kuhn-Tucker Conditions 
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     author = {Ginchev, Ivan and Ivanov, Vsevolod I.},
     title = {First-Order {Conditions} for {Optimization} {Problems} with {Quasiconvex} {Inequality} {Constraints}},
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Ginchev, Ivan; Ivanov, Vsevolod I. First-Order Conditions for Optimization Problems with Quasiconvex Inequality Constraints. Serdica Mathematical Journal, Tome 34 (2008) no. 3, pp. 607-618. http://geodesic.mathdoc.fr/item/SMJ2_2008_34_3_a5/