Nonsmooth calculus, minimality, and monotonicity of convexificators.

*(English)*Zbl 0956.90033From the introduction: The authors introduce the notion of a convexificator which is a closed set, but is not necessarily bounded or convex. The significance of noncompact convexificators is that they allow applications of convexificators to continuous functions. A continuous function that admits a locally bounded convexificator at a point becomes locally Lipschitz at the point. They show that, for a locally Lipschitz function, most known subdifferentials, such as the subdifferential of Clarke, Michel-Penot. Ioffe-Morduchovich, and Treiman are convexificators. Moreover, these known subdifferentials of a locally Lipschitz function may often contain the convex hull of a convexificator. From the point of view of optimization and applications, the descriptions of the optimality conditions and calculus rules in terms of convexificators provide sharp results. The convex hull of convexificators are used to express results, such as extremality properties and mean-value conditions. However, the nonconvexity aspect of convexificators or of their extensions to vector functions turns out to be useful in other applications, such as generalized monotonicity characterizations or generalized Newton methods for nonsmooth equations. A characterization of quasiconvexity of a continuous function is given in terms of convexificators.

A continuous function may have several convexificators at a point. A minimal convexificator at a point is one that does not contain any other convexificator at the point. The question of finding conditions for minimal convexificators of a continuous function and also the question of guaranteeing uniqueness of minimal convexificators has, thus for, remained open. In Section 3, the authors answer this question by presenting conditions in terms of the set of extreme points for minimal convexificators and unique minimal convexificators.

A continuous function may have several convexificators at a point. A minimal convexificator at a point is one that does not contain any other convexificator at the point. The question of finding conditions for minimal convexificators of a continuous function and also the question of guaranteeing uniqueness of minimal convexificators has, thus for, remained open. In Section 3, the authors answer this question by presenting conditions in terms of the set of extreme points for minimal convexificators and unique minimal convexificators.

##### Keywords:

upper convex approximations; lower concave approximations; nonsmooth analysis; extremality; mean-value conditions; quasi-convexity; quasimonotonicity; minimal convexificators
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\textit{V. Jeyakumar} and \textit{D. T. Luc}, J. Optim. Theory Appl. 101, No. 3, 599--621 (1999; Zbl 0956.90033)

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