In this paper, we first present a new important property for Bouligand tangent cone (contingent cone) of a star-shaped set. We then establish optimality conditions for Pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of Bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

in this paper, we first present a new important property for bouligand tangent cone (contingent cone) of a star-shaped set. we then establish optimality conditions for pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.

As already done for the matrix case in [6, p.256], [1, Thm. 6.1, p.1872] and [10, Thm. 3.2] we give a parametrization of the Bouligand tangent cone of the variety of tensors of bounded TT rank. We discuss how the proof generalizes to any binary hierarchical format. The parametrization can be rewritten as an orthogonal sum of TT tensors. Its retraction onto the variety is particularly easy to co...

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, equivalent characterizations of Bouligand tangent cone, Clarke cone and their corresponding normal cones set derived. Secondly, by using cones, four types stationary points GSCO given: TB-stationary point, NB-stationary TC-stationary point NC-stationary which used to characte...

We obtain a characterization of the proximal normal cone to prox-regular subset Riemannian manifold and some properties Bouligand tangent cones these sets are presented. Moreover, we show that on an open neighborhood set, metric projection is locally Lipschitz it directionally differentiable at boundary points set. Finally, necessary condition for curve be minimizing in set derived.

We prove a sufficient condition for a time-dependent closed set to be viable with respect to a delay evolution inclusion governed by a strongly-weakly u.s.c. perturbation of an infinitesimal generator of a C0-semigroup. This condition is expressed in terms of a natural concept involving tangent sets, generalizing tangent vectors in the sense of Bouligand and Severi. Mathematics Subject Classifi...

Given a set K in a Banach space X, we define: the tangent set, and the quasi-tangent set to K at ξ ∈ K, concepts more general than the one of tangent vector introduced by Bouligand (1930) and Severi (1931). Both notions prove very suitable in the study of viability problems referring to differential inclusions. Namely, we establish several new necessary, and even necessary and sufficient condit...

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Som...

Since the early 1970's there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is \best" has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper it is shown that there are an innnite number of tangent cones with some of the nicest properties of Clarke's co...

In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lipschitzian, or co...