Kutateladze, S. S. Infinitesimals and calculus of tangents. (Russian) Zbl 0643.46054 Tr. Inst. Mat. 9, 123-135 (1987). The reviewed paper contributes to the “infinitesimal non-smooth analysis”, being created by S. S. Kutateladze in recent years. Non- smooth analysis (or, as it is called also, subdifferential calculus) is based, roughly speaking, on an idea of approximating a mapping (or, more generally, a correspondence, i.e., a multi-valued mapping) at a point of a (topological) vector space, by means of sets of special form, cones. Among those one may mention the Hadamard cone, Clarke cone, hypertangent cone etc. It was discovered earlier by the author himself that all known types of approximating cones can be defined in terms of Non-Standard Analysis, which leads not only to an essential simplification and unification within this system of concepts, but also to the appearence of new types of cones. [see, e.g., the author’s paper: Infinitesimal tangent cones, Sib. Mat. Zh. 26, No.6, 67-76 (1985)]. In the present paper three new types of cones are constructed for any subset F of a vector space X endowed with a so-called vector topology, any point x’\(\in X\) and any (exterior) set \(\Lambda\) of positive real infinitesimals, these three are denoted \(C\ell_{\Lambda}(F,x')\), \(Ha_{\Lambda}(F,x')\), \(Bo_{\Lambda}(F,x')\). (If one poses \(\Lambda =\mu ({\mathbb{R}})_+\), then those cones coincide with classical ones: \(C\ell (F,x')\), etc.) Algebraical, topological and geometrical (convexity) properties of those cones are investigated carefully. In the concluding section new notions are applied to the theory of subdifferentiation of a composition of two correspondences. A typical result states that under certain conditions imposed on a mapping T:X\(\to Y\), the following inclusion holds for any F, x’ and \(\Lambda\) as above: \(T(C\ell_{\Lambda}(F,x'))\subset C\ell_{\Lambda}(T(F),Tx')\), etc. Reviewer: V.Pestov Cited in 1 Review MSC: 46S20 Nonstandard functional analysis 46G05 Derivatives of functions in infinite-dimensional spaces 49J52 Nonsmooth analysis 90C55 Methods of successive quadratic programming type Keywords:monads; infinitesimal non-smooth analysis; subdifferential calculus; correspondence; multi-valued mapping; Hadamard cone; Clarke cone; hypertangent cone; Non-Standard Analysis; vector topology; positive real infinitesimals; subdifferentiation of a composition of two correspondences × Cite Format Result Cite Review PDF