zbMATH — the first resource for mathematics

Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes. (French) Zbl 0567.49008
In Section 1 the authors introduce a new notion of a strict radial upper directional derivative for a function \(f: X\to {\bar {\mathbb{R}}}\) (where X is a topological vector space): \[ f^{\square}(a,x):=\sup_{w\in X} \limsup_{(t,r)\to (0_+,f(a)),\quad r\geq f(a+tw)}t^{-1}[f(a+tw+tx)- r]. \] The corresponding subdifferential is given by \[ \partial^{\square}f(a):=\{x^*\in X^*| <x^*,x>\leq f^{\square}(a,x)\quad for\quad all\quad x\in X\} \] and is always contained in the generalized gradient of Clarke (this inclusion is often strict). The authors give some calculus rules for this subdifferential and a multiplier rule for a mathematical programming problem. In Sections 2 and 3 the authors examine the notions of a prototangent cone and an interiorly prototangent cone as well as corresponding subdifferentials. The prototangent cone is a closed convex cone which contains Clarke’s tangent cone and is contained in the classical tangent cone. Due to many misprints these sections are difficult to understand (the paper contains no proofs). More information about this subject is given in the paper by the second author [”Nondifferentiable Optimization: Motivations and Applications”, Lect. Notes Econ. Math. Syst. 255, 41-54 (1985)].
Reviewer: M.Studniarski

49J50 Fréchet and Gateaux differentiability in optimization
46G05 Derivatives of functions in infinite-dimensional spaces
49K27 Optimality conditions for problems in abstract spaces
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
90C30 Nonlinear programming
26E20 Calculus of functions taking values in infinite-dimensional spaces