zbMATH — the first resource for mathematics

Nonsmooth calculus in finite dimensions. (English) Zbl 0633.46043
New calculus rules are given for the Clarke generalized gradient \(\partial f\) of a general lower semicontinuous function \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\cup \{\pm \infty \}\). These include rules for computing \(\partial f\) when \(f=f_ 1+f_ 2\circ F\) in cases where \(f_ 1\), \(f_ 2\) are l.s.c. and F is either stricly differentiable or isotone. Basing their analysis upon the generalized upper directional derivative \(f^{\uparrow}(x;v)\), the authors obtain the weakest conditions to date under which such formulas are valid; their method also readily yields conditions for equality in the formulae. Among the consequences of the new calculus rules are new necessary conditions for constrained mathematical programming problems, new formulae for the calculus of (Clarke) normal and tangent cones, and all the known results for the finite-dimensional subgradient calculus of convex functions. Other forms of generalized derivatives (contingent, Ursescu, etc.) are discussed, and there is a wealth of insightful commentary.
Reviewer: P.Loewen

46G05 Derivatives of functions in infinite-dimensional spaces
49J52 Nonsmooth analysis
90C30 Nonlinear programming
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
Full Text: DOI