On Nonconvex Subdifferential Calculus in Banach Spaces

We study a concept of subdifferential for general extended-real-valued functions defined on arbitrary Banach spaces. At any point considered this basic subdifferential is a nonconvex set of subgradients formed by sequential weak-star limits of the so-called Fréchet ε-subgradients of the function at points nearby. It is known that such and related constructions possess full calculus under special geometric assumptions on Banach spaces where they are defined. In this paper we establish some useful calculus rules in the general Banach space setting.

[1]  E. B. Leach,et al.  A note on inverse function theorems , 1961 .

[2]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[3]  R. Rockafellar.  Directionally Lipschitzian Functions and Subdifferential Calculus , 1979 .

[4]  A. Ioffe.  Approximate subdifferentials and applications. I. The finite-dimensional theory , 1984 .

[5]  R. Rockafellar.  Extensions of subgradient calculus with applications to optimization , 1985 .

[6]  A. Kruger.  Properties of generalized differentials , 1985 .

[7]  Jonathan M. Borwein,et al.  Proximal analysis and boundaries of closed sets in Banach space, Part I: theory , 1986 .

[8]  Jay S. Treiman,et al.  Clarke’s gradients and epsilon-subgradients in Banach spaces , 1986 .

[9]  A. Ioffe.  Approximate subdifferentials and applications 3: the metric theory , 1989 .

[10]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[11]  F. Clarke.  Methods of dynamic and nonsmooth optimization , 1989 .

[12]  A. Ioffe.  Proximal Analysis and Approximate Subdifferentials , 1990 .

[13]  L. Thibault.  On subdifferential of optimal value functions , 1991 .

[14]  B. Mordukhovich.  Sensitivity analysis in nonsmooth optimization , 1992 .

[15]  Philip D. Loewen.  Optimal Control Via Nonsmooth Analysis , 1993 .

[16]  Philip D. Loewen,et al.  Optimal Control of Unbounded Differential Inclusions , 1994, SIAM Journal on Control and Optimization.

[17]  Lionel Thibault,et al.  A note on Fréchet and approximate subdifferentials of composite functions , 1994, Bulletin of the Australian Mathematical Society.

[18]  B. Mordukhovich.  Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis , 1994 .

[19]  B. Mordukhovich.  Discrete Approximations and Refined Euler--Lagrange Conditions forNonconvex Differential Inclusions , 1995 .

[20]  Lionel Thibault,et al.  Extensions Of Subdifferential Calculus Rules in Banach Spaces , 1996, Canadian Journal of Mathematics.