On the theory of subdifferentials

Abstract

The theory presented in the paper consists of two parts. The first is devoted to basic concepts and principles such as the very concept of a subdifferential, trustworthiness and its characterizations, geometric consistence, fuzzy principles and calculus rules, methods of creation of new subdifferentials etc. In the second part we study certain specific subdifferentials, namely, subdifferentials associated with bornologies, their limiting versions and metric modifications. For each subdifferential we verify that the basic properties discussed in the first part are satisfied and prove calculus rules for two main operations: summation and partial minimization. Separate sections are devoted to the Fréchet and limiting Fréchet subdifferentials and to the approximate G-subdifferential. For the last two new definitions are given which lead to a certain unification and simplification of analysis. The sum rule for these two subdifferentials is proved with the so-called "linear metric qualification condition", so far the most general. In the last section we briefly discuss how other operations reduce to the two mentioned basic operations and give the corresponding calculus rules (with suitable versions of the metric qualification conditions).

Quadratic growth and critical point stability of semi-algebraic functions

“…Gsubdifferential, generalized gradient). We do not need a general definition here: an interested reader may look into [14]. The important point is that most of them can be effectively used only in certain classes of Banach spaces.…”

Subdifferential Stability and Subdifferential Sum Rules

“…For example, in [15], r S (f ) is denoted ∧ S (f ) (more or less as in [3]) and is called stabilized infimum; the usual infimum inf S f is declared robust when it is equal to r S (f ); a pointx achieving this value, i.e. f (x) = r S (f ), is called a robust minimizer (more or less as in [10]). In general, r S (f ) < inf S f for arbitrary lsc f : additional conditions (so-called qualification conditions) are required to have the equality r S (f ) = inf S f .…”

Outer limits of subdifferentials for min–max type functions

“…Theorem 3. Let f : X → R be as in (11), and in addition assume that for every i ∈ I the function f i is piecewise affine, i.e. ∂f (x).…”