Extensions of Fréchet \(\varepsilon\)-subdifferential calculus and applications
The authors present calculus rules using the limiting Fréchet \(\varepsilon\)-subdifferentials of lower semicontinuous functions and the limiting Fréchet \(\varepsilon\)-normals of closed sets in Banach spaces, especially in Asplund spaces. In the first part of the paper, a calculus formula for the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon p(\overline u)\) of the marginal function \[ p(u)= \min_{x\in F(u)} \varphi(u, x) \] (where \(\varphi: U\times X\to \mathbb{R}\cup\{+\infty\}\) is a lower semicontinuous function and \(F: U\rightrightarrows X\) is a set-valued mapping with closed graph) is given in terms of the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon \varphi(\overline u,\overline x)\) and the limiting Fréchet \(\varepsilon\)-normal \(\widehat N(\text{graph }F,\cdot)(\overline u,\overline x)\). In the second part, the authors present a calculus rule for the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon(g\circ F)(\overline x)\) of the composite function \(g\circ F: X\to \mathbb{R}\cup \{+\infty\}\) (where \(F: X\to Y\) is a strictly (compactly) Lipschitzian mapping and \(g: Y\to \mathbb{R}\cup\{+\infty\}\) is a lower semicontinuous function) in terms of the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon(y^*\circ F)(\overline x)\) where \(y^*\) is contained in the limiting Fréchet subdifferential \(\widehat\partial g(F(\overline x))\). In the last part of the paper, the results are used to derive necessary optimality conditions for the approximate solutions of the general nonsmooth constraint optimization problem \[ \min f(x)\qquad\text{s.t. }F(x)\in -S, \] where \(f: X\to \mathbb{R}\) is a Lipschitz function, \(F: X\to Y\) is a strictly compactly Lipschitzian mapping and \(S\subset Y\) is a nonempty closed convex cone.
limiting Fréchet \(\varepsilon\)-subdifferentials
scientific article; zbMATH DE number 1763457
scientific article