Sub- and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions

Envelopes $\sup_{\gamma\in\Gamma}f_{\gamma}(x)$ or $\inf_{\gamma\in\Gamma}f_{\gamma}(x)$ of parametric families of functions are typical non-differentiable functions arising in non-smooth analysis, optimization theory, control theory, the theory of generalized solutions of first-order partial differential equations, and other applications. In this survey formulae are obtained for sub- and supergradients of envelopes of lower semicontinuous functions, their corresponding semicontinuous closures, and limits and $\Gamma$-limits of sequences of functions. The unified method of derivation of these formulae for semicontinuous functions is based on the use of multidirectional mean-value inequalities for sets and non-smooth functions. These results are used to prove generalized versions of the Jung and Helly theorems for manifolds of non-positive curvature, to prove uniqueness of solutions of some optimization problems, and to get a new derivation of Stegall's well-known variational principle for smooth Banach spaces. Also, necessary conditions are derived for $\varepsilon$-maximizers of lower semicontinuous functions.
Bibliography: 47 titles.