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**Sub- and supergradients of envelopes, semicontinuous closures, and limits of sequences of functions.**
*(English.
Russian original)*
Zbl 1248.49019

Russ. Math. Surv. 67, No. 2, 345-373 (2012); translation from Usp. Mat. Nauk 67, No. 2, 157-186 (2012).

Summary: 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 nonsmooth 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 nonsmooth 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.

### MSC:

49J52 | Nonsmooth analysis |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49Q20 | Variational problems in a geometric measure-theoretic setting |

52A35 | Helly-type theorems and geometric transversal theory |

58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |