## 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
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