We consider a sequence of convex integral functionals <$EF sub s ~:~W sup {1,~p } ( OMEGA sub s )~symbol О~roman bold R> and a sequence of weakly lower semicontinuous and, in general, non-integral functionals <$EG sub s ~:~W sup {1,~p } ( OMEGA sub s )~symbol О~roman bold R>, where <$E{ OMEGA sub s>} is a sequence of domains in <$E{ roman bold R} sup n> contained in a bounded domain <$EOMEGA~symbol <172>~{ roman bold R} sup n> (<$En~symbol У~2>) and p >> 1. Along with this, we consider a sequence of closed convex sets <$EV sub s ~=~{v~symbol <174>~W sup {1,~p } ( OMEGA sub s )~:~M sub s (v)~symbol Г~0> a.e. in <$EOMEGA sub s>}, where <$EM sub s> is a mapping from <$EW sup {1,~p } ( OMEGA sub s )> to the set of all functions defined on <$EOMEGA sub s>. We establish conditions under which minimizers and minimum values of the functionals <$EF sub s ~+~G sub s> on the sets <$EV sub s> converge to a minimizer and the minimum value of a functional on the set <$EV~=~{v~symbol <174>~W sup {1,~p } ( OMEGA )~:~M(v)~symbol Г~0> a.e. in <$EOMEGA>}, where M is a mapping from <$EW sup {1,~p } ( OMEGA )> to the set of all functions defined on <$EOMEGA>.

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