The Clarke tangent and normal cones to decomposable sets in Lebesgue spaces

Abstract

Let $(T,\Sigma,\mu)$ be a complete, $\sigma$-finite measure space, let $Y,Z$ be a separable Banach spaces, and let $S : T \rightrightarrows Z$ be a measurable multifunction with nonempty closed values. We study the relation between the Clarke tangent cone to the decomposable set $\mathsf{Sel}_p(S):=\{x \in L^p(T,Z) : x(t) \in S(t) \hspace{0.1cm} \text{a.e.}\}$ and the $L^p$-selections of the pointwise Clarke tangent cones. We prove that $$ \widehat{T}_{\mathsf{Sel}_p(S)}(x) = \{v \in L^p(T,Z) : v(t) \in \widehat{T}_{S(t)}(x(t)) \hspace{0.1cm} \text{a.e.} \} \hspace{0.2cm} \text{for any $p \in [1, \infty)$}, $$ More generally, we consider this problem in $L^p(T,Y) \times L^r(T,Z), p,r \in [1,\infty),$ and denote $S : T \rightrightarrows Y \times Z$ and $\mathsf{Sel}_{p,r}(S):=\{(x,y) \in L^p(T,Y) \times L^r(T,Z) : (x(t),y(t)) \in S(t) \hspace{0.1cm} \text{a.e.}\},$ The global-to-pointwise inclusion is shown to hold for all exponents unconditionally, whereas the reverse inclusion is obtained from an equi-integrable correction condition. Consequently, if $p,r$ are finite it holds $$\widehat{T}_{\mathsf{Sel}_{p,r}(S)}(x,y)=\mathsf{Sel}_{p,r}\left\{t \mapsto \widehat{T}_{S(t)}(x(t),y(t))\right\} \hspace{0.2cm} \text{for any $(x,y) \in \mathsf{Sel}_{p,r}(S).$}$$ Whereas, if $(p,r)=(1,\infty)$ the equality holds if $\mathsf{Sel}_{\infty,1}(S)$ satisfies a variational correction condition (\textbf{V}). Condition shown to hold whenever $S(t):=Gr F(t,\cdot),$ where $F : T \times Y \rightrightarrows Z$ is measurable in $t$ and $F(t,y) \subset F(t,x)+ \ell(t) ||y-x|| \overline{\textbf{B}}_{Z}.$ Applications are given to nonsmooth optimization with pointwise constraints, graphs of Nemytskii operators, and a multiplier rule for nonconvex integral programs is derived.