AI 中文总结
研究在特定测度空间和巴拿赫空间下，\(\mathsf{Sel}_{p}(S)\)的克拉克切锥是否可通过\(S\)值的逐点克拉克切锥的\(L^p\)选择得到，给出\(1 \leq p < \infty\)时的肯定答案及相关结论，并应用于相关优化问题。
AI 中文摘要
设\((T,\Sigma,\mu)\)是完备的\(\sigma -\)有限测度空间，\(Z\)是可分巴拿赫空间，\(S: T \rightrightarrows Z\)是具有非空闭值的可测多值函数。对于\(1 \leq p\leq \infty\)，考虑\(\mathsf{Sel}_{p}(S)\)。研究\(\mathsf{Sel}_{p}(S)\)的克拉克切锥相关问题，给出\(1 \leq p < \infty\)时的主要结果及\(p = \infty\)时的部分结果，还推导了相关应用。
英文摘要
Let $(T,Σ,μ)$ be a complete, $σ$-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.
