Inheritance properties on cone continuity for set-valued maps via scalarization (Q6621982)

The paper investigates inheritance properties of continuity in set-valued maps through scalarization functions. Additionally, the paper provides counter examples of the complement part that is the half-missing part of the results of the following paper:\N[K. Ike, M. Liu, Y. Ogata, T. Tanaka, ``Semicontinuity of the composition of set-valued map and scalarization function for sets'', J. Appl. Num. Optim. 1 267--276 (2019)].\N\NThe described features are obtained via a generalization of both concepts of cone continuity for set-valued maps and ordinary lower and upper semicontinuities for real-valued functions. For the reader's convenience we recall the definition of scalarization map. Let \(Y\) be a real topological vector space. Let \(\theta_Y\) be the zero vector in \(Y\). Denote by \(P(Y)\) the set of all nonempty subsets of \(Y\). Let \(C\) be a convex cone in \(Y\) with \(\mbox{int}(C) \neq \emptyset\) and \(\theta_Y \in C\). Accordingly, we can define a preorder \(\leq_C\) on \(Y\) induced by \(C\) as follows: let \(y_1,y_2 \in Y\) \[ y_1 \leq_C y_2 \quad \Longleftrightarrow \quad y_2-y_1 \in C. \]\N\NFor \(A,B \in P(Y)\), we define the following six types of binary relations on \(P(Y)\).\N\begin{itemize}\N\item[(i)] \(A \leq_C^{(1)} B \quad \Longleftrightarrow \quad \forall \, a \in A, \; \forall \, b \in B, a \leq_C b\)\N\item[(ii)] \(A \leq_C^{(2L)} B \quad \Longleftrightarrow \quad \exists \, a \in A, \mbox{s.t.} \; \forall \, b \in B, a \leq_C b\)\N\item[(iii)] \(A \leq_C^{(3L)} B \quad \Longleftrightarrow \quad \forall \, b \in B, \; \exists \, a \in A, \mbox{s.t.} \; a \leq_C b\)\N\item[(iii)] \(A \leq_C^{(2U)} B \quad \Longleftrightarrow \quad \exists \, b \in B, \mbox{s.t.} \; \forall \, a \in A, a \leq_C b\)\N\item[(iv)] \(A \leq_C^{(3U)} B \quad \Longleftrightarrow \quad \forall \, a \in A, \; \exists \, b \in B, \mbox{s.t.} \; a \leq_C b\)\N\item[(v)] \(A \leq_C^{(4)} B \quad \Longleftrightarrow \quad \exists \, a \in A, \; \exists \, b \in B, \mbox{s.t.} \; a \leq_C b\)\N\end{itemize}\N\NFor each \(j =1,2L,3L,2U,3U,4\) the scalarization functions are defined as \[ \begin{array}{l} I_C^{(j)}(A,V,d)=\inf\{t \in \mathbb{R} : A \leq_C^{(j)} V+t d \} \\ S_C^{(j)}(A,V,d)=\inf\{t \in \mathbb{R} : V+t d \leq_C^{(j)} A \} \end{array} \] for any \(A,V \in P(Y)\) and \(d \in Y\).