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\fancyhead[HCO]{8th Int. Conf. on Mathematics and Informatics,
	September 9-10, 2021, T\^argu Mure\c s, Romania}
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\begin{center}{\Large\bf
Necessary Optimality Conditions for Generalized Weak Optimal Solutions in Set-Valued Optimization 
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\begin{center}{\large\bf\noindent
Anca Grad
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Department of Mathematics,\\ Faculty of Mathematics and Computer Science, \\Babe\c s Bolyai University of Cluj-Napoca
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anca.grad@ubbcluj.ro
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\vspace*{7mm}
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The specialized literature abounds in approaches of tackling the notion of an optimal solution for set-valued optimization problems, of the form:
        $$\left\{\begin{array}{l}\min F(x)\\G(x)\in C\end{array}\right.$$
wehere $F:X\rightrightarrows Y$ and $G:X\rightrightarrows Z$ are set-valued functions.
Among first such attempts we mention the so-called scalar approach, where  a certian poin $x_0\in X$ is said to be an optimal solution,  if there exists $y_0\in F(x_0)$ such that $y_0$ is a minimal point of the set $\bigcup_{x\in X\\G(x)\in C} F(x)$.  Another approach, the so-called {\it the natural criteria} in set-valued optimization was first introduced by D. Kuroiwa in 1998 and it basically considers $x_0\in X$ as an optimal solution, if the set $F(x_0)$ is minimal, when compared the other sets $F(x)$, in terms of a set-inclusion relation, defined by the means of a convex cone.

    Continuing Kuroiwa's idea, we introduce a new weak set-relation, with the help of the notion of the quasi-relative interior of a convex cone. In terms of that, we are able to deliver a new optimality notion of a weak solution of a set-valued optimization problems. This notion is accompanied by necessary optimality conditions. As well, we introduce a set-valued dual and a duality theorem. I began this path of research within \cite{2.} and \cite{3.}
    
     Let $A$ and $B$ belong to $\mathcal P_0(Y)$ and $K\subseteq Y$ be a nonempty, pointed convex cone. Then we write:
             \begin{itemize}
                \item[\rm{(a)}]$A\unlhd^l_{\rm qi  K} B \text{ if } B\subseteq A+\rm qi  K\quad and \quad A\unlhd^u_{\rm qi  K} B \text{ if } A\subseteq B-\rm qi  K$.
                 \item[\rm{(b)}] an {\bf ${\rm l-Min_{\rm qi}}$-efficient set} of $\mathcal S$, if for each set $B\in \mathcal S$ satisfying
                        $$ B\unlhd^l_{\rm qi  K} A, \text{ the relation }  A\unlhd^l_{\rm qi  K} B \text{ holds}.$$
                             \item[\rm{(c)}]  an {\bf ${\rm u-Min_{qi}}$-efficient set} of $\mathcal S$, if for each set $B\in \mathcal S$ satisfying
                        $$ B\unlhd^u_{\rm qi  K} A, \text{ the relation }  A\unlhd^u_{\rm qi  K} B \text{ holds}.$$
             \end{itemize}
             
   An extension of the classical Fenchel theorem can be stated in this case, as it is seen below: let $x_0,x_1\in \rm dom   F$, and  let $T\in \mathcal L(X,Y)$ be such that
                $F(x_1)-Tx_1\in -F^*_{\rm qi _K}(T).$ Then
                \begin{itemize}
                \item[\rm{(a)}] If  $F(x_0)-Tx_0{\unlhd}^l_{\rm qi _K} F(x_1)-Tx_1$, then $F(x_1)-Tx_1\unlhd^l_{\rm qi _K} F(x_0)-Tx_0$.
                \item[\rm{(b)}] If $F(x_0)-Tx_0{\unlhd}^l_{\rm qi _K} F(x_1)-Tx_1$, then $F(x_1)-Tx_1\sim^l F(x_0)-Tx_0$.
                \end{itemize}
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 \bibitem{1.}
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 D. Kuroiwa, The natural criteria in set-valued optimization, \textit{RIMS Kokyuroku}  \textbf{2049,}  (1998) 86--92.
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\bibitem{2.}
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A. Grad, \textit{Generalized Duality and Optimality Conditions}, Mega, 2010.

\bibitem{3.}
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A. Grad, A generalized interior approach to constrained set-valued duality, \textit{Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity} \textbf{12,} (2014), 37--56
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