This section works with the optimality conditions for problem (P) in the sense of the set criterion. Firstly, let us recall the concepts of set relations.
Definition 4.1
see [5]
Let A and B be two nonempty sets of Y. We write \(A\leqslant^{\mathrm{l}} B\), if for all \(b\in B\) there exists \(a\in A\) such that \(a\leqslant b\). Here, ‘\(\leqslant^{\mathrm{l}}\)’ is called a lower relation.
Remark 4.2
The above defined lower relation is equivalent to \(B\subset A+C\) and it is the generalization of the order induced by a pointed convex cone C in Y, in the following sense: \(a\leqslant b\) if \(b\in a+C\).
Definition 4.3
see [14]
Let A and B be two nonempty sets of Y. We write \(A< ^{\mathrm{lw}} B\), if \(B\subset A+\operatorname {int}C\). Here, ‘\(< ^{\mathrm{lw}}\)’ is named a lower weak relation.
Definition 4.4
see [14]
Let \(\mathcal{A}\) be a family of subsets of Y.

(1)
\(A\in\mathcal{A}\) is named a lower minimal of \(\mathcal {A}\), if for each \(B\in\mathcal{A}\) such that \(B\leqslant^{\mathrm{l}} A\), it satisfies \(A\leqslant^{\mathrm{l}} B\);

(2)
\(A\in\mathcal{A}\) is called a lower weak minimal of \(\mathcal{A}\), if for each \(B\in\mathcal{A}\) such that \(B< ^{\mathrm{lw}} A\), it satisfies \(A< ^{\mathrm{lw}} B\).
The relation ‘\(\leqslant^{\mathrm{l}}\)’ determines in \(\mathcal{A}\) the equivalence relation:
$$ A\sim B \quad\Leftrightarrow \quad A\leqslant^{\mathrm{l}} B\quad \mbox{and}\quad B \leqslant^{\mathrm{l}} A $$
whose classes it is denoted by \([A]^{\mathrm{l}}\). Analogously the relation ‘\(< ^{\mathrm{lw}}\)’ determines in \(\mathcal{A}\) an equivalence relation whose classes are written \([A]^{\mathrm{lw}}\).
In the sense of a set criterion, the problem (P) can be rewritten in the next form:
$$ \mbox{(SP)}\quad \textstyle\begin{cases} \min_{\leqslant^{\mathrm{l}}}& \{F(x)\}\\ s.t.& x\in S\subset X. \end{cases} $$
In this case, a lower weak minimizer of the problem (SP) is a pair \((\bar{x}, F(\bar{x}) )\) such that \(\bar{x}\in S\) and \(F(\bar{x})\) is a lower weak minimal of the family of images of F, i.e. the family
$$ \mathcal{F}= \bigl\{ F(x): x\in S \bigr\} . $$
We say that \(F(\bar{x})\) is a lower weak minimal of (SP) instead of \(\mathcal{F}\).
Example 4.5
Let \(S=[0,1]\subset\mathbb{R}\), \(Y=\mathbb {R}^{2}\) and \(C=\mathbb{R}^{2}_{+}\). Let \(F_{0}: [0,1]\rightarrow2^{\mathbb {R}^{2}}\) be a setvalued map and defined by
$$ F_{0}(x)= \textstyle\begin{cases} \{(y_{1},y_{2})\in\mathbb{R}^{2}: (y_{1}1)^{2}+(y_{2}1)^{2}=1\}, & x=0,\\ [(x,x+1),(1,x+1)],& 0< x< 1,\\ [(0,1),(1,0)],& x=1, \end{cases} $$
where \([y,y']\) with \(y,y'\in\mathbb{R}^{2}\) denotes the line segment, i.e.
\([y,y']=\{\lambda y+(1\lambda)y': 0\leq\lambda\leq1\} \). Considering the following problem (SP)_{0}:
$$ \mbox{(SP)}_{0} \quad \textstyle\begin{cases} \min_{\leqslant^{\mathrm{l}}}& \{F_{0}(x)\}\\ s.t.& x\in[0,1]. \end{cases} $$
By computing, we can derive that \((1, F(1))\) is the lower minimizer of (SP)_{0} and \((x,F(x))\) with \(0< x \leq1\) is the lower weak minimizer of (SP)_{0}.
Definition 4.6
see [19]
Let \(F(\bar{x})\) be a lower weak minimal of (SP). \(F(\bar{x})\) is named strict lower weak minimal of (SP), if there exists a neighborhood \(U(\bar{x})\) of x̄ such that \(F(x)\nless^{\mathrm{lw}} F(\bar{x})\), for all \(x\in U(\bar{x})\cap S\). Then x̄ is called a strict lower weak minimum of (SP).
Lemma 4.7
see [19]
Let
\(\bar{x}\in S\). Then
\(F(\bar {x})\)
is a lower weak minimal of (SP) if and only if for each
\(x\in S\)
one of the conditions below is fulfilled:

(1)
\(F(x)\in[F(\bar{x})]^{\mathrm{lw}}\).

(2)
There exists
\(\bar{y}\in F(\bar{x})\)
such that
\((F(x)\bar{y} )\cap(\operatorname{int}C)=\emptyset\).
Definition 4.8
see [19]
(Domination property). It is called that a subset \(A\subset Y\) has the Dweak minimal property, if for all \(y\in A\) there exists a weakly minimal a of A such that \(ay\in(\operatorname{int} C)\cup\{0\}\).
Lemma 4.9
Let
\(\bar{x},x\in S\)
and
\(\bar{y}\in F(\bar {x})\). If
\(F(x)\nless^{\mathrm{lw}} F(\bar{x})\), \(\operatorname{WMin}[F(\bar{x}),C]=\{ \bar{y}\}\)
and
\(F(\bar{x})\)
has the
Cweak minimal property, one has
$$ \bigl(F(x)\bar{y} \bigr)\cap (\operatorname{int}C )=\emptyset. $$
(4.1)
Proof
Assuming that (4.1) does not hold, that is,
$$ \bigl(F(x)\bar{y} \bigr)\cap (\operatorname{int}C )\neq\emptyset. $$
Then there exist \(y\in F(x)\) and \(d\in\operatorname{int}C\) such that \(y\bar {y}=d\). So, it yields \(\bar{y}=y+d\) and \(\bar{y}\in F(x)+\operatorname{int}C\). On the other hand, it follows from \(\operatorname{WMin}[F(\bar{x}),C]=\{\bar {y}\}\) and the Cweak minimal property of \(F(\bar{x})\) that, for all \(\bar{y}'\in F(\bar{x})\), there is \(d'\in\operatorname{int}C\cup\{0\}\) such that
$$ \bar{y}'=\bar{y}+d'\in F(x)+\operatorname{int}C. $$
Therefore, we derive that
$$ F(\bar{x})\subset F(x)+\operatorname{int}C, $$
which means \(F(x)<^{\mathrm{lw}} F(\bar{x})\). This contradicts the hypothesis. □
Theorem 4.10
Let
x̄
be a strict lower weak minimum of (SP). If
\(\operatorname{WMin}[F(\bar{x}),C]=\{\bar{y}\}\)
and
\(F(\bar {x})\)
has the
Cweak minimal property, then
$$ \bigl(\overline{D}F(\bar{x},\bar{y}) (x) \bigr)\cap (\operatorname{int}C )=\emptyset,\quad \textit{for all } x\in S. $$
(4.2)
Proof
Because x̄ is a strict lower weak minimum of (SP), we see from Definition 4.6 that there exists a neighborhood \(U(\bar{x})\) of x̄ such that \(F(x')\nless^{\mathrm{lw}} F(\bar{x})\) for all \(x'\in U(\bar{x})\cap S\). Let \(x\in S\). Assuming that \(y\in \overline{D}F(\bar{x},\bar{y})(x)\) then there exist \((t_{n})\rightarrow 0^{+}\), \((x_{n})\rightarrow\bar{x}\) and \(y_{n}\in F(\bar{x}+t_{n} x_{n})\) such that
$$ \frac{y_{n}\bar{y}}{t_{n}}\rightarrow y. $$
Therefore for large enough n, we can get \(\bar{x}+t_{n} x_{n}\in U(\bar {x})\cap S\) which verifies \(F(\bar{x}+t_{n} x_{n})\nless^{\mathrm{lw}} F(\bar{x})\). Thus, it follows from Lemma 4.9 that
$$ \bigl(F(\bar{x}+t_{n} x_{n})\bar{y} \bigr)\cap( \operatorname{int}C)=\emptyset, \quad\mbox{for large } n. $$
(4.3)
Let us prove that \(y\notin\operatorname{int}D\). Otherwise, one has
$$ y_{n}\bar{y}\in\operatorname{int}C,\quad \mbox{for large } n. $$
Noticing that \(y_{n}\in F(\bar{x}+t_{n} x_{n})\), we derive
$$ y_{n}\bar{y}\in \bigl(F(\bar{x}+t_{n} x_{n}) \bar{y} \bigr)\cap(\operatorname {int}C),\quad \mbox{for large } n. $$
This is a contradiction to (4.3). So, we derive (4.2), as desired. □
Theorem 4.11
Let
x̄
be a strict lower weak minimum of (SP). If
\(\operatorname{WMin}[F(\bar{x}),C]=\{\bar{y}\}\)
and
\(F(\bar {x})\)
has the
Cweak minimal property then
$$ \underline{D}F(\bar{x},\bar{y}) (x)\cap(\operatorname{int} C)=\emptyset,\quad \textit{for all } x\in T(S,\bar{x}). $$
(4.4)
Proof
Based upon Theorem 4.10, we see that equation (4.2) holds. The rest of the proof can be followed by similar arguments to that of Theorem 3.3. □
Theorem 4.12
Let
S
be convex set and
F
be
Cpseudoconvex at
\((\bar{x},\bar{y})\in\operatorname{graph}(F)\). If for every
\(x\in S\)
we have
$$ \bigl(\overline{D}F(\bar{x},\bar{y}) (x\bar{x}) \bigr)\cap (\operatorname {int}C )= \emptyset, \quad\forall x\in S, $$
then
x̄
is a lower weak minimum of (SP).
Proof
Suppose that x̄ is not a lower weak minimum of (SP), it follows from Lemma 4.7 that there exists \(x'\in S\) such that for each \(\bar{y}'\in F(\bar{x})\) there is \(y'\in F(x')\) with
$$ y'\bar{y}'\in\operatorname{int} C. $$
(4.5)
Since F is Cpseudoconvex at \((\bar{x},\bar{y})\), we derive from Lemma 2.4 that
$$ F\bigl(x'\bigr)\bar{y}\subset\overline{D}F(\bar{x},\bar{y}) \bigl(x'\bar{x}\bigr)+C. $$
Therefore, we obtain
$$ y'\bar{y}\in\overline{D}F(\bar{x},\bar{y}) \bigl(x' \bar{x}\bigr)+C. $$
(4.6)
Combining (4.5) with (4.6), we get
$$ \bigl(\overline{D}F(\bar{x},\bar{y}) \bigl(x'\bar{x}\bigr)+D \bigr)\cap(\operatorname {int}C)\neq\emptyset, $$
furthermore,
$$ \overline{D}F(\bar{x},\bar{y}) \bigl(x'\bar{x}\bigr)\cap( \operatorname{int}D)\neq \emptyset, $$
which is a contradiction. □
Remark 4.13
By comparing the results derived in Section 3, it can be found that the optimality for the set criterion and vector criterion, with the suitable conditions, possesses the same forms in terms of Studniarshi derivatives.