Let *S* ⊂ *X* be a nonempty set, *F* : *S* → 2^{Y} and G : *S* → 2^{Z} be two set-valued maps. We consider the following set-valued optimization problem:

$$\text{(SOP)}\text{\hspace{0.17em}}\{\begin{array}{cc}\text{minimize}& \text{F(x)}\hfill \\ \text{subject\hspace{0.17em}to}& G(x)\cap (-E)\ne \varnothing ,\hfill \\ & x\in S.\hfill \end{array}$$

Let *Ω* = {*x* ∈ *S* : *G*(*x*)∩(-*E*) ≠ ∅}. We begin with giving a necessary optimality condition in type of separating sets for a local minimizer of (SOP).

#### Lemma 3.1

(see [21]). *If* $\overline{z}\in -E$, $z\in -\mathrm{int}(\mathrm{cone}(E+\overline{z}))$, $\frac{1}{{t}_{n}}({z}_{n}-\overline{z})\to z$ *and t*_{n} → 0^{+}, *then z*_{n} € -*intE for large n*.

#### Theorem 3.2

*Let* $(\overline{x},\overline{y})\in \mathit{\text{graph}}(F)$ and $\overline{z}\in G(\overline{x})\cap (-E)$. *Suppose that* $(\overline{x},\overline{y})$ *is local weak minimizer of (SOP). Then for all* $x\in T(S,\overline{x})$, *it holds that*

$$Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x)\cap -(\mathrm{int}D\times \mathrm{int}(\mathrm{cone}(E+\overline{z})))=\varnothing .$$(3.1)

#### Proof

Because $(\overline{x},\overline{y})$ is a local weak minimizer of (SOP), we get there exists $U\in U(\overline{x})$ such that $(F(\mathrm{\Omega}\cap U)-\overline{y})\cap -\text{int}D=\varnothing $. This follows that

$$(F(\Omega \cap U)+D-\overline{y})\cap -\text{int}D=\varnothing .$$(3.2)

We proceed by contradiction. Suppose that (3.1) does not hold. Then there exist $x\in T(S,\overline{x})$, *y* ∈ *Y* and *z* ∈ *Z* such that

$$(y,z)\in Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x)\cap -(\text{int}D\times \text{int}(\text{cone}(E+\overline{z}))).$$

Hence, it yields from Definition 2.1 that there exist sequences *t*_{n} → 0^{+}, *x*_{n} → *x*, (*f*,*g*) ∈ CF((*F* × *G*)_{+}) such that

$$(y,z)=\underset{({t}_{n},{x}_{n})\to ({0}^{+},x)}{\mathrm{lim}}\frac{(f\times g)(\overline{x}+{t}_{n}{x}_{n})-(\overline{y},\overline{z})}{{t}_{n}},\text{\hspace{0.17em}with}\text{\hspace{0.17em}}(f\times g)(\overline{x})=(\overline{y},\overline{z}),$$(3.3)

and

$$(y,z)\in -(\text{int}D\times \text{int}(\text{cone}(E+\overline{z}))).$$(3.4)

From (3.3) and (3.4), for large *n* we can get

$$f(\overline{x}+{t}_{n}{v}_{n})-\overline{y}\in -\text{int}D.$$(3.5)

On the other hand, it follows from Lemma 3.1 that

$$g(\overline{x}+{t}_{n}{v}_{n})\in -\text{int}E,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for\hspace{0.17em}large\hspace{0.17em}}n.$$

Hence, for large *n*, we have $f(\overline{x}+{t}_{n}{v}_{n})\in {F}_{+}(\overline{x}+{t}_{n}{v}_{n})$, $g(\overline{x}+{t}_{n}{v}_{n})\in {G}_{+}(\overline{x}+{t}_{n}{v}_{n})\cap -E$ and $\overline{x}+{t}_{n}{v}_{n}\in \Omega \cap U$. We derive from (3.5) that

$$f(\overline{x}+{t}_{n}{v}_{n})-\overline{y}\in ({F}_{+}(\Omega \cap U)-\overline{y})\cap -\text{int}D.$$

This contradicts to (3.2).

#### Theorem 3.3

(Fritz-John type). *Let* $(\overline{x},\overline{y})\in \mathit{\text{graph}}(F)$, $\overline{z}\in G(\overline{x})\cap (-E)$ *and* $(\overline{x},\overline{y})$ *be a local weak minimizer of (SOP). Suppose that* ($((F-\overline{y})\times G)$ *is nearly D* × *E-subconvexlike map on S around $\overline{x}$. Then there exists* (*y*^{*}, *z*^{*}) ∈ (*D*^{*} × *E*^{*})\{(0,0)} *such that for all* $(y,z)\in Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x)$, $x\in T(S,\overline{x})$,

$${y}^{*T}y+{z}^{*T}z\ge 0,$$(3.6)

*and*

$${z}^{*T}\overline{z}=0.$$(3.7)

#### Proof

Because $(\overline{x},\overline{y})$ is a locally weak minimizer of (SOP), we get that there exists ${U}_{0}\in U(\overline{x})$ such that

$$((F\times G)(S\cap {U}_{0})-(\overline{y},0))\cap -\text{int}(D\times E)=\varnothing .$$(3.8)

Since $((F-\overline{y})\times G)$ is nearly *D* × *E*-subconvexlike map on *S* around $\overline{x}$, it follows from Definition 2.11 that for above *U*_{0} there exists $\overline{U}\in U(\overline{x})$ such that *Ū* ⊂ *U*_{0} and $((F-\overline{y})\times G)$ is nearly *D* × *E*-subconvexlike map on *S* ∩ *Ū*. Denote

$$H(x):=(F(x)-\overline{y})\times G(x),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}x\in S.$$

Thus, it implies that *H* is nearly *D* × *E*-subconvexlike map on *S* ∩ *Ū*, and we get from (3.8) that

$$H(S\cap \overline{U})\cap -\text{int}(D\times E)=\varnothing .$$

Hence, It yields from Lemma 2.12 that there exists (*y*^{*}, *z*^{*}) ∈ (*D*^{*} × *E*^{*})\{(0,0)} that such

$${y}^{*T}(y-\overline{y})+{z}^{*T}z\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}(y,z)\in F\times G(S\cap \overline{U}).$$(3.9)

Taking $y=\overline{y}$ and $z=\overline{z}$, we obtain ${z}^{*T}\overline{z}\ge 0$. Noticing that $\overline{z}\in -E$, we have ${z}^{*T}\overline{z}\le 0$. Hence, we obtain (3.7). Furthermore, it yields from (3.7) and (3.9) that

$${y}^{*T}(y-\overline{y})+{z}^{*T}(z-\overline{z})\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}(y,z)\in F\times G(S\cap \overline{U}).$$(3.10)

Now, for $x\in T(S,\overline{x})$ and $(y,z)\in Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x)$, we shall prove that (3.6) holds. In fact, it follows from the definition of limit set of (*F* × *G*)_{+} at $\overline{x}$ in the direction $x\in T(S,\overline{x})$ with respect to $(\overline{y},\overline{z})$ that there exist *t*_{n} → 0^{+}, *X*_{n} → *X*, (*f*, *g*) ∈ CF((*F* × *G*)_{+}) and (*y*_{n}, *z*_{n}) → (*y*, *z*) such that for all *n*

$${y}_{n}\in \frac{f(\overline{x}+{t}_{n}{x}_{n})+D-\overline{y}}{{t}_{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{n}\in \frac{g(\overline{x}+{t}_{n}{x}_{n})+E-\overline{z}}{{t}_{n}}.$$(3.11)

For large *n*, we can get $\overline{x}+{t}_{n}{x}_{n}\in S\cap \overline{U}$. It yields from (3.11) that there is (*d*_{n}, *e*_{n}) ∈ *D* × *E* such that

$${y}_{n}=\frac{f(\overline{x}+{t}_{n}{x}_{n})+{d}_{n}-\overline{y}}{{t}_{n}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{n}=\frac{g(\overline{x}+{t}_{n}{x}_{n})+{e}_{n}-\overline{z}}{{t}_{n}}.$$

For large *n*, we derive from (3.10) and *y*^{*T} *d*_{n} + *z*^{*T} *e*_{n} > 0 that

$${y}^{*T}(\frac{f(\overline{x}+{t}_{n}{x}_{n})+{d}_{n}-\overline{y}}{{t}_{n}})+{z}^{*T}(\frac{g(\overline{x}+{t}_{n}{x}_{n})+{e}_{n}-\overline{z}}{{t}_{n}})\ge 0,$$

which shows that *y*^{*T} *y*_{n} + *z*^{*T} *z*_{n} ≥ 0. Taking *n* → +∞, we obtain that *y*^{*T} *y* + *z*^{*T}*z* ≥ 0. This completes the proof.

#### Example 3.5

*Let X* = *Y* = ℝ^{2}, *Z* = ℝ, $D={\mathbb{R}}_{+}^{2}$, *E* = ℝ_{+} *and S be defined by*

$$S:=\{(x,y)\in {\mathbb{R}}^{2}:\text{\hspace{0.17em}}x\ge -2,y\ge 0\}\cup \{(x,y)\in {\mathbb{R}}^{2}:\text{\hspace{0.17em}}x\ge 0,y\ge -2\}.$$

*The set-valued mappings F* : *S* → 2^{Y} and G : *S* → 2^{Z} are defined by F(*x*, *y*) = *S and G*(*x*, *y*) = 0 *for all* (*x*, *y*) ∈ *S*. *Taking* $\overline{y}=(-2,0)$, *we derive that* $((F-\overline{y})\times G)$ *is a nearly D × E-subconvexlike map on S since* $cl[cone(((F-\overline{y})\times G)(S)+D\times E)]$ *is convex set. However*, (*F* × *G*) *is not a nearly D* × *E-subconvexlike map on S because cl*[*cone*((*F* × *G*)(*S*) + *D* × *E*)] *is not convex*.

It is well known that optimality condition of Kuhn-Tucker type can be derived from that of Fritz-John type by adding a suitable constraint qualification. Next, we present a necessary optimality condition in Kuhn-Tucker type, which is implied from Theorem 3.3 by giving the local generalized Slater constraint qualification for the constraint set-valued map.

#### Theorem 3.6

(Kuhn-Tucker type). *Let* $(\overline{x},\overline{y})\in \text{graph}(F)$ and $\overline{z}\in G(\overline{x})\cap (-E)$. *Suppose that* $(\overline{x},\overline{y})$ *is a local weak minimizer of (SOP) and* $((F-\overline{y})\times G)$ *is nearly D* × *E-subconvexlike map on S around* $\overline{x}$. *If for all* $U\in U(\overline{x})$, *there exists* $\widehat{x}\in S\cap U$ *such that* $G(\widehat{x})\cap -\mathrm{int}E\ne \varnothing $, *then there exists y*^{*} ∈ *D*^{*}\{0} *and z*^{*} ∈ *E*^{*} such that (3.6) and (3.7) hold for all $(y,z)\in Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x)$, $x\in T(S,\overline{x})$.

#### Proof

Since the conditions of Theorem 3.3 are fulfilled, we get that there exists (*y*^{*}, *z*^{*}) ∈ (*D*^{*} × *E*^{*})\{(0,0)} such that (3.6) and (3.7) hold for all $(y,z)\in Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x)$, $x\in T(S,\overline{x})$. Hence, it is only necessary to prove *y*^{*} ≠ 0. From the proof of Theorem 3.3, there is $\overline{U}\in U(\overline{x})$ such that

$${y}^{*T}y-{y}^{*T}\overline{y}+{z}^{*T}z\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}(y,z)\in F\times G(S\cap \overline{U}).$$

If *y*^{*} = 0, then *z*^{*} ≠ 0 and

$${z}^{*T}z\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}z\in G(S\cap \overline{U}).$$(3.12)

By the assumption, with this *Ū*, there exists $\widehat{x}\in S\cap \overline{U}$ such that $G(\widehat{x})\cap -\text{int}E\ne \varnothing $. This illustrates that there exists $\widehat{z}\in G(\widehat{x})\cap -\text{int}E$ such that *z*^{*T}ẑ < 0. This is a contradiction to (3.12).

Under the assumption of nearly cone-subconvexlike property of $Y{(F\times G)}_{+}(\overline{x},(\overline{y},\overline{z}))$, we can obtain the next result.

#### Theorem 3.7

*Let* $(\overline{x},\overline{y})\in \text{graph}(F)$ *and* $\overline{z}\in G(\overline{x})\cap (-E)$. *Suppose that* $(\overline{x},\overline{y})$ *is a local weak minimizer of (SOP) and* $Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))$ *is nearly D* × *E-subconvexlike map on* $T(S,\overline{x})$. *Then there exists* (*y*^{*}, *z*^{*}) ∈ (*D*^{*} × *E*^{*})\{(0,0)} *such that (3.6) and (3.7) hold for all* $(y,z)\in Y{(F\times G)}_{+}(\overline{x},(\overline{y},\overline{z}))(x)$, $x\in T(S,\overline{x})$.

#### Proof

Since $(\overline{x},\overline{y})$ is a local weak minimizer of (SOP), we derive from Theorem 3.2 that

$$Y{(F\times G)}_{+}(\overline{x},(\overline{y},\overline{z}))(x)\cap -(\text{int}D\times \text{int}(\text{cone}(E+\overline{z})))=\varnothing .\text{\hspace{0.17em}}\text{for\hspace{0.17em}all}\text{\hspace{0.17em}}x\in T(S,\overline{x})$$

On the other hand, since $Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))$ is nearly *D* × *E*-subconvexlike map on $T(S,\overline{x})$, it yields from Lemma 2.10 and $E\subset \text{cone}(E+\overline{z})$ that $Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))$ is nearly $D\times \text{cone}(E+\overline{z})$-subconvexlike map on $T(S,\overline{x})$. Hence, there exists $({y}^{*},{z}^{*})\in ({D}^{*}\times {(\text{cone}(E+\overline{z}))}^{*})\backslash \{(0,0)\}$ such that

$${y}^{*T}y+{z}^{*T}z\ge 0,\text{\hspace{0.17em}}\text{for\hspace{0.17em}all\hspace{0.17em}}(y,z)\in Y{(F\times G)}_{+}(\overline{x},(\overline{y},\overline{z}))(x),\text{\hspace{0.17em}}x\in T(S,\overline{x}).$$

Since ${(\text{cone}(E+\overline{z}))}^{*}\subset {E}^{*}$, we get that *z*^{*} ∈ *E*^{*} and (3.6) holds. For (3.7), since $\overline{z}\in -E$ it is clearly that ${z}^{*T}\overline{z}\le 0$. In addition, we derive from ${z}^{*}\in {(\text{cone}(E+\overline{z}))}^{*}$ that ${z}^{*T}(e+\overline{z})\ge 0$ for all *e* ∈ *E*. Taking *e* = 0, we have ${z}^{*T}\overline{z}\ge 0$. Thus, ${z}^{*T}\overline{z}=0$.

At the end of this section, we present a sufficient optimality condition for a local weak minimizer of (SOP). This result and Theorem 3.6 will be applied to duality in next section.

#### Theorem 3.8

*Let* $(\overline{x},\overline{y})\in \text{graph}(F)$, $\overline{z}\in G(\overline{x})\cap (-E)$ *and for all X* ∈ *S*

$$((F\times G)(x)-(\overline{y},\overline{z}))\subset Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x-\overline{x}).$$(3.13)

*If there exists y*^{*} ∈ *D*^{*}\{0} and z^{*} ∈ *E*^{*} such that (3.6) and (3.7) hold, then $(\overline{x},\overline{y})$ *is a local weak minimizer of* (*SOP*).

#### Proof

Suppose that $(\overline{x},\overline{y})$ is not a local weak minimizer of (SOP), then for all $U\in U(\overline{x})$ there is *x* ∈ *Ω* ∩ *U* such that

$$(F(x)-\overline{y})\cap -\text{int}D\ne \varnothing .$$

Hence, there are *y* ∈ *F*(*X*) and *z* ∈ *G*(*x*) ∩ -*E* such that

$$y-\overline{y}\in -\text{int}D.$$(3.14)

It yields from the condition (3.13) that

$$(y-\overline{y},z-\overline{z})\in ((F\times G)(x)-(\overline{y},\overline{z}))\subset Y{(F\times G)}_{+}(\overline{x};(\overline{y},\overline{z}))(x-\overline{x}).$$

So, we get from (3.6) that

$${y}^{*T}(y-\overline{y})+{z}^{*T}(z-\overline{z})\ge 0.$$

Noticing that *z*^{*T}*z* ≤ 0 and (3.7) holds, one has

$${y}^{*T}(y-\overline{y})\ge -{z}^{*T}z+{z}^{*T}\overline{z}=-{z}^{*T}z\ge 0.$$

This contradicts to (3.14).

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.