The ρ(mathbb{C})A(mathbb{C})

We introduce the notion of ρ(mathbb{C})A(mathbb{C}) of set-valued maps as a generalization of (mathbb{C})A(mathbb{CS})(mathbb{VM}) which was introduced by Das et al. [3442] and Treanţă et al. [43]. For (rho =0), we have the usual notion of cone convex set-valued maps introduced by Borwein [33].

Definition 3.1

Let U and V be real normed spaces, (Wsubseteq U) be an arcwise connected, (mathrm{u}_{1}, mathrm{u}_{2}in W), (ein operatorname{int}(Omega )), and (mathfrak{F}: U to 2^{V}) be a set-valued map with (W subseteq operatorname{dom}(mathfrak{F})). Then (mathfrak{F}) is said to be ρ-Ω-arcwise connected (ρ-Ω-A(mathbb{C})) with respect to e on W for (mathrm{u}_{1}), (mathrm{u}_{2}) if there exists (rho in mathbb{R}) such that

$$ (1 – upeta ) mathfrak{F}(mathrm{u}_{1}) + mathrm{u} mathcal{F}( mathrm{u}_{2}) subseteq mathrm{u}bigl( mathcal{H}_{mathrm{u}_{1}, mathrm{u}_{2}} (upeta )bigr) + rho upeta ( 1 – upeta ) Vert mathrm{u}_{1} – mathrm{u}_{2} Vert ^{2} e + Omega , $$

(2)

(forall upeta in [0, 1]).

Remark 3.1

If (rho > 0), then (mathfrak{F}) is said to be strongly ρ-Ω-arcwise connected ((mathbb{S}rho )-Ω-A(mathbb{C})); if (rho = 0), we have the usual notion of Ω-arcwise connectedness; and if (rho <0), then (mathfrak{F}) is said to be weakly ρ-Ω-arcwise connected ((mathbb{W}rho )-Ω-A(mathbb{C})). Obviously, (mathbb{S}rho )-Ω-A(mathbb{C}) Ω-arcwise connectedness (mathbb{W}rho )-Ω-A(mathbb{C}).

Further, we construct an example of ρ(mathbb{C})A(mathbb{CS})(mathbb{VM}), which is not cone arcwise connected.

Example 3.1

Let (U =mathbb{R}^{2}), (V = mathbb{R}), (Omega = mathbb{R}_{+}), and

$$ W = biggllbrace mathrm{u} = (mathrm{u}_{1}, mathrm{u}_{2}) : mathrm{u}_{1} + mathrm{u}_{2} geq frac{1}{4}, mathrm{u}_{1} geq 0, mathrm{u}_{2}geq 0 biggrrbrace subseteq U.$$

Define

$$ mathcal{H}_{mathrm{u},acute{mathrm{u}}}(upeta ) = bigl(1 – upeta ^{2}bigr) upeta + upeta ^{2} acute{mathrm{u}},$$

where (mathrm{u}=(mathrm{u}_{1},mathrm{u}_{2})), (acute{mathrm{u}}=( acute{mathrm{u}}_{1}, acute{mathrm{u}}_{2})), and (upeta in [0,1]). Clearly, W is an arcwise connected set. Define a set-valued map (mathfrak{F} : mathbb{R}^{2} to 2^{mathbb{R}}) as follows:

$$ mathfrak{F}(mathrm{u}) = textstylebegin{cases} [0,4], & mathrm{u}_{1} + mathrm{u}_{2} geq frac{1}{2}, mathrm{u}_{1}neq 3 mathrm{u}_{2}, mathrm{u}= ( mathrm{u}_{1}, mathrm{u}_{2}), \ [5,9],& text{otherwise}.end{cases} $$

We choose (mathrm{u}=(1,0)), (acute{mathrm{u}}=(0,1)), and (upeta = frac{1}{2}). Then

$$ mathcal{H}_{mathrm{u},acute{mathrm{u}}} biggl( frac{1}{2} biggr) = biggl( frac{3}{4}, frac{1}{4} biggr)$$

and

$$begin{aligned} frac{1}{2} mathfrak{F}(1, 0)+frac{1}{2} mathfrak{F}(0, 1) & = frac{1}{2} [0,4] + frac{1}{2} [0,4] \ & = [0,4] nsubseteq [5,9] +mathbb{R}_{+} \ & = mathfrak{F} biggl( frac{3}{4}, frac{1}{4} biggr) + mathbb{R}_{+}. end{aligned}$$

Hence (mathfrak{F}) is not (mathbb{R}_{+})-arcwise connected. On the other hand, by considering (rho = -2) and (e = 5), we get that

$$ (1- upeta ) mathfrak{F} (1, 0)+upeta mathfrak{F}(0, 1) = (1- upeta ) [0,4] + upeta [0,4] = [0,4]$$

and

$$ mathfrak{F}bigl(mathcal{H}_{mathrm{u}, acute{mathrm{u}}} (upeta )bigr) + rho upeta ( 1 – upeta ) Vert mathrm{u} – acute{mathrm{u}} Vert ^{2} e = mathfrak{F} bigl( 1 – upeta ^{2}, upeta ^{2}bigr)- 20 upeta ( 1 – upeta ).$$

For (upeta neq 0.5), we have

$$ mathfrak{F} bigl(1 – upeta ^{2}, upeta ^{2}bigr) = [0,4].$$

So,

$$begin{aligned} ( 1 – upeta ) mathfrak{F}(1, 0)+upeta mathfrak{F}(0, 1) + 20 upeta ( 1 – upeta ) & = [0,4] + 20 upeta ( 1 – upeta ) \ & subseteq [0,4] + mathbb{R}_{+} = mathbb{R}_{+}. end{aligned}$$

For (upeta = frac{1}{2}), we have

$$ mathfrak{F} bigl(1-upeta ^{2}, upeta ^{2}bigr) = mathfrak{F} biggl( frac{3}{4}, frac{1}{4} biggr)= [3,5].$$

So,

$$begin{aligned} ( 1 – upeta ) mathfrak{F}(1, 0)+upeta mathfrak{F}(0, 1) + 20 upeta (1 – upeta ) & = [0,4] + 5 \ & = [5,9] subseteq [5,9]+ mathbb{R}_{+}. end{aligned}$$

Consequently, (mathfrak{F}) is ((-2))(mathbb{R}_{+})-A(mathbb{CS})(mathbb{VM}) with respect to 5 on W for ((1, 0)), ((0, 1)).

Theorem 3.2

Let U, V be real normed spaces, (Wsubseteq U) be arcwise connected, (ein operatorname{int}(Omega )), and (mathfrak{F}: U to 2^{V}) be ρ-Ω-A(mathbb{C}) with respect to e on W. Let (acute{mathrm{u}}in W) and (acute{mathrm{v}}in mathfrak{F}(acute{mathrm{u}})). Then

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where

$$ acute{mathcal{H}}_{acute{mathrm{u}}, mathrm{u}}(0+) = lim_{ upeta to 0+} frac{ mathcal{H}_{acute{mathrm{u}}, mathrm{u}}( upeta ) – mathcal{H}_{acute{mathrm{u}}, mathrm{u}}(0)}{upeta},$$

assuming that (acute{mathcal{H}}_{acute{mathrm{u}}, mathrm{u}}(0+)) exists for all (mathrm{u}, acute{mathrm{u}}in W).

Proof

Let (mathrm{u}in W). As (mathfrak{F}) is ρ-Ω-A(mathbb{C}) with respect to e on W, we have

$$ (1-upeta ) mathfrak{F}(acute{mathrm{u}}) + upeta mathfrak{F}( mathrm{u}) subseteq mathfrak{F}bigl( mathcal{H}_{acute{mathrm{u}}, mathrm{u}}( upeta )bigr) + rho upeta (1 – upeta ) Vert mathrm{u} – acute{mathrm{u}} Vert ^{2} e + Omega ,$$

(forall upeta in [0, 1]). Let (mathrm{v}in mathfrak{F}(mathrm{u})). Consider a real sequence ({upeta _{n}}) with (upeta _{n} in (0, 1)), (n in mathbb{N}), such that (upeta _{n}to 0+) when (nto infty ). Suppose (mathrm{u}_{n} = mathcal{H}_{acute{mathrm{u}}, mathrm{u}} ( upeta _{n})) and

$$ mathrm{v}_{n} = (1 – upeta _{n}) acute{mathrm{v}} + upeta _{n} mathrm{v} – rho upeta _{n} ( 1 – upeta _{n}) Vert mathrm{u} – acute{mathrm{u}} Vert ^{2} e.$$

Therefore, (mathrm{v}_{n}in mathfrak{F}( mathrm{u}_{n}) + Omega ). It is clear that

$$ {mathrm{u}_{n} = mathcal{H}_{acute{mathrm{u}}, mathrm{u}} ( mathrm{u}_{n}) to mathcal{H}_{acute{mathrm{u}}, mathrm{u}} (0) = acute{ mathrm{u}},}$$

(mathrm{v}_{n}to acute{mathrm{v}}), when n tends to ∞,

$$ frac{mathrm{u}_{n} – acute{mathrm{u}}}{ upeta _{n}} = frac{ mathcal{H}_{acute{mathrm{u}}, mathrm{u}} (upeta _{n}) – mathcal{H}_{acute{mathrm{u}}, mathrm{u}} (0)}{ upeta _{n}} to acute{mathcal{H}}_{acute{mathrm{u}}, mathrm{u}}(0+),$$

when n tends to ∞, and

$$ frac{mathrm{v}_{n} – acute{mathrm{v}}}{ upeta _{n}} = mathrm{v} – acute{mathrm{v}} – rho ( 1 -upeta _{n}) Vert mathrm{u} – acute{mathrm{u}} Vert ^{2} e to mathrm{v} – acute{mathrm{v}} – rho Vert mathrm{u} -acute{mathrm{u}} Vert ^{2} e,$$

when (nto infty ). Therefore,


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which is true for all (mathrm{v}in mathfrak{F}(mathrm{u})). Hence,

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Hence the theorem follows. □

Formulation of the main problem

Let U, (V_{1}), (V_{2}), and (V_{3}) be real normed spaces and (Omega _{1}), (Omega _{2}), and (Omega _{3}) be solid pointed convex cones in (V_{1}), (V_{2}), and (V_{3}), respectively. Let A be an arbitrary set and W be a nonempty subset of U. Suppose that

$$ mathfrak{F} : Utimes A to 2^{V_{1}}, qquad mathfrak{G} : U times A to 2^{V_{2}}$$

are set-valued maps and (mathrm{p} : Utimes A to V_{3}) is a single-valued map with

$$ W times A subseteq operatorname{dom}( mathfrak{F} ) cap operatorname{dom}( mathfrak{G}).$$

We consider a parametric (mathbb{S})(mathbb{VPOP}s) (1), where u is the state variable and a is the parameter. The feasible set Š of problem (1) is defined by

$$ check{S} = bigl{ ( mathrm{u}, mathrm{a}) in Wtimes A : mathfrak{G}( mathrm{u}, mathrm{a})cap (-Omega _{2}) neq emptyset text{ and } mathrm{p}(mathrm{u}, mathrm{a})=0 bigr} .$$

The minimizer and weak minimizer of problem (1) are defined in the following ways. A point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})in U times A times V_{1}), with ((acute{mathrm{u}}, acute{mathrm{a}}) in check{S}) and (acute{mathrm{v}}_{1} in mathfrak{F}(acute{mathrm{u}}, acute{mathrm{a}})), is called a minimizer of problem (1) if there exists no point ((mathrm{u}, mathrm{a}, mathrm{v}_{1}) in Utimes A times V_{1}), with ((mathrm{u}, mathrm{a}) in check{S}) and (mathrm{v}_{1}in mathfrak{F}(mathrm{u}, mathrm{a})), such that

$$ mathrm{v}_{1} – acute{mathrm{v}}_{1} in – Omega _{1} setminus {theta _{V_{1}}},$$

and is called a weak minimizer of problem (1) if there exists no point

$$ (mathrm{u}, mathrm{a}, mathrm{v}) in Utimes A times V_{1},$$

with ((mathrm{u}, mathrm{a}) in check{S}) and (mathrm{v}_{1} in mathfrak{F}(mathrm{u}, mathrm{a})), such that (mathrm{v}_{1} – acute{mathrm{v}}_{1} in – operatorname{int}(V_{1})).

Sufficient optimality conditions

Let

$$ (mathrm{u}, mathrm{a}),(acute{mathrm{u}}, mathrm{a}),( mathrm{u}, acute{ mathrm{a}}),(acute{mathrm{u}}, acute{mathrm{a}}) in Utimes A, $$

(acute{mathrm{v}}_{1} in mathfrak{F}(acute{mathrm{u}}, acute{mathrm{a}})), and (acute{mathrm{v}}_{2} in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}})). Throughout the paper, we use the following assumptions:

$$begin{aligned}& mathfrak{F}(mathrm{u}, mathrm{a}) – mathfrak{F}( acute{ mathrm{u}}, mathrm{a}) subseteq Omega _{1}, \& mathfrak{G}(mathrm{u}, mathrm{a}) – mathfrak{G}( acute{mathrm{u}}, mathrm{a}) subseteq Omega _{2}, \& mathfrak{F}(mathrm{u}, acute{mathrm{a}}) – acute{mathrm{v}}_{1} subseteq -Omega _{1}, \& mathfrak{G}(mathrm{u}, acute{mathrm{a}}) – acute{mathrm{v}}_{2} subseteq -Omega _{1}, \& mathrm{p}(mathrm{u}, acute{mathrm{a}}) + mathrm{p}( acute{mathrm{u}}, mathrm{a}) in -Omega _{3}. end{aligned}$$

(3)

We now prove the following lemma which assists in establishing the sufficient KKT optimality conditions of the parametric (mathbb{S})(mathbb{VPOP}s) (1).

Lemma 3.3

Let W be an arcwise connected subset of U and ((acute{mathrm{u}}, acute{mathrm{a}})in Utimes A) with (acute{mathrm{v}}_{1} in mathfrak{F}(acute{mathrm{u}}, acute{mathrm{a}})), (acute{mathrm{v}}_{2} in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}})), and (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})geq 0). Let (ein operatorname{int}(Omega _{1})), (e^{prime}in operatorname{int}(Omega _{2})), and (e^{prime prime}in operatorname{int}(Omega _{3})). Suppose that (mathfrak{F}(cdot, acute{mathrm{a}}) : U to 2^{V_{1}}) is (rho _{1})-Ω-A(mathbb{C}) with respect to e, (mathfrak{G}(cdot, acute{mathrm{a}}) : Uto 2^{V_{2}}) is (rho _{2})-Ω-A(mathbb{C}) with respect to (e^{prime}), and (mathrm{p}(cdot, acute{mathrm{a}}) : U to V_{3}) is (rho _{3})-Ω-A(mathbb{C}) with respect to (e^{prime prime}) on W. Assume that the contingent epiderivatives


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exist and the Gâteaux derivative (mathrm{p}^{prime}(cdot, acute{mathrm{a}})(acute{mathrm{u}})) exists. If equations in (3) are satisfied, then we have








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(4)

(forall (mathrm{u}, mathrm{A})in Wtimes A).

Proof

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(5)

As (mathfrak{G}(cdot, acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{2})-Ω-A(mathbb{C}) with respect to (e^{prime}) on W and (acute{mathrm{v}}_{2}in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}})), we have

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(6)

Again, as (mathrm{p}(cdot, acute{mathrm{a}}) : U to V_{3}) is (rho _{3})-Ω-A(mathbb{C}) with respect to (e^{prime prime}) on W, we have

$$ mathrm{p}(mathrm{u}, acute{mathrm{a}}) – mathrm{p}( acute{mathrm{u}}, acute{mathrm{a}}) in mathrm{p}^{prime}(cdot, acute{mathrm{a}}) (acute{mathrm{u}}) bigl( acute{mathcal{H}}_{ acute{mathrm{u}}, mathrm{u}}(0+) bigr) + rho _{3} Vert mathrm{u} – acute{mathrm{u}} Vert ^{2} e^{prime prime}+ Omega _{3}. $$

(7)

Hence, from Eq. (5), we have








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(8)

By Eq. (3), we have

$$begin{aligned}& bigllangle mathrm{v}_{1}^{ast}, mathfrak{F}( mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{1} bigrrangle geq bigllangle mathrm{v}_{1}^{ast}, mathfrak{F}(acute{ mathrm{u}}, mathrm{a}) – acute{mathrm{v}}_{1} bigrrangle , \& bigllangle mathrm{v}_{2}^{ast},mathfrak{G}( mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{2} bigrrangle geq bigllangle mathrm{v}_{2}^{ast}, mathfrak{G}(acute{ mathrm{u}}, mathrm{a}) – acute{mathrm{v}}_{2} bigrrangle , \& bigllangle mathrm{v}_{1}^{ast}, mathfrak{F}( mathrm{u}, acute{mathrm{a}})-acute{mathrm{v}}_{1} bigrrangle leq 0, bigllangle mathrm{v}_{2}^{ast}, mathfrak{G}( mathrm{u}, acute{mathrm{a}}) – acute{mathrm{v}}_{2} bigrrangle leq 0, end{aligned}$$

and (langle mathrm{v}_{3}^{ast}, mathrm{p}(mathrm{u}, acute{mathrm{a}}) + mathrm{p}(acute{mathrm{u}}, mathrm{a}) rangle leq 0). By assumption, we have (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})geq 0). Therefore,

$$begin{aligned} &bigllangle mathrm{v}_{1}^{ast}, mathfrak{F}( mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathfrak{G}( mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{2} bigrrangle \ &quad geq bigllangle mathrm{v}_{1}^{ast}, mathfrak{F} (mathrm{u}, acute{mathrm{a}}) – acute{mathrm{v}}_{1} + mathfrak{F}( acute{mathrm{u}}, mathrm{a}) – acute{mathrm{v}}_{1} bigrrangle \ &qquad {} + bigllangle mathrm{v}_{2}^{ast}, mathfrak{G}( mathrm{u}, acute{mathrm{a}}) – acute{mathrm{v}}_{2} + mathfrak{G}( acute{mathrm{u}}, acute{mathrm{a}}) – acute{ mathrm{v}}_{2} bigrrangle \ &qquad {} + bigllangle mathrm{v}_{3}^{ast}, mathrm{p}( mathrm{u}, acute{mathrm{a}}) – mathrm{p}(acute{mathrm{u}}, acute{ mathrm{a}}) + mathrm{p}( acute{mathrm{u}}, mathrm{a}) bigrrangle . end{aligned}$$

(9)

Consequently,








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(10)

It completes the proof of Lemma 3.3. □

We establish the sufficient KKT optimality conditions of the parametric (mathbb{S})(mathbb{VPOP}s) (1) under contingent epiderivative and ρ(mathbb{C})A(mathbb{C}) assumptions.

Theorem 3.4

(Sufficient optimality conditions)

Let W be an arcwise connected subset of U and ((acute{mathrm{u}}, acute{mathrm{a}})in Utimes A), with ((acute{mathrm{u}}, acute{mathrm{a}})in check{S}),

$$ acute{mathrm{v}}_{1} in mathfrak{F}(acute{mathrm{u}}, acute{ mathrm{u}}), qquad acute{mathrm{v}}_{2} in mathfrak{G}( acute{ mathrm{u}}, acute{mathrm{a}}) cap (-Omega _{2}),$$

and (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})geq 0). Let (ein operatorname{int}(Omega _{1})), (e^{prime}in operatorname{int}(Omega _{2})), and (e^{prime prime}in operatorname{int}(Omega _{3})). Suppose that (mathfrak{F}(cdot, acute{mathrm{a}}): U to 2^{V_{1}}) is (rho _{1})(Omega _{1})A(mathbb{C}) with respect to e, (mathfrak{G}(cdot, acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{2})(Omega _{2})A(mathbb{C}) with respect to (e^{prime}), and (mathrm{p}(cdot, acute{mathrm{a}}) : U to V_{3}) is (rho _{3})(Omega _{3})A(mathbb{C}) with respect to (e^{prime prime}) on W. Assume that the contingent epiderivatives


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)(
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and


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a
´
)(
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,

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)

exist and the Gâteaux derivative (mathrm{p}^{prime}(cdot, acute{mathrm{a}})(acute{mathrm{u}})) exists. Suppose that the conditions of Lemma 3.3hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) for some

$$ bigl(mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr) in Omega _{1}^{+} times Omega _{2}^{+} times Omega _{3}^{+},$$

with (mathrm{v}_{1}^{ast}neq theta _{V_{1}}) and

$$ rho _{1} bigllangle mathrm{v}_{1}^{ast}, ebigrrangle + rho _{2} bigllangle mathrm{v}_{2}^{ast}, e^{prime}bigrrangle + rho _{3} bigllangle mathrm{v}_{3}^{ast}, e^{prime prime} bigrrangle geq 0, $$

(11)

such that








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(12)

(forall (cdot, mathrm{a}) in Wtimes A), and

$$ bigllangle mathrm{v}_{3}^{ast}, acute{mathrm{v}}_{2}bigrrangle =0, $$

(13)

then ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})) is a weak minimizer of problem (1).

Proof

Suppose that ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})) is not a weak minimizer of problem (1). Then there exist ((mathrm{u}, mathrm{a})in check{S}) and (mathrm{v}_{1}in mathfrak{F}(mathrm{u}, mathrm{a})) such that (mathrm{v}_{1} <acute{mathrm{v}}_{1}). As

$$ mathrm{v}_{1}^{ast}in Omega _{1}^{+} setminus {theta _{V_{1}}},$$

(langle mathrm{v}_{1}^{ast}, mathrm{v}_{1} – acute{mathrm{v}}_{1} rangle < 0). As ((mathrm{u}, mathrm{a})in mathrm{a}), there exists

$$ mathrm{v}_{2} in mathfrak{G}( mathrm{u}, mathrm{a})cap (- Omega _{2}).$$

So, (langle mathrm{v}_{2}^{ast},mathrm{v}_{2} rangle leq 0) as (mathrm{v}_{2}^{ast}in Omega _{2}^{+}). Since (langle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2} rangle = 0), we have

$$ bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} – acute{mathrm{v}}_{2} bigrrangle = bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} bigrrangle leq 0.$$

Therefore,

$$ bigllangle mathrm{v}_{1}^{ast}, acute{mathrm{v}}_{1} – acute{mathrm{v}}_{2} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} – acute{mathrm{v}}_{2} bigrrangle < 0. $$

(14)

As the conditions of Lemma 3.3 hold at ((acute{mathrm{u}}, acute{mathrm{u}}, acute{mathrm{u}}, acute{mathrm{u}}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})), from Eqs. (4), (11), and (12), we have

$$ bigllangle mathrm{v}_{1}^{ast}, mathfrak{F}( mathrm{u}, mathrm{u})- acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2}^{ast}(mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{2} bigrrangle geq 0.$$

Hence,

$$ bigllangle mathrm{v}_{1}^{ast}, mathrm{v}_{1}-acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} – acute{mathrm{v}}_{2} bigrrangle geq 0,$$

which contradicts (14). Consequently, ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})) is a weak minimizer of problem (1). □

Wolfe type dual

We consider a Wolfe type dual (15), where (mathfrak{F}(cdot, acute{mathrm{a}})) and (mathfrak{G}(cdot, acute{mathrm{a}})) are contingent epiderivable set-valued maps and (mathrm{p}(cdot, acute{mathrm{a}})) is a Gâteaux derivable single-valued map, where (acute{mathrm{a}}in A).




maximize





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,




subject to










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(15)

(forall (mathrm{u}, mathrm{a}) in Wtimes A), (acute{mathrm{u}}in W), (acute{mathrm{a}}in A), (acute{mathrm{v}}_{1}in mathfrak{F}(acute{mathrm{u}}, acute{mathrm{a}})), (acute{mathrm{v}}_{2}in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}})), (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}}) geq 0),

$$ bigl(mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr) in Omega _{1}^{+} times Omega _{2}^{+} times Omega _{3}^{+},$$

and (langle mathrm{v}_{1}^{ast}, erangle =1).

Definition 3.5

A point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) satisfying all the constraints of (15) is called a feasible point of problem (15). The feasible point

$$ bigl(acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr)$$

of problem (15) is called a weak maximizer of (15) if there exists no feasible point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, tilde{mathrm{v}}_{1}^{ast}, tilde{mathrm{v}}_{2}^{ast}, tilde{mathrm{v}}_{3}^{ast})) of (15) such that

$$ bigl(mathrm{v}_{1} + bigllangle tilde{mathrm{v}}_{2}^{ast}, mathrm{v}_{2} bigrrangle ebigr) – bigl(acute{ mathrm{v}}_{1} + bigllangle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2} bigrrangle ebigr)in operatorname{int}(Omega _{1}).$$

We prove the duality results of Wolfe type of problem (1). The proofs are very similar to Theorems 3.103.12, and hence omitted.

Theorem 3.6

(Weak duality)

Let W be an arcwise connected subset of U, ((acute{mathrm{u}}_{0}, acute{mathrm{a}}_{0}) in check{S}), ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) be a feasible point of problem (15), and (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})geq 0). Let

$$ ein operatorname{int}(Omega _{1}),qquad e^{prime}in operatorname{int}( Omega _{2}), qquad e^{prime prime}in operatorname{int}(Omega _{3}).$$

Suppose that (mathfrak{F} (cdot, acute{mathrm{a}}) : U to 2^{V_{1}}) is (rho _{1})(Omega _{1})A(mathbb{C}) with respect to e, (mathfrak{G}(cdot, acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{2})(Omega _{2})A(mathbb{C}) with respect to (e^{prime}), and (acute{mathrm{a}}(cdot, acute{mathrm{a}}) : U to V_{3}) is (rho _{3})(Omega _{3})A(mathbb{C}) with respect to (e^{prime prime}) on W. Assume that the contingent epiderivatives


D

F(,
a
´
)(
u
´
,

v
´

1
)

and


D

G(,
a
´
)(
u
´
,

v
´

2
)

exist and the Gâteaux derivative (mathrm{p}^{prime}(cdot, acute{mathrm{a}})(acute{mathrm{u}})) exists. Suppose that the conditions of Lemma 3.3hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) and (17) is satisfied. Then

$$ mathfrak{F}(mathrm{u}_{0}, mathrm{a}_{0}) – bigl( acute{mathrm{v}}_{1} + bigllangle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2}bigrrangle ebigr) subseteq V_{1} setminus -operatorname{int}(Omega _{1}).$$

Theorem 3.7

(Strong duality)

Let ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})) be a weak minimizer of problem (1) and (acute{mathrm{v}}_{2}in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}})cap (-Omega _{2})). Assume that for some

$$ bigl(mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr) in Omega _{1}^{+} times Omega _{2}^{+} times Omega _{3}^{+},$$

with (langle mathrm{v}_{1}^{ast}, erangle =1), Eqs. (12) and (13) are satisfied at the point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})). Then ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is a feasible solution for problem (15). If the weak duality Theorem 3.6between (1) and (15) holds, then the point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is a weak maximizer of problem (15).

Theorem 3.8

(Converse duality)

Let W be an arcwise connected subset of the space U and ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) be a feasible point of problem (15) with (langle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2}rangle geq 0) and (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})geq 0). Let (ein operatorname{int}(Omega _{1})), (e^{prime}in operatorname{int}(Omega _{2})), and (e^{prime prime} in operatorname{int}(Omega _{3})). Suppose that (mathfrak{F} (cdot, acute{mathrm{a}}) : Uto 2^{V_{1}}) is (rho _{1})(Omega _{1})A(mathbb{C}) with respect to e, (mathfrak{G}(cdot, acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{2})(Omega _{2})A(mathbb{C}) with respect to (e^{prime}), and (mathrm{p}(cdot, acute{mathrm{a}} ) : Uto V_{3}) is (rho _{3})(Omega _{3})A(mathbb{C}) with respect to (e^{prime prime}) on W. Assume that the contingent epiderivatives


D

F(,
a
´
)(
u
´
,

v
´

1
),
D

G(,
a
´
)(
u
´
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v
´

2
),

exist and the Gâteaux derivative (mathrm{p}^{prime}(cdot, acute{mathrm{a}})( acute{mathrm{u}})) exists. Suppose that the conditions of Lemma 3.3hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) and (17) is satisfied. If ((acute{mathrm{u}}, acute{mathrm{a}})in check{S}), then ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{2})) is a weak minimizer of (1).

Mond–Weir type dual

We consider a Mond–Weir type dual (16), where (mathfrak{F}(cdot, mathfrak{F})) and (mathfrak{G}(cdot, acute{mathrm{a}})) are contingent epiderivable and (mathrm{p}(cdot, acute{mathrm{a}})) is a Gâteaux derivable single-valued map, where (acute{mathrm{a}}in A).




maximize 





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(16)

(forall (mathrm{u}, mathrm{u}) in Wtimes A), (langle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2} rangle geq 0), (acute{mathrm{u}}in W), (acute{mathrm{a}}in A), (acute{mathrm{v}}_{1} in mathfrak{F}(acute{mathrm{u}}, acute{mathrm{a}})), (acute{mathrm{a}}in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}})), (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}}) geq 0),

$$ bigl(mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr) in Omega _{1}^{+} times Omega _{2}^{+} times Omega _{3}^{+},$$

with (langle mathrm{v}_{1}^{ast}, erangle =1).

Definition 3.9

A point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) satisfying all the constraints of problem (16) is called a feasible point of (16). The feasible point is called a weak maximizer of problem (16) if there exists no feasible point ((mathrm{u}, mathrm{a}, mathrm{v}_{1}, mathrm{v}_{2}, tilde{mathrm{v}}_{1}^{ast}, tilde{mathrm{v}}_{2}^{ast}, tilde{mathrm{v}}_{3}^{ast})) of (16) such that (mathrm{v}_{1}- acute{mathrm{v}}_{1} in operatorname{int}(Omega _{1})).

Theorem 3.10

(Weak duality)

Let W be an arcwise connected subset of U, ((mathrm{u}_{0}, mathrm{a}_{0}) in check{S}), ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) be a feasible point of problem (16), and (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})geq 0). Let

$$ {ein operatorname{int}(Omega _{1}), qquad e^{prime}in operatorname{int}(Omega _{2}),qquad e^{prime prime}in operatorname{int}( Omega _{3}).}$$

Suppose that (mathfrak{F}(cdot, acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{1})(Omega _{1})A(mathbb{C}) with respect to e, (mathfrak{G}(cdot, acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{2})(Omega _{2})A(mathbb{C}) with respect to (e^{prime}), and (mathrm{p}(cdot, acute{mathrm{a}}) : U to V_{3}) is (rho _{3})(Omega _{3})A(mathbb{C}) with respect to (e^{prime prime}) on W. Assume that the contingent epiderivatives


D

F(,
u
´
)(
u
´
,

v
´

1
)

and


D

G(,
a
´
)(
u
´
,

v
´

2
)

exist and the Gâteaux derivative (mathrm{p}^{prime}(cdot, acute{mathrm{a}})(acute{mathrm{u}})) exists. Suppose that the conditions of Lemma 3.3hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})). Assume that

$$ rho _{1} +rho _{2} bigllangle mathrm{v}_{2}^{ast}, e^{prime}bigrrangle + rho _{3} bigllangle mathrm{v}_{3}^{ast}, e^{prime prime} bigrrangle geq 0. $$

(17)

Then (mathfrak{F}(mathrm{u}_{0}, mathrm{a}_{0}) – acute{mathrm{v}}_{1} subseteq V_{1} setminus – operatorname{int}(Omega _{1})).

Proof

We prove the theorem by the method of contradiction. Suppose that for some

$$ mathrm{v}_{1}^{circ}in mathfrak{F}( mathrm{u}_{0}, mathrm{a}_{0}), qquad mathrm{v}_{1}^{circ}- acute{mathrm{v}}_{1} in – operatorname{int}(Omega _{1}).$$

Therefore, (langle mathrm{v}_{1}^{ast}, mathrm{v}_{1}^{circ}- acute{mathrm{v}}_{1} rangle < 0) as (theta _{V_{1}} neq mathrm{v}_{1}^{ast}in Omega _{1}^{+}). Again, since ((mathrm{u}_{0}, mathrm{a}_{0})in check{S}), we have

$$ mathfrak{G} (mathrm{u}_{0}, mathrm{a}_{0})cap (- Omega _{2} ) neq emptyset ,$$

and (mathrm{p}(mathrm{u}_{0}, mathrm{a}_{0}) =0). We choose

$$ mathrm{v}_{2}^{circ}in mathfrak{G}( mathrm{u}_{0}, mathrm{a}_{0}) cap (-Omega _{2}).$$

So, (langle mathrm{v}_{2}^{ast}, mathrm{v}_{2}^{circ}rangle leq 0) as (mathrm{v}_{2}^{ast}in Omega _{2}^{+}). Again, from the constraints of (16), we have (langle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2} rangle geq 0). Therefore,

$$ bigllangle mathrm{v}_{2}^{ast},mathrm{v}_{2}^{circ}- acute{mathrm{v}}_{2} bigrrangle = bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2}^{circ}bigrrangle – bigllangle mathrm{v}_{2}^{ast}, acute{ mathrm{v}}_{2} bigrrangle leq 0.$$

Hence,

$$ bigllangle mathrm{v}_{1}^{ast}, mathrm{v}_{1}^{circ}- acute{mathrm{v}}_{1} bigrrangle +bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2}^{circ}- acute{mathrm{v}}_{2} bigrrangle < 0. $$

(18)

As the conditions of Lemma 3.3 hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})), from Eqs. (4), (17) and the constraints of (16), we have

$$ bigllangle mathrm{v}_{1}^{ast}, mathfrak{F} bigl( mathrm{v}_{1}^{circ}, mathrm{a}bigr) – acute{ mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathfrak{G}(mathrm{u}_{0}, mathrm{a}) – acute{ mathrm{v}}_{2} bigrrangle geq 0.$$

Hence,

$$ bigllangle mathrm{v}_{1}^{ast}, mathrm{v}_{1}^{circ}- acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2}^{circ}- acute{mathrm{v}}_{2} bigrrangle geq 0,$$

which contradicts (18). Therefore,

$$ mathfrak{F}(mathrm{u}_{0}, mathrm{a}_{0}) – acute{ mathrm{v}}_{1} subseteq V_{1}setminus – operatorname{int}(Omega _{1}).$$

It completes the proof of the theorem. □

Theorem 3.11

(Strong duality)

Let ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})) be a weak minimizer of problem (1) and (acute{mathrm{v}}_{2}in mathfrak{G}(acute{mathrm{u}}, acute{mathrm{a}}) cap (-Omega _{2})). Assume that for some

$$ bigl(mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr) in Omega _{1}^{+} times Omega _{2}^{+} times Omega _{3}^{+},$$

with (langle mathrm{v}_{1}^{ast}, erangle = 1), Eqs. (12) and (13) are satisfied at the point

$$ bigl(acute{mathrm{u}}, acute{mathrm{a}},acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast}bigr).$$

Then ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is a feasible solution for problem (16). If the weak duality Theorem 3.10between (1) and (16) holds, then the point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is a weak maximizer of (16).

Proof

As Eqs. (12) and (13) are satisfied at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})),








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(19)

(forall (mathrm{u}, mathrm{a}) in Wtimes A), and (langle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2} rangle =0). As ((acute{mathrm{u}}, acute{mathrm{a}})in check{S}), (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}})=0). Hence, ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is a feasible solution for (16). Suppose that the weak duality Theorem 3.10 between problems (1) and (16) holds and the point ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is not a weak maximizer of problem (16). Let

$$ bigl(mathrm{u}, mathrm{a}, mathrm{v}_{1}, mathrm{v}_{2}, tilde{mathrm{v}}_{1}^{ast}, tilde{ mathrm{v}}_{2}^{ast}, tilde{mathrm{v}}_{3}^{ast}bigr)$$

be a feasible point for (16) such that (acute{mathrm{v}}_{1} – mathrm{v}_{1} in -operatorname{int}( Omega _{1})). It contradicts the weak duality Theorem 3.10 between (1) and (16). Consequently, ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) is a weak maximizer for (16). □

Theorem 3.12

(Converse duality)

Let W be an arcwise connected subset of U, (mathrm{p}(acute{mathrm{u}}, acute{mathrm{a}}) geq 0), and ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) be a feasible point of problem (16). Let

$$ ein operatorname{int}(Omega _{1}),qquad e^{prime}in operatorname{int}(Omega _{2}),qquad e^{prime prime}in operatorname{int}(Omega _{3}).$$

Suppose that (mathfrak{F}(cdot, acute{mathrm{a}}) : U to 2^{V_{1}}) is (rho _{1})(Omega _{1})A(mathbb{C}) with respect to e, (acute{mathrm{a}}(cdot,acute{mathrm{a}}) : U to 2^{V_{2}}) is (rho _{2})(Omega _{2})A(mathbb{C}) with respect to (e^{prime}), and (acute{mathrm{a}}(cdot, acute{mathrm{a}}) : U to V_{3}) is (rho _{3})(Omega _{3})A(mathbb{C}) with respect to (e^{prime prime}) on W. Assume that the contingent epiderivatives


D

F(,F)(
u
´
,

v
´

1
)

and


D

G(,
a
´
)(
u
´
,

v
´

2
)

exist and the Gâteaux derivative (mathrm{p}^{prime}(cdot, acute{mathrm{a}})(acute{mathrm{u}})) exists. Suppose that the conditions of Lemma 3.3hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})) and (17) is satisfied. If ((acute{mathrm{u}}, acute{mathrm{a}})in check{S}), then ((acute{mathrm{u}}, acute{mathrm{a}},acute{mathrm{v}}_{1})) is a weak minimizer of (1).

Proof

Suppose that ((acute{mathrm{u}}, acute{mathrm{a}},acute{mathrm{v}}_{1})) is not a weak minimizer of problem (1). Then there exist ((mathrm{u}, mathrm{a})in check{S}) and (mathrm{v}_{1} in mathfrak{F}(mathrm{u}, mathrm{a})) such that (mathrm{v}_{1} < acute{mathrm{v}}_{1}). As (mathrm{v}_{1}^{ast}in Omega _{1}^{+} setminus {theta _{V_{1}} }), (langle mathrm{v}_{1}^{ast}, mathrm{v}_{1} – acute{mathrm{v}}_{1} rangle < 0). As ((mathrm{u}, mathrm{a})in check{S}), there exists

$$ mathrm{v}_{2}in mathfrak{G}(mathrm{u},mathrm{a}) cap (-Omega _{2}).$$

So, (langle mathrm{v}_{2}^{ast}, mathrm{v}_{2} rangle leq 0) as (mathrm{v}_{2}^{ast}in Omega _{2}^{+}). By the constraints of (16), we have (langle mathrm{v}_{2}^{ast}, acute{mathrm{v}}_{2}rangle geq 0). Therefore,

$$ bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} – acute{mathrm{v}}_{2} bigrrangle = bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} bigrrangle – bigllangle mathrm{v}_{2}^{ast}, acute{ mathrm{v}}_{2}bigrrangle leq 0.$$

Indeed,

$$ bigllangle mathrm{v}_{1}^{ast}, mathrm{v}_{1} – acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} – acute{mathrm{v}}_{2} bigrrangle < 0. $$

(20)

As the conditions of Lemma 3.3 hold at ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1}, acute{mathrm{v}}_{2}, mathrm{v}_{1}^{ast}, mathrm{v}_{2}^{ast}, mathrm{v}_{3}^{ast})), from Eqs. (4), (17) and the constraints of (16), we have

$$ bigllangle mathrm{v}_{1}^{ast}, mathfrak{F}( mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathfrak{G}(mathrm{u}, mathrm{a}) – acute{mathrm{v}}_{2}bigrrangle geq 0.$$

Hence,

$$ bigllangle mathrm{v}_{1}^{ast}, mathrm{v}_{1} -acute{mathrm{v}}_{1} bigrrangle + bigllangle mathrm{v}_{2}^{ast}, mathrm{v}_{2} – acute{mathrm{v}}_{2}bigrrangle geq 0,$$

which contradicts (20). Consequently, ((acute{mathrm{u}}, acute{mathrm{a}}, acute{mathrm{v}}_{1})) is a weak minimizer of problem (1). □

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