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The vector complementary problem and its equivalences with the weak minimal element in ordered spaces. (English) Zbl 0712.90083
Let (X,C) and (Y,P) be ordered Banach spaces with a nonempty interior int P. We define the weak dual cone with respect to P as $$C_ p^{w+}=\{\ell \in L(X,Y):$$ ($$\ell,x)\nless 0$$, all $$x\in C\}$$. Let T be a map from X to L(X,Y). The following problem, $\text{find $$x\in C,$$ such that }(T(x),x)\ngtr 0,\;T(x)\in C_ p^{w+},$ may be called the vector complementarity problem. We prove the equivalence of the vector complementarity problem, the vector variational inequality, the vector extremum problem, the weak minimal element problem, and the vector unilateral minimization problem in ordered spaces.
Reviewer: Guangya Chen

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 49J40 Variational inequalities 90C29 Multi-objective and goal programming 90C48 Programming in abstract spaces
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