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Vector variational inequality and its duality. (English) Zbl 0809.49009
Some problems concerning the vector variational inequalities in Banach spaces are studied. If \(X\) is a Banach space, \((Y,P)\) is an ordered Banach space such that the cone \(P\) is a convex, closed set with non- empty interior, \(C\subset X\) is a closed, convex set, \(T: X\to L(X,Y)\) and \(f: X\to Y\), then the variational inequality under consideration is: Find \(x_ 0\in C\) such that \[ \langle T(x_ 0), x- x_ 0\rangle\not< f(x_ 0)- f(x)\quad\text{for all } x\in C.\tag{1} \] Here \(\langle\cdot,\cdot\rangle\) denotes the duality mapping and the weak order relation “\(\not<\)” is defined by \[ y\not< x\Leftrightarrow x- y\not\in\text{int }P.\tag{2} \] Next, some properties of \(T\), such as: \(v\)- hemi-continuity, \(v\)-monotonicity, \(v\)-coercivity, and condition (L) are defined.
If in (1) the right-hand-side is replaced by \(0\) and \(T\) is continuous, \(v\)-coercive, then the solution of (1) exists. Some special cases are also discussed.
At the end, the dual inequality to (1) is introduced and some interconnections between the solutions of the inequality (1) and its dual are proved. Some applications to multicriteria optimization problems are presented, too.

49J40 Variational inequalities
90C29 Multi-objective and goal programming
46A40 Ordered topological linear spaces, vector lattices
Full Text: DOI
[1] Giannessi, F., Theorems of alternative, quadratic programs and complementarity problems, (), 151-186 · Zbl 0484.90081
[2] Chen, G.-Y.; Chen, G.-M., Vector variational inequality and vector optimization, (), 408-416
[3] Chen, G.-Y.; Craven, B.D., A vector variational inequality and optimization over an efficient set, ZOR-meth. models op. res., 34, 1-12, (1990) · Zbl 0693.90091
[4] Chen, G.-Y.; Yang, X.Q., The vector complementary problem and its equivalences with the weak minimal element in ordered spaces, J. math. analysis applic., 153, 136-158, (1990) · Zbl 0712.90083
[5] Yang, X.Q., Vector complementarity and minimal element problems, J. optim. theor. appl., 77, 483-495, (1993) · Zbl 0796.49014
[6] Mosco, U., Implicit variational problems and quasi-variational inequalities, (), 85-156 · Zbl 0346.49003
[7] Ekeland, I.; Temam, R., Convex analysis and variational problems, (1976), North-Holland Amsterdam
[8] Mosco, U., Dual variational inequalities, J. math. analysis applic., 40, 202-206, (1972) · Zbl 0262.49003
[9] Wanka, G., On duality in the vectorial control-approximation problem, ZOR-meth. models op. res., 35, 309-320, (1991) · Zbl 0744.90078
[10] Sawaragi, Y.; Nakayama, H.; Tanino, T., Theory of multiobjective optimization, (1985), Academic Press New York · Zbl 0566.90053
[11] Yang, X.Q., A Hahn-Banach theorem in ordered linear space and its applications, Optimization, 25, 1-9, (1992) · Zbl 0834.46006
[12] Jameson, G., Ordered linear spaces, () · Zbl 0196.13401
[13] Chen, G.-Y., Existence of solutions for a vector variational inequality. an extension of hartman-Stampacchia theorem, J. optim. theor. applic., 74, 445-456, (1992) · Zbl 0795.49010
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