Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions. (English) Zbl 1205.90279

This paper applies a scalarization approach of Konnov for studying strong and weak vector variational inequalities with set valued mappings. More precisely, let \(X\) be a real Banach space, \((Y,P)\) an ordered Banach space induced by a closed convex and pointed cone \(P\). Let \(L(X,Y)\) be the set of continuous linear mappings from \(X\) to \(Y\), and let \(K\subset X\) be nonempty, closed and convex. Given \(T:X\to 2^{L(X,Y)}\) a set-valued mapping, the authors consider two kinds of vector variational inequalities as follows.
The Strong Vector Variational Inequality: Find \(x^*\in K\) such that \[ \exists\,\,t\in T(x^*) : \langle t^*,y-x^* \rangle \not< 0,\;\;\forall\;y\in K. \] The Weak Vector Variational Inequality: Find \(x^*\in K\) such that \[ \forall\;y\in K,\,\exists\,\,t\in T(x^*) : \langle t^*,y-x^* \rangle \not< 0. \] The authors consider several kinds of strong and weak variational inequalities and their gap functions. Under certain conditions, they prove the equivalence between the scalar and vector version of these problems and analyze the relations among their corresponding gap functions. They apply their results to derive error bounds for gap functions. Finally, they obtain some existence results of global error bounds for gap functions and derive characterizations of global and local error bounds fo these gap functions.


90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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[1] Giannessi, F.: Theorems of the alternative, quadratic programs, and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.), Variational Inequality and Complementarity Problems, pp. 151–186. Wiley, Chichester (1980) · Zbl 0484.90081
[2] Chen, G.Y.: Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992) · Zbl 0795.49010
[3] Chen, G.Y., Goh, C.J., Yang, X.Q.: On gap functions for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, pp. 55–72. Kluwer Academic, Dordrecht (2000) · Zbl 0997.49006
[4] Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005) · Zbl 1104.90044
[5] Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer Academic, Dordrecht (2000) · Zbl 0952.00009
[6] Hadjisavvas, N., Schaible, S.: Quasimonotonicity and pesudomonotonicity in variational inequalities and equilibrium problems. In: Crouzeix, J.P., Martinez-Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity, pp. 257–275. Kluwer Academic, Dordrecht (1998) · Zbl 0946.49005
[7] Konnov, I.V.: A scalarization approach for vector variational inequalities with applications. J. Global Optim. 32, 517–527 (2005) · Zbl 1097.49014
[8] Konnov, I.V., Yao, J.C.: Existence of solutions for generalized vector equilibrium problems. J. Math. Anal. Appl. 233, 328–335 (1999) · Zbl 0933.49004
[9] Li, J., He, Z.Q.: Gap functions and existence of solutions to generalized vector variational inequalities. Appl. Math. Lett. 18, 989–1000 (2005) · Zbl 1079.49006
[10] Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) · Zbl 0688.90051
[11] Yang, X.Q., Yao, J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002) · Zbl 1027.49003
[12] Yang, X.Q.: On the gap functions of prevariational inequalities. J. Optim. Theory Appl. 116, 437–452 (2003) · Zbl 1027.49004
[13] Huang, N.J., Li, J., Yao, J.C.: Gap functions and existence of solutions to a system of vector equilibrium problems. J. Optim. Theory Appl. 133, 201–212 (2007) · Zbl 1146.49005
[14] Li, J., Huang, N.J.: Implicit vector equilibrium problems via nonlinear scalarisation. Bull. Austral. Math. Soc. 72, 161–172 (2005) · Zbl 1081.49008
[15] Li, J., Huang, N.J.: An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems. J. Global Optim. 39, 247–260 (2007) · Zbl 1128.49008
[16] Li, S.J., Teo, K.L., Yang, X.Q., Wu, S.Y.: Gap functions and existence of solutions to generalized vector quasi-equilibrium problems. J. Global Optim. 34, 427–440 (2006) · Zbl 1090.49014
[17] Mastroeni, G.: Gap functions for equilibrium problems. J. Global Optim. 27, 411–426 (2003) · Zbl 1061.90112
[18] Luo, X.D., Luo, Z.Q.: Extension of Hoffman’s error bound to polynomial systems. SIAM J. Optim. 4, 383–392 (1994) · Zbl 0821.90113
[19] Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normal spaces. SIAM J. Optim. 12, 1–17 (2002) · Zbl 1040.90041
[20] Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. A 79, 299–332 (1997) · Zbl 0887.90165
[21] Wu, Z.L., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program. Ser. A 92, 301–314 (2002) · Zbl 1041.90053
[22] Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997) · Zbl 0878.49006
[23] Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. Ser. A 86, 433–438 (1999) · Zbl 0937.49003
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