The vector complementary problem and its equivalences with the weak minimal element in ordered spaces.

*(English)*Zbl 0712.90083Let (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 |

##### Keywords:

ordered Banach spaces; weak dual cone; vector complementarity; vector variational inequality; vector extremum problem; weak minimal element problem; vector unilateral minimization
PDF
BibTeX
Cite

\textit{G. Chen} and \textit{X. Yang}, J. Math. Anal. Appl. 153, No. 1, 136--158 (1990; Zbl 0712.90083)

Full Text:
DOI

##### References:

[1] | Benson, H.P., Efficiency and proper efficiency in vector maximization with respect to cones, J. math. anal. appl., 93, 273-289, (1983) · Zbl 0519.90080 |

[2] | Browder, F., Existence and approximation of solution of nonlinear variational inequality, (), 1080-1086 · Zbl 0148.13502 |

[3] | Borwein, J.M., Generalized linear complementary problems treated without fixed point theory, J. optim. theory appl., 43, 343-356, (1984) · Zbl 0518.90087 |

[4] | {\scChen Guang-Ya and Chen Ging-Min}, Vector variational inequality and vector optimization, in “Lecture Notes in Econom. and Math. Systems,” Vol. 285, pp. 408-416, Springer-Verlag, New York/Berlin. |

[5] | Cottle, R.W.; Pang, J.S., On solving linear complementary problem as linear programs, Math. programming stud., 7, 88-107, (1978) · Zbl 0381.90072 |

[6] | Cryer, C.W.; Dempster, A.H., Equivalence of linear complementary problems and linear program in vector lattice Hilbert space, SIAM. J. control optim., 18, (1980) · Zbl 0428.90077 |

[7] | Borwein, J.M., Multivalued convexity and optimization: A unified approaches to inequality and equality constraint, Math. programming, 13, 183-199, (1977) · Zbl 0375.90062 |

[8] | Ding, G.-G., Introduction to Banach spaces, (1984), Sci. Press China |

[9] | Ekeland, I., Nonconvex minimization problems, Bull. amer. math. soc., 1, 442-474, (1974) |

[10] | Elster, K.H.; Nehse, R., Optimality condition for some non-convex problems, (), 1-9 |

[11] | Giannessi, F., (), 151-186 |

[12] | Jahn, J., Existence theorem in vector optimization, J. optim. theory appl., 50, 397-406, (1986) · Zbl 0577.90078 |

[13] | Jameson, G., Ordered linear spaces, () · Zbl 0196.13401 |

[14] | Karamardian, S., Generalized complementary problem, J. optim. theory appl., 8, 161-168, (1971) · Zbl 0218.90052 |

[15] | Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. math. anal. appl., 97, 151-201, (1983) · Zbl 0527.47037 |

[16] | Riddell, R.C., Equivalence of nonlinear complementary problems and least element problems in Banach lattice, Math. oper. res., 6, (1981) · Zbl 0494.90077 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.