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Generalization of an existence theorem for complementarity problems. (English) Zbl 1288.90104

Summary: We present a new notion of exceptional \(d\)-regular mapping, which is a generalization of the notions of exceptional regular mapping and \(d\)-regular mapping. By using the new notion, we establish a new existence result for complementarity problems. Our results only generalize Karamardian’s and Zhao’s existence results (Theorem 3.1 in [S. Karamardian, Math. Program. 2, 107–129 (1972; Zbl 0247.90058)], Theorem 3.8 in [P. H. Harker et al., “Finite dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithm and applications”, Math. Program. 48B, 161–220 (1990)], Theorem 4.1 in [Y. B. Zhao and G. Isac, J. Optim. Theory Appl. 105, No. 1, 213–231 (2000; Zbl 0971.90093)], Theorem 3.1 in [Y. Zhao, Oper. Res. Lett. 25, No. 5, 231–239 (1999; Zbl 0955.49004)]. In our analysis, the notion of a new generalized exceptional family of elements for complementarity problems plays a key role.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
55M25 Degree, winding number
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References:

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