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Alternative theorems for nonlinear projection equations and applications to generalized complementarity problems. (English) Zbl 1047.49014
In recent years, the so-called exceptional families of elements (EFE) have been frequently used to study the existence of solutions for the complementarity problem; see for instance [{\it G. Isac}, {\it V. Bulavski} and {\it V. Kalashnikov}, J. Glob. Optim. 10, No. 2, 207--225 (1997; Zbl 0880.90127)]. The same method has been generalized very recently to variational inequality problems. The present paper introduces a further generalization of EFE to nonlinear projection equations (NPE), i.e., to equations of the form $h(x)=\Pi_{K}\left( g(x)-f(x)\right)$ where $f,g,h$ are functions from $\Bbb{R}^{n}$ to itself, and $\Pi_{K}$ is the orthogonal projection to a subset $K$ of $\Bbb{R}^{n}$. It is shown that, under suitable assumptions, either NPE has a solution or there exists an EFE. Applications are then given to special cases of NPE such as the generalized complementarity problem. In case the functions involved are continuous, these results generalize considerably previously known ones.

##### MSC:
 49J40 Variational methods including variational inequalities 47J20 Inequalities involving nonlinear operators 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 90C48 Programming in abstract spaces
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##### References:
 [1] Clarke, F. H.: Optimization and nonsmooth analysis. (1983) · Zbl 0582.49001 [2] Cottle, R. W.; Pang, J. S.; Stone, R. E.: The linear complementarity problem. (1992) · Zbl 0757.90078 [3] Cottle, R. W.; Yao, J. C.: Pseudomonotone complementarity problems in Hilbert space. J. optim. Theory appl. 75, 281-295 (1992) · Zbl 0795.90071 [4] Ferris, M. S.; Pang, J. S.: Engineering and economic applications of complementarity problems. SIAM rev. 39, 669-713 (1997) · Zbl 0891.90158 [5] Hadjisavvas, N.; Schaible, S.: Quasimonotone variational inequalities in Banach space. J. optim. Theory appl. 90, 95-111 (1996) · Zbl 0904.49005 [6] Harker, P. T.; Pang, J. S.: Finite-dimensional variational inequalities and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. program. 48, 161-220 (1990) · Zbl 0734.90098 [7] Isac, G.: Complementarity problems. Lecture notes in mathematics 1528 (1992) · Zbl 0795.90072 [8] Isac, G.; Bulavski, V.; Kalashnikov, V.: Exceptional families, topological degree and complementarity problems. J. global optim. 10, 207-225 (1997) · Zbl 0880.90127 [9] Isac, G.; Obuchowska, W. T.: Functions without exceptional family of elements and complementarity problems. J. optim. Theory appl. 99, 147-163 (1998) · Zbl 0914.90252 [10] Karamardian, S.: Complementarity problems over cones with monotone and pseudo-monotone maps. J. optim. Theory appl. 18, 445-454 (1976) · Zbl 0304.49026 [11] Karamardian, S.; Schaible, S.: Seven kinds of monotone maps. J. optim. Theory appl. 66, 37-46 (1990) · Zbl 0679.90055 [12] Kojima, M.; Megiddo, N.; Noma, T.; Yoshise, A.: A unified approach to interior point algorithm for linear complementarity problems. Lecture notes in computer sciences 538 (1991) · Zbl 0745.90069 [13] Lloyd, N. Q.: Degree theory. (1978) · Zbl 0367.47001 [14] Megiddo, N.: A monotone complementarity problem with feasible solutions but no complementarity solutions. Math. program. 12, 131-132 (1977) · Zbl 0353.90084 [15] Moré, J. J.: Classes of functions and feasibility conditions in nonlinear complementarity problems. Math. program. 6, 327-338 (1974) · Zbl 0291.90059 [16] Ortega, J. M.; Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. (1970) · Zbl 0241.65046 [17] Pang, J. S.; Yao, J. C.: On a generalization of a normal map and equation. SIAM J. Control optim. 33, 168-184 (1995) · Zbl 0827.90131 [18] Smith, T. E.: A solution condition for complementarity problems with an application to spartial price equilibrium. Appl. math. Comput. 15, 61-69 (1984) · Zbl 0545.90094 [19] Yao, J. C.: Variational inequality with generalized monotone operator. Math. oper. Res. 19, 691-705 (1994) · Zbl 0813.49010 [20] Yao, J. C.: Multivalued variational inequality with K-pseudomonotone operators. J. optim. Theory appl. 83, 391-403 (1994) · Zbl 0812.47055 [21] Zhao, Y. B.: Exceptional family and finite-dimensional variational inequality over polyhedral convex sets. Appl. math. Comput. 87, 111-126 (1997) · Zbl 0912.49009 [22] Zhao, Y. B.: Existence of a solution to nonlinear variational inqualites under generalized positive homogeneity. Oper. res. Lett. 25, 231-239 (1999) · Zbl 0955.49004 [23] Zhao, Y. B.; Han, J. Y.: Exceptional family of elements for a variational inequality problem and its applications. J. global optim. 14, 313-330 (1999) · Zbl 0932.49012 [24] Zhao, Y. B.; Han, J. Y.; Qi, H. D.: Exceptional families and existence theorems for variational inequality problems. J. optim. Theory appl. 101, 475-495 (1999) · Zbl 0947.49005