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The generalized order complementarity problem. (English) Zbl 0795.90073

Summary: Given an ordered Banach space \((E,\mathbb{K})\) and \(m\) functions \(f_1,f_2,\dots,f_m: E\to E\), the generalized order complementarity problem associated with \(\{f_ i\}\) and \(\mathbb{K}\) is to find \(x_0\in \mathbb{K}\) such that \(f_i(x_0)\in \mathbb{K}\), \(i= 1,\ldots,m\), and \(\Lambda(x_0, f_1(x_0), \ldots,f_m(x_0))= 0\). The problem is shown to be equivalent to several fixed-point problems and equivalent to the order complementarity problem studied by J. M. Borwein and M. A. H. Dempster [Math. Oper. Res. 14, No. 3, 534–558 (1989; Zbl 0694.90094)] and the first author [Contemp. Math. 72, 139–155 (1988; Zbl 0672.47040)]. Existence and uniqueness of solutions and least-element theory are shown in the spaces \(C(\Omega,\mathbb{R})\) and \(L_p(\Omega,\mu)\). For general locally convex spaces, least-element theory is derived, existence is proved, and an algorithm for computing a solution is presented. Applications to the mixed lubrication theory of fluid mechanics are described.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C20 Quadratic programming
90C48 Programming in abstract spaces
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