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On a generalization of a normal map and equation. (English) Zbl 0827.90131

Let f and g be two given mappings from n into itself. Let K be a nonempty closed convex set in n . The generalized normal map (GNM) is defined to be the mapping h: n n where h(x)=g(x)-π k (g(x)-f(x)) for x n and π k is the projection operator onto k under the Euclidean norm. Associated with the GNM, h(x)=0 is the generalized normal equation.

First, Robinson introduced the class of normal maps to describe a certain nonsmooth equation and derived various properties of these maps when the underlying set is a convex polyhedron. When K is the polyhedral and g is the identity map, then h is the normal map, but this normal map is not quite the same as Robinson’s normal map.

The authors show that the generalized normal equations provide a compact representation for quasi-variational inequalities, and a generalized normal equation is equivalent to some complementarity problem. The authors use degree theory to establish some existence results for a generalized normal map to have a zero and discuss their applications. They also apply a recent sensitivity theory for a parametric smooth equation studied recently by J. S. Pang to investigate the stability of a generalized normal equation at a given solution.

MSC:
90C30Nonlinear programming
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C31Sensitivity, stability, parametric optimization