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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 $\bbfR^n$ into itself. Let $K$ be a nonempty closed convex set in $\bbfR^n$. The generalized normal map (GNM) is defined to be the mapping $h: \bbfR^n\to \bbfR^n$ where $h(x)= g(x)- \pi_k(g(x)- f(x))$ for $x\in \bbfR^n$ and $\pi_k$ 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.

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