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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 ${ℝ}^{n}$ into itself. Let $K$ be a nonempty closed convex set in ${ℝ}^{n}$. The generalized normal map (GNM) is defined to be the mapping $h:{ℝ}^{n}\to {ℝ}^{n}$ where $h\left(x\right)=g\left(x\right)-{\pi }_{k}\left(g\left(x\right)-f\left(x\right)\right)$ for $x\in {ℝ}^{n}$ and ${\pi }_{k}$ is the projection operator onto $k$ under the Euclidean norm. Associated with the GNM, $h\left(x\right)=0$ 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:
 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 90C31 Sensitivity, stability, parametric optimization