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

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

### MSC:

 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C31 Sensitivity, stability, parametric optimization
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