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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 \(\mathbb{R}^n\) 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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