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Normal maps induced by linear transformations. (English) Zbl 0777.90063
Summary: We study a certain piecewise linear manifold, which we call the normal manifold, associated with a polyhedral convex set, and a family of continuous functions, called normal maps, that are induced on this manifold by continuous functions from \({\mathbf R}^ n\) to \({\mathbf R}^ n\). These normal maps occur frequently in optimization and equilibrium problems, and the subclass of normal maps induced by linear transformations plays a key role.
Our main result is that the normal map induced by a linear transformation is a Lipschitzian homeomorphism if and only if the determinant of the map in each \(n\)-cell of the normal manifold has the same (nonzero) sign.

MSC:
90C30 Nonlinear programming
49J52 Nonsmooth analysis
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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