Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1

In this paper, we discuss the smoothing of an implicit function defined by a nonsmooth underdetermined system of equations F(y,z) = 0. We apply a class of parametrized smoothing methods to smooth F and investigate the limiting behavior of the implicit function solving the smoothed equations. In particular, we discuss the approximation of the Clarke generalized Jacobian of the implicit function when F is piecewise smooth. As an application, we present an analysis of the generalized Karush-Kuhn-Tucker conditions of different forms for a piecewise-smooth equality-constrained minimization problem.

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