Abstract. In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others.

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In the present paper we prove that if the data of a parametric linear optimization problem are smooth, the solution map admits a local smooth selection "almost" everywhere. This in particular shows that the set of points where the marginal function of the problem is nondifferentiable is nowhere dense.