Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs

In this paper, we establish the equivalence between the half‐space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions. As consequences, we prove that the corresponding marginal function is differentiable and the solution map admits a differentiable selection.