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.
- Parametric semiclosed polyhedra
- Piecewise linear program
- Smooth representation
ASJC Scopus subject areas
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics