Graphs of minimal point mappings of parametric optimization problems appear in the definition of feasible sets of bilevel optimization problems and of semi-infinite optimization problems, and the intersection of multiple such graphs defines (generalized) Nash equilibria. This paper shows how minimal point graphs of nonconvex parametric optimization problems can be written with the help of purely box-constrained problems with additional parameters. This yields a superset of the graph, which coincides with it under mild assumptions. We specify our results to the setting of generalized Nash equilibrium problems.
The presented box-constrained reformulation allows to construct approximations of the graphs of minimal point mappings by branch-and-bound methods. We provide corresponding numerical results in a separate paper.