分野別セミナー

【講演要旨】Constrained optimization is a problem of minimizing objective functions within the feasible set that is described by the system of equalities and inequalities of constraint functions.  A fundamental tool for characterizing solutions is the Karush–Kuhn–Tucker (KKT) condition, which requires the existence of suitable Lagrange multipliers. In unconstrained optimization, this reduces to the familiar first-order condition, which every local minimizer satisfies. In constrained problems, by contrast, the existence of multipliers does not automatically follow from local minimality. This fact is precisely what motivates constraint qualifications: they are assumptions placed only on the constraint system, ensuring the validity of the KKT condition at all local minimizers. In this talk, we first introduce a classification result on the map-germs that appear in generic parameter families of constrained functions, obtained by applying techniques from singularity theory. We then explain when the map-germs arising in this classification satisfy several well-known constraint qualifications.