Summary: In this work we study the parametric family of the minimization problems \(f(b,x)\to \min\), \(x\in X\), on a complete metric space \(X\) with a parameter \(b\) which belongs to a Hausdorff compact space \({\mathcal B}\). Here \(f(\cdot,\cdot)\) belongs to a space of functions on \({\mathcal B}\times X\) endowed with an appropriate uniform structure. We show that for a generic function \(f(\cdot,\cdot)\) the minimization problem has a solution for all parameters \(b\in {\mathcal B}\). This result and its extensions are obtained as realizations of a generic variational principle.