The authors present an overview of some recent progress in the theory of optimization problems with perturbations of the form \[ \underset{x\in X}{\text{Minimum}} f(x,u),\qquad \text{subject to }x\;(u), \] where \(X\) is a Banach space and the perturbation parameter \(u\) can be a scalar, a finite-dimensional vector or an element of a metric space \(U\). Mainly, methods based on upper and lower estimates of the objective function of the perturbed problem are considered. Some illustrations are given by computing the equilibrium position of a chain that is almost vertical or horizontal.