A notion of boundedly ϵ‐lower subdifferentiable functions is introduced and investigated. It is shown that a bounded from below, continuous, quasiconvex function is locally boundedly ϵ‐lower subdifferentiable for every ϵ>0. Some algorithms of cutting plane type are constructed to solve minimization problems with approximately lower subdifferentiable objective and constraints. In those algorithms an approximate minimizer on a compact set is obtained in a finite number of iterations provided some boundedness assumption be satisfied.
