This study proposes novel optimization methodologies for the optimization of problems that reveal non-physical step discontinuities. More specifically, it is proposed to use gradient-only techniques that do not use any zeroth order information at all for step discontinuous problems. A step discontinuous problem of note is the shape optimization problem in the presence of remeshing strategies, since changes in mesh topologies may - and normally do - introduce non-physical step discontinuities. These discontinuities may in turn manifest themselves as non-physical local minima in which optimization algorithms may become trapped. Conventional optimization approaches for step discontinuous problems include evolutionary strategies, and design of experiment (DoE) techniques. These conventional approaches typically rely on the exclusive use of zeroth order information to overcome the discontinuities, but are characterized by two important shortcomings: Firstly, the computational demands of zero order methods may be very high, since many function values are in general required. Secondly, the use of zero order information only does not necessarily guarantee that the algorithms will not terminate in highly unfit local minima. In contrast, the methodologies proposed herein use only first order information, rather than only zeroth order information. The motivation for this approach is that associated gradient information in the presence of remeshing remains accurately and uniquely computable, notwithstanding the presence of discontinuities. From a computational effort point of view, a gradient-only approach is of course comparable to conventional gradient based techniques. In addition, the step discontinuities do not manifest themselves as local minima.