We present an inverse engineering topological-axiomatic approach applicable to nonimaging optical design. The reflective and/or refractive surfaces of the optics are sequentially modified within a given parametrization scheme and a constraint set until performance objective global optimality, evaluated upon a system radiometric model is achieved. This formalism permits the study of non-concave, re-entrant, and piecewise continuous reflector and lens configurations that can constructively exploit multiple reflections for maximal energy transfer, beam shaping, or irradiance redistribution. We derive, under a single-reflection approximation, a new variational principle for constructing axially symmetric reflector forms that maximizes energy transfer and can be extended to provide optimal beam shaping. Our derivation accounts for generally shaped sources with arbitrary radiance distributions and for reflection losses. We compare known edge-ray designs with our solutions and find that the 3D CPC concentrator and the 3D involute CPC reflector (operated in reverse as a projector) can be improved upon. We present a projective design, employing a spherical source which makes use of source re-energization through retroreflection. This design achieves a beam radiance greater than that of the (naked) source and requires a re-entrant component, if a requirement for continuity of the reflective surface is imposed.